Opposition in Plato's Lysis

Opposition in Plato’s Lysis

τίς γὰρ οἶδεν; (221A)

Who the fuck knows?

Plato and Aristotle can be credited, among other things, with introducing what we would recognize as technical terminology into philosophy. But, as is the case with most beginnings, their efforts were not entirely organized. They frequently used a single term with several different meanings or used several terms with largely the same meaning. They also relied heavily on the nuances of Greek declensions and verbal moods to convey differences in meaning. Moreover, the terminological thesaurus differs between Plato and Aristotle and neither one is internally consistent. Their English translators turn mere confusion into chaos by often translating a single term in many different ways or using one English word for several different Greek terms. One of the most egregious mistranslations comes from H.N. Fowler, the Loeb translator of the Parmenides who renders Plato's substantive names for abstract entities - which should read schematically as "The F itself" or even simply “F-ness” - as "the absolute F," leaving the impression that Plato is much more mystical than he already is. The upshot is that it is nearly impossible to write coherently about Platonic metaphysics without keeping careful track of the original terms and doing one's best to maintain a one-to-one correspondence in the choice of representative English alternatives.

Appearances to the contrary, however, this essay is not meant to be a piece of historical or philological scholarship. For that reason I don’t review all the literature on the meaning of Greek philosophical terms, nor do I enter into controversies about those meanings that do not serve my immediate fuckosophical purposes. One other note: I use and mention words written in the Greek alphabet indifferently without signaling mention by way of quotation marks. I simply didn’t want to clutter the text with excess diacritical symbols and context should make clear whether the Greek term is being used or mentioned.

On the face of it Plato’s Lysis is utterly mysterious. Socrates and some friends are in a bath house and one of his companions becomes the target of some good natured ribbing because of a homosexual passion he has conceived for an adolescent of their acquaintance. The lover has tried to gain the favor of his beloved by composing and performing poetry that sings the praises of the young man but he has not succeeded in wooing him away from another young companion. Socrates undertakes to instruct the lover in the proper form of courtship which consists in a somewhat obtrusive exercise of the maieutic method about a puzzling topic that appears to make underhanded references to the love triangle (At least the lover seems to think that his suitability as a lover will be demonstrated if a certain conclusion is reached). That topic is whether things that are similar to or like each other entertain some sort of special relation with each other. Likewise Socrates considers whether things that are utterly opposed to each other - not just different or unlike but absolutely contrary (The Greek is ἐναντίος and my preferred translation of this term for Plato’s (not Aristotle’s) philosophical use is “contrapositive”) - maintain some sort of consistent relation to each other whatever the specific contrapositivity may be. The problem seems to be that unlikes must not like each other since they are unlike, but, on the other hand, they must like each other since they are alike in being unlike. Socrates conducts the young man, Lysis, through a blizzard of hypotheses and refutations of those hypotheses until the discussion breaks down in total confusion and the boys are summoned away by their tutor. This being what most though not all Plato scholars would consider an early dialogue, the hasty exit of the boys seems to function as a perhaps not insignificant placeholder for what would later be the recounting of a philosophical myth.

We are bound to think that part of the confusion stems from aspects of Athenian culture that require a good deal of footnoting as well as to references to a personal situation between the real life characters that was familiar to Plato’s readers. And we are, of course, in the right, at least partially. But Plato, I dare to assert, wrote for the ages; he wasn’t interested in the intrigues of a closed coterie unless they could lead his readers to general truths about the world and the proper way to lead one’s life. The question is what general truths this little vignette is supposed to make appear. The answer requires first rooting out those specifics of Hellenic philosophical culture that are no longer part of our own.

Opposition and its conjunct notions of otherness and contrapositivity are, with one significant exception, a concern unique to the Greeks among philosophers. A problematic so prominent among many Presocratics and throughout Plato’s works, barely appears after Aristotle’s logical definition of various types of propositional opposition in the Prior Analytics and conceptual opposition in other texts, perhaps because those definitions seemed to clear up the issues that had so bothered his predecessors. Once the groundwork was laid in the first part of the Organon subsequent books like the Topics mostly applied and ramified the fundamental linguistic turn. Plato, it should be mentioned, set the stage for Aristotle’s logical reconstruction of opposition in the Philebus, the Timaeus and The Statesman where opposition is reduced to or significantly amplified by the – for Plato – more satisfactory models of objective standards of measure with respect to inanimate objects and statecraft with respect to animate entities.

As regards inanimate objects at least Geach (p. 93) opines that “It was the lack (of the notion of intensity) that brought Greek science to a dead end: the Greeks tried to work with pairs of opposites suggested by ordinary language – heavy and light, right and left, hot and cold etc.  – which turned out scientifically useless.” Apparently Aquinas solved the whole issue by coming up with a notion of intensive magnitude thereby setting the stage – at least in Geach’s mental universe – for the glories of modern science. All I can say is nice try. Obviously Plato already introduced the notion of degree (admittedly expressed in terms of multiples and not a series of integers) with the clear intent of disposing of the notion of contrapositives or ἐναντίοι. As I noted, we see this in The Statesman and the Philebus. Moreover, whatever may have been the origin of measurement in empirical science, the chances that its source lay in a scholastic doctrine of degree are fairly slim.

It remains an open question whether the discovery of contrapositives actually serves any philosophical or scientific purpose. Still, distinguishing between complementarity and contrapositivity can help clarify our thinking. When George H. Smith (pp. 7-8) pointed out that atheists are not people who believe that the statement that god does not exist is true, but are rather people who do not believe that the statement that god exists is true, he is in fact distinguishing between contrapositives and complements. Atheism is the complement of religious belief, not its contrapositive. It is interesting that the same analysis can be stated in terms of the scope of a logical operator such as negation or the belief operator. A complementary class in this case is expressed by the word “not” falling outside the scope of the belief operator and a contrapositive class is expressed by the negation falling inside its scope. This is significantly not a universal mark of the difference between contrapositivity and complementarity since, for example, the contrapositivity of the Guelphs and the Ghibellines or hot and cold are unrelated to logical scope.

Nevertheless opposition as an explanatory principle or conceptual matrix in Greek philosophy comes to an abrupt halt of sorts with Aristotle. Something like the Presocratic tool of opposition did survive in the physical observational sciences – a pursuit that Parmenides and then Plato (who, as becomes evident in the Lysis, was in full critical cry against opposition if not similarity or likeness as an explanation of anything at all) denigrated by consigning observation through the senses to the realm of φαίνεσθαι. There is just the chance that the interpretation of electromagnetic force, to take just one example of opposition in natural science, in terms of polarities – a concept so central to modern physics – can trace its genealogy, by way of alchemists and physicians like Galen, if not exactly a conceptual affinity, to Greek notions about opposites. We may indeed find a lingering effect of Presocratic science in Galen and the oppositional theory of humors (Cf. Moulton pp. 16 and pp. 52-53), some practices of Hippocrates or even homeopathic medicine. There is perhaps a distant echo (not an influence) of Presocratic theories of oppositional exchange and the belief that objects can neither be created nor destroyed in the First Law of Thermodynamics and the concept of the preservation of mass/energy. Whatever relation there may be to later developments in the physical sciences may be as much due to a certain operational ease introduced by oppositional categories when we try to put some order into our observations as to literal textual influence.

The exception I referred to above is Hegel who clearly saw so much in the Presocratic problematic that he made the discovery and analysis of pairs of opposites the centerpiece of his philosophy. Like some Presocratics he made opposition and the interaction of opposites into an important principle with respect to the physical as well as the human and social worlds. A Hegel inspired viewpoint would later be adopted by Sartre and, to a large degree à leur insu, by non-philosophical writers addressing issues like racism. Sartre ignored the world of physics and limited his analysis of opposition to a way of understanding human motives and actions. This tendency in Presocratic philosophy, such as it was, was also abruptly cut short by Plato, for Plato does not explain all social conflict on the basis of the phenomenological or psychological generalization that otherness breeds hostility (The xenophobic references in the dialogues are largely dramatic and incidental; and, of course, Plato’s great hope for realizing his political theories lay with the ostensibly “other” Syracusians). One of the driving forces behind Plato’s work is the intent to undermine the conceptual framework of opposition as a way of identifying Ideas or describing their interaction.

As long as we’re on the topic of modern theories of opposition: the heart of the Hellenic problematic of opposition relates to the concept of opposition in structuralist phonetics - where opposition is properly between a given phoneme and its complement within a domain, roughly the domain of articulable sounds for a natural language that could be substituted for that phoneme in the language - only by the wildest stretch of the post-modernist imagination. However, there are few limits to Deleuze’s imagination. In Différence et Répétition (page references. Cf. also p. 303) he frequently holds out differentialism in phonetics as a potentially alternative model to Platonic essentialism (and pretty much the rest of philosophy. However, pp. 262 ff. he argues that structuralist phonology doesn’t go far enough. It appears that Deleuze, at least in this passage, understands opposition solely in terms of complementarity and appears unaware of oppositional contrapositivity as it appears in the Lysis. A proper discussion of the issue of opposition and essentialism in Deleuze requires a separate essay.) Derrida (pp. 118 and 128) uses a broader concept of structuralist opposition to discuss opposition in the Phaedrus but he does not in his essay address the cosmological oppositions proposed by the Presocratics. His oppositions relate more to the structure of argument and thought. He also does not address the issue of whether opposites can like or dislike each other, which is what is under discussion in the Lysis. Bollack and Wismann do bring up cosmological issues but do not argue convincingly for a deconstructionist take on opposition in Heraclitus. It is in the pre-deconstructionist structural anthropology of Lévi-Strauss that we find an analysis of oppositional thinking that is relevant to the kind of non-complementary or contrapositive opposition at issue in Plato and much other Hellenic thought. It is striking that, if Lévi-Strauss’ model is valid, then there is a striking analogy between early Greek thought and the unwritten means of classifying natural and social phenomena in so-called primitive societies. Lévi-Strauss, however, constantly refers to the oppositional structures he describes as logical. As we shall see, post-Aristotelian logic including its modern variants has no means for representing contrapositive opposition, since it has not developed an operation for effectively deriving  one term of a contrapositive pair from another in the same way that the operation of negation within a domain is a method for deriving a complement from its complementary opposite. This is not a limitation of classical logic. It simply means that contrapositive pairs are not distinguishable by known logical operations or general semantic categories. But more on opposition in structuralism below.

One consequence of this cultural distance is that we have to think ourselves back into the Hellenic mindset in order to understand what the problem was. My own reasons for doing so are partly historical and philological; Hegel notwithstanding, a proper understanding of the Greek problematic of opposition may not contribute a great deal to philosophical concerns we consider important today, aside from idiotic non-starters like “goodness logically requires evil” (Cf. Mackie’s articles in Martin & Monnier, pp. 61-96). Still, there is value in clearing up such logical confusions and I am surprised at the fact that many philosophers remain unaware of the fact that contrapositivity constitutes a distinct category in the grammar and semantics of the natural languages (Anscombe (p. 9) uses the attractive term “antipode” in her discussion of Aristotle, but she doesn’t seem to perceive the logical thicket the concept of an antipode or contrapositive involves (This may be the only time that “Anscombe” and “attractive” appear in a sentence together)). Plato doesn’t completely resolve the confusion between contrapositivity and complementarity (although The Sophist comes within sight of land, of which more below). But he did successfully set aside most talk of opposition in his finished metaphysics and cosmology even with the concessions to Empedocles. In addition, Socrates makes one of the clearest distinctions between complementarity and contrapositivity in Philebus 43D ff. with his doctrine of three lives (Cf. also 33E for the distinction between forgetting, on the one hand, and not remembering because there was never an original perception, on the other). That doctrine comes down to observing that feeling pleasure and not feeling pain are not the same thing. We should recall that Burke made the same distinction.

A kind of solution appears in Aristotle and even then the fallacy of attributing logical relations to pairs of contrapositive concepts was not fully laid to rest. In De Soph. Elen. XXV 180a 32 ff. Aristotle also reprises the distinction and consequent solutions hinted at in The Sophist in terms whose clarity and straightforwardness Plato could only aspire to. Unfortunately Aristotle squanders any clarity this passage affords in much of the rest of his philosophy. There is a fuller treatment of Aristotle below.

There is also an erotic/aesthetic interest in taking up this subject. After all, Plato disapproved of any fucking whose goal was not breeding (between humans, that is; apparently fucking the Idea of the Good was OK). Untangling the arguments of the Lysis is fuckosophy in its pure form, a thrilling experience in understanding how a cunning mind, however misguided and indeed repulsive on so many issues, works. I like butt fucking a smokin’ whore as much as the next guy. But you have to take a break occasionally and contributing to the downfall of Platonic moralism provides the appropriate sort of relaxation.

Plato framed the debate of the intellectual tradition we call philosophy and we still address ontological issues in terms he originally set. Morally and aesthetically he is about as disgusting as anyone who was not an actual genocidal tyrant. In that regard he was so effectively hosed by Nietzsche that, despite the best efforts of scholars like Cornford, Schuhl, Vlastos et al., his reputation has never recovered – and it doesn’t deserve to be. Platonic realism, on the other hand, the ontological doctrine, devoid in some versions of ethical implications, has flourished sporadically in the last century or so, attributed or self-attributed appropriately or not to Frege and Gödel, mathematicians like the Russian Name Worshippers and artists such as Mondrian (although Any Platonism on Mondrian’s part is open to doubt). Is there some sort of intimate connection between Plato’s aesthetics, ethics and politics and his proto-logic and ontology such that one leads inevitably to the other? Obviously Plato thought so and his views were loosely held together by the realist theory of Forms. Can we junk Platonic philosophy without losing altogether whatever may be valuable in the idea of valid argument and perhaps lapsing into the darkness of dogmatism and religion? Interesting questions.

A Note on Terminology. One of the central concepts in the Lysis is ϕιλία (alternatively ϕιλότης) and there is no single English word – not “affection,” not “friendship,” not “love,” most assuredly not romantic or erotic love – that covers the multiplicity of meanings of ϕιλία. (Vlastos in “Love in Plato,” pp. 3-4 esp. ftn. 4, provides a clear basic distinction of the meanings of ϕιλία and ἔρως that can serve as a frame of reference for understanding the skein of what Plato meant by these terms.) Note that in the Republic III 402-403 Socrates specifically says that man-boy love should be non-sexual in the physical sense.  But if you try to render ϕιλία by different terms as the occasion warrants, you end up reducing Socrates’ argument to gibberish since it depends on the assumed application of a single term and a single meaning to different contexts of use. The situation with terms for what we call love contrasts strongly with the situation regarding words such as λόγος or the numerous conjugational variants on εἰμί that are so important in Greek philosophy. It is nearly impossible to translate λόγος and εἰμί sensibly without using a variety of English words. However, if you do that, then you lose the connections of meaning that drive the argument not only in Plato but in Parmenides and Aristotle as well, to name only the most obvious cases (Geach and Anscombe did admirable work in clearing up the numerous mistranslations). With respect to love, however, Greek philosophers used several terms many of which are derived from different roots. In approximate order of “sexiness” some of these are: ἀγάπη, ϕιλία, ϕιλότης, ἔρωτα or ἔρως (or, as a modifier of ϕιλία or μίγεσθαι, ἔρωτι) and Ἀφροδίσια or ἀφροδίσιοι. Nevertheless, the earliest relevant usage of ϕιλότης is Hesiod’s ϕιλότητι μιγεῖσα (Theogony 125) and ϕιλότητος ἐφιμέρου (Ibid. 132; cf. also 177, 306, 333 and passim.) where the sexual meaning is added precisely by ϕιλότης. We can conclude that the meaning is specifically sexual from the context, one of generation without reference to anthropomorphic affectivity. And Aeschylus used the semantic unit φιλ-  with sexual overtones: e.g. φιλήτωρ in Agamemnon 1446. Conversely in Choepheroi 595-601 Aeschylus uses ἔρως and ἔρωτας in the sense of passion without a unique sexual meaning or allusion. And in Agamemnon 1478 ἔρωϛ appears to mean potentially non-sexual lust, although Aeschylus’ use may be sarcastic with an analogical sexual implication. It may also be that the phrase κακοῦ…ἔρωτος in Seven Against Thebes (687-688) infuses the passionate rush to war with a sexual feeling. In Symposium 205D Diotima suggests  meanings for ἔρως such as we are more apt to associate with ϕιλία. Her thought is that ἔρως takes on a sexual cast by way of a kind of synecdoche. Unsurprisingly several centuries later Plotinus stripped ἔρως of its carnality by stating that we can mystically ἐρωτικώτεραι the invisible (I.6.4 l 20). Interpretations and translations of Empedocles that that do not go beyond understanding ϕιλότης as a kind of sexless affection may very well miss the point and miss also the moral basis for Plato’s distaste for Empedoclean cosmology (For the “carnal” sense of ϕιλότης cf. Bollack Empédocle, p. 277-278 ftn. 5). Ἀφροδίσια almost always involves fucking. As a kind of adjunct to the Ἀφροδίσια side, we find Plato using terms such as ἀκολασία (Republic BK X 609B) and ἀπληστία (Republic BK IX 590B) that pretty much mean whoring around with hetairai and other unsavory creatures. Phrases such as these reflect Plato’s personal attitude to Cyprian sex and is far from being shared by all other Greeks (although we only need to glance at Pausanias’ distinctions between types of aphroditic love in The Symposium to realize that Plato was not an isolated innovator either. We might say that Plato and certain members of his coterie were the Athenian equivalent of Ralph Reed and his Family Research Council). Γαμεῖσθαι, which is not at issue here, just means “to get married,” but it is used in Sophocles’ Antigone ((l 594) in a negative and sexual sense. In Topics VI vii 146a 10  and VII I 152b 9-10 Aristotle uses the appropriately modest term συνουσία when he talks about fucking, although I am not sure whether this usage was current in the 6th and 5th centuries BCE. It is interesting to speculate on what implications such a compound can have on whether οὐσία is best translated as “essence” or “being” J. With Aristotle ϕιλία becomes for all intents and purposes void of sexual content. Interestingly relations between men and women are not discussed at all in his chapters on friendship in the Nicomachean Ethics. Aristotle seems psychologically unable to deal with the complex coexistence of attraction and conflict, possession and self-surrender, cooperation and hierarchy that characterizes sexual relations. It will not do to object that the Greeks in general did not have an institution like that of courtly romantic love and so they did not know many of its emotional components. The myths of Circe and Medea should be enough to belie that notion. That the Middle Ages and the Renaissance had an inkling of Aristotle’s inadequacy in this respect is pointed up in the legend of Phyllis lovingly illustrated by Hans Baldung Grien:Hans Baldung Grien- Phyllis and AristotleThere appear to be a few Easter eggs scattered though this jocular woodcut. Note the boobs decorating the vase in the niche and the barely distinguishable pair of titties poking through the windows of the house next to the turret in the distance. Grien’s is the only depiction I have found that shows Phyllis naked, but I venture to opine that this particular Phyllis would have done well to keep her clothes on. There is no etymological relation between the words ϕιλία and “Phyllis” although that does not mean that the satirists would not have associated the two. Even more interesting is Phyllis’ identity as an Indian princess in the legend. One wonders how aware Europeans were of the flourishing of Aristotelian logic in India.

The words ἕτερος and ἐναντίος can be univocally translated with not as much loss. However, some indication of the actual Greek term is needed to capture how Greek philosophers elided from one meaning to another in the use of these words. For simplicity’s sake I will as much as possible make a practice of using the standard dictionary entry form of a Greek noun or verb notwithstanding the grammatical context of the English sentence in which the word appears. Occasionally a declensional or conjugational variant will be employed. It is extremely important to know, however, that Aristotle used these terms, especially ἐναντίος, with completely different meanings from those in Plato. Not realizing that will land you quickly in a very dark wood.

In conversational and poetic Greek ἐναντίος is a spatial term meaning “opposite” or “facing” as in the Iliad Bk VI, l. 247 where the chambers of Priam’s daughters are described as ἐναντίοι or facing across a court the chambers of his sons. In the Odyssey Bk XIII l. 226 ἐναντίος is used to describe Odysseus approaching and facing Athena, and in Sappho’s beautiful lyric φαίνεταί μοί (No. 31 p. 78) a man is described as sitting across from the woman the poet addresses. (A later use with approximately the same spatial sense of “being across from” can be found in the Anecdota Bekkeri , Cf. Bollack Empédocle, II, p. 117.)  Plato uses ἐναντίος in this spatial sense in Euthydemus 278E. Plotinus (Enn. I.6.3 l. 12) uses ἐναντίος in a sort of quasi-spatial sense, although it is unclear whether the φύσις in question is opposite to what perceives it or to its form. Aristophanes in The Symposium (190A) also uses ἐναντίος in a roughly spatial sense although, unlike the chambers in Homer and the people in Sappho who face towards each other, Aristophanes’ creatures’ faces are turned away from each other. In all these passages “opposite” applies to things and not to concepts. People and objects are opposite to each other; that is, they face each other or turn their backs to each other. To call a single thing opposite is incomplete since opposition is a two-part relation. A more complete expression would say that it is “opposite to a” where “a” stands for something in space and time; taken together they are opposite to each other. It is noteworthy that this use of ἐναντίος is different from the use in the Presocratics and later philosophers not only in that it has a primarily spatial connotation. The philosophical use characterizes things that are opposite or rather contrapositive because they fall under contrapositive concepts. Fire is for some Presocratics ἐναντίος to water and bits of fire are ἐναντίοι to bits of water. The concept fire and the concept water are why a bit of fire and a bit of water are ἐναντίοι. However there is no concept other than the spatial opposition itself that induces us to call two facing physical objects “opposite.” Odysseus and Athena are not described as falling under contrapositive concepts. Still, non-philosophical examples of a non-spatial sense of opposition may be missing simply for lack of opportunity.

Plato frequently uses words meaning opposition with both philosophical and non-philosophical significance. In The Republic (Bk. III 397 C and D; cf. also Bk. III 400 B-401 A) ἐναντίος appears conversationally but with a non-spatial meaning to mean opposite as when a mimetic and a non-mimetic speaker are opposite kinds of speakers. (Cf. also the summary statement in Bk. III 402 C where, incidentally, the contrasting term to ἐναντία is αδελφά.) In Menexenus 238C ἐναντίος is used conversationally to mean the opposite of a noble or καλὴ πολιτεία. Plotinus (Enn. I.4.2 l. 29) contrasts pleasure to reason as ἐναντίῳ γένει although there is no reason to believe he meant anything more technical than an informal difference in kind. The Seventh Letter 342A uses a spatial meaning of ἐναντίος metaphorically when he describes an argument as unavoidable. An interesting sense of ἐναντίος appears in Laws Bk. I 647a and Bk. III 689B where it means “acting against” or “opposing” (ἐναντιῶμαι). But in the same passage (6789D) Plato uses ἐναντίος to mean conceptual contrapositivity (The significance of this will become clear below). In Bk. VII 816 D-E Plato asserts that both sides of the contrapositives comedy and tragedy should be learned so that people can learn not to mix them in performance. In Laws Bk. VIII 835C ἐναντία λέγων means to “to oppose” presumably with an emphasis on verbal opposition. Incidentally, ἐναντία is used substantively in the passage to meaning opposing arguments. It is also noteworthy that what is opposed is unbridled lust (ἐπιθυμία). In Bk. VIII 836D-E τούτου τοὐναντίον is correctly translated as “on the contrary” (Cf. also the Republic 490 C.). In The Republic Bk X 604B opposing impulses in men are qualified as ἐναντίας and in 607C ἐναντιώσεως means enmity. In The Republic Bk VI 487 B-C Plato uses ἐναντίον and ἐναντιοῦσθαι synonymously with ἀντιλέγω to mean contradict or argue against. In the Timaeus 65 B τοὐναντίον characterizes opposite effects on the senses in a purely descriptive sense. Even though these opposite effects are most likely contrapositive, the logical issues surrounding contrapositivity are not raised. In Timaeus 83 E ἐξ ἐναντίων refers to opposite kinds of food but it is not specified what opposite foods are supposed to be. In Timaeus 87 D- E ἐναντίος is used to characterize differences between souls and bodies. Ἕτερος gets thrown in when Plato needs a term for differences between types of differences. (Vlastos pp. 321-322 ftn. 10 mistakenly asserts that for Plato ἐναντία are non-intersecting concepts, i.e. the classes they specify do not intersect. Since he provides no argument or textual evidence for this surprising thought, I don’t know what to make of it other than that he may not have understood that there is a stronger sort of opposition between good and bad or dryness and wetness than there is between snow and hot, to cite his example.)

Aristotle picked up on some of the conversational uses I listed in Plato. For example in Topics I xi 104b 14 and 24 ἐναντίος is used in the sense of opposing arguments in a debate. Also in Politics 1341a 24 VIII vi 5 he uses ἐναντίον to qualify stimulating entertainment as a use for music as opposed to its educational use.

Socrates’ Arguments in the Lysis

Let’s turn to the actual dialogue. The rapid fire stream of argument is as bewildering and slippery as anything in the Parmenides.

1. The first argument arises somewhat abruptly after Socrates questions Lysis and his boyfriend Menexenus about their personal qualities in the course of which questioning Menexenus is suddenly called away. It starts at 207D ff.: Assume that parents love (φιλεῖ) their sons, i.e. that Lysis’ parents love Lysis. Then they want him to be as happy as possible (εὐδαιμονέστατον). Assume further that no man who is enslaved (δουλεύων) or who cannot do whatever he wants (ἐπιθυμῇς) is happy. Lysis’ parents want to bring about his happiness (προθυμοῦνται– the idea that they would be eager to bring about his happiness is unexceptionable but it is not essential to the argument as a pure string of syllogisms). Therefore they do not or should not try to prevent Lysis from doing whatever he wants.

If, however, Lysis’ parents do prevent (κωλύουσιν) him from doing many things, then their behavior or attitude is not consistent with behavior implicitly prescribed by the second assumption above. Lysis attempts to resolve the conflict by saying that he is not yet of age (ἡλικίαν ἔχω). Socrates answers that there are things that Lysis’ parents allow him to do (Presumably if age were the only qualification, they would not allow him to do anything without direction). Lysis qualifies his response by saying there are things he understands (ἐπίσταμαι and later φρονεῖν) better than his parents even though he is not of age. Therefore we are permitted to do as we please in matters we have some understanding or expertise about, but we are prevented from doing as we please in matters where we have no such understanding.

At this point Socrates advances a reason why people allow us the freedom to do what we want in matters in which we have expertise. That is, we are useful (χρήσιμος) to them by way of our expertise. One consequence of this view that is not stressed is that Lysis’ parents end up just like everyone else in their affection for Lysis. Their love too, assuming that the argument to this point is valid, is purely self-interested and disappears in cases where Lysis lacks an expertise they value. Inverting (contraposing in the Aristotelian sense) this assumption and generalizing on the inversion, Socrates concludes people do not love (φιλήσει) us or count us as friends (φίλοι) in matters in which we are useless (ἀνωφελεῖς) to them. Presumably being useless is equivalent to or a consequence of lacking φρονεῖν.  Therefore Lysis’ parents do not love him whenever (ὅσον ἄν) he is useless (ἄχρηστος). The conclusion is that Lysis needs to become wise or smart (σοφός), and therefore useful and good (with respect to others), so everybody will love him and “become intimate” (οἰκεῖοι) with him. The skipped steps are obvious but it is worth noting that Socrates equates being σοφός with ἐπίσταμαι and φρονεῖν. Socrates concludes by observing that Lysis cannot think too much of himself (μεγαλόφρων) as long as there are areas where he needs a teacher and is therefore still lacking in understanding (ἄφρων).  This conclusion plays on various derivatives of φρονεῖν and introduces the next passage where Socrates all but admits his argument was not serious (although much of the argument is akin to later Platonic doctrines of usefulness and love. Vlastos has argued that utility lies at the bottom of Plato’s views, at all stages of his writing, concerning the nature of love between humans. In this light Socrates’ faulty third argument in the Lysis is actually consistent with Plato’s mature views. See below.). His intention, rather, was to tie Lysis up in knots and therefore bring him down a notch. It was dialectic in the sense of having more to do with personal relations than abstractly valid argument. He says he wanted to illustrate to Hippothales, Lysis’ admirer, the superiority of humbling argument over glorifying poetry as a way of winning over the object of one’s desires.

Despite the fact that Socrates is not completely serious, it is nevertheless not so easy to explain the reason why this apparently so specious argument is faulty. Obviously one fallacy is identifying being happy with being allowed to do whatever one wants. Presumably Lysis’ parents are able to contrast happiness here and now with happiness in the future and in the circumstances in question, since they know the relative value of each, are able to prefer happiness in the future over the immediate gratification of one’s wishes. More importantly, when Socrates concludes that Lysis’ parents love him to the extent that he is useful to them (still mistakenly equating happiness with being allowed to do something), i.e. because he is expert at some activity they find valuable, he skirts the insight that certain words, despite lexical appearances and setting aside quibbles to the effect that all meaning is contextual, do not have a complete and unified meaning by themselves, i.e. in order to be properly understood their meaning must be qualified by some adjunct term or other. In Socrates’ phrase parental φιλία involves a semantic hole that needs filling. Lysis’ parents “love” him to the extent that (ὅσον ἄν) he is useful to them. Ὅσον ἄν requires a qualifier that must be understood if the concept of love in question is to be properly understood. Socrates argues that usefulness in a respect is just such a qualifier. But obviously usefulness in a respect has its own hole that must be filled. The respect needs to be specified. The chain of semantic specification runs: (a) I love him… (b) I love him in that… (c) I love him in that he is useful in respect of… (d) I love him in that he is useful in respect of his knowledge about…. (e) I love him in that he is useful in respect of his knowledge of horses. This sense of φιλία may be perhaps more faithfully rendered by the phrase “value for...” If I say I value someone I am likely to be asked why I value him. “Value” so understood requires a filler: I value a man for his economic expertise or for his horsemanship or I value a woman for her beauty or her skill at dialectic. There may be a different sense of love or value that is not incomplete in this way. That sort of love would perhaps be unqualified in the psychological sense as well. Moreover, it may even not compel the conclusion that the beloved would be allowed to do whatever he wants as in cases where unqualified love for him conflicts with unqualified love for another, a sibling perhaps (a conundrum similar to the conflict of rights in political theory), or in cases where the lover believes that the beloved’s proposed course of action will lead to his harm and unhappiness.

It is worth noting that in the apocryphal and most likely post Aristotelian dialogue Hipparchus Socrates, in the course of introducing sets of opposites like gain and loss, also introduces the idea that the term ἐναντίος must be qualified in order to avoid contradiction. Things are not just ἐναντίοι, they are ἐναντίος in a certain respect (ἐναντίος κατά).

Incomplete terms indeed play a prominent role in Aristotle. In Categories VII the theory of πρός τι is in effect a theory that certain one-place predicates are incomplete. This relates to the response that some of the conundra about contrapositives are really due to incomplete predicates. And in Soph. Elen. XIII 173b 42 ff. Aristotle proposes to solve certain sorts of fallacies by invoking incomplete predicates: πρός τί.

The recognition of incomplete terms, I assert by way of forecast, will contribute to a solution of the problem of the predicability of existence. For “exists” in most (and I emphasize most; the exceptions are significant) of its usages is just such an incomplete term. A special case is where “exists” is expressed by logical operators such as the existential quantifier; that sort of expression wears its incompleteness on its face, as Wittgenstein might say. A perhaps more informative expression than “incomplete term” would be “ellipsis” or “elliptical expression.” “Exists” in most of its usages is elliptical.

2. It is impossible to restate the flow of this argument (212B ff.) without using several different English words for the semantically anarchic term φιλία. But in that case much of the argument will be lost since, for example, it would make little sense to say that you are not a horse fancier if your horse doesn’t love you. Accordingly I will insert φιλία and its cognates (φιλέω, φιλῶν, φιλούμενος, φίλος, ἀντιφιλεῖσθαι and various constructs such as φίλιπποι) into my own text with the understanding that φιλία has a kaleidoscope of meanings that include erotic love, romantic or sentimental love (or something very much like it), friendship, fondness for, and a fancy for or a liking of. I will also mostly ignore declensional and conjugational niceties and simply use the simplest version that contains the grammatical root. It is worth noting that μισέω is a proper contrapositive not of each meaning of φιλέω, but of the senses that involve romantic love or friendship only. A horse might hate a given horse fancier, but this is not the same contrapositive relation of a boy who hates his suitor. The horse’s emotions do directly contrast with the dispositions of an equine hobbyist who in addition practices bestiality and loves the hating horse. Aristotle understood this difference of meaning when he distinguished between φιλία as fancying (wine) and φιλία as “friendship” in Eth. VIII ii 3 ff. However, he goes too far, in the course of establishing his criteria for true friendship, when he asserts that there is no commonality of meaning at all. One result is that Aristotle’s doctrine of friendship does not adequately address the issues raised in the Lysis, a dialogue he does not even reference. More on Aristotle and friendship below.

In a relationship of φιλία, there is a φιλῶν and a φιλούμενος. Which of these is a φίλος: the φιλῶν, the φιλούμενος or both or neither? Menexenus’ initial response is that both are φίλοι. But a φιλούμενος might not ἀντιφιλῇ his φιλῶν. He might even μισέω him. This introduces three types of asymmetry. The φιλῶν may not be the φίλος of the φιλούμενος. The φιλούμενος may not be the φίλος of the φιλῶν. Or else it could be that neither is a φίλος. Menexenus opts for the last type to characterize lack of reciprocity or ἀντιφιλία, i.e. where the parties are not equally φιλῶν. In that instance neither is a φίλος to the other. (Nietzsche considered reciprocity so little necessary that he called Gegenliebe a Remedium amoris (Morgenröte IV 415).)

Then Socrates brings up examples where a lack of ἀντιφιλία is not due to μισέω, a genuine contrapositive relation, but is due to simple complementarity such as where a horse or a newborn neither φιλέω nor μισέω the φιλῶν. He cites two different sorts of examples. The horse may simply be indifferent to the φίλιππος although able, from a certain perspective, to φιλέω or μισέω him; this is certainly the case with the newborn child and his parents. To expect ἀντιφιλία on the part of γυμνασίδιοι for the φιλογυμνασταί is a simple category mistake.

Socrates points out that, if Menexenus’ second position is correct, then the poet Solon was wrong to write that children, horses and hounds could be friends to a happy man. But his objection is misconceived. The previous argument did not conclude that horses, hounds or children could never be φίλοι to a φιλῶν. Menexenus only addressed the situation where one or the other did not in fact ἀντιφιλῇ the φιλῶν, i.e. where reciprocity is both relevant and missing. However, Menexenus agrees that the poet could not be mistaken, so Socrates concludes that the φιλούμενος has to be a φίλος to the φιλῶν whether the reciprocal feeling be love or hate (ἐάν τε φιλῇ ἐάν τε καὶ μισῇ, 212D). Nevertheless Socrates concludes and Menexenus also agrees that, if the poet is right, the φιλούμενος is a φίλος of the φιλῶν whether the φιλούμενος exhibits ἀντιφιλία or not, or even μισέω the φιλῶν.

Socrates proceeds to draw the parallels with μισέω. It is worth noting that ἐχθρός, which is supposedly the parallel to φιλούμενος is not semantically or conjugationally related to μισέω as are the words φιλούμενος and φιλέω. Nevertheless Socrates concludes that, by the same reasoning the μισούμενος is an ἐχθρός, while the μισῶν is not unless there is a reciprocal feeling. The somewhat paradoxical result is that a φίλος can μισέω a φιλῶν or an ἐχθρός φιλέω a μισῶν. It could even turn out that someone is a φίλος to his ἐχθρός and vice versa. However, Socrates argues that this last conclusion is absurd and Menexenus agrees. So Socrates tries the other alternative, namely that the φιλῶν is a φίλος to the φιλούμενος and the μισῶν is an ἐχθρός to the μισούμενος. But this option leads to the same sort of absurd conclusion. The paradoxical upshot is that neither a φιλῶν nor a φιλούμενος can be a φίλος. However Socrates, ignoring the fact that this was precisely Menexenus’ second position, draws the further unwarranted conclusion that a φιλῶν and a φιλούμενος cannot be φίλοι even where there is ἀντιφιλία (μήτε οἱ φιλοῦντές τε καὶ φιλούμενοι, 213C). At least it is not warranted by the absurdity of the situation where a φιλῶν or a φιλούμενος would be a φίλος where there is no reciprocal feeling. So Socrates asserts, and Lysis abruptly agrees, that they have found something wrong with each alternative, the conclusion being that there must be something wrong with the way the question was posed.

It is worth noting at this stage how this argument fits into the dramatic structure of the dialogue. Socrates frames his inquiry by observing that the one thing he always desired (ἐπιθυμέω) above all else was to have φίλοι. But his is not an ordinary desire. His desire for a good φίλος is ἐρωτικός, a term which can signify love and passionate love quite generally, but has an unmistakable sense of bodily sexual contact particularly in Plato (Cf. The Symposium 186C where ejaculation and defecation don’t seem to be clearly distinguished). The use of erotic language is likely a flirtatious lure since Socrates’ long term goal is always to guide man/boy φιλία in the direction of the older lover instructing the boy in σοφία (meaning both knowledge and worldly wisdom) and virtue.

It is also noteworthy that in his list of anyone who may qualify as an unrequited φιλῶν, Socrates includes (212D) philosophers in case Sophia does not return the proffered φιλία. This is not a problem for the fuckosopher whose first words are always “Face down, Ass Up!” So Sophia doesn’t ἀντιφιλῇ in face to face reciprocity? Fuck her up the butt!

3. This long chain of argument (213D ff.) addresses anew the question of what sorts of things can be φίλοι either reciprocally or not. But it dispenses with framing the question around the φιλῶν and the φιλούμενος. Rather, Socrates introduces other distinguishing traits of φίλοι. It is at this stage that he introduces the notion of ὅμοιος which will in the course of the argument bring up the counter concept of ἐναντίος. Socrates’ first attempt enunciates a premise drawn from a line in Homer to the effect that god always brings like and like together (ὡς αἰεὶ τὸν ὁμοῖον ἄγει θεὸς ὡς τὸν ὁμοῖον, Odyssey Bk 17 218). Homer is cited in this instance as a sort of authority on the basis of the belief that the poets are “fathers” and guides toward σοφία, i.e. wisdom or knowledge.

It is worth pointing out that Socrates’ citation of Homer is disingenuous to say the least since the line in question comes from the mouth of Melanthius the quisling-like shepherd and amounts to nothing more than a not very credible sneer directed at Eumaeus and Odysseus: “So birds of a feather flock together.” More à propos is Empedocles’ doctrine that ὅμοιοι “come together” both cosmologically and socially. Indeed since much Presocratic thought is organized along the principles of likeness (ὁμοιότης) and contrapositivity (ἐναντιότης), much of the following line of argument must more or less explicitly be directed against Presocratic ontology and cosmology with Empedocles as the specific target. The strategy is to show through a multiplicity of counterexamples that likeness and contrapositivity are irrelevant to φιλία. However, as in the Parmenides, no definite conclusion is reached; rather an entire conceptual framework is dismantled piece by piece. And the destruction of the first piece lies in an implicit pairing of Empedocles and Melanthius. (For the attraction of likes or ὁμοῖοι as part of the Empedoclean theory of the creation of the elements from the original sphere (and therefore the real target of Socrates’ arguments in the Lysis) cf. Bollack, Empédocle I, p. 50 and 51 esp. ftn. 3. Aristotle adds the attraction is to a loved (i.e. beloved) γέννα.)

So - however doubtful the Homeric authority may be - Socrates adopts as his initial premise that φίλοι are distinguished by being ὅμοιοι. This premise is immediately challenged by a counterexample. The wicked (πονηροί) are presumably alike in their wickedness, but they cannot be considered φίλοι. Socrates’ exact argument is that the closer one worthless man comes to another the more hateful (ἐχθίων) he becomes and, being hateful, he will harm the other man. And someone who harms another is not a φίλος of the other.

An alternative formulation, Socrates opines, is that the good (ἀγαθοί) are both ὁμοίοι and φίλοι, while the bad (κακοί) are neither ὁμοίοι nor φίλοι. He in fact throws in the intriguing but scarcely defensible observation that bad men are not even like themselves since they are unbalanced (ἐμπλήκτους) and unsteady (ἀσταθμήτους). Going to this extreme in the characterization of the bad actually preserves the validity of the entire the initial premise (assuming, of course, that unsteady things cannot be alike in their unsteadiness; but in that case unsteadiness is not a single concept but an indefinite number of different concepts applied to different individuals), namely that ὁμοίοι are φίλοι in virtue of their ὁμοιότης, a point that Socrates fails to notice in his conclusion that the initial premise must apply only to the good and not to the bad. Moreover he draws the perhaps laudable but utterly counterintuitive conclusion that φίλοι can only be found among the good (οἳ ἄν ὠσιν ἀγαθοί – The grammar of this phrase indicates that goodness is a minimal qualification for being a φίλος; not all good men are φίλοι). It is counterintuitive because it assumes that a bad man must always do harm to everybody without exception and so cannot have a single φίλος (Presumably he does direct harm even to himself). A contributing factor to the confusion is that Socrates has made no attempt to define φιλία outside of the initial identification with ὁμοιότης, an identification to which he has already found an apparent exception. He has not proposed a sense of φιλία that permits a rational evaluation of whether ὁμοίοι are φίλοι or not. The exclusion of bad men because they are ἀδικοῦντας backs into a kind of negative qualification of φίλοι, viz. a man cannot be a φίλος and ἀδικών at the same time. And, as already noted, because of the identification of φιλία with ὁμοιότης, Socrates’ point is counterintuitive in another way, namely in the consequence that κακοί cannot be ὅμοιοι to each other, not even in their badness.

But Socrates finds difficulties with this position also. Recalling the previous discussion with Lysis to the effect that his parents only φιλέω him to the extent that he was χρήσιμος to them, he wonders whether ὅμοιοι are indeed χρήσιμος to each other. Socrates’ argument is that, if indeed two individuals are ὅμοιοι, then the one can bring no advantages to the other that the latter does not already possess.  Every individual is self-sufficient with respect to his ὅμοιοι. Accordingly ὅμοιοι do not bring ὠφέλεια or ἐπικουρία to each other. That being the case, they cannot be ἀγαπῷ to each other. But without ἀγάπη, there is no φιλία either (The unannounced appearance of what was destined to become the celebrated notion of ἀγάπη adds nothing to the argument. Either it is synonymous with φιλία, in which case Socrates is stating a tautology. Or else it means something else, in which case Socrates doesn’t tell us what it means.) So, in this light ὅμοιοι could not possibly be φίλοι. The same argument applies to those who are good, and it does not help to claim that the good can be φίλοι, for, if two individuals are both good, they are accordingly ὅμοιοι and thus cannot be φίλοι for the reasons just stated as to why no ὅμοιοι can be φίλοι. So, this objection denies φιλία even to the ἀγαθοί, not just the κακοί, because the argument is based solely on their ὁμοιότης.

The fallacies in this line of reasoning are patent. First there is no reason to assume, as Socrates does throughout, that being useful or χρήσιμος is a necessary condition of φιλία (assuming anything but an ad hoc definition of φιλία).  Secondly, there is no reason to assume that ὅμοιοι are completely useless to each other. For one thing the sort of ὁμοιότης this presumes seems to involve some sort of identity. Socrates does not consider the idea that there may be degrees or kinds of ὁμοιότης. The idea lurking in the background is presumably that Menexenus and Lysis are ὅμοιοι while Hippothales and Lysis are not. But it is difficult to see how Menexenus could in no way be useful to Lysis. Indeed, if the doctrine that a φιλῶν should instruct his φιλούμενος in moral behavior is valid, then such instruction would seem to be a way of being χρήσιμος. According to the argument under consideration, however, Hippothales cannot provide such instruction since as an unlike he is by definition useless to Lysis. Menexenus, on the other hand, need not since he and Lysis already possess all the same qualities. Finally the idea that someone is good hardly implies that he is self sufficient; the two concepts seem at best to be tangentially related unless Socrates understands the concept of ὁ ἀγαθός as necessarily including a measure of self-sufficiency.

It is also worth noticing how the language of what we would call romantic or sentimental love if not necessarily erotic love creeps into Socrates’ descriptions of φιλία or conditions for φιλία: ἀγάπη, ποθενοί, χρείαν αὑτῶν ἔχουσι, περὶ πολλοῦ ποιῆται, ἐρῶντα. All of these terms encroach semantically on the concept of φιλία, since, if they were abstracted out, Socrates’ ad hoc identification of φιλία first with likeness (presumably  ὁμοιότης ) and then with χρήσιμος would not be affected. They add something to the meaning of φιλία that would not be there without them. Likewise it would appear that concepts such as longing for, cherishing, making much of and the rest would certainly be comprehensible without a distinct and perhaps encompassing concept such as φιλία. The rhetorical implication, however, is that people who engage in “romantic” behavior such as longing for etc. cannot be ὅμοιοι with the objects of their longing even in the restricted sense of both being ἀγαθοί.

Many of the problems Socrates uncovers in the doctrine of love arising from similarity can be traced to the fact that the notion of ὅμοιος or similarity is elliptical. In a different context Aristotle discussed at least two meanings of likeness or similarity in terms of the respect in which we regard things as alike (De An. 2 5 417a 15 ff.).

4. The specific argument in the context of which Plato introduces his examples of contrapositives (215C) is the following: 1) Like cannot ϕιλéω like. In support of this assertion Socrates cites an unnamed source who quotes Hesiod. Socrates adds as an illustrative explanation that individuals who pursue the same profession cannot like each other but must be filled with “envy, contention and hatred” towards each other. Interestingly enough Boethius seems to have held the contrary view, assuming he meant the same thing as Socrates in speaking of similarity and dissimilarity (Quoted in Eco p. 62). 2) Socrates continues to quote the unnamed source to the effect that friendship or love could only be found between things that are unlike. The reason is that unlikes need each other. One is τροϕἠ for the other. A series of unlike things or, more accurately, ἐναντίοι are presented as (rather unconvincing) examples. Dryness loves wetness for example.

But Socrates immediately moves to refute this contention, at least as a generalization, citing ἀντιλογικοί as possible sources. The gist of the refutation is a set of counterexamples. Evil, for example, cannot like good. Socrates immediately attacks the refutation and so forth, but I will leave the outline of the entire argument for below. Right now my interest is to establish the discursive context in which the examples of ἐναντίοι appear.

This argument introduces a concept and a set of examples that make the Lysis interesting and worth analyzing. The concept is ἀνομοιότατα or “things most unlike” which is equivalent perhaps to τὸ ἐναντώτατον further down in the paragraph. The import of the superlative is not to compare degrees of unlikeness; rather it is clearly meant as an equivalent to what I call contrapositivity, for Socrates makes use of a grammatical distinction between ὅμοιος and ὁμοιότατα and likewise between (presumably) ἀνόμοιος and ἀνομοιότατος (equivalent to ἐναντίος and ἐνατώτατος) that had not appeared in the preceding arguments where likeness alone was at issue; likeness seems to be introduced here in order to accommodate the notion not simply of complementarity but of utter contrast or opposition. The sense of contrapositivity is carried ambiguously in the Lysis sometimes by ἐναντίος and sometimes by ἐναντώτατος. But ὁμοιότατος and ἀνομοιότατος are not semantically equivalent in this passage. Whereas, as we saw, ἀνομοιότατος carries the sense of utter opposition and does not admit in any significant way of degrees, ὁμοιότατος is a true grammatical superlative constructed on the comparative idea that some things are more alike than others. The tipoff to this asymmetry is that there is no really clear meaning to ὁμοιότατος except perhaps self-identity. The upshot is that, while Socrates is reaching about for the concept of contrapositivity, he does so in a muddled fashion much like the founders of the internet who introduced the now largely useless notion of www to segment the novel http protocol. The muddle has consequences for it may influence Plato’s thinking about contrapositivity. By casting contrapositivity as the grammatical superlative of ἐναντίος, Plato implies that there is a quasi-logical (i.e. grammatical) way to define a contrapositive to a given concept. But there is no grammatical connection. “Contrapositive” is not the superlative of “opposite.” The two are different words that, rigorously defined, may end up meaning the same thing. There may consequently be room to believe that this muddle seriously disqualifies Plato’s “refutation” of Empedocles. In any event, Socrates spends no time trying to analyze the new conceptual framework (presumably because he is simply trying to recollect what someone else said). Rather he proceeds immediately to his examples in order to elucidate the meaning of contrapositivity. Of course, in The Sophist the term of choice becomes properly ἐναντίος which is contrasted with ἕτερος, a vocabulary Aristotle would retain but redefine in the Organon. It is in this early dialogue, nevertheless, that, as far as I can tell, Plato first seriously addresses the philosophical issues surrounding contrapositivity which would be partially resolved in the important distinction in The Sophist between complementarity and contrapositivity and the insight that a Form can partake of its contrapositive, or, to use a more nominalist, non-Platonic formulation, a concept can actually be qualified by its contrapositive. There is a valuable insight in the concomitant glimpse of the structure of higher order predication. However, the reason Socrates addresses contrapositives here is implicit in his examples: wet and dry, cold and hot, bitter and sweet, sharp and blunt, κενὸν πληρώςεως and πλῆρες κενώσεως. These oppositions formed the warp and woof of Presocratic cosmology, so, however inexplicitly, Socrates is actually exploring the logic behind those important cosmological theories. Not just exploring the logic, he is attacking the validity of the concept they exemplify. By use of multiple arguments, counterexamples and retractions, Socrates reduces the notion of contrapositivity to the sheerest nonsense. The consequence is that a cosmology conceived in terms of contrapositives like wet and dry and their interaction is shown to be logically incoherent or at least irreducibly vague.

Historically Socrates’ attack enjoyed mixed success. Aristotle partially redefined contrapositivity as a property of propositions. He may have understood Empedocles’ love and strife as simple possibly non-oppositional forces accounting for specific ccases of mixture and dispersal (Cf. Bollack Empédocle I, p. 47 ftn 5). Nevertheless something like Empedoclean contrapositivity lived on in ancient medicine (Cf. Bollack Empédocle, pp.262-263 ftn. 5) and in Epicureanism (Cf. Lucretius De rerum natura V 437-438 and Bollack Empédocle pp. 175-177 and 190. Note also on p. 190 the ideas that opposites heal each other). Most tellingly, despite his mockery of Empedocles in the Lysis and later in The Symposium, Plato himself would later adopt elements of Empedoclean contrapositivity in his own cosmology (Cf. among other examples, Bollack Empédocle pp. 232-233 ftn. 7 and p. 268 ftn. 3). This development adds some weight to the view that the Lysis is an early dialogue, dating from a period when Plato may still have been under the influence of Socrates, although the anti-Empedoclean polemic was if anything sharpened in the transitional dialogue The Symposium.

How important is the distinction adumbrated in the Lysis and the historical event by which it fell by the wayside? Philosophers to this day make the conceptual error of not distinguishing clearly between complementarity and contrapositivity and many supposedly valid doctrines fall prey to the error.

Let’s turn to the argument. Because of the successful counterexample to his initial premise, Socrates tries adopting the opposite premise, namely that ὅμοιοι and ἀγαθοί are so little friends that they are the most hostile (πολεμιώτατοι) to each other. In support of this premise he quotes Hesiod to the effect that ὁμοιότατα are filled with envy contention and hatred toward each other. But ἀνομοιότατα do experience friendship. The examples are the poor who φιλέω the rich, the weak the strong, the sick who φιλέω doctors and the ignorant (τὸν μὴ εἰδότα) the knowledgeable (τὸν εἰδότα). The reference to Hesiod is more than just a literary flourish, for the passage Socrates quotes (Works and Days ll. 24 ff.) is precisely one where Hesiod introduces the notion of ἔρις or rather distinguishes between good and bad strife. Ἔρις became part of Heraclitus’ cosmology and Empedocles would introduce the related notion of νεἶκος. Thus Socrates’ examination and rejection of a basis in likeness of Hesiodic good strife could very well be meant to target the cosmological explanatory principles of love and strife as they appear in the Presocratics. There is also a glancing blow at Hesiod himself. The good strife of ὁμοιότατα such as that of potters for potters will be replaced in The Statesman with a different model for personality types within a polity. Consider also: The Republic Bk. I iii 329a on affinity of like to like; Laws Bk. IV 716C where Plato says immoderate likes show no attraction to each other or to their opposites although, in the possible softening of his view I alluded to above, he suggests that moderate likes can and should; and Phaedrus 255B where in a very confusing sentence with lots of negatives Plato seems to affirm the argument that evil cannot be friends with evil but good can be with good. Or rather he states more hesitantly that this is possible. “For it was never decreed by fate that bad should be a friend of bad nor that good should not be a friend of not good.” I will return to the later dialogues below.

By way of aside, as long as we are quoting poets, there’s no reason for not throwing that notorious Heraclitean, Marlowe into the mix:

…this strife of hers, like that Which made the world, another world begat Of unknown joy.

The reason for Socrates’ new hypothesis refers back to the previous criterion for φιλέω, i.e . ἐπικουρία which is equivalent to being χρήσιμος. One affective term, ἀγάπη, used to describe the φιλία of ἀνομοιότατα is also lifted from the previous argument. The next set of examples shift from a Hesiodic to a Presocratic, especially Empedoclean reference, although it is quite clear that the Presocratics viewed cosmology the way they did because of the Hesiodic heritage. These examples are the physical contrapositives I listed above.

Of course this premise is no more satisfactory than the first. In opposition to the unnamed source whose words Socrates claimed to recollect in defense of this premise, he now refers to unnamed ἀντιλογικοί or skilled disputants as immediately perceiving its defects. The primary counterexample consists of a hating thing (τὸ ἐχθρόν) and a φίλον. The question is thereby skirted as to whether ἔχθρα and φιλία themselves are ἐνανατιώτατα since Plato does not distinguish between specific hating/loving things, hating/loving things in general and the hating/loving thing as such (a sort of Platonic Idea of hating/loving). As in The Sophist, one import of this move is that concepts are very nearly regarded as second order since an implicit question can arise of whether φιλία itself can be φίλος to something else. Other counterexamples are the just and the unjust, the temperate (σῶφρον) and the profligate (ἀκολάστος) and the good and the bad. The conclusion is that neither ὅμοιοι nor ἐναντίοι can be φίλοι either to each other or to their opposites.

5. The seemingly obvious alternative (216C ff.) is that φίλοι can be recruited from among the class of things that are neither good nor bad. By suggesting this possibility Socrates recognizes the important concept of complementarity (Or perhaps it “reveals” itself to him – φαεῖσθαι? εὑρίσκειν? Socrates’ rather more innocent formulation is δοκεῖ μοι). Appropriately enough, Socrates calls the good, the bad and the neither good nor bad (viz., the complement of the class that contains both the good and the bad) γένη foreshadowing Aristotle. We are actually dealing with three complementary classes here - although Socrates only explicitly mentions the last one - namely the not good, the not bad and the neither good nor bad. The not good is not equivalent to evil, namely to an ἐναντίος (Incidentally there is no reason that there cannot be more than one ἐναντίος to a concept. For example evil and disastrous can both be regarded as ἐναντίοι of good). Rather it is a complement (in an informal Boolean sense of “complement”) or what both Plato and Aristotle would call ἕτερος (conceptually ἕτερος to be absolutely precise) when they are being terminologically consistent. The idea that a complement can be a φίλος to something suggests objections such as had already been made to the candidacies of ὅμοιοι and ἐναντίοι. Also we need to be clear as to whether we mean that only the neither good nor bad can be φίλοι or that all the neither good nor bad are φίλοι. Assuming the first alternative as Socrates appears to do, being neither good nor bad is a necessary condition for being a φίλος, and, assuming the second, being a φίλος is a necessary condition for being neither good nor bad. And if being neither good nor bad is no more than a necessary condition of being φίλος, then it is possible that some neither good nor bad things are not φίλοι. But unless being neither good nor bad is also a sufficient condition for being a φίλος, this alternative is at best a partial explanation of what it means to be φίλος. The obvious converse equivalents of the two possibilities are that all φίλοι are neither good nor bad and that only φίλοι are neither good nor bad. It should also be observed that the unqualified notion of complementarity can be very broad. A piece of quartz is presumably neither good nor bad. Does that mean it is a better candidate to be a φίλος than a good or evil human being?

I have stated this alternative without reference to a possible object for the ϕιλὀτης of the neither good nor bad. Socrates however puts it more specifically by suggesting that the neither good nor bad is a φίλος of the good. This alternative is not really subject to argument at this point.  Lysis simply signals his befuddlement and Socrates admits that he also is dizzied by the course of the argument. For, assuming the validity of his counterexamples, he seems to have shown that none of the following can be φίλοι to either themselves or to their ἐναντίοι: the like, the unlike, the good, the bad and very possibly the neither good nor bad. Socrates uses the all important term ἀπορία to describe this situation implying that his argument has the force of a Zeno style paradox.

By way of aside Socrates throws out the possibility that the beautiful (τὸ καλόν) is a φίλος and that indeed the beautiful is the good. This suggestion is somewhat gratuitous from the strict standpoint of the argument thus far. But it does serve to add extra intuitive content (empirical or phenomenological, you might say) to our understanding of the concepts of ἀγαθός and φίλος. Socrates has already “proved” that the good is not φίλος to the good and the bad is not φίλος to the bad. Assuming nothing can be φίλος to the bad and having shown that the neither good nor bad cannot be φίλος to something to which it is ὅμοιος, then the sole remaining alternative is that the neither good nor bad is φίλος to the good/ beautiful and to that alone. It is ambiguous whether the abstract terms employed at this stage of the argument mean an individual thing characterized by that term (“the good” means “a good thing”) or whether they are meant to stand for abstract entities, a sort of dress rehearsal for the full blown Theory of Forms. The same ambiguous use of abstract referring terms occurs in Theaetetus 190 B specifically in the context of whether a thing qualified in a certain way or alternatively the abstract quality itself can be qualified by an ἐναντίος (Can something be beautiful in an ugly way or can beauty be ugly? These are different questions). (Cf. in this regard Cornford, pp. 118-119. On p. 185 Cornford clarifies his repeated claim that Plato deals with the Forms themselves and the relations between the Forms. He does not mean that Plato explicitly addresses or even recognizes order distinctions such as Husserl dealt with or Russellian type distinctions. Plato does not come up with a theory to the effect that calling beauty ugly or beautiful or saying that nothingness exists may or may not be a type or order error. Rather Plato is concerned with the definability of a Form understood as a species and in particular a lowest level “indivisible species” (p. 186).  The problem Plato addresses is the lexical definition of lowest level species; lexical definitions are subject to either circularity or indefinite regress. Oddly Cornford’s rendition of Plato, possibly enticed by that temptress of a term, γένη, is Aristotelian in the extreme. It may perhaps benefit from heed paid to Vlastos’ observation that Plato’s views are expressed ontologically and not in a framework that we have come to recognize as logical. Things partake of Forms, which is not the same as being subsumed under a concept. In this framework a Form can partake of a complementary Form for the simple reason that it does not partake of itself. Plato lights on this solution to the paradox of the existence of non-being at the end of The Sophist and thereby anticipates Aristotle’s logical solution.)

Taking this as his new assumption, namely that the neither good nor bad is friendly to the good/beautiful and to that alone, Socrates proceeds to introduce a general notion of reason or cause into the equation. The operative term is διά which can mean a kind of immediate causation as where Hesiod says that various immortals fuck διὰ…Ἀφροδίτην. In the present context it also functions as a reason for a certain attitude or behavior. Healthy men are not friends to doctors διὰ τὴν ὑγίειαν. But sick men are friendly to doctors διὰ τὴν νόσον. However, a sick man (or his body) cannot φιλεῖν a doctor once it has become bad (γενέσθαι ἀυτὸ κακὸν) since it has already been proved that κακοί cannot φιλεῖν good things. Of course it is doubtful whether a sick man is κακὸς in quite the same way that what causes his illness or his illness itself is κακὸς, if indeed he is κακὸς at all. Accordingly Socrates introduces a distinction between a distinct "something" causing a state or an apparent state in a body (for example) and the body being τοιοῦτον, i.e. relevantly the same as whatever causes its state. Only when a “cause” (διὰ τὴν…) is present in a body “in a particular way” (κατά τινα τρόπον παρῇ) can the body be τοιοῦτον as the cause. Thus at one stage sick men are not evil like their sickness because the sickness is not yet present in them in the relevant way and we can say that they φιλεῖν the good, viz. medicine or doctors, without contradicting the previous argument.

This does seem to be somewhat of an ad hoc solution but Socrates applies it immediately to an important issue which in other contexts he would formulate as part of a solution to philosophical problems surrounding goodness and φιλία. By this reasoning, he says, wise or knowledgeable men (σοφοί) cannot be φίλοι of σοφία (φιλοσοφεῖν). Also the ignorant (ἔχοντες…ἄγνοιαν) cannot φιλοσοφεῖν, because their very ignorance makes them κακοί. Observation confirms that the ἀγνώμονες and ἀμαθεῖς in fact do not. Socrates again invokes the ad hoc solution to the effect that certain men are ignorant but are not yet bad because they are aware that they are ignorant. These are the ones who truly φιλοσοφοῦσιν. In all cases (κατὰ τὴν ψυχὴν, κατὰ τὸ σῶμα καὶ πανταοῦ) only the neither good nor bad φιλεῖν only the good and that is because (διὰ) the presence (παρουσία) in them of something bad.

6. At this stage Socrates again introduces the notion of reasons for φιλία. Recalling his discussion with Menexenus about parental love, Socrates uses the terms “for the sake of” (ἕνεκα) and “because of” (διά). As before Socrates doesn’t actually present any arguments to the effect that φιλία cannot be simply disinterested (And certainly there is no logical compulsion to require a reason for φιλία interpreted as “loves” as in the example of parental love. Socrates’ earlier qualifications of parental love in terms of the usefulness of the children to the parents were, as he presented them, no more than possible reasons for certain sorts of parental love – not part of the grammar or semantics of the word “love.” Interpreted as the verb “values,” however, on some understandings of “to value,” φιλία is incomplete and requires an answer to the question “Values for what?” in order to be fully understood.). Rather he simply asks Lysis to agree to the view that φιλία of one thing is always for the sake of a second thing and because of another thing. Nevertheless Socrates does pursue a reductio argument against the necessity for a reason by examining troublesome examples and pursuing the consequences of assuming that there always has to be a reason for φιλία.

Socrates’ example of a reason for φιλία is a man’s φιλία for a doctor. We φιλεῖν doctors for the sake of health and because of illness. This example is a bit hard to fathom for reasons that may not be entirely related to cultural differences.  Plato also doesn’t distinguish between doctor fanciers in the sense of horse fanciers and those who simply approve of doctors because of their healing powers (or indeed an individual patient who φιλεῖν a specific doctor for the sake of his personal cure). As a result the latter get lumped together with horse fanciers. (The distinct concept of doctor lovers in the sense of doctor groupies or in the yet different sense of those who love doctors for their money (Are you listening, Bree?) probably exhibits a much lower degree of instantiation in real life but is nevertheless a lexically legitimate parsing.) Regardless, Socrates’ assertion is that a body that is not sick is neither good nor bad and so fits the criterion that concluded the last argument to the effect that only things that are neither good nor bad could φιλεῖν and the only things they could φιλεῖν are good things. Doctors and medicine count as good things. Disease is obviously a bad thing and health a good thing (at least to the diseased or healthy individual himself in most instances; the disease of a brutal dictator might count as a good thing to his victims; indeed a masochist might consider his own disease to be a good thing). So, just as a man is a φίλος to medicine, he is also a φίλος to health. In this case both the first thing (i.e. medicine) and “that for the sake of which” (i.e. health) are good and both are φίλοι (ostensibly only as φιλούμενοι). Socrates implicitly generalizes on this example and very nearly concludes that in every case φιλία is a three way relationship always involving a “for the sake of” which is also a φίλος.

That conclusion is just a stalking horse, however, for Socrates immediately launches into two objections to its validity. The first is that a man and health, on this understanding, are φίλοι since they are φίλοι to each other. As φίλοι, however, they are ὅμοιοι and he has already shown that ὅμοιοι cannot be φίλοι. (Note the criss-cross of first and second order predication). The second objection is that the conclusion leads to infinite regress. For it must also be for the sake of something else that a man is φίλος to health, and so he must be φίλος to that other thing and for the sake of yet another thing and so on. Infinite regresses, both Socrates and Lysis agree, are utterly exhausting. A better way of putting it might be to observe that they have very little value in explaining a phenomenon or concept, since the answer to the question “for the sake of what?” in this case is never complete.  So Socrates invites Lysis to hypostatize a beginning (ἀρχὴ; to translate ἀρχὴ as “first principle” is way too Aristotelian), an initital link in the chain of φίλοι that is the ultimate φίλος (πρῶτον φίλον) – a φίλος not for the sake anything else and for whose sake we are φίλοι with all our other φίλοι in the regress.

In the Philebus (12 E ff.) Plato considers the oppositeness (ἐναντίος) of certain things like ἀκολασταίνοντα and σωφρονοῦντα (12D) but quickly asserts this is more properly a case of simple dissimilarity orunlikeness (ἀνόμοια – a crucial distinction based on a term that doesn’t appear in the Lysis) and further asserts that only their sources are really opposite. The following pages constitute a dialectic of the like and the unlike which is resolved by saying that unlike items such as different kinds of pleasure are alike in that they both have a common element (ταὐτόν, 13B), e.g. pleasure or goodness. The idea of opposite sources as a qualifier of opposition is the reverse side, so to speak, of the idea of “for the sake of which” in the Lysis.

Using a very strange if portentous example of sons, hemlock, wine and cups and the rather more reasonable example of gold and silver about which we make much of (περὶ πολλοῦ ποιούμεθα), Socrates argues that all those things we consider φίλοι for the sake of something else are not really φίλοι just as we don’t value gold for itself but only, presumably, as currency for the acquisition of things we really value. The only real φίλος is the unspecified ultimate φίλος. Furthermore, assume that this ultimate φίλος is the good (τὸ ἀγαθόν). The good is a φίλος because of the bad (since it apparently is a curative (φαρμακός) for the bad). Socrates invites Lysis to imagine a world without anything bad in it. But then the good would be of no use (οὐδεμίαν χρείαν ἔχει) and so presumably would disappear the reason for its being a φίλος. Assuming the tripartite structure of friend, reason and cause, the good under those circumstances could not be a φίλος, however counterintuitive that conclusion may be. At this point Socrates crosses himself up since he begins (220D) quite reasonably by saying that the bad functions like illness in the structure of φιλία, viz. that the bad is a cause (διά) of our φιλία with the good. But he seems to forget that ἕνεκα and διά mean two different things because he interprets that to mean that the good is a φίλος for the sake of (ἕνεκα) the bad. The conclusion without the mistake would be that, if we assume that there is no reason for the good to be a φίλος and if we assume that all φίλοι must be so for a reason, then the good cannot be a φίλος. That is sufficient to disqualify the three part (or even two part) structure of reasons for φιλία, because good is clearly a φίλος and remains a φίλος even in the absence of reasons or causes for its being so. Socrates does not entertain the possibility we would have thought to be a natural in Platonic philosophy that the good is the ultimate φίλος (πρῶτος φίλος), but he does seem to be in a hurry to reach his conclusion at this stage.

If reasons do not provide an explanation for the occurrence of φιλία, perhaps there may be another explanation (ἄλλη τις αἰτία). At this point Socrates observes that we can have desire (ἐπιθυμία) even in the absence of the bad. We can be hungry or thirsty in a world without the bad and under those circumstances those desires could be considered as solely beneficial to us (Apparently βλάπτειν under such circumstances is not κακόν). Desire, Socrates asserts, is neither good nor bad, so it is possible to desire even in a world where the bad does not exist. Accordingly Socrates suggests ἐπιθυμία itself as an alternative explanation. On this basis he who desires (ἐπιθυμοῦν) is a φίλος to the person desired (τούτου…ἐπιθυμεῖ).

This stage of the argument consists in the sudden irruption of an entirely new set of concepts that hadn’t been entertained so far. In the first place, Socrates calls ἐπιθυμία a cause (αίτιον) of friendship in a sense not quite the same as ἕνεκα or διά. Desire is not a good state like health for the sake of which a person has φιλία; nor is it something bad like illness that compels us to φιλεῖν someone else in order to get rid of it. Rather the state of desire seems to be either a psychological mechanism that explains why someone should φιλεῖν another person or, as more explicitly stated by Socrates, a justification for a φιλούμενος to return the φιλία of the φιλῶν.

In Laws Bk VIII 836E-837B Plato appears to revisit the issues in the Lysis, viz. the types of affectionate relation between equivalents (ἴσος not ὅμοιος) and opposites (ἐναντίος). He provides clarification of his exact meaning in this dialogue of φιλία, ἐπιθυμία and ἐρώτοι. Although he mentions the three formulations, the Stranger quite clearly states they are (to him at least; there is no guarantee that anything the Stranger says in Laws is meant to be consistent with what Socrates says in the Lysis) just one word (ἓν ὄνομα) and that, whatever this word means, that is the meaning of all three formulations. So, although the passage distinguishes between physical and spiritual affection, the formulations are not sorted according to that distinction. Echoing Socrates, the Stranger distinguishes between affection or valuing between equivalents and the same phenomenon between opposites. However, unlike Socrates he places an uncompromising positive spin on the former and an equally uncompromising negative spin on the latter. Even the value the poor place on the rich is terrible and bitter and seldom reciprocated. A clue to the distance of this passage from the Lysis lies in the choice of examples. Could the Stranger argue convincingly that the value a sick man places on a doctor is terrible and bitter? Affection between equivalents, on the other hand is gentle and forever reciprocated. The Stranger then makes another move that has no precedent in the Lysis. He identifies physical affection (by which he clearly means butt fucking - μίμησιν τοῦ θήλεος 836E) with the affection of opposites. He makes his claim without support of argument and possibly expects us to understand that the penis is ἐναντίος to the vulva, or in this case the butt hole. But by that reasoning instructing in virtue and learning virtue (Admittedly no reference is made to instruction in this passage; the operative state is living together in chastity) are also ἐναντίος. In addition, if the man and the boy are both in heat, the Stranger’s logic should lead us to the conclusion that they are ἴσος in that respect. These differences could mean that Laws is inconsistent with the Lysis or that that it represents a further development of Plato’s views and a modification of what was implicit in the Lysis or else that it draws conclusions that aren’t mentioned in the Lysis but are compatible with any views expressed in the earlier dialogue. In any event, we have to be careful in comparing the early and late Plato.

There appears to be a general presupposition among scholars who take notice of the Lysis is that its subject is φιλία glossed as friendship and that the conclusions of this dialogue are the same as or at least consistent with conclusions made more explicit in Laws and Aristotle’s Nicomachean Ethics. In point of fact I find most of the standard literature on the Lysis, which invariably deals with the nature of “friendship” (Grote being a notable exception), uninspiring in the extreme. Levin provides as good a bibliography of this literature as any (For one example of this cf. below on “Ethical Issues in the Lysis”). My reading is that the actual or primary subject of the Lysis is the conceptual structure of Presocratic cosmology and ontology and, in dealing with this subject, issues of love and friendship appear more by way of example, perhaps to lend some human drama to what could very quickly become a dry subject. Aristotle was particularly prone to this sort of practice. We often witness him using important and controversial philosophical issues as examples to illustrate purely logical points in the Organon and elsewhere.

In Books VIII and IX of the Nicomachean Ethics Aristotle discusses issues surrounding ϕιλία that are very similar to those raised in the Lysis. However, as I just noted, Plato’s main concern was “logical.” He wanted to tease out the relations involved in likeness and opposition. In that case ϕιλία occurs more by way of an – albeit important – example and illustration. Needless to say, If Vlastos is right in that for Plato all intra-human ϕιλία is ultimately based on utility and the only real ϕιλία is with the Form of the Good (Note Aristotle’s sarcastic comment on a possibly less than completely friendly relation with the good itself in VIII vi 4), then Plato would not have agreed with Aristotle’s view that reciprocal uninterested ϕιλία sits at the top of some sort of hierarchy. It is difficult in Aristotle, here as elsewhere, to determine whether he is just reporting his contemporaries’ opinions and presuppositions on the matter of ϕιλία and subjecting them to a conceptual examination, or whether he is espousing a possibly contrarian doctrine in his own name. Notice in VIII iv 6 and VIII viii 7 Aristotle repeats a move familiar from his logic, physics and metaphysics when he applies logical analysis to ethical concepts, presumably derived from Plato. Note also in VIII v 4 Aristotle uses a term that never or rarely occurs in Plato - ϕίλησις.

Another concept Socrates throws in practically without introduction is ἔρων or more specifically erotic or physical love (221 B).  Not only is its appearance sudden and unexplained, the very concept doesn’t seem to mesh well with parental φιλία or φιλία for a doctor or φιλία for horses. (Well, it could mesh with parental and horse φιλία in a society tolerant of incest and bestiality; but it does seem totally irrelevant to a discussion of the reasons stated in the dialogue, viz. self-interested reasons, for φιλεῖν a doctor or medicine unless erotic pleasure is construed as a self-interested reason.) However incongruously, Socrates proceeds to list ἐπιθυμία, φιλία and ἔρως in a row (221 E) as related if not identical concepts. The point seems to be to enable him to return to the not irrelevant dramatic context that spurred the discussion. Hippothales does not just φιλεῖν Lysis, he ἐρᾷ him. So at this stage it appears that the φιλούμενος must return the φιλία of the ἔρων (the erotic lover) because of the ἔρων’s desire. With that conclusion, it further appears that Socrates has accomplished the task he assigned himself for Hippothales’ sake, namely that of wooing Lysis by means of argument and not poetry. And the reactions of the trio to this stage of the argument indicate that they understood its implications.

But Socrates does not stop there, nor should he since this conclusion is just as specious as the others. He brings up two new words (ἐνδεές and ἀφαιρῆται) which may both signify the single concept of want or lack or deprivation. This concept clearly recalls the standing discussion of reasons. It functions in much the same way as ἕνεκα. That is, it explains the choice of a particular individual as a φιλούμενος. The much more mysterious and difficult concept of οἰκεῖός also makes its appearance.  Two things are οἰκεῖοί if they are appropriate to each other or fitting or belong together. One might even say they are destined for each other. This is certainly how Socrates understands it when he speaks of things as φύσει πῃ οἰκεῖοί, i.e. united by some sort of natural bond. Accordingly, just as the idea of deprivation takes the place of “for the sake of,” the idea of a natural bond takes the place of similarity. Φίλοι are οἰκεῖοί, not ὅμοιοι. The way is prepared for uniting Hippothales and Lysis with the conclusion that someone who desires must, because of a deficiency that can only be filled or satisfied by the object of his desire, be united by a natural bond to that object. But furthermore the concept of οἰκεῖός is reflexive and reciprocal; the ἔρων and φιλούμενος are bonded naturally. Therefore the object (in a nod to the actual social situation referred to as a παιδικός (222 B)) must return the φιλία of his suitor. (This is a point where Hoerber is grossly inaccurate and perhaps mendacious. The natural bond Socrates hints at is with Hippothales and not with Menexenus or with Lysis’ parents who have dropped out of the discussion altogether. The reciprocity is not some sort of supreme virtuous meeting of the minds, such as the Athenian Stranger might advocate and Aristotle describe, but rather a fit between Hippothales’ desire and some quality in Lysis that matches Hippothales’ desire, somewhat like the match between a nut and a bolt or a penis and an asshole. For more on Hoerber cf. below.)

I don’t have to spell out the faults of this line of argument. Socrates in fact finds several on his own which undercut the case for Hippothales that he had apparently tried so carefully to build. If, Socrates argues, there is no difference between being similar (ὅμοιος) and being “fit for each other” (οἰκεῖός), then being fit for each other is subject to the same objections he had already brought against similarity. It is not clear how this could be the case without robbing the concept of being fit for each other of all of its original intuitive meaning, but the assumption Socrates entertains does seem to be that οἰκεῖός is strictly synonymous with ὅμοιος. The argument is that naturally bonded things or persons are so much alike that neither one can provide the other what he needs and so the relation does not live up to the criterion of usefulness. If, on the other hand, they differ in meaning, then the concept of being fit for each other falls into the following alternatives:  either the good is φίλος to everything or φιλία is a relation that can hold only between things that are equally good, bad or indifferent. Yet these alternatives have already been shown to be unacceptable.

Socrates concludes with a profession of complete befuddlement that reads not unlike the end of the Parmenides.  These contraries such unity do hold, Only to flatter fools, and make them bold!

Socrates proves all sides of the question of whether likes love likes or unlikes (I use the verb “love” in this instance because the English word “like” would make the phrase even more confusing than it already is). In this respect he occupies exactly the position of the sophist he criticized in so many other loci. From another perspective he may be said to act as a sort of negative sophist since what he actually does is disprove all sides of that question. It appears that Socrates’ goal is to cast doubt on the entire conceptual framework of contrapositivity as a way of explaining affective relations such as ϕιλότης. He goes so far as to name the framework (216B) κατὰ τὴν ἐναντιότητα. In a broad sense the issue is whether thinking κατὰ τὴν ἐναντιότητα is valid at all.

There is good reason to believe that Socrates’ performance is as much meant for comic effect as to draw any serious philosophical conclusions. The image it conjures up is one of the lovers looking on in growing consternation while trying to fondle each other behind their rivals’ backs. And, even though the conclusion is consistent with the view that true love or something approaching true love expresses itself in a desire for the moral improvement of the beloved that is expressed throughout Plato’s writings, the idea that moral improvement could result from the sort of dialectical display Socrates puts on, derives from rather shaky assumptions about the psychology of daily life, to paraphrase Jeeves. Speaking personally, I have found that trying to impress a hot babe by showing off my debating and analytical skills in any setting where that wasn’t the point of the exercise is to court disaster. She is just as likely to feel resentfully inferior or turn judgmental (in our culture – or biology? – where males assume the role of performers for the approval and sexual selection of the female attendees, this has become a female specialty). I’ve found a much sounder strategy is to have someone sing your praises intellectual or otherwise behind your back despite the possibility of the John Alden syndrome. Ultimately anything you do by way of intellectual performance is liable to work only in cases where she has already selected you from out the herd for a good fuck.

So is there any point at all to this apparently silly dialogue? Or at least a point that addresses issues that philosophers today, or at the very least classical philosophers today, see as relevant? After all what does it mean to question whether things that are similar or dissimilar or utterly similar or totally opposed should love each other or not? Why do we care? The Lysis belongs to that group of mostly early dialogues that are almost entirely negative - that are concerned with criticizing and refuting existing cosmological and philosophical systems in such a way as to set the stage for Plato’s own theories. The Lysis in particular is concerned with criticizing Presocratic concepts such as likeness and opposition and theories of relations between things that rely on affective notions such as ϕιλότης and νεἶκος. Plato’s purpose is to replace the reigning cosmologies and theories of human interaction based on likeness and dissimilarity with his own construct and also to show the logical incoherence of applying those affective notions to pairs of similars and opposites. And even though he would walk back some of the criticism in the post-Symposium dialogues, the negative Socrates influenced dialogues still constitute a needed prolegomenon to the mature theory. In order to understand the implications of Plato’s argument let’s briefly review the relevant doctrines from his predecessors, the Presocratics.

Where Did the Doctrine of Opposites and its Psychological and Social Gloss Come From?

The nature of similar or like (not identical) and opposite (not different) things and how they interact was an issue in Greek science and philosophy before Plato. As is the case with the nature of “all being,” the Presocratics can be distinguished among each other according to the distinct positions they staked out regarding the nature of similarity and opposition (Cf. Reinhardt pp. 238-241 for an interesting historical genealogy of opposition and opposites among the Presocratics).


The initial statement of the Greek problematic of opposites appears by most accounts in Anaximander. The idea may be that what we recognize as logical difficulties in the concept of everything (τὸ πᾶν) (Cf. The Pseudo-Plutarchean Stromateis 2, DK 12A10 and also Kirk’s juxtaposition of two passages in Aristotle, p. 110) can be resolved by dividing everything into its component parts, namely the like and the opposite (Both terms are vague at this stage and they are not true complements). Anaximander’s move strikingly foreshadows Plato’s own. The highest Forms as they emerge at the end of The Sophist are devised for what appear to be the same reasons.

This view needs to be qualified, however, in two ways. First the naming of opposites, as distinct from their conceptualization as opposites, occurs as early as Hesiod’s tale of the coupling of earth and sky in the Theogony – assuming of course that earth and sky really were viewed as opposites (Hesiod obviously does not use the sort of classifying language employed by philosophers). It is notable that the interaction of earth and sky in Hesiod has not only theogonical but also cosmogonical consequences. If the early φυσικοί were indeed less interested in the origins of the gods than they were in the origin of the physical universe, Anaximander at least appears to have imported the conceptual framework of the interaction of opposites as a way of explaining physical events, particularly generation and destruction. (Bollack (Empédocle I  p. 32 ftn. 9) associates the opposites in the Presocratics with their theories of becoming, i.e. physical or cosmological creation and change.) The Hesiodic source suggests that the idea of interaction and the possible role that opposition plays in interaction was already a part of Greek religion, for it is reasonable to suppose that these ideas were simply codified by Hesiod from pre-existing myth. Furthermore, the sexual interaction in Hesiod is not carried over into Anaximander’s account. It would take the ingenious Empedocles to revive the sexual terminology which Plato accepted only half-heartedly and mostly in order to defuse the sexual overtones.

The second qualification is that much of what we know about Anaximander comes from Aristotle, Theophrastus and Simplicius, none of whom took excessive care to cite his texts directly. It is difficult to state what Anaximander’s actual ideas were with any clarity since most of their renditions are permeated with Aristotelian terms and concepts in such a way that the line between reporting and reconstruction is never clearly drawn. Obviously Aristotle had begun to lay out clear distinctions between logic, physics and metaphysics that are not apparent in Anaximander. Specifically no distinction between complements and contrapositives was made until at least Aristotle’s vastly confused attempts (and indeed, some of his distinctions being logical and not ontological, they not entirely relevant to Anaximander). For that reason what appear to be logical conclusions are drawn about the nature of physical phenomena, such as the thought that fire is not just opposed in nature to water but that fire and water together (or some like combination of the basic elements) comprise the entirety of physical phenomena presumably because they are the complementary components of the entirety of τὸ ἄπειρον. Contrapositives are treated as if they have a stronger logical relationship to each other than they actually do. More importantly for this discussion, we cannot really tell whether the ideas attributed to Anaximander would have been recognized as such by Plato or that Plato would have even recognized that his own treatment of opposition hearkened back to a conceptual framework originating in Anaximander. Nevertheless, if Aristotle is at least partially correct in his history of philosophy, then Anaximander is indeed the source of the problematic of opposition as a theoretical and not a mythological set of ideas. And he also gives the reason why opposition should be considered an important issue, for it provides a solution to problems in Thales’ groundbreaking theoretical cosmology.

Yet even though Anaximander speaks of things we regard as opposites, he never in actual quoted material used an abstract term for opposite or opposition. Indeed abstract terms only made a first hesitant appearance with Empedocles and weren’t catalogued until Plato. Aristotle (Cf. De gen. et corr. B5, 332A19, Phys. A4 187A12, Phys. Γ 5, 204b22, Phys. A4 187a20 and Simplicius Phys. 24, 21) uses the words ἐναντιότητος and ἐναντίων in speaking of Anaximander. But he regards an opposite in this case as a στέρησις or privation, a gloss which is more consistent with his own logical theory of complementarity than a Presocratic way of thinking or talking. Moreover, the comments in Aristotle’s Phys. Γ are better applicable to Empedocles than Anaximander and may very well mistakenly assimilate τὸ ἄπειρον with Parmenides’ τὸ ἕν. In this regard Kirk’s caution (pp. 119-120 and 121) to the effect that the imposition of abstractions like ἐναντιότης on Anaximander’s concrete terminology is a Peripatetic addendum is probably justified.

In order to understand the significance of opposites for Anaximander we need to start with a somewhat crude but serviceable restatement of what may have been Thales’ ideas. The doctrine of interest is that water is everything. Thales, of course, never said that water is everything. Indeed, lacking actual quotations, we have no evidence that he said anything at all. Our sources are entirely doxographical commentary. Aristotle attributed two views to Thales. The first is that the earth rests on water, an empirical theory. The second is that water is the ἀρχή of everything else.  This is the view that could be interpreted as an ontological generalization. We don’t know whether Thales meant that other things that don’t look like water are water despite appearances or that all things, though different from water, could spontaneously revert to their watery state or that water is the cause or material source of things that are by nature different from water. But either of the first two meanings constitutes a universal ‘pataphysical claim and the third one may as well. It is impossible to say for sure whether the other Presocratics also held what would be Aristotle’s views of what Thales taught. But there does seem to be some evidence that they proposed other candidates for the universal “substance.” In this light Anaximander’s was τὸ ἄπειρον (Kirk 103 includes the relevant passages from Simplicius, Hippolytus and the Pseudo-Plutarchean Stromateis; cf. also Augustine De civitate dei VIII,2), a term that is significant in that it was the only candidate that did not function as a name for a physical element as well. The fact that τὸ ἄπειρον is also ambiguous and could be taken to mean both spatial and sequential (numerical) infinity (or indeed continuity in the sense of mathematical density (Cf. Graham and Kantor, Naming Infinity p. 21)) and conceptual indeterminacy raises an important and complex issue in the history of Greek philosophy, an issue whose satisfactory discussion would lead us too far afield in this context. Anaximander’s argument begins with the assertion that fire cannot be reduced to water because the two are “opposite.” The apparent reasoning is that, if fire were really water, fire would not exist at all. To generalize on that argument: no specific element (στοιχεῖον) could be universal because every specific element has an opposite. Rather, specific substances and their opposites are aspects of τὸ ἄπειρον (It is unclear whether this only means they are physically caused by or generated from τὸ ἄπειρον or whether it also means that their concepts are subsumed under the concept of τὸ ἄπειρον; it could also mean that the concepts involved in the opposition are indeterminate and that the indeterminacy in these examples of opposites renders any definition or even description of τὸ πᾶν indeterminate as well), the only “cosmological” principle (ἀρχή) or phenomenon (ϕύσις) or underpinning (τὸ ὑποκείιμενον -  in Phys. 24,21 Simplicius uses this ultra Aristotelian term) that does not admit of an opposite. Apparently the concept of opposition is not generated by an argument; rather it is a presupposition of the purported argument against Thales’ aquatic monism. That fire is opposite to water and therefore cannot be “reduced” to water since opposites cannot be reduced to one another is stated as a fact (and not as the conclusion of a deduction) in order to serve as a counter to the universal assertion that everything could be reduced to water. It was simply obvious to Anaximander that fire could not. (If Anaximander really employed these concepts and really argued in this way, he operated at a level of philosophical sophistication that was well beyond his contemporaries - even, to a degree, Plato and the Pythagoreans. Which is one reason why the historical Anaximander probably didn’t. Kirk (p. 130) states succinctly that the nature of τὸ ἄπειρον was probably not clearly defined in Anaximander’s own mind, to which I might add that an important source of the confusion may be the lack of a clear distinction between semantic indefiniteness and physical or mathematical (serial and/or recursive) infinity. Aristotle introduces another source of vagueness by not distinguishing between the meaning of “indefinite” (ἄπειρος) as “cannot be defined” and indefiniteness as a feature (perhaps the only feature) of an otherwise unspecified ὑποκείμενον underlying the στοιχεῖα. Ultimately even Aristotle failed to grasp the significance of this critique of a universalizing ontology or ‘pataphysical theory (as opposed to a theory of only the physical world which, assuming that the physical world was not everything, would not get dragged into logical problems) that he attributed to Anaximander. For our purposes the ultimate source of Plato’s concern with ἐναντίοι as a philosophical issue, and the association of this issue with physical substances, lies in Anaximander.

We should keep in mind that none of the later commentators actually use terms for opposition in specific reference to Anaximander. Simplicius does use the phrase τίσιν ἀλλήλοις (103 Ibid.) which implies reciprocity if not outright opposition. So the relation to Plato’s discussion of ἐναντίοι is indirect but, I believe, correct, again assuming that the Aristotelians’ beliefs about Anaximander were shared by Plato avant la lettre. Indeed, Kirk (p. 118) does gloss τίσιν ἀλλήλοις   as “opposed substances.” On the other hand, in his general history of the Presocratic φυσικοί including Anaximander, Aristotle actually employs the term ἐναντιότης (De gen. et corr. B5, 332a19 and Phys. Α4, 187a 12). He seems to believe that τὸ ἄπειρον comes from adding ἐναντιότης to one of the στοιχεῖα. In the Physics he mentions Anaximander by name in saying that ἐναντίοι are separated out from “the One” (τὸ ἕν). The implication is that the One is a gloss on or an alternative to Anaximander’s τὸ ἄπειρον. In Phys. Γ5, 204b22 and Γ4, 203b15 a doctrine of cosmic contrapositives (The actual context is an argument from the impossibility that any one of a pair of opposites understood as contrapositives or ἐναντίοι could be ἄπειρος since by definition its counterpart would be destroyed, i.e. would not exist) is attributed to Anaximander by Aristotle without an exact quote. Aristotle’s reason is compelling although it involves a mish mash of physical terms and logical argument: If any one side of a pair of opposites (such as water) were all that existed, i.e. be ἄπειρος, then it would “destroy” its contrapositive counterpart. Phys. Γ4, 203b15 is subtly different in that it presumably takes τὸ ἄπειρον to be an “infinite” (in the sense of inexhaustible or unending as opposed to the logical or, more properly speaking, semantic sense of indeterminate, although this passage could also be interpreted as making a semantic point) something from which (other) things appear (ὅθεν ἀφαρεῖτα boldly and misleadingly translated as “source” by Kirk). The idea behind both is that a totalizing concept such as Thales’ water cannot explain the existence of something opposite (ἐναντίος) in nature to itself or explain the (quite different) phenomenon of coming to be and disappearance.

The claim that the views expressed in Aristotle’s Physics are actually what Anaximander meant is supposed to be supported by a direct quote in Simplicius (Phys. 24, 21). However, the sole justification for calling the beginning of the fragment a quoted text is Simplicius’ qualification of it as “rather poetic” (ποιητικωτέροις).  Moreover, even if it does contain Anaximander’s own words, no mention is made of opposites. Indeed it may be, as Kirk observes (p. 114), that Aristotle was setting Anaximander up as a sort of straw man for easy refutation. Whatever the outcome of the debate concerning Anaximander’s actual view, what is important from the standpoint of a history of philosophy is that no doctrine of opposites is actually stated verbatim by Anaximander as opposed to being attributed to him by later commentators in suspiciously Aristotelian terminology. Nevertheless many of the ancients and their commentators believed that the concept of cosmological opposites originated with Anaximander and that belief may have been current as early as Heraclitus’ active lifetime. So, if Heraclitus himself did not first articulate this doctrine, somebody before him had to be the source. (Hippocrates’ reference to τοῖς παλαιοῖς is no more definitive than the comments from the doxographical sources.)

It may also be that Anaximander actually identified the opposites that arose from τὸ ἄπειρον. The general belief (summarized by Kirk pp. 119 & 133) seems to have been that he  considered pairs of opposites to include at least hot/cold and dryness/moisture, and that he adopted these opposites from the prevailing cosmological framework of Greek religion as exemplified in Hesiod. Although Kirk raises the possibility that cosmological concepts so abstract may be Aristotelian overlays, the Lysis itself testifies to the fact that they were in philosophical use prior to Aristotle. Kirk's quoted source is Heraclitus by way of Hippocrates (de carnibus 2 ibid.), so the retrospective attribution of these pairs to Anaximander is, like the attribution of the concept of opposition in general, unavoidably speculative.

Actual likes and opposites were certainly central to the philosophies of both Heraclitus and Parmenides as well Anaximander. But an important caution that cannot be reiterated often enough is that neither Anaximander nor his successors actually used terms for likeness or opposition. Rather they identified entities – hot and cold or fire and water – that Plato would come to regard as examples of what he identified as opposition. Ultimately the codification of opposites into inclusive categories such as ὅμοιοι and ἐναντίοι would have to await Empedocles (Kirk p. 119 speaks of Empedocles as "codifying" what his predecessors said about opposition).

It is interesting that Simplicius (121) regards opposition as an alternative physical theory to the Aristotelian theory of a substratum.  Seen in this way it may have some contemporary viability. Could there be some sort of dynamic of conceptual or physical opposition that could serve even today as a viable alternative to Aristotelian ideas about matter or substrate and form or other candidates for a general ontological framework, such as Plato’s theory of Forms? Simplicius’ suggestion also gives another more “material” way of understanding the concept of ἄπειρος. It may also mean that a substratum (Anaximander evidently had Thales’ water in mind, but the argument could be applied to the generalized concept of a ὑποκείιμενον understood as the material ὕλη of a thing) for physical change alternatively either cannot be specified or simply does not exist. Understood in purely material terms, however, this view forfeits its logical justification (although there is nothing to prevent the logical and the material interpretations from coexisting, if not actually interpenetrating). (This interpretation would also have a decidedly negative impact on Quine’s solution of the Heraclitus river problem by way of distinguishing definite and indefinite singular terms. Cf. Word and Object, pp. 116 and 171.)

Anaximander’s opposites include coming to be and destruction themselves, an obvious pair for any doctrine that applies opposites to cosmogony although no Presocratic, indeed no philosopher until Hegel as far as I can tell, speculated about how the abstract entities coming to be and destruction themselves might interrelate. The idea of a "second order" theory has to await Empedocles for a rough adumbration. Anaximander does seem to regard some of the elements as opposite to each other. However, the opposition is not bipolar (Cf.  Aristotle, Physics, Γ5 204b22) but many valued, since it just seems to mean that different elements can have differing and mutually exclusive properties. If anything it is the properties (dry/wet or hot/cold, for example) that are the true Anaximandrean opposites.

The other theme that made its first appearance in a philosophical context in Anaximander was the use of anthropomorphic language (the source again probably being Hesiod), viz. the language of human social interaction, to describe physical events. The evidence, again a bit slim, relies on the terms chosen by post-Theophrastian commentators. Simplicius (Phys. 24,17) speaks of the physical destruction (φθορά) of existing things (τοῖς οὖσι) as a kind of temporal retribution or penalty (δίκη) for their mutual injustice (τίσιν ἀλλήλοις τῆς ἀδικίας). Hippolytus adds (DK 12A11) that τὸ ἄπειρον is (presumably) undying (αἴδιον) and does not age (ἀγήρω). This latter characterization implies that Anaximander’s understanding of τὸ ἄπειρον was (largely?) ontological, indeed biological, and not logical.  Yet if these comments are in any way historically accurate, then Anaximander set the stage not only for continued use of anthropomorphic metaphors by other Presocratics but also for Plato's elision from physical or cosmological opposition to opposing human personality types and social categories.

It is worth explaining what I mean here when I say “logical” since I will quite often refer to a distinction between the logical, the ontological and the epistemological, a distinction that obviously did not appear in the writings of Plato and the Presocratics, and also because many philosophers, particularly those in the ordinary language tradition, call “logical” a number of views and observations that they make which are better characterized as grammatical, semantic or the products of a phenomenology of the meanings of certain words. Of course, “logical” in this sense is related to “logical” in the sense of having to do with deductive logic since it is the meanings of words that make non-purely formal deductions possible. In a broader sense the meanings of words also make possible purely formal deductions if we agree that these are based on contextual definitions of logical connectives. But it is perhaps wiser to limit the term “logical” to treatments of terms we currently regard as contextually defined by the axioms of a system. There is a properly logical dimension to some issues of opposition that arise because the terms at issue are defined by formal systems. Both complementarity (i.e. specific otherness) and contrapositivity as Aristotle understands them are more or less clearly defined in the Prior Analytics. Moreover, modern logic has concepts that are quite similar and certainly genetically related to Aristotle’s concept of complementarity as expressed by the term ἕτερος. Complementarity is embodied in the notion of a complementary set or class and expressed by the logical operator negation. Aristotle’s understanding of ἐναντίος, however, does not capture the perhaps irreducibly ambiguous concept of contrapositivity as Plato tended to use it or as we use it when we say that good and evil or black and white are contrapositive concepts. “All porn stars are babes” and “No porn stars are babes” are contrapositive propositions (where propositions are meaningful sentences that can be uttered) in the system of the Prior Analytics. But only the lesbo left would consider porn stars and babes to be contrapositives in the way that good and evil are supposed to be contrapositives.(Link to Mackie)).


Heraclitus developed the doctrine of cosmological opposites in a number of ways. First he specified heat and air (αἰθὴρ) or, more pointedly, fire as at least one member of ostensibly two pairs of opposites (Hippocrates, de carnibus 2). (According to Aristotle Empedocles had one overarching opposition between fire and the other elements (Cf. Bollack, Empédocle pp. 82-85. Aristotle also ties together the opposites in all the Presocratics.)) He also provided many more examples of opposites although, like Anaximander, he did not use a word meaning “opposite” (Heraclitus uses ἀντίον (DK B120) in the standard spatial sense of, for example, heavenly bodies facing each other. This would not be particularly important except that Heraclitus is a philosopher and one whose m.o. is to make seemingly indifferent statements of ultimately great significance. Bollack and Wismann (pp. 330-331) in fact interpret opposition in terms of their version of Heraclitean “reconciliation.” The rising and the setting of the sun are supposed to contain within themselves the otherwise irreconcilable opposites of day and night). Examples of pairs of opposites attributed to Heraclitus by later sources include drinkable/undrinkable, up/down, disease/health, hunger/satiety, lack/satiety, fatigue/rest, living/dead, awake/asleep, young/old, whole/not whole, bringing together/separating, in tune/out of tune, day/night, winter/summer and war/peace. The attribution to Heraclitus of these pairs is often periphrastic, Heraclitus being one of the Hippocratean παλαιοῖς. Night and day are mentioned by Heraclitus in DK B57 (Hippolytus, Refutation of All Heresies, IX, 10, 20). This passage is most likely a quote and not a paraphrase and, although no term for opposition is used, night and day certainly do look to us like opposites in some sense.  This passage is also significant in that it contains a criticism of Hesiod. It provides a direct textual link between the philosophical preoccupation with opposition and the role of opposition in the Hesiodic rendition of Greek religion. Heraclitus also introduced the concept of ἔρις or strife (also a notion that appears prominently in Hesiod) as an apparent emendation of Anaximander’s injustice. Following upon that change, Heraclitus considered the result of the strife between opposites to be justice (δίκη) or a sort of recompense or mutual balance. Justice means that opposites change into their opposites instead of simply being destroyed as in the putative Anaximandrean scheme. If this is indeed what Heraclitus meant, then the logical basis for opposition as a philosophical issue largely disappears unless opposites can somehow be construed as successive cyclical definitions of everything. Barnes provides a different take on this which I examine below.

Other opposites that appear in the actual fragments (Although the fragments appear authentic, we must always be cautious about the authenticity of anything cited in patristic and even Stoic literature) are as follows. Life/death and birth/death are contrasted (DK B36, Clement of Alexandria, Stromata, VI, 17, 2.), as are mortality/immortality (DK B62 , Hippolytus, Refutation of All Heresies, IX, 10, 6 and 76, Marcus Aurelius, Meditations, IV, 46), day/night, winter/summer, war/peace, satiety/hunger (DK B67, Hippolytus, ibid. IX, 10, 8), waking/sleeping and young/old (DK B88, Plutarch, On Superstition, 3, 166 C). DK B36, however, expresses something more complicated than a simple relation of opposites. Death is indeed mentioned, but life is not. Moreover death is a process whereby three items, viz. water, soul and earth, are transformed into one another. Both life and death are mentioned in DK B76 and both mediate a transformational process involving four items, viz. fire, earth, water and air. It is an open question what the relationship of soul (ψυχή) in DK B36 is to fire and/or air in DK B76. Immortality and mortality are also mediated by life and death in DK B62 although obviously not in the same way, for immortality and mortality do not turn into each other. No mediating term between opposites is mentioned in DK B88. Indeed life and death, which had mediated the transformation of opposites into each other in the other fragments, are themselves cited here as changing in to one another. Satiety/hunger along with disease/illness and fatigue/rest appear in DK B111 (Stobaeus, Anthology, III, 1, 177). These pairs are mediated by pleasantness (qualified as good). However, the other half of the mediating pair, the half that would effect a reverse transition, presumably unpleasantness, is not mentioned. Cold/hot appear in DK B126 (John Tzetzes, Commentary on the Iliad, p. 126). So does what appears to be one other opposition, viz. wet/dry, although that opposition is presented by two different pairs of terms (ὑγρὸν/αὐαίνεται and καρφαλέον/νοτίζεται) where the καρφαλέον introduces a sense of water seeking, i.e. parched, to the more straightforward sense of dry in ὑγρὸν. Another peculiarity of this fragment is the representation of one member of a pair by a noun and the other member by a passive reflexive verb, e.g. “Cold is warmed” or “Cold warms itself.” The mediation comes about by the agency of the terms themselves and not through an external agency. DK B48 (Etymologicum magnum, Article: βιός) also mentions life/death and, like DK B36, does so in a ramified way. Life/death is associated with a new opposition, name/function (ὄνομα/ἔργον) and both are used to make a statement about the archer’s bow. The somewhat unreliable Hippolytus cites the contrast between straining apart (διαφερόμενον) and harmonizing (ὁμολογέει) in DK B51 (Hippolytus, Refutation of All Heresies, IX, 9, 2).

Barnes (pp. 69 ff.) links the many instances of opposites cited by or attributed to Heraclitus with fragments like DK B10 and DK B50 that assert that everything is one in order to attribute to him a doctrine that (a) opposites are identical or that (b) things can have opposite qualities. (a) and (b) are obviously quite different although Barnes (p. 70 (3)) appears to assimilate them or at least attribute both to Heraclitus. Understood a certain way, which is the understanding of many if not most Heraclitus scholars, both (a) and (b) violate the Law on Non Contradiction. (The huge exception is Heidegger (Cf. Vorträge und Aufsätze, esp. p. 268). Although Heidegger clearly does not equate das Selbe or ταὐτόν with either indiscernability or numerical identity (Ibid. pp. 234 and 241), I have little doubt he would regard Barnes’ entire discussion as just another example of Oxonian Alltäglichkeit. Bollack & Wismann also attribute tolerable contradictions to Heraclitus (Cf. e.g. pp. iv and viii) mostly because they don’t regard the fragments as constituting actual affirmative statements to which any obvious truth or falsity can be attributed. The best discussion of the problem of contradiction in Heraclitus is Reinhardt’s in pp.208 ff. where he derives Heraclitus’ doctrine from the Eleatic problem of contradiction and seems to assume if not explicitly state a solution by distinguishing between a cosmological issue of change, which does not involve contradiction, and the Parmenidean onto-theological problem, which does involve a logical contradiction as stated but does not apply to “the world of appearance” which is the actual world for Heraclitus. That said, Reinhardt pads his case a bit by throwing in the phrase “ahead of abstract logic” in his translation of Fr 55. (p. 213)).

Certainly we can’t have that. I will temporarily ignore (a). It deals with or appears to deal with second order entities a discussion of which will raise more problems than needed at this point and because lack of clarity on the part of Heraclitus (and Plato) as to the difference between (a) and (b) allow us to assume that at least they meant (b). Barnes argues that (b) violates the Law of Non-Contradiction for the Heraclitean opposites for the most part only if our understanding of those concepts rests on a “fallacy of equivocation” (p. 72) and a “fallacy of dropped qualification” (p.73). In this he is correct. Personally I express this by saying that the terms of the Heraclitean opposites are incomplete, or, linguistically speaking, they are expressed by incomplete predicates. A simple example is the concept of beauty. Just to say something is beautiful is not fully meaningful. We have to add qualifiers that answer the questions “to whom” and “in what respect” for our assertion that something is beautiful to be understood. Think of some politician, for example, calling a particularly slimy maneuver “just beautiful.” The same holds true for Heraclitean opposites like cold/hot not only as secondary qualities (i.e. qualities essentially linked to being apprehended in a certain way) but also as something that inheres in the cold or hot object independently of our apprehension (identified by Barnes as the object’s real essence).

(In Topics II xi 115b ll 29-35 Aristotle shows himself quite aware of the need to qualify some assertions in terms of respect when he distinguishes between unqualified (ἁπλῶς) and qualified (He uses the terms τισὶ and προστιθείς) attributions. But in this passage he seems unaware that qualification solves some of the apparent paradoxes of attributing opposite qualities to objects. (But cf. De Soph. Elen. passim). In Anal. Post. II xvii Aristotle gives an example of the sort of term that I call incomplete (Aristotle calls it ὁμώνυμα or equivocal) and that needs qualification by respect to be fully understood. His example is “similar” or ὅμοιος. Aristotle does not heed his own caution in Topics II vii 113a 20 ff. where he says that it is impossible for contrapositive concepts to apply simultaneously to a single thing. His example is not illuminating since it gets involved in the toils of Platonic ontology, but we could very well assert that a parked car is both at rest and in motion (because of the rotation of the earth). Aristotle’s point would have been valid if he had just negated the expression “at rest” instead of proposing a contrapositive. Cf. also Topics V iv 133b 33-36 for a remark about difference in a respect. He is addressing the issue of attaching a property to a thing vs. the same thing differentiated in some way. So it is not entirely relevant to the different issue of ambiguities involving opposition. But the idea of “in a respect” is basically similar. Aristotle’s clearest statement of this point is in De Soph. Elen. V 166b 38 – 167a 21. Cf. also V 166b 29-36 in the same passage.)

Aside from these considerations, however, Barnes further opines that Heraclitus may have in some respects violated the Law of Non Contradiction in asserting that opposites are one (pp. 79-81). I don’t think that has to be the case, however, for, as we know (link to my Mackie essay), “opposite” can mean either complement or contrapositive. The complement of hot is not-hot. One of its contrapositives could be cold. If the word “cold” is understood as not-hot, then the doctrine that things are both hot and cold does indeed involve a contradiction. However, if cold is distinguished from lukewarm or warm, for example, and understood as contrapositive to hot, then the contradiction is not so clear. A lover may feel both cold and hot but not both hot and not hot. He or she may shiver and sweat at the same time but cannot sweat and not sweat at the same time. We can assert that contrapositive properties simultaneously characterize an object most of the time because of equivocation as in the case of a good father also being an evil thief. However, such equivocations are not possible for complements understood in the sense of elementary Boolean algebra as “belonging to the complementary set of…” On this understanding, the assertion that something is simultaneously hot and not hot is always contradictory (pace French post-strucuralism and certain developments in the logic of quantum physics) whereas the assertion that something is simultaneously hot and cold need not be as in cases of Barnesian equivocation such as “…as a specimen of feminine beauty” and “…as a girlfriend.”

Plato makes largely the same point in Theaetetus 152 B where his qualifiers involve relativizing a state of affairs to whoever feels or doesn’t feel cold. Plato does not suggest that there can be other kinds of qualifiers and, for that reason, does not appear to recognize that the non-mutual exclusivity of contrapositive terms has as much to do with the nature of things and the language we use to describe them as much as it does with our perceptions of things (Cf. Cornford pp. 33-36; his background of the issue implies that the issue of opposites was framed entirely in terms of human perceptions and for that reason lacked a preliminary analysis of the grammar of the terms. But cf. 152 C).  Barnes’ hasty declaration that Heraclitus proposes a self contradictory theory (If indeed the theory of the unity of opposites is what Heraclitus really had in mind – Cf. Heidegger Vorträge und Aufsätze pp. 210 ff esp. p. 213. who proposes a totally different though perhaps incorrect interpretation, but at the very least wisely cautions us that the elliptical nature of this fragment allows of no sure interpretation. Heidegger’s caution can be extended to Raven’s suggestion that “Heraclitus had…shown that one of a pair of opposites cannot exist without the other (p. 381).” This is unsatisfactory for several reasons. For example, the meaning of “exists” is not clear and the notion of Heraclitus showing or proving anything is dubious.) is a result of not clearly making this distinction, which is not captured by the Aristotelian Square of Oppositions alone (That’s why Aristotle wrote about ambiguity also, stupid). His statement that Heraclitus’ opposites “are patently contrary, and patently thought of as such (p.79)” is just wrong. They are ἐναντίοι and not ἕτεροι, and contradiction inevitably obtains only if they are ἕτεροι. I have an intimation that the apprehension of contrapositives is a valid form of insight and may lead to the discovery of novel philosophical and other truths. That remains to be seen. What is certain is that complementarity can be given logical precision either via Boolean functions or through the contextual definition of negation in even the most elementary of first order logical theories. Contrapositives to this point, however, are logically untamed. After all what would an all encompassing contextual definition of the contrapositivity of north and south, the Guelphs and the Ghibellines, the Hatfields and the McCoys, mollusca and echinodermata or vaginal sex and anal sex even look like? As I mentioned, contrapositives like good and evil are different from propositional ἐναντίοι as Aristotle mostly understands that term in his logic. The propositions “Socrates is good” and “Socrates is evil” cannot be rendered à la Aristotle into logical ἐναντίοι or contraries such as “All men are good” and “No men are good.” If Aristotle’s intent was to capture contrapositives like hot and cold or good and evil by way of an analysis of propositional truth, obviously he did not accomplish that goal. (Contemporary philosophers sometimes use the verb “contrapose” as a synonym for deducing via modus tollens (Cf. e.g. Wiggins, p. 117) This is different from but related to Aristotle’s usage and completely unrelated to my concept of contrapositivity.)

The problem with Barnes’ reconstruction of Presocratic doctrines in deductive form is that, as usual, all the action is usually in the premises of the deductions. He makes an important contribution to textual analysis since he helps clarify arguments and positions whose confused presentation might lead to hasty adoption or rejection. But eventually we will arrive at premises that can’t be established by deduction alone and whose validity or lack thereof may not be intuitive or obvious.

Some of Heraclitus’ oppositions are significant in more ways than one. We might (with great caution) consider Heraclitus’ ταὐτό in DK B88 to be a precursor of Empedocles’ ὁμοίωσις. DK B10 (Ps. Aristotle, On the World, 5. p. 396b20) – very likely a paraphrase although Kirk (p. 191), Fink and Heidegger (Heraklit, p. 216) all treat it as a literal quotation – contrasts the whole and the not whole, the drawn together and the drawn apart, the harmonious and the discordant and the one and the all. This fragment if genuine is a remarkable foreshadowing of Aristotle and important for the reconstruction of a Heraclitean “system.”  However that may be, our interest here is that the terms cited could serve as more examples of opposites mentioned by Heraclitus. What we consider to be non physical abstractions were, in a manner not entirely clear to our way of thinking, assimilated with physical and social phenomena not only by Heraclitus but by most if not all the Presocratics. To regard them as simply having made ontological category errors such as would be savaged by Plato in dialogues like the Lysis (Kirk refers to without quite endorsing this view, pp. 188-189 ibid.) may in fact close off a sympathetic reading and deny us valuable insights their philosophies may contain.

The reconstruction of his views regarding cosmological opposition remains largely based on doxographical paraphrases by later commentators. Not a few modern historians propose something amounting to a Heraclitean system based on the literary shards. Kirk’s is as follows.

Strife or war is Heraclitus’ metaphor for the dominance of change in the world. It is obviously related to the reaction between opposites; most kinds of change (except for e.g. growth, which is the accretion of like to like), it may be inferred, could be resolved into change between opposites. At all events, change from one extreme to the other might seem to be the most radical possible. The ‘war’ which underlies all events, and is responsible for different and indeed opposed conditions of men and for the fate after death…is called δίκη, the ‘indicated way’ …or the normal rule of behavior. This must be a deliberate amendment of Anaximander’s dictum that things pay retribution to each other for the injustice of their alternate encroachments in the process of natural change. Heraclitus points out that if strife – that is, the action and reaction between opposed substances – were to cease, then the victor in every contest of extremes would establish a permanent domination, and the world as such would be destroyed. (pp. 195-196)


Kirk seems to echo Nietzsche’s nuanced comments (pp. 825-826) which identify struggle or ἔρις (Ringen, Streit) with justice (Gerechtigkeit). It is the fleeting and successive dominance of one of a pair of opposites over its opposite that is justice. Reconciliation is not an end state but the process of struggle itself.

If this account of certain kinds of change were a purely physical description dependent on observation, it would at the very least be heavily theory laden. Aristotle would, perhaps misleadingly, criticize Heraclitus’ account of sameness of opposites as logically contradictory. An Aristotle-style solution of the contradiction would be to consider specific examples of opposites as state changes of an underlying substance. Heidegger warns (p. 255) that a solution like that imposes an Aristotelian metaphysics on Heraclitus which he most certainly did not have in mind and to which he might have in advance proposed a compelling alternative. Nevertheless, both Heidegger and Fink concur that this fragment asserts that contrasting and conflicting things such as Heraclitus cites go together or come together (zusammengehen, p. 257).

Once again picking up on Nietzsche, Kirk makes a kind of case (p. 213) for the interweaving of moral (if not logical) and physical concepts on the grounds that the generally accepted Greek ethics, advocating as it does balance between behavioral extremes, finds a rational justification in the thesis that the physical world behaves in exactly the same way. Thus stated the argument has nothing more than historical interest, but it does provide an example of a felt need for rational justification among the Presocratic philosophers that would distinguish their way of thinking from the authoritarianism (Do it because god says so) and consequentialism (Do it or god will get ya!) of mythical and religious dogmas. Fink (p. 205) is more hesitant and suggests that Heraclitus’ words are Kennzeichnungen – close enough to metaphorical or picture language to merit a caution that they are not intended to be metaphors – addressing the same or very nearly the same issues we do when we use abstract or physical words. Along this line of interpretation we might regard metaphorical language as half way between literal physical language and the language of abstractions such as Parmenides Plato would introduced.

Some grammatical evidence against a reconciliation interpretation such as Kirk’s could be drawn from Fink (p. 75) who points out that Heraclitus’ intent in DK B57 could not have been to state that night and day are one. That does not prevent Fink from suggesting an interpretation, somewhat similar to Kirk’s, which centers around a higher unity of being that encompasses oppositions like day and night. Heidegger, however, wholeheartedly proposes his own version of a reconciliation in Vorträge und Aufsätze, p. 268.

Reinhardt (pp. 192 ff.) sees a religious purpose in Heraclitus’ doctrine of opposites. Or rather, a philosophical foundation for believing in survival after death. The idea is that, just as everything else in the cosmos does not really come to an end, but simply changes into its opposite, so for the living human death is not a complete end but simply a change into a contrapositive state, which state in turn will eventually change back to life. In change the human soul participates in the recurrent transformations of the elements. Reinhardt’s take on Heraclitus’ river fragments overturns the canonical interpretation of Plato and Aristotle and of many modern logicians pursuing the issue of identity. For Reinhardt the point is not that there is never a self identical river. Rather the conclusion Heraclitus draws is that the river retains its identity across continual change of the water in the river. On its face, this position is remarkably Aristotelian and, allowances made for his more ramified ontological distinctions, Quinean. Unfortunately, if this is indeed Heraclitus’ intent and conclusion, I find it much less interesting than Aristotle’s “mistaken” interpretation. The Heraclitus who advocates the impossibility of stable definable objects and the impossibility of Aristotle’s own ontology pursues a much more important and complex ontological analysis than the Heraclitus who was merely Aristotle’s forerunner.

As I already mentioned, Heraclitus does not use any Greek term for “opposite” (notably not ἐναντίος) in any of his quoted fragments. Aristotle, Hippolytus, Stobaeus and the pseudo-Plutarch provide examples of the way “opposites” can interact, but no actual word for “opposite” appears in the fragments themselves.  Aristotle’s term διαφερόμενον (de mundo, 5,396b20)  is one of a pair of opposites and not a  general term for opposition, although it may function that way in Hippolytus Ref. IX, 9, I.  Τἀναντία in Ref, IX, 10, 8 appears to be a parenthetical gloss by Hippolytus. The Eudemian Ethics (H i, 1235a25) does use the phrase ἐναντίων ὄντων. Kirk (p.201) notes that Heraclitus does not analyze his three cosmic components of earth, water and fire into opposites.

Clearly Kirk’s interpretation should not be taken as canonical. Heidegger (in Fink, p. 25) states that Heraclitus is not interested in opposition (much less the reconciliation of opposites) although his viva voce comment most likely focused on Hegel’s understanding of opposition and contradicts the views in Vorträge und Aufsätze, p. 268 cited above. As far as the literal texts are concerned, Heidegger has a point; he may also be right when he implies that the Aristotelianizing rewriting of Heraclitus was a patristic corruption. Yet in other passages Heidegger does say that Heraclitus works with terms like fire and water that are at the very least contrasting. And Fink not only stresses opposition (Gegensätze, pp. 133-138, 151 and 155 and Entgegengesetzungen, p. 151), he also proposes, albeit speculatively, something like a reconciliation interpretation of Heraclitean cosmology (Cf. Fink’s concept of das Unterschiedene and its Selbigsein and Zusammengehörigkeit, pp. 191 ff).

Bollack and Wismann (p.30) propose an understanding of Heraclitus’ writings on opposites (specifically the opposition of day and night) that appears to assimilate opposition to a type of Hegelian dialectical reasoning. This is not entirely out of the question since Hegel’s method of argument is at least partly based on his understanding of Heraclitus. And in this way they may give a logic-like structure to our understanding of contrapositives. The idea boils down to the fact that day, for example, may be a contrapositive of night, but they are the same or “one” in that they are both distinct particulars. This is the logical or conceptual core of the reconciliation that Kirk and others present as no more than a cosmological doctrine. However, the Bollack and Wismann analysis obviously applies to any arbitrarily chosen particulars, whether or not they are opposites or contrapositives. There is nothing special about day and night, least of all the fact that they are or may be contrapositive to each other, that makes them the same or one. In an interpretation of DK B57 Bollack and Wismann situate the reconciliation view of Heraclitus in the context of a response to Hesiod’s cosmology which was supposed to maintain a radical separation of opposites. Their view is that this fragment contests Hesiod not so much because he failed to recognize the (presumably dialectical) unity among distinct (not necessarily contrapositively qualified) objects but because he failed to recognize the diversity of particulars in the unified whole. If this is the sense of Heraclitus’ objection then that objection could probably be extended to Anaximander’s cosmology as I described it above.

Ultimately I do not see the reconciliation view in any of its forms as incompatible with the distinct view, more traditionally attributed to Heraclitus, that objects do not remain self-identical even over infinitesimally short durations of time. We could by way of intentional equivocation characterize a river as simultaneously both small and large or cold and hot (reconciled struggle of opposites) and also maintain the impossibility of stepping into the same river twice (anti-essentialism). Conversely, even if there were good reasons, observational or logical, for rejecting the doctrine of opposites and their possible reconciliation, those reasons would not form a basis for defending essentialism and the idea of persisting identity of objects across continuous change.

The Pythagoreans

Opposition does not disappear from the Greek philosophical mindset in the period between Heraclitus and Empedocles. However, its importance is reduced in favor of a consideration of the single universal that supposedly embraces oppositional pairs. The Pythagoreans focused on numbers (ἀριθμοί). Parmenides equated a particular “number,” the one (τὸ ἕν), with “being” (τὸ ὄν). Both instances of quotation marks in the previous sentence are meant to be scare quotes.

 Aristotle (Metaphysics A5, 985b23) attributes to one school of Pythagoreans (and to Alcmaeon of Croton) a theory of ten principles that are expressed by corresponding (κατὰ συστοιχίαν) pairs. These are: limit/unlimited, odd/even, one/plurality, right/left, male/female, resting/moving, straight/curved, light/darkness, good/bad, and square/oblong. Aristotle calls them τὰς ἐναντιότητας. He makes an intriguing distinction between definite (διωρισμένας) and random (τυχόυσας) ἐναντιότητας, but doesn’t explain what he means. All he says is that the Pythagorean opposites are definite and Alcmaeon’s are indefinite (ἀδιορίστως) or random. At the very least he seems to mean that the Pythagorean contrapositives are limited to their magic number of ten, while Alcmaeon’s were not so structured. If we look at Aristotle’s examples of indefinite contrapositives – white/black, sweet/bitter, good/bad, great/ small – some appear to be more “concrete” than some of the examples in the Pythagorean table which evidence a genuine irruption of abstraction into philosophical thought. So definiteness might have some sort of relation to abstractness. But even this is not clear. Good and bad, for example, seem less concrete than male and female. Aristotle calls the Pythagorean pairs principles (ἀρχάι) from which he concludes that they are also supposed to be causes (αἰτίας) presumably of all things in the physical and human universe. If the view he attributes to the Pythagoreans is historically accurate and if we understand “cause” in any of the senses in which Aristotle uses the term, then he complains rightly that no explanation is given of how the contrapositives could function as causes of physical substances. (Cf. also Metaphysics A5, 985b2.)

Another set of more relevantly cosmological opposites are singled out in Physics Δ6, 213b22 and in Strobaeus (quoting Aristotle, Anth. 1,18,1c). These are the void (τὸ κενόν) vs. the heavens (οὐρανός), the unlimited (τὸ ἄπειρον) vs. the heavens and oneness (ἕνα) vs. any one of time, breath or the void. The actual word ἐναντίος is not used in either passage and it is not clear whether these are meant to be opposites in any of the previous senses or not, although the passage cited above does include limited/unlimited as a pair of opposites. The idea might be to describe how opposition comes to be realized in actual cosmological events.

Another use of ἐναντίος in a Pythagorean context is the theory of oppositions in the soul (De Anima A3, 407b20). The viewed expressed is that the soul and body are composed (συγκεῖσθαι) of opposites that are synthesized and blended in ἁρμονία. It is possible that this theory in its original Pythagorean version constitutes the background of Plato’s discussion of opposites as principles of psychological and social explanation.


Alcmaeon (DK B286, Aetius, v, 30, 1) speaks of the opposites wet/dry, hot/cold and bitter/sweet, again without using any term for opposite or opposition. What is interesting about this passage, however, is that it seems to be a purely “empirical” observation to the effect that illness is caused by an imbalance between opposites in the body. A man can become ill because he is too hot, for example. So, on the one hand, Alcmaeon’s view eschews the broader speculations of the more philosophically inclined Presocratics, and on the other, it shows the value of a conceptual framework in structuring our observations and explanations.


Zeno is famous in college classrooms for introducing the world to philosophical hijinks with his proofs that the impossible is necessary. What is important for us, however, is that any number of Zeno’s paradoxes are based on pairs of opposites and notions of likeness and unlikeness. Barnes’ discussion (p. 237) of Zeno’s method is as good as any.  The basic pair of opposites that drive the argument is like (ὅμοια)/unlike (ἀνόμοια). Zeno clearly regards this opposition as one of complementarity and not in need of qualification (Aristotle might characterize Zeno’s conception of likeness as ἁπλῶς). Given this basic, complementary unqualified pair Zeno can go on to prove that some ontological (to use the term loosely) views entail contradictions. For example, the view that there are many things in the world. For (and this is my suggestion for filling textual lacunae in Zeno’s reasoning) if there is more than one thing, they are alike in being (among the) many and they are unlike in that they are many (and presumably not the same, i.e. alike).  Barnes suggests other possibly plausible but rather vague ways in which members of the class of the many could be both like and unlike (Suggesting that similarity or likeness could result from the fact that two things both exist is a classic example of out of the frying pan into the fire). Because his suggestions are vague Barnes doesn’t really spell out the fallacy, opting instead to state that the apparent contradictions are “harmless” and have no measurable effects on philosophical sleep habits. In fact the unraveling of this paradox is ready to hand and scattered throughout Aristotle’s logical treatises especially the Topics and On Sophistical Refutations. There is no such thing as likeness and unlikeness ἁπλῶς. These terms are, as I would call them, incomplete predicates (Contrast this with identity). In order to say that something is like or unlike something else you need to specify the respect in which the two are like or unlike. If you don’t do that, whatever you say makes no (or not enough) sense because it is incomplete. Under these conditions distinct things are alike in respect of (Barnes’ phrase is “in so far as”) being members of a class of more than one thing and unlike in that they are numerically different. Which is not a contradiction. In any event Zeno’s argument is nothing to write home about. And it should be clear by now that Socrates’ dialectic of likes liking likes in the Lysis is a (possibly satirical or possibly exemplificatory of Aristotle’s implied view that Socrates could be as sophistical as the best of them) application on Zenonic reasoning to concepts initially proposed by Zeno.

Another opposition raised by Zeno is large (μεγάλα)/small (μικρά). As far as Zeno’s actual argument is concerned (Cf. Simplicius Phys. 139 8, 140 29 and 141 1; Aristotle’s comments in De Gen et Corr. I 2 are also relevant although their focus is on Empedocles and the atomists), the opposition is between a very restricted case of large and small, viz. between infinitely large (ἄπειρα τὸ μέγεθος) and infinitely small, i.e. having no size at all (μηθὲν ἔχειν μέγεθος). Once again Zeno’s aim is to prove that, if there are a plurality of things, then each thing is both infinitely large and infinitely small, and since these contrapositives cannot compatibly characterize the same thing, there is no plurality, i.e. there are not many things (and so there is presumably only one thing). I don’t know if anyone has noticed that many of Zeno’s arguments also apply to Parmenides’ One and also to the sole thing in the world. So if his intention is to prove a version of Parmenides according to which there is just one thing in the world, a kind of ontological Robinson Crusoe, then he shoots himself in the foot (Cf. Barnes p. 240 “…the antinomy works impartially against…monism”).

Personally I am of the opinion that Zeno's arguments about the many both having and not having magnitude are too vague for anyone except classical scholars to worry about. The confusion, I believe, arises from the fact that Greek concepts such as στιγμή approached tantalizingly (Shall we say asymptotically?) close to what we would call an infinitesimal. But the Greeks had not developed a mathematics of infinitesimals (alternatively, classical discussions of infinite division lacked the related notions of limit and convergence, as Barnes himself argues, pp. 251-252) and so found it difficult to conceive of parts of a physical object converging to an infinitesimally small point (Aristotle’s distinction between potential and actual infinity was another early attempt to deal with the issues involved). Barnes' attempt at a reconstruction (pp. 240-242) is heroic but flawed. Without getting into the details, I shall just observe that he states as facts a couple of rather ambiguous assertions. In the first place, it is not clear that a (cinematic) light shone on a screen adds nothing (presumably physical (n.b. Aristotle's later emendation); without this qualification we could say it adds an image and indeed there is no guarantee that images, as light energy convertible with mass, are not actually physical) to the screen. Questions of relativistic mass aside, doesn't the light add an admittedly small amount of momentum? And why shouldn’t momentum either be or have μέγεθος? Barnes’ paraphrase of μὴ ἔχοι μέγεθος…καὶ πάχος  as “…has neither magnitude, nor mass, nor bulk…” is an egregious mistranslation interpolating at least one concept (mass) that, in its modern sense, which Barnes relies on for his light on a screen example, was completely beyond the ken of Presocratic science. Also Barnes’ ontological distinction between two dimensional and three dimensional objects begs the question as to whether idealized geometric objects such as points and screens (Screens are geometrical planes? Not bloody likely) are measurable or not. Part of Zeno’s argument turns on the assumption that a στιγμή is not measurable. But that doesn’t mean that a plane isn’t measurable. If a screen is measurable, then the light projected on it should be measurable as well, at least in respect of length and width. Secondly it is not at all obvious that shadows exist. Shadows as commonly understood are undefined in physics. And even in ordinary language we can reasonably assert that shadows exist only if other kinds of absence exist also. Existence is the Big Kahuna of 'pataphysical concepts and invoking it to analyze a Neoplatonist's rendition of a peculiar Presocratic paradox lands us right back in the philosophical fire (perhaps this would be pleasing to Nietzsche and assorted Zoroastrians). To be fair to Barnes, he does recognize that ontological issues (specifically the issue of whether magnitude is a necessary condition of existence) lie at the heart of Zeno's problem. The paradox of the infinitely largely and infinitely small, while an amusing brain teaser (whose long term heritage includes differential calculus), doesn’t add much to the large/small contraposition and so we can look past it in this context.

Zeno’s other oppositions include: rest/motion and finite/infinite. He does not treat the one/many opposition in the same way as the others since his overall goal is to argue that there is no such thing as the many (which we could understand to mean there is only one thing in the world). The paradoxes involving the other oppositions are meant to show the contradictory consequences of assuming that there is such a thing as the many. Curiously Barnes makes only passing reference to Plato’s extensive discussion of the one/many paradox in the Parmenides (on which more below). He does allude to the Phaedrus (261D) where Socrates makes a passing reference to Zeno, and revisits briefly the theme of the Lysis when he mentions ἐναντίοι and the paradoxes of likes and unlikes. Zeno is thrown in with the sophists and (implicitly) compared unfavorably to those who know how things really are (literally “what each entity is”).

To return to our subject, let me reiterate that many of the oppositions raised by Socrates in the Lysis and the basic framework of understanding at least some cases of opposition in terms of like/unlike can be traced back to Zeno. This is not without consequences such as will be detailed in Plato’s late metaphysical dialogues. From the standpoint of the development of Plato’s own doctrines we should regard the Phaedrus as a stage between the Lysis and the Parmenides in Plato’s continuing hostile examination of the Eleatic paradoxes.


There can be very little question but that opposition played a role in Parmenides’ thought if only because contrasting terms appear in actual quoted fragments. However, Parmenides’ attitude toward a cosmology (more properly speaking a structured ontology) based on oppositions is highly critical. Indeed the aporia that result from the interaction of opposites and how they relate to τὸ ἕν is one of the central themes of Plato’s eponymous dialogue.

The most instructive rendition is in Simplicius (Phys. 30, 14 B8). At issue (l. 53) is what Parmenides calls μορφάς, i.e. forms or configurations. According to the δόξας…βροτείας or beliefs of mortal men configurations always come in structurally (δέμας) oppositional pairs (τἀντία). Their distinguishing marks (σήματ’) are what differentiate configurations from each other (χωρὶς ἀπ’ ἀλλήλων). Parmenides could have understood this distinction as one of contrapositivity or some other less well defined sense of opposition. He almost certainly doesn’t mean complementarity. By way of the term αὐτός Simplicius attributes this doctrine to Parmenides, but the rest of the fragment, as one would expect, immediately proceeds to criticize it. However, Parmenides’ criticism may be solely from the standpoint of the so-called Way of Truth. As far as the apparent world is concerned dualism is a “valid” description of phenomena and their ἀρχαί. This is J.E. Raven’s reading (p, 281). Parmenides is supposed to criticize this view or δόξα as badly mistaken and part of what he calls the Way of Opinion. From the standpoint of the Way of Truth there is only one configuration (τῶν μίαν). However, Reinhardt (Cf. esp. p. 18) argues that Parmenides does not consider his description of the apparent world to be anything but accurate. Rather, the contrapositives that are generated by an ultimate duality between Eros and the children of the night interact in such a way as to explain the visible cosmos (Cf. also p. 24, “…als Wesen aller Dinge sprang ihm überall der Gegensatz heraus”). This standpoint, he argues, is not too different from Empedoclean physics. The important point for our purposes is that Parmenides addresses the terminological matrix of opposition inherited from Anaximander, whatever his position may be with respect to that matrix. The opposites actually cited by Parmenides are fire (πῦρ) and night (νύκτ’) to which Simplicius adds disquotationally light and darkness, fire and earth, dense and rare as well as, strikingly, sameness and difference (ταὐτὸν καὶ ἕτερον).

In Fragment 8 l. 53, where Parmenides calls the pairs τἀντία and χωρὶς ἀπ’ ἀλλήλων, he asserts that, according to the dualistic belief, each side of the pair is identical with itself ἑωυτῷ… τωὐτόν but different from its opposite ἑτέρῳ μὴ τωὐτόν, which he rephrases and κατ’ αὐτὸ τἀντία. That the duality sameness and difference (ταὐτὸν καὶ ἕτερον) should, according to Simplicius, itself be listed as a pair of opposites represents a clear leap in conceptual levels. Cosmic phenomena like fire and earth can be self identical and different from each other. But also those μορφάι themselves, i.e. sameness and difference, which are assigned (σήματ’ ἔθεντο) to opposing manifestations like fire and night, are in turn self identical and different from each other. It is statements like this that introduce higher order conceptualization into philosophy.

Parmenidesi himself uses the terms ἑτέρῳ and τἀντία in this passage to which Simplicius adds the heavily Aristotelian ἀντίθεσις in the phrase πρώτην ἀντίθεσιν. Simplicius continues the assimilation to Aristotle by glossing the text to mean that opposition of configurations establishes as far as mortal belief is concerned the primary elements (ἀρχὰς …στοιχειώδεις) of created things.

Fragment 12 throws in an opposition between male and female. This may be significant to Plato’s social view of opposition. Socrates may have applied the opposition between sameness and difference to the male/female opposition in order to treat homosexuality as equal to and just as legitimate as heterosexuality. One consists of a relation between ὁμοίοι and the other as a relation between ἐναντίοι. Incidentally by boys on the right and girls on the left (Fragment 17), Parmenides may have meant a position in the womb that determined gender (Cf. Bollack II pp. 238-239). However, if that is indeed Parmenides’ meaning, I wonder on what kind of observations he could have based his assertion.

Theophrastus’ comments (de sensu I ff.) on the Presocratics and sense perception (αἰσθήσεως) are theoretically interesting even if historically suspect because of the heavily Aristotelian flavor of the disquotational comments.  The idea seems to be that Parmenides viewed sense perception as a relation between similar things (ὁμοίοι) although the opposites (ἐναντίοι) hot and cold appear to play a role in the production of thoughts.


The most detailed proponent of a thoroughgoing doctrine of anthropomorphic cosmological opposites was Empedocles who seems to have been Plato’s direct target in the relevant passages from the Lysis. If the historical narrative to the effect that Anaximander proposed a vision of permanent opposition between contrapositives based on the paradox he perceived in a single explanatory or reductive concept encompassing everything that is, a vision that was modified by his successors into some form of ultimate cosmic and perhaps social reconciliation, is true, then Empedocles can be regarded as having resurrected the inclination toward pure unreconciled opposition or at the very least an evolution of opposites that did not end in some sort of stasis.

Undoubtedly Empedocles’ views on opposition are the most sophisticated and nuanced among the Presocratics (It can be argued that Anaxagoras was talking about something else. See below). For one thing he introduces the notion of likeness as a counterpart to that of opposition and treats the two as basic terms (unlike Zeno for whom like/unlike was one opposition among others even though that opposition further qualifies some of the things qualified by some of the others). In his hands the world is structured not just by opposites and their interaction, but by a complex interweaving of opposites and “likes” (ὅμοιοι). In addition Empedocles replaces the single relation between ontological “beginnings” or “principles” (ἀρχαί) and the cosmos with a dual structure. As an alternative to Anaximander’s (and Heraclitus’) singular φθορά or δίκη Empedocles states that the ἀρχαί are themselves potentially related by one of two distinct principles, ϕιλότης or νεἶκος. The negative force of νεἶκος largely replaces the idea of ἔρις found in the pre-Empedocleans and ϕιλότης introduces or re-introduces in a philosophical context important sexual associations. Semi-sexual interaction hearkens strongly back to Greek religion as represented by Hesiod and appears to skip over the intermittent trend toward abstraction from Thales onwards (But cf. Parmenides Fragments 12 and 17).  Terms such as ϕιλότητι μιγεῖσα, ἔρις and νεῖκος appear frequently in Hesiod’s Theogony and they all reappear as either universalizing concepts or explanatory principles in Presocratic philosophy. Aristotle (Phys. VIII I 252a ff.) discusses attraction and repulsion, φιλία and νεἶκος, in Empedocles without anthropomorphizing them. But that doesn’t mean he understands these terms in a modern physicalist sense. The example of human attraction and repulsion in ll. 30 ff. is enough to show that and to retain an echo of the sexual element in Empedocles’ cosmogony.

The structure of thought that informed Presocratic philosophy is fully captured by Empedocles. The attitude is (1) that there is a science of all things (Thales), (2) the only way to pursue this science is via oppositions, or, to use a more modern term, opposing concepts (Anaximander), and (3) it is important to investigate how these opposing concepts relate (Heraclitus). Aristotle’s telling comment in Met. III ii l. 21 might help to clarify the role of opposition in Greek philosophy as a whole: Only if the ἀρχαί are ἐναντία can there be a single universal science dealing with them all. (This is a fairly gnomic statement and may only be Aristotle’s rendition of a prevailing view and not reflect his own attitude.)

Simplicius (Phys. 25, 21, DK31 A 28) attributes to Empedocles a doctrine of direct interaction between the principles of φιλία (Simplicius’ term. We should always keep in mind that φιλία if not ϕιλότης is closer in meaning to friendship or fondness in many post-Platonic philosophers and any sexual overtones have more to do with a kind of romantic sexual love. Φιλότης is the Homeric and Hesiodic term of choice when referring to physical sex and Empedocles’ use of this term rather than the milder φιλία as the name of one of his ἀρχαί is not without significance. A gap separates φιλία in this sense from Sappho’s more naturalistic use of that and related terms to mean romantic cum sexual love (A sampling includes Frs. 15, 16 , 23 and 31) or the blunt expression ϕιλότητι μιγεῖσα (Cf. e.g. Theogony l. 125) – to fuck – that appears in Hesiod and Homer. Cf. also Parmenides’ μιγῆν (Fragment 12) and νεἶκος in relation to what he, or, more accurately, Simplicius, calls the cosmological elements (στοιχεῖα), viz. fire, air, water and earth.  Bollack (Empédocle, p. 287) presents an interesting view of why Empedocles uses a selection of words (Cf. below on στοργή) to express the idea of love or attraction. He sees in Empedocles a doctrine that no single term can express the essence of anything, much less one of the fundamental forces. These essences are not singular and static. They are multi-faceted and constantly changing.

Empedocles never uses a generic term for “opposite” or “opposition” and as far as I can tell he uses the term ὅμοιος just once (DK B62, Simplicius Phys. 381, 31). Among his commentators, Theophrastus (DK 31A 86, de sensu 9) contrasts ὅμοιος not with ἐναντίος but with ἀνόμοιος (though this may say more about Theophrastus than any of the Presocratics. In the Anaxagorean fragment DK 59A 92 he uses ἀνόμοιος and ἐναντίος indifferently to contrast with ὅμοιος). So it is something of a leap to state outright that Empedocles’ doctrine was that likes or ὁμοίοι are related by ϕιλότης and opposites by νεἶκος. However, the pairs related by νεἶκος that he selects are intuitively opposite to each other and the outright general statement is what all his successors and commentators, Plato in particular, believed he taught. So lacking a better interpretation, we can, albeit hesitantly, attribute this view to Empedocles.

In an actual quoted text (DK B 6, Aetius I,3,20) what Simplicius had referred to using the Aristotelian term στοιχεῖα are called the πάντων ῥιζώματα or the roots of everything. This sort of parallel between terms leads us to assume an Aristotelianizing emendation on Simplicius’ part. The roots are also personified. In DK B6 they are called by the names of gods: Zeus, Hera, Aidoneus and Nestis. Change in Empedocles’ account is the product of a mixing and shifting of these roots. We should always keep in mind how concrete the language of the Presocratics – Anaximander, the Pythagoreans and Parmenides partly excepted – was; any striking abstract terms in later commentaries are usually inspired by Aristotle. Along the same lines ϕιλότης or the force that unites the roots is very probably identified with Κύπρις or Aphrodite in DK B 128 (Porphyrius, de abstinentia II, 21) although the account refers to religious practices and is not historical in nature. Finally, in at least one passage (DK B 122, Plutarch, de tranq. an. 15, 474B), clear opposites are referred to using personifications such as Χθονίη and Ἡλιόπη. But Presocratic views are philosophy or cosmology, not myth. For example, in DK B29 (Hippolytus Ref. VII, 29) Empedocles mostly likely subscribes to a criticism of naïve anthropomorphism that dates back to Xenophanes.

There is another way of understanding Empedocles’ use of the language of personification (and consequently the emotional content such language carries). A hint comes from intriguing art historical interpretations of Greek vase decoration. Boardman (p. 239) has shown how women, often identified as deities or subdeities, are depicted on pottery as signs for emotions. For example, in order to show that a scene is happy, instead of representing the actors of the scene as happy, the goddess of happiness, Eudaimonia is pictured as part of the scene.  It is a pictographic language which is an alternative to the direct representation of emotion by way of expressive drawing of faces and gestures. Significantly other examples cited by Boardman include the goddess Dike as an icon for justice and Eris as an icon for strife.  (Why always women? Boardman opines that it is because the words naming the concepts alluded to always have feminine endings. But in that case the icon for νεἶκος would have to be male. And what about neuter nouns?) Something akin to this kind of pictography might also be at work in the anthropomorphic personifications of the Presocratic philosophers and that might help explain invocations of the gods by a culture that, at least among the educated, no longer believed in the real existence of the gods. If there is anything to this parallel, it provides a clue to the “non-conceptual” thinking of some of the Presocratics. The personal names do not stand for real divinities. Rather they are a pictographic mode of representation for what we would consider to be abstractions. Whether the use of these verbal pictographs is solely due to an inability to think conceptually (Reinhardt (p. 24 and pp. 67-68) asserts that Parmenides, and presumably Empedocles as well, stood on a tipping point between thinking exclusively in terms of concrete material objects, materiality being a status legitimately applicable to Greek gods, and conceptual thinking. His idea is that mythological and poetic expression is not external to their way of thinking; they simply didn’t have the abstract words that later philosophers would develop and so they expressed their doctrines in terms that would seem to us overly concrete.) such as would be replaced by Plato’s superior method of reifying abstractions and using abstract names to stand for his new ontic realm, or whether they embody a concrete mode of thinking different from and inaccessible to thought based on conceptual abstractions (such as Heidegger seems to suggest), is an open question.

J.E. Raven (p. 330) raises an interesting issue on the basis of DK31 A 28. He takes this passage to mean that Empedocles endorsed or assumed a concept of “material reality” – a position that has two consequences: (1) τὸ πᾶν refers exclusively to material reality (“he is…unable to imagine any form of existence other than spatial extension….”) which Raven distinguishes from “the abstract”; (2) the personified language of proper names and psychological motivations Empedocles uses in describing the agency of love and strife is itself nothing more (and nothing less) than a description of “material reality.” Raven’s approach is certainly interesting but it is most likely misguided. There is no evidence that the Presocratics had any concept approaching our concept of spatial extension that is the heritage of post-Cartesian, post Newtonian cosmology. Indeed even the Euclidean understanding of space was not codified until the 4th century BCE. Indeed it may be hard to make sense of such a concept unless you already understand something like Platonic ideas or forms by way of contrast. In addition the quoted passage is from Simplicius who uses decidedly Aristotelian terms such as ἀρχαί, σωματικὰ and στοιχεῖα that are non-Empedoclean in the extreme.  Ultimately there is a difference between not making a clear distinction between events in the physical cosmos and abstract logical (or indeed social) distinctions, on the one hand, which is the most accurate way to characterize Empedocles, and simply regarding everything as material, on the other. It is also wise not to confuse the concrete language of the other Presocratics with the materialism we associate with Democritus and Sextus Empiricus. It is clearly a different type of concreteness that is best understood in contrast to Platonic and Aristotelian abstractions. (Heidegger most likely would consider atomism to be a type of abstractionism also.)

At the other extreme, while Parmenides and the Pythagoreans continue to spark interest among philosophers, Empedocles has been beloved of poets from Hölderlin to Arnold (admittedly not the longest stretch of time). The affinity may be that the Romantics interpreted Empedocles’ doctrine as stating that material nature was infused with something like romantic love. There is a bit of paradox about that. One might speculate that, if Empedocles had to choose between ϕιλότης as love in the romantic sense and ϕιλότης as something like magnetic attraction, he probably would have opted for the latter. For that reason Jarry showed some insight in depicting an Empedocles whose adventures on Aetna were motivated by scientific curiosity (“Il va, - pour arracher au volcan son secret.” p. 118). Indeed the Romantics might be faulted with an assumption analogous to Raven’s materialistic assumption. There is no reason to believe that Empedocles or the other Presocratics or indeed the Greek poets meant anything similar to Romantic love when they used terms such as φιλία or ϕιλότης.

A philosophical translation of these all important terms for Greek thought could stress either an anthropomorphic or a mechanistic take on their meaning, a distinction that was foreign to Empedocles. Hence ϕιλότης could be liking, friendliness, personal attraction or even love in some sexual sense related to Empedocles’ way of thinking. Likewise νεἶκος closely approaches the standard translation as strife but, in order to provide a more direct contrast with one sense of love, it could be glossed dispositionally as hostility. On the other hand, it is not completely excluded that ϕιλότης and νεἶκος could be re-interpreted by us as something like mechanical attraction and repulsion. The fact that Empedocles did not make this kind of distinction, which we see as almost obvious, helps us understand how he could anthropomorphize cosmological events, categories and principles. It also provides the basis for the parallel he assumes and may assert between human emotions and cosmological forces. It certainly accounts for the parallelism if not necessarily consistency between On Nature and the Purifications.

The roots of everything (a phrase whose meaning includes both a classificatory or categorical and a causal sense) are supposed to explain physical change in a way that an opposition of being and non-being or an original unity à la Parmenides cannot. Empedocles does not in the extant fragments give a reason (An understanding of λόγοι as arguments or reasons make their first real appearance, pace Zeno, with the character of Socrates in the Platonic dialogues and serves as the criterion for distinguishing philosophy from rhetoric and poetry) for preferring his alternative but we can speculate, on the basis of Plato’s Parmenides and Sophist, that one reason could be that an all-embracing concept like being or water leads to paradox (More on this below). Thus, although this may not have been explicit in Empedocles’ thinking, it gives us a reason for preferring his cosmology to Parmenides’. Notably, as we saw, this is the same sort of reasoning that may have led Anaximander to replace Thales’ naïve monism with the idea that the ultimate principle is ἄπειρος. On the other hand, much of what Empedocles actually says - especially with respect comingto be, destruction and change - occurs in the context of cosmological and biological observations. Plato may have erred in taking Empedocles' theory only as a logical insight not involving generalization from observation.

Empedocles’ roots can be added to and subtracted from each other in order to form objects in the visible world even though and perhaps because no one of them is characterized as universal or “underlying”. There is no substrate or equivalent to an Aristotelian ὑποκείιμενον or ὕλη. Although the four “elemental” roots as a whole in effect comprise “the all” or the entirety of the physical universe (undistinguished, as noted above, from what we would call the universe of logically possible objects), they are not explicitly described as such, presumably, as I noted in the previous paragraph, to avoid contradiction or paradox (although, if Empedocles’ universe of discourse is limited to physical objects, paradox can be avoided in other easier ways). The “principles” or, to speak physically, the forces (As noted above, neither “principle” nor “force” are Empedoclean terms; DK B 17 uses grammatical cases to signify agency. It is the ever accommodating Simplicius who uses the term ἀρχὴ, usually translated as “principle,” in describing Empedocles’ doctrine of love and hostility (DK 31 A 28, Phys, 25, 21 (426)) that bring about the mixing and separation of the elements are ϕιλότης and νεἶκος. DK B 17 gives an indication of how sexual Empedocles thought ϕιλότης really was. “She” is called Aphrodite and, interestingly enough, Γηθοσύνη (Joy), an affect given a proper name and accordingly anthropomorphized. She is “inborn in mortal limbs (θνητοῖσι νομίζεται ἔμφυτος ἄρθροις)”. In DK B 62 (Simplicius, Phys. 381, 31. Cf also  DK B 26, Simplicius Phys. 33, 21 and DK 31 A 37, Aristotle Met. A4, 985a23) Empedocles applies the principles of love and hostility to human interaction, with a likely stress on human sexuality (Cf. especially Aetius’ summary in DK 31 A 72 (Aetius V, 19, 5) which explicitly cites female beauty (τῆς εὐμορφίας τῶν γυναικῶ) as exciting the movement of sperm (τοῦ σπερματικοῦ κινήματος)). It is interesting that Socrates, in his refutation of Empedocles in the Lysis ignores the sexual aspect of the theory, even though there is an underlying sexual tension in the dramatic situation. As we know Plato goes so far as to substitute the more anodyne φιλία for ϕιλότης.

With love and hostility as agents and various entities such as fire, water, earth and air (sometimes bearing the names of gods and demigods) as roots, Empedocles articulates a rather complete theory of generation (ἐπιγίγνεται) and corruption (ἀπολήγει) or simply change. The elements are the background material. They may be timeless. The closest Empedocles comes to saying they are anything like eternal is to say they are “of like age (ἡλικα γένναν ἔασι).” The specific manifestations we encounter are the result of one or the other element “prevailing (κρατέουσι)” by way of the “whirling (ἑλισσομένην)” agency of Love/Aphrodite/Joy. Empedocles’ actual term for referring to fire/water and earth/air as a group is πλεόνοι (“the manifold” or “the many”) divided from the one (τὸ ἕν). Is his doctrine of the many roots interacting through the agency of love and hostility (attraction and repulsion) an improved version of the Parmenidean doctrine of the One? It looks to me more like a direct slap down.

In the Metaphysics (B4,1000b12; cf. also De Gen. et Corr.II 7) Aristotle notes, perhaps with surprise, that Empedocles does not mention a cause for change (τῆς μεταβλῆς αίτιον), further highlighting that the concept of αίτιον is specifically Aristotelian and burdened with the gloss Aristotle gives it. Aristotle’s phrase for Empedocles’ view is that change is not causal but rather that these things ὅτι οὕτως πέφυκεν, which is best translated as “just happen that way” and not by Raven’s “are so by nature.” In fact it is more accurate to say that Empedocles had his own terms that prefigured both ἀρχὴ and αίτιον, viz. ϕιλότης  and νεἶκος. They don’t of course mean the same thing as Aristotle’s terms and the difference is the key to Empedocles’ own more naturalistic theory. Nevertheless, sexual attraction and conflict may on their own sufficiently satisfy all the logical requirements Aristotle proposes in his arguments for the necessity of a first mover. The issue then is not whether Empedocles articulated an Aristotelian style physical theory involving causality as Aristotle understands it, which he obviously did not, but in what way the concepts of ϕιλότης  and νεἶκος are related to that of αίτιον and indeed, if we are interested on a unified history of science, to the Newtonian concepts of attraction, repulsion and force.

In addition to the theory of interaction of portions of the cosmological manifold under the influence of ϕιλότης and νεἶκος, there is the general theory of attraction of like to like (ὅμοιοι) (Cf. Bollack’s remarks in Empédocle I pp. 110-111. Also Aristotle discusses Empedocles’ view that likes attract vs. the view that opposites attract (For the relevant citations cf. Bollack Empédocle II pp. 62-65.)).  At an even greater level of sophistication the two interactive principles are themselves related in a complex either self-reflexive or alternatively second order way. (Love and Strife may themselves be opposites Cf. Bollack’s quote of and comments on Simplicius pp.155-156.) Empedocles may have in mind a principle of the attraction of like to like in a quoted passage (DK B 109, Aristotle, Met. B4, 1000b6; cf. also Raven pp. 340 ff.), which interestingly enough uses the family friendly στοργή instead of ϕιλότης. In this passage we “see” (ὀπώπαμεν) the cosmological roots together with each other (water with water etc.). The reason may be that they are likes (ὅμοιοι) though this is not stated. But affection itself is also “seen” with affection and hostility with hostility even though these are agents for the interaction of the roots and accordingly different in kind from the roots. We “see” στοργή with στοργή and νεἶκος with νεἶκος. The agents that push together like to like and pull apart what is unlike - affection and hostility themselves - are subject to their own force. The self reflexive interpretation of this concept is that love loves itself and hostility is hostile to itself - rather inappropriate if we consider love and hostility to be abstractions. The alternative interpretation is that Empedocles meant that loving things are attracted to each other and hostile things are attracted to each other. Understood in this way, the phrase becomes a generalization of the principles Empedocles had just applied to the roots. This latter interpretation (and probably the self-reflexive interpretation too) is open to Socrates’ attacks in the Lysis. Obviously things that are hostile to each other are both hostile. Therefore they are likes. Therefore they should love each other. Therefore they are not hostile (at least to each other – This is the weakness in Socrates’ refutation) etc. More properly speaking, it is unclear how, when a thing loves one thing and is hostile to another thing, it could be said to be only a loving thing or only a hostile thing.

The idea that opposites and likes could relate to each other in two ways that are themselves opposites appears to be an Empedoclean innovation. Empedocles, as I noted, introduced his own vocabulary to describe the relations between opposites and likes. Νεἶκος is presumably equivalent to ἔρις, but it is not clear whether they mean exactly the same thing and, if they differ, where they differ. (Bollack and some German commentators translate νεἶκος as “hate.” But isn’t there difference between νεἶκος and hatred (presumably μῖσος)? We will recall that Socrates in the Lysis freely contrasted φιλία and μῖσος. So part of the issue in interpreting the Lysis then becomes how much meaning Empedocles’ terms share with Plato’s.) Φιλότης (alternatively φιλία or στοργή) is a conceptual as well as a terminological innovation. In Heraclitus ἔρις more or less resolved into δίκη and τάξις, but these are not real opposites to some sort of attraction principle. And Anaximander had no term that wholeheartedly prefigured ϕιλότης in the way that τῆς ἀδικίας prefigured νεἶκος.

In DK B 62 (Simplicius, Phys. 381, 31) Empedocles seems to speak of human generation in describing a chthonic origin occurring as complete forms (i.e. forms not in need of development) of what are presumably proto-people (Intriguingly the word εἴδεος appears for what may be the first time in a philosophical context, providing a potential source for Plato’s use of the term). What the fragment seems to say is that the fire portion of the newly arisen forms “sends them (the forms) forth (ἀνέπεμπε) each desiring to reach its like (θέλον πρὸς ὅμοιον ἱκέσθαι)” (Presumably the desired likes are of the forms and not of fire, which is a grammatically possible reading. Cf. also DK B 26, Simplicius Phys. 33, 21 and DK 31 A 37, Aristotle Met. A4, 985a23.) Raven (p. 340) cites Aetius to note that “the principle of attraction of like to like” accounts for nutrition and growth as well as pleasure and pain, a connection Plato picks up on in the Lysis. Since the agent of such a desire is presumably ϕιλότης, it is reasonable to suppose that Empedocles is giving an account of the attraction between the sexes. And, since these primitive forms are very unformed, possessing neither limbs nor voice nor human organs, this account along with others (e.g. DK 31 A72 and DK B 61) might also have been a basis for Aristophanes’ theory in The Symposium. If that is the case, then Aristophanes might have been a Platonic stalking horse. Once again the real target may very well have been Empedocles and Plato’s reluctance to cite him by name could indicate how serious an opponent Empedocles was or how hostile Plato was to Empedocles at least in The Symposium. (For confirmation of this surmise cf. Bollack Empédocle, p. 200 ftn. 5.)

Along the same lines Theophrastus (DK 31 A 86, de sensu 9) attributes a theory of cognition (περὶ φρονήσεως καὶ ἀγνοίας) and sensations to Empedocles based on his broader theory of the mutual attraction of likes and repulsion of unlikes. In the actual quoted text Empedocles does mention both cognition, on the one hand, and pleasure and pain, on the other. Presumably the things from which (ἐκ τούτων) knowledge and pleasure are harmonized and constructed are those things that are alike. Pain and ignorance as well as misperception arise from opposites. Theophrastus uses the term ἐναντίος throughout his exposition and in at least one instance (par 8) uses the Aristotelian term ἀντικειμένος as if it were synonymous. Versions of this theory appear throughout Greek philosophy. It certainly seems that the doctrine of homoiosis was an all-purpose principle serving to explain several different sorts of phenomena. Incidentally it wasn’t until the Italian Renaissance and Francesco Patrizzi that cognition was described in patently sexual terms as coitus, at least as far as I have been able to discover.

However, Theophrastus applies to biology a contrary view which he also attributes to Empedocles, namely that, due to the effects of climate and some inherent nature of different types of body, likes are harmful to each other and contrapositives beneficial (De Causis plantarum 1,21,5; p. 42, 13 ff. cited by Bollack Empédocle II pp. 244-245.)

Empedocles’ theory formed the basis for a kind of universal “science” almost in the sense that we understand the term today. The roots, the distinction between like and unlike and the agency of love and hostility apply to cosmological events including the apparent creation and destruction of things in the observable world. They also apply to what we would consider to be the more specialized realm of biology and to the workings of the mind and to human history and pre-history.

Nevertheless, as I noted with respect to Anaximander and Parmenides, logical problems suggest themselves immediately in cases where a philosopher tries to articulate a definition of “everything there is” and even, though less obviously so, in cases where he propounds a principle or theory explaining the existence or character of everything there is. Anaximander may have been aware of these problems; Plato certainly was (In The Sophist, for example, Empedocles is listed along with other Presocratics as supporting a version of the possibly paradoxical doctrine of a relation between the one and the many (Cf. Bollack Empédocle I pp. 122-124)). Empedocles was a very empirically minded philosopher. His focus was on the cosmos and on the human and biological world. However, he did show a glancing interest in the logical problem in a couple of surviving fragments (Who knows what may have disappeared?). DK B 13 (Aetius 1,18,2) and DK B 14 (Pseudo-Aristotle, MXG 2, 976b24) make very much the same point, namely that the concept of a thing that is not included in everything is a non-starter, or, to put a Platonic gloss on the matter, self contradictory (Let us define a self-contradictory concept informally as a concept that, when predicatively asserted of something as a property of that thing, results in a statement that entails its own negation) or at the very least leads to some wildly self contradictory conclusions (Cf. Raven p. 324 on the parallels and the very likely influence of Parmenides on Empedocles). DK B 13 states that “the all” is neither empty nor over full. I will hazard an interpretation of this passage to the effect that the number of things that we include in everything is just right. Nothing is left over (Yes, I know Plato and Hegel would go to town on whether Nothing really is left over or not) and it is not empty, i.e. it includes things. If my interpretation is correct then Raven’s translation (p. 323) is misleading and ultimately incorrect: οὐδέ τι τοῦ παντὸς does not mean “Nor is any part of the whole” especially since the concept of part was much discussed in Parmenidean philosophy and presumably, if Empedocles meant to refer to parts, he would have used the term; also there is a distinct Greek word for “whole,” namely ὅλα. Indeed if Empedocles were talking just about parts of the whole, this comment and DK B 14 would make no sense. οὐδέ τι rather means “nothing” syncategorematically understood, or, more properly, “in no way.”  So the phrase in DK B 13 should be translated “Nothing about the all is empty or over full” or else “In no way is everything taken together empty or over full.” Similarly for τοῦ παντὸς δ’ οὐδὲν in DK B 14. Ultimately and independently of Empedocles, interpretation these logical problems admit of a resolution, as I will try to show later.

Like so many others after him, however, (armies of benighted theologians aside, Locke and Voltaire are glaring examples from the so-called Enlightenment) Empedocles apparently did not understand that you cannot (or you can’t without great risk) draw ontological (in his case specifically cosmological) conclusions from logical distinctions or from truths about the nature of language. If DK B 11 (Plutarch, adv. Colot. 12 1113C) and DK B 12 (Pseudo-Aristotle, MXG 2, 975bI) are meant as conclusions to be drawn from the general “logical” principle stated in DK B 13 and DK B 14, then Empedocles did just that. DK B 11 and DK B 12 are early appearances of the by now well worn assertion that nothing can come (γίγνεσθαι) from nothing (So the things in the universe have to come from something – a proto version of Aristotle’s first cause theory) and nothing can (or cannot? lol) be destroyed (a play on the ambiguous meaning of “nothing” as a syncategorematic term vs. a singular referring term). For a situation like that simply can’t be contrived (ἀμήχανόν of which Raven’s much too epistemological “inconceivable” is probably as good a faute de mieux as any). If we interpret this to mean that it is “inconceivable” that something can just appear (in either the physical or perhaps also some other non physical universe), then it is obviously wrong. I am conceiving it right now (as Hume did before me). First I conceive the universe as it is. Next I conceive a universe that includes a Jenna Jameson clone who is madly in love with me and whose existence adds to the sum total of mass/energy in the universe (assuming appropriate adjustments in the laws of conservation of mass/energy). The second universe would be a biological sensation and the realization of all my hopes and dreams. But it is not a logical contradiction (despite whatever the firm beliefs of the real Jenna may be). This is a case of coming to be ex nihilo. But if in fact a Jenna Jameson who is madly in love with me were to suddenly appear in the universe as a rearrangement of existing matter, just as an Aristotelian statue comes to be as the result of a rearrangement of bronze, that would be a case of something or someone just appearing that would not require us to perform a radical adjustment in our understanding of the conservation of mass/energy.

It is just possible that Empedocles is making a different point. DK B 12 uses the phrase ἔκ τε γὰρ οὐδάμ’ ἐόντος which means “from something that in no way exists.” If Empedocles meant that nothing can come to be from something that isn’t, he is of course right. Something that doesn’t exist (Take god as an example) pretty much can’t do anything. However, taken at face value DK B 11 draws the erroneous logical conclusion.

The concepts of otherness and likeness and the interactions between like and different things was an ongoing concern for Presocratic cosmology before it became subject to logical and ontological (Heidegger would probably qualify this historically as “metaphysical”) treatment. If we interpret these terms as cosmological concepts in the modern sense of scientific physics we find it difficult to empathize with their interest for the Presocratics since those concepts seem to be incidental in post Newtonian physics. Polarities are simply a way to describe observed phenomena rather than part of an explanatory theory. And, of course, in the terminology of modern physics it is unlikes that attract and likes that repel. Nevertheless the concepts of attraction and repulsion, as well as positive and negative charges and opposite valences are analogous to the types of oppositions that the Presocratics observed in nature. And, to the extent that these phenomena play a role in chemical reactions and even physical state change, there is an eerie similarity to the way the “roots,” combine and recombine in Empedoclean cosmogony. Accordingly we might with justice regard the homoiotic terminological apparatus as simply a historical fact; if we assume this attitude, then we should not be concerned with whatever justification it may have so much as the circumstances of its devising and elaboration and the influence it has had on those currents of Greek thought – the logical and the ontological – that are of continuing relevance.

With respect to the issue of likes and opposites Plato, at least initially, operated within a terminological matrix (Out of respect for Heidegger’s insightful reading I will try not to refer to these terms as concepts. And Quine’s ontological Puritanism would also have us avoid talk of concepts when we want to make strictly regimented philosophical points. However,I think there is an understanding of the word “concept” that is looser than understanding it as a Platonic abstraction or Hegelian Begriff. This looser understanding allows us to use this valuable word even in discussions of mythology much less Presocratic thought. Of course, the notion of a term as in “terminological matrix” might very well have as much metaphysical baggage as the notion of a concept) he inherited partly from Heraclitus and mostly from Empedocles (probably via Archelaus) and which was central to Greek philosophical thought at that time. It is a matrix in which like and unlike things as well as (assuming that Empedocles really did make the second order claim I alluded to above) likeness, unlikeness, opposition, liking and disliking (love and hostility) all interact (In the Lysis he does not make the dialectical point, such as he would later make in The Sophist with respect to being and non-being, that love and hostility are themselves opposites or ἐναντίοι). Does he stake out a position in this terminological matrix; likewise does he criticize or accept any of the doctrines of his predecessors? In the Lysis Socrates seems to confine himself to the critical view that both liking and dislike can be proved and disproved to apply to likes (i.e. things that are alike), opposites and to things that are neither alike nor opposite. The result as it gets worked out over the course of all of Plato’s writings is that the matrix of opposition and similarity is mostly dropped. It is supplemented by a completely different matrix, one that uses notions such as appearance and reality (Paradoxically this could be viewed as Plato’s essential opposition, the contrast that embraces and neutralizes all the others) and, in the Timaeus, the doctrine, inherited from Anaxagoras, of creation by an anthropomorphic craftsman. Even so, a refined version of opposition appears in the Timaeus and resurfaces with a vengeance in Aristotle’s physical theory. In addition, in Plato the technique of differentiation or analysis (διαϕέρειν), as practiced in great detail in The Sophist, appears to supplement and indeed largely replace traditional cosmological explanantion by opposition.

Socrates’ argument in the Lysis forces together two distinct strands in Presocratic philosophy. The first is the roughly logical discussion of the relation of contrapositives as it appears in Heraclitus and Parmenides (Yes, Martin, I know that the Presocratics did not literally separate out a distinct pursuit called logic; I’m doing it for them.). The second is Empedocles’ theory of the (psychologically worded) mutual relations between likes and unlikes. Plato jumbles Empedocles’ cosmological (and social) theory with the slowly emerging theory of contrapositives or ἐναντίοι as a logical theory such as would come to fruition in Aristotle. The ultimate recognition in The Sophist that non-existence could exist even though things that are characterized as non-existing cannot without contradiction be said to exist (How can I attribute this conclusion to The Sophist given Cornford’s restrictive translation of the passages in question? Tune in next year to find out.) is based on his important discovery that there is a difference between concepts and things that fall under concepts. Although Plato would couch this difference in what we would call ontological language, his doctrine of Forms and things that partake of Forms lays the groundwork for Aristotle’s distinction between particulars and qualities and correspondingly between the subject and the predicate of a proposition.  I suspect it is this distinction between concepts and things that fall under concepts that leads Heidegger to consider Plato as a metaphysical and the Presocratics, Heraclitus in particular, as pre-metaphysical thinkers.


Opposites are mentioned by Anaxagoras according to the commentators. But, with the exception of his doctrine of perception, he offers no theory of the interaction of opposites as opposites. Moreover, the assumption that he adhered to a doctrine of the mutual attraction of likes is tenuous. That principle seems to have been replaced by a more generalized notion of mixing. For these reasons there isn’t really anything in Anaxagoras for the Lysis to attack. Plato’s attitude toward Anaxagoras is expressed in the Phaedo.

Simplicius (Fragment 4) attributes mention of a handful of what his predecessors had designated as opposites to Anaxagoras. These are moist/dry, hot/cold and bright/dark. DK B16 adds dense/ethereal. Simplicius also uses terms signifying likeness and difference. These are ἐοικότων and ἕτερον. The term ἐναντίος is not used. DK B4 does not mention any relation of attraction or repulsion between these likes and opposites. On the contrary, the opposites at least are simply different and, furthermore, capable of mixing together (σύμμιξις) in a whole (σύμπαντι) that is undifferentiated precisely because its components are different from each other. Instead of attraction and repulsion the cause of mixing and therefore of generation may be rotation. Empedoclean elements (Is this the implication of the phrase γῆς πολλῆς?) and seeds are apparently also included in the mixture. Opposites and substances may be portions of the seeds in a way that Raven considers to be a reduction in status for both the opposites and for Empedoclean substances. DK59 A41 may be about an attraction between likes, but the term actually used by Simplicius (συγγενῆ) was not used by or attributed to Anaxagoras’ predecessors.

I think we can safely ignore the concept of homoeomories in this context not only because our focus is on opposition and not likeness, but also because, as a glaringly Aristotelian term of art, it might not have been used by Anaxagoras at all (Cf. Raven pp. 386-388). Based on the evidence we would not be unjustified in extending this argument to the traditional attribution of a doctrine of opposites in the Presocratic mode to Anaxagoras. True, he does mention items we recognize as Heraclitean opposites and he does contrast these with non-oppositional natural substances. But since he does not, at least in the quoted portions of the extant literature, use terms meaning opposition or likeness (aside from the controversial συγγενῆ), we may speculate that he was not overly concerned with the nature of opposition (or indeed likeness) as factors in the explanation of the behavior of seeds or natural or composed substances. Moreover he does not appear to occupy himself with the logical relation between likeness and opposition at all. Like Empedocles and the atomists he seemed to be mostly concerned with the physical world and theories immediately applicable to it and not with the more speculative aspects of the theories he devised. Of course, Plato’s use of his notion of mind as a creative force would swing the pendulum sharply back towards Parmenidean style speculation.

Anaxagoras does broaden the contrapositive basic components of “everything” from Empedocles’ four roots to include, well, everything. This is likely the sense of his famous statement that everything is part of everything (DK B12). But the modification seems to imply, if we can rely on terminological evidence, that Anaxagoras did not believe he was talking about the same thing at all. Three sets of terms are in play. The first belongs to Empedocles who speaks of ῥιζώματα (roots) and ὁμοίοι (likes). The second is the set of terms Aristotle uses to talk about both Empedocles and Anaxagoras indifferently. His terms are στοιχεῖα (elements), ἀρχαί (principles), αἰτία (causes), ἐναντίοι and, specifically with reference to Anaxagoras, ὁμοιομερῆ. Anaxagoras’ own terms are σπέρματα (seeds) and ἕτεροι (complements or alternatively different things). The fact that Anaxagoras did not use the same terms as Empedocles to refer to roughly equivalent concepts indicates that he thought the objects of his theory were quite distinct, and they were. Secondly, Anaxagoras’ seeds do not maintain the critical relation of contrapositivity or ἐναντιότης. They are not contrapositives, they are complements; not opposites but other. Some like hot and cold match Empedoclean ἐναντίοι, but what Anaxagoras seems to regard as important about them is not their contrapositivity so much as the fact that, like other non oppositional χρήματοι, they underlie fully formed entities. Barnes (pp. 320-322) argues that, while Anaxagoras included opposites along with what he calls “stuffs” (and Raven “natural substances”) in his ontology of nature, this was a mistake stemming from his substantialization of properties. In this passage Barnes does not address the status of opposites as abstract second order entities.

In DK B12, again from Simplicius, Anaxagoras seems to collapse the opposites (and perhaps Empedocles’ roots as well (Cf.  Raven, p. 374)) into a mixture of everything except one thing (τὰ μὲν ἄλλα παντὸς μοῖραν μετέχει). The one other thing is νοῦς or mind. Everything (or almost everything) are qualified as ἕτερος and mind is ὅμοιος. Raven suggests that the concept of mind replaces the ostensibly equally anthropomorphic forces of attraction and hostility from Empedocles, although, if that is the case, Anaxgoras does no more than offer an alternative cosmological picture or model. He does not argue for its superiority over Empedocles’ model. Aristotle’s comparison of Anaxagoras and Empedocles in the Physics (A4, 187a23) also focuses on generation in the physical world of cosmic and biological events. However, one purpose of the Lysis might be precisely to produce logical conundra to undermine Empedocles and so possibly imply an argument for the Anaxagorean model. It is possible that (partially on the basis of the term μίαν in DK B4) Anaxagoras thought that his model also incorporated Parmenides’ the One in its proper place as νοῦς. If that were so, he wouldn’t convince Parmenides’ himself whose logical arguments would not tolerate a separation of the One from anything else. Interestingly enough Aristotle appears to assert that for Empedocles the constituents of the world were limited to στοιχεῖα since he claims that Anaxagoras supplemented the στοιχεῖα with the opposites (and everything else).

As in so many other cases when it comes to interpreting Greek culture, Nietzsche (p. 863) in a few words delivers one of the more insightful observations about Anaxagoras when he compares his homoiomeries (to use Aristotle’s term) to Anaximander’s τὸ ἄπειρον. He points out that the homoiomeries are qualitatively completely indeterminate. He also states that they are quantitatively indeterminate, although that is possibly contradictory since the very notion of individual particles is quantitative. The “dust-like” homoiomeries are numerically distinct, which is presumably a quantitative notion. It is some idea of quantity that distinguished Anaxagoras’ Urmischung from τὸ ἄπειρον. Nietzsche compares the two Presocratic notions in terms of how they fit into a cosmological or physical theory. It is in this context that he suggests that Anaxagoras’ chaos is superior to τὸ ἄπειρον. I suppose he means that it provides a better, richer explanation of the composition of the universe. If that is the case, then underlying Nietzsche’s comparison is a deeper distinction between these basic concepts. Anaximnder’s notion, I propose, is motivated by what we would recognize as a logical conundrum. As I suggested above, it puts a label on the fact that any statement that predicates a purportedly universal concept of everything is contradictory. Anaxagoras’ notion homoiomeries, on the other hand, is cosmologically motivated and is meant to provide a model for understanding the constitution of the physical universe. Accordingly or statements about it are not subject to objections of logical contradiction.

The terms ἐναντίος and ὅμοιος do appear in DK59 A92. But that passage is more or less an outlier because it deals with perception and sensation and the relation between opposites here is not in any obvious sense one of attraction and repulsion or justice or strife. That is, it is not the kind of metaphysical and cosmological relation addressed by his predecessors. Rather the idea is that perceived opposites cause pain in the perceiver, a view that is on the surface of things strictly biological.

While Anaxagoras may recognize a difference between contrapositivity and otherness (He uses the term ἐναντίοι but his puzzle about the origin of hair (Fragment 10), for example, envisions only things that are not hair and not anything opposite or contrapositive to hair), the logical issues regarding everything that bothered Parmenides and regarding opposition that appear in the Lysis do not arise.

Nietzsche (p. 864-867) seems to describe an Anaxagoras who fits opposites into a cosmogonical motion based on likes alone. That is, opposition is not, as in some of the other Presocratics, an explanatory primitive or ἀρχὴ in the way that likeness is a primitive. Instead opposition is itself explained by a cosmogony one of whose ἀρχαί is likeness. Incidentally in Nietzsche’s version Anaxagoras’ account of the universe in motion is both mechanistic and goal directed. That is, the cosmos is tending toward an end state implicit in the process of change and motion, but the motion and the end state are not the work of an anthropomorphic νοῦς.


Socrates’ teacher may have restored Empedocles’ elements or roots to a more prominent position than they enjoyed Anaxagoras’ overall cosmology. He may also have given a “mechanistic” gloss on the interaction of opposites (Cf. DK60 A10 and 18; Raven’s term is “automatic”) in place of the anthropomorphic language of human-like intentions preferred by almost the entire tradition. If that is the case, then there would be little surprise that the central role of an anthropomorphic mind in the cosmology of the Timaeus is incompatible with Archelaus’ doctrines.

Philosophical Issues Concerning Opposition

The Significance of Opposition and Likeness for Greek Philosophers

One thing should stand out from this summary of the historical background of the philosophical concept of opposition. At least one purpose of the Lysis was to undermine the conceptual foundations of Empedocles’ cosmology and biology and possibly also any social theory he may have held – a program that was pursued further in The Symposium. Anaxagoras may have been the victim of collateral damage to the extent that he actually held traditional views about likeness and opposition - which supposition, because of the difference of his terminology, is not immune from doubt. In any event many of Plato’s objections to Anaxagoras were already rehearsed in the Phaedo and Plato would retain the concept of νοῦς in a strengthened version in his own cosmology.

The reference to Empedocles should be no cause for surprise. Even Lamb footnotes Empedocles (p.43). Nevertheless our mini history may give some idea of how far the rejection of the conceptual framework of likeness and opposition, on the one hand, and attraction and repulsion, on the other, extends. It is Empedocles’ system, or more accurately Empedocles’ terminological/conceptual matrix that set the stage for Plato’s drama. Of course, Empedocles’ philosophy did not appear in a vacuum. In attacking Empedocles, Socrates attacks a way of thinking that winds its way through much of Presocratic thought and in doing so he strikes, wittingly or not, at the heart of Greek religion. Almost every cosmology of the Presocratic tradition is affected in one way or another including the Hesiodic version. The Lysis, along with the other mostly negative pre-Symposium dialogues, sets the stage for Plato’s eventual proposal to overturn the tradition and replace it with a cosmology and ontology (and effectively a religion) of his own devising. It is a warm up for the tetralogy stretching from the Parmenides to the Timaeus and Plato’s own theory of opposition that would be elaborated through the later dialogues but most completely in the Timaeus. Nevertheless, we can regard Plato’s purely negative early dialogues up to Diotima’s speech in The Symposium as capable of standing on their own and providing an intelligent critique of existing philosophical doctrines without necessarily specifying an alternative.

Furthermore, what hasn’t been noticed, or, in my opinion, sufficiently emphasized is the vagueness of the concept of opposition both in Plato and the Presocratics. Understanding that vagueness will help us make some perhaps hitherto unremarked distinctions regarding the concept of opposition and even unearth a few formal properties. So, now that we have an idea of what Socrates was up to in the Lysis, it may be worthwhile to examine just how convincing Plato’s attack on the conceptual coherence of the inherited framework is before we turn to the fate of opposition in Plato’s own system.

Different areas of interest can be sorted out in the Lysis: 1) The very idea of opposition. 2) Types of opposition - complementarity and contrapositivity. 3) How items characterized by opposing concepts relate to each other. 4) Whether the concepts of opposition that apply to the physical world parallel (or are the same as) the concepts of opposition that apply to the emotional or social world.

The idea of opposition as a way of understanding the world has, as we saw, its roots in Hesiodic religion. I speculated above that Anaximander introduced this idea into ontology or general cosmology as a way of avoiding problems in Thales style reductive systems, problems that we would regard as conceptual or logical. The opposite to opposition, namely likeness, did not appear as a significant factor in Presocratic philosophy until Empedocles. Historically opposition and likeness arose somewhat spontaneously as concerns for Greek philosophers. They never explained what they meant by those concepts or the importance of creating theories of opposition. The Presocratics did not define opposition or likeness. Rather they moved straight to creating theories based on those concepts (Again I’m using the sort of terminology Heidegger deplored. Whatever). Moreover, for the most part the Presocratics did not use a term for opposition at all. They spoke of the world in terms of pairs that later philosophers would regard as exemplifying what later philosophers called opposition. Much the same could be said about likeness. So when Plato comes to deal with opposition and likeness in the Lysis, he has in hand concepts that his predecessors had left undefined but which, probably because of familiarity, all the characters of his dialogue appear to understand intuitively. In very uncharacteristic fashion Socrates never asks for a definition of opposition or likeness. Indeed he is the primary culprit in the unrestrained use of those concepts.

By “alike” Plato partly means similar and partly identical. How does identity enter the fray? The thought is entertained as to whether second order entities such as goodness or liking like (the English word play obviously does not exist in Greek) themselves (More on this in a moment). This commingling of similarity and identity in the concept of likeness adds confusion to vagueness resulting from lack of clear (actually any) definitions.

Plato does not exclude complementarity from the meaning of “opposite” even though there is another term (ἕτερος) that more accurately translates as “complementary.” As with so many other terms, Plato’s use of ἐναντίος is not consistent. Specifically he can use it to mean both complementarity and contrapositivity as the occasion requires. In Euthydemus 278 A, for example, ἐναντίως implies complementarity and not contrapositivity in that the opposites are knowledge and non-knowledge: ἐπὶ τῷ τε είδότι καὶ ἐπὶ τῷ μή. But in 281D the apparent meaning is contrapositivity. In The Sophist Plato (if not Cornford) would eventually recognize the distinction. Otherwise Socrates throughout mingles complements and contrapositives. These two relations are not at all the same.

The confusion between complement and contrapositive persists to our time. I pointed out elsewhere that this confusion defeats the goddist argument that good logically presupposes evil. Likewise Nietzsche pp. 837-838 credited Parmenides with introducing logical thought in response to Anaximander’s concept of ἄπειρος. The Parmenidean world is divided into positive qualities and their ἐναντίοι. According to Nietzsche ἐναντίοι correspond to “negations” of the positive qualities. Parmenides designates these two halves as seiend and nicht-seiend, to use Nietzsche’s German. Obviously the sense of ἐναντίος here is “complementary” even though Parmenides’ actual examples are in fact contrapositives. The best that can be said is that Parmenides introduced something like logic into philosophy but that his very first step incorporated a conceptual ambiguity.

Another example is Strawson who (Logico-Linguistic Papers p. 81) appears to confuse complements with contradictories. The only reason the confusion is not patent is that he doesn’t specify the exact meaning of supposedly complementary terms. His first pair, for example is happy/unhappy. As we have seen the true complement of happy is not-happy. Stones and numbers belong to the set complementary to the set of happy things as long as we do not understand being happy as a state applicable only to humans and some animals. They do not belong to a contradictory or contrapositive set that includes Raskolnikov among others. Strawson fails to make this distinction, but his failure has no bearing on the point of his article which is to explain by way of classes the asymmetry of subjects and predicates through negation.

As it turns out defining opposition is no easy task. The first step, viz. distinguishing clearly the types of opposition, otherness or complementarity (ἕτερος) and contrapositivity (ἐναντίος), is fairly straightforward. And we can impose a Boolean understanding of complementarity on Plato without contravening most of his actual uses of the term ἕτερος. The problems arise with contrapositivity under which the examples raised in the Lysis fall. Plato never tells us what it is about hot/cold, good/bad, loving/hating etc. that makes these pairs similar to one another and similar in virtue of being contrapositive. When Socrates assimilates contrapositivity to unlikeness he is simply conceptually confused. Nevertheless we can fill in the blanks by specifying a very schematic group of basic properties (for which see below). It is worth noting that the concept ‘opposite’ understood in some sense other than complementary is not a polar concept to like. Cobblers and archers are unlike, but we can’t without further qualification call them opposites.

The theory of interaction of opposites in terms of love and strife or attraction and repulsion suffers from the same deficiencies. Empedocles does not define ϕιλότης or νεἶκος just as nobody defined likeness or opposition. So when Socrates turns to undermine the conceptual coherence of those terms he continues to deal with terms for which an understanding is presumed but never made explicit. As a result Socrates can pretty much say what he likes about the attraction or repulsion of likes and opposites and rely only on his hearers’ intuitions – or unquestioning pedagogic acquiescence – for agreement. I don’t at this point see how illuminating formal properties can be specified for the love or conflict of opposites and likes in general in such a way as to set the stage for rigorous philosophical assessment. Barring any valid psychological or sociological observations the best we can do is note the parallel appearance of those notions in a social context and in Greek cosmology and its heritage as well as their eventual use by modern science for the physical universe in a well defined but much diluted version. But never say never.

In the Lysis Plato elaborates his views on contrapositives in the middle of the apparently unlikely discussion of whether things characterized contrapositively can ϕιλéω each other. In this case the concept of ἐναντίος has its own contrapositive, namely likeness or ὅμοιος. The issue is whether contrapositive or like things can ϕιλéω similar or contraposed things or not at all or both. (As I noted above, there is an ambiguity in the word “like” that exists in English but not in Greek. Similarly ϕιλéω in Greek just as aimer in French can mean both love and a non-amorous preference for another person or thing analogously to the toddler’s profession of love for cookies. For the French analogy think of the difference between je l’aime bien and la bien-aimée.) In what relation of opposition the abstractions like and opposite themselves stand to each other is not clear. Socrates raises that issue but does not pursue it. Some but not all of the confusion will eventually be cleared up in the Theory of Forms in The Sophist. Nevertheless, we might be tempted to say that we didn’t know that hot and cold were even acquainted much less in a position to like each other. I for one was of the opinion that they moved in different circles. I won’t address the second order issues here. The topic at hand is the concept of contrapositivity and the relations between first order qualities demands a separate treatment.

Socrates is not completely off his nut. His background reference is, as I have been emphasizing, more than likely the Empedoclean world view which tends to attach personality traits to what we benighted moderns would consider the abstract or the inanimate. Another possibility is that Empedocles attached a specialized philosophical and less anthropomorphic meaning to φιλία. In that case it is reasonable to assume that Socrates engaged in deliberate misunderstanding when he used examples from the personal lives of his friends to make Empedocles look ridiculous and criticize the idea of attraction or repulsion among all opposites, including presumably physical or cosmological opposites.

Moreover, Socrates’ argument is dialectical (in the Hegelian peritropic sense) when he discusses whether like can love the like or the unlike love the unlike: Things that are unlike are alike in the sense that they are both unlike. Therefore if you believe that the like cannot love the like but that the unlike can love the like, you contradict yourself because the unlike loves the unlike but they are like in that both are unlike. It should be noted that the same argument does not hold for things that are simply like. Or if likes like each other and unlikes dislike each other, unlikes must also like each other since they are like in both being unlikes. Socrates’ intent is most likely to subject the Empedoclean conceptual scheme to a Parmenidean miasma of inescapable self contradiction

The argument in the Lysis is extremely intricate and apparently needlessly so. But, as I noted above, I am inclined to think that the experience of reading the Lysis is meant at least partly as an elaborate joke and the tortured nature of the argument is what structures the joke. Many of Plato’s early dialogues not having to do with the trial and death of Socrates are in fact verbal hijinks perhaps parodying the fashionable rage for clever argument that Plato associated with the sophists. He as much as said even as late as the Cratylus that the etymological fireworks featured in that dialogue weren’t meant to be taken seriously even though it ends with the discovery of the all important doctrine of the direct knowledge of things with no veil of language or εὕρεσιν τῶν ὄντων (436a). In the same way the via dolorosa of specious argument that Socrates drags us through in the Lysis has at least one important conclusion: there is something wrong with the conceptual structure of opposition and similarity, especially when it is associated with behavior involving strife and attitudes like the one the Greeks summed up in the overworked concept of ϕιλία. Socrates tries to demonstrate that conceiving the social world as structured by opposition leads to absurdity. His attack is not head on, nor could it be since opposition and similarity constituted a fundamental categorical framework within which Greek thoughts, beliefs and apparently social practices were firmly anchored. Since all thought was shaped by those concepts, there was little room to stand outside them and mount a critique. Plato’s strategy is consequentialist: Accept the conceptual framework and see where it leads. The Lysis does not delve into cosmology, but the implications are clear.

Socrates does not conclude as he does in so many other dialogues that true ϕιλία consists in teaching the ϕιλούμενος, in any relationship involving ϕιλία, σοφία and a knowledge of the good, teaching him, that is, that the interest of a man in love should consist not so much in corporal sexual activity, presumably fucking, as in encouraging the younger object of his feelings to avoid disreputable behavior and become an active and valuable member of his community. The Lysis takes a different tack. The first is to show that the “courtship” strategy should not consist in writing lyric odes to the glorification of the ϕιλούμενος. Rather the φιλῶν, should subject his target to the rigors of philosophical argument, which will have the double benefit of instructing him in the life of the mind and reducing rather than magnifying his self esteem. The second is to conclude at least penultimately that it is not similarity or ὁμοιότητα that should serve as a basis for true ϕιλία; rather the basis is one party feeling a desire or ἐπιθυμία, for the other. The assumption, although this is not quite clear in the context of the dialogue, is that what made Lysis and Menexenus similar and distinguished Lysis from Hippothales was age. But Hippothales felt a perhaps genuine desire for Lysis and it was due to this factor, not spurious similarity, that Lysis owed Hippothales his ϕιλία.

Phenomenological Ruminations on Opposition

If the practice of phenomenology is cleared of its pretensions to reach necessary or a priori conclusions by means of introspection, it turns into doing field work without actually going out into the field. There is nothing wrong with this. In fact mathematicians and some theoretical physicists, not to mention novelists, do it all the time. It is particularly à propos when the observations to be made are about my own feelings or my own understanding of terms in a natural language I speak. So let’s free our minds from the by now somewhat tired Hellenic examples of hot/cold etc. and look for other examples of what we would consider to be contrapositives in order to see what they might teach us. As with all primary observations, the detection of contrapositivity doesn’t admit of long chains of proof. On the contrary, they should be “obvious” to anyone to whom they are presented and who understands the language in which they are presented. Someone may just not “see” the contrapositives listed below and the best I could do would be to try to explain why I regard them as contrapositives. And, of course, individual examples I present may not be contrapositives. I might have characterized them erroneously.

Here are a couple that come to mind right off the bat: placid/agitated, indifferent/interested (where “indifferent” means “having considered the issue and doesn’t care what the outcome may be” and not not-interested in the way that Juliet’s bed is not-interested in her sex with Romeo), hydrolysis/dehydration, oxidation/reduction. I’m sure that with a minimum of effort the reader can come up with many more.

Social categories provide more examples:  In the Protagoras 323D Plato refers to the social advantages (τὰ καλὰ) and their contrapositives, but he identifies the latter only as τούτων κακά without giving a positive definition of what these contrapositives might be. Let’s step out on our own and consider the oppositional pairs Guelphs/Ghibellines, Democrats/Republicans or Whigs/Tories. In a three party system such as existed and perhaps continues to exist in the UK contrapositivity may not apply. Are the Conservatives the contrapositives of Labor, of the Liberals or of a combination of the two?

On the other hand, Empson’s example of (propositions about) a ruled population logically entailing (opposing propositions about) a ruler (7 Types of Ambiguity p. 195) is actually a case of disguised complementarity. When we “know” the ruled, we mentally restrict our domain of reference.

Artistic schools and styles also admit of contrapositivity: Stravinskians/Schoenbergians, drawing/color, Ingres/Delacroix, analytical philosophy/continental philosophy. Another set are comprised in Wölfflin’s categories for understanding Renaissance and Baroque art: linear/malerisch, Fläche/Tiefe, geschlossene Form/offene Form, Vielheit/Einheit, Klarheit/Unklarheit where unclear is not just not-being-clear but rather a specific kind of visual organization. It’s kind of vague as to whether modernism and post modernism are true contrapositives because it is vague as to what the post modernists mean by modernism. They almost always include formalist criticism and Abstract Expressionist art. But they usually pull in the entire avant-garde from Manet to Pollock as well. Quite often they just mean anything they don’t like especially if they can accuse it of being capitalist, imperialist, racist, sexist, exploitative and, of course, bourgeois. Nietzsche’s famous Gegensatz of the Apollonian and Dionysian sets off a train of constitutive contrapositives: dream/reality, image/intoxication and individual/unity to name only a few. Baudelaire appears to have developed a world view based on complementary contrasts as exemplified by the opposites line/color and artist/bourgeois (although both oppositions are nuanced in that he talks about bourgeois artists (e.g. p. 414) and considers line to be an abstraction from color). Claude Pichois provides an interesting summary of Baudelaire’s doctrine of contrasts in a footnote to his critical edition of the Paris “Salon of 1846” (pp. 1294-1295). Equally noteworthy is Baudelaire oppositional classification of types of women as examples of the “contrastes, qui gouvernent l’ordre morale et l’ordre physique” (p 19): honest women/actresses, loose women (filles)/stupid women, love/ beef stew (pot au feu). Hey, I’m just the messenger!

Sporting rivalries comprise an interesting set of contrapositives since they are so fluid. Yale/Harvard continues to flourish in the minds of some. But, even though at one time we could say Yankees/Dodgers, probably no more. That rivalry has been replaced with Yankees/Red Sox. Steelers/Cowboys has disappeared without new rivalries taking its place. The USC Trojans have more than one rivalry indicating that contrapositivity need not be exclusive. (In the Protagoras 332D ff. Hippocrates asserts and Protagoras agrees that ἐναντίοι must be exclusive. There can only be one and not many opposites to a given concept. He does not defend this assertion and the USC example seems to contradict it. Indeed Hippocrates himself provides what is supposed to be a paradoxical example of multiple contrapositives on the way to reaching the dubious conclusion that wisdom and temperance are the same thing, when he asserts that wisdom and temperance could both be considered contrapositives to ἀφροσύνη. Whatever we may think about wisdom and temperance, I think we may safely maintain that UCLA and Notre Dame are not the same thing. So I will ignore his assertion, and indeed the proof of the conclusion that the wise must be temperate, as invalid. The assertion may be based on a confusion between complementarity and contrapositivity or else to some ambiguity in the notions of wisdom and temperance.) These examples of contrapositivity are certainly not transitive or reflexive and it most likely not symmetrical either. Consider losing teams who consider last year’s champion to be their most bitter rival. The fluidity of sports rivalries indicates that contrapositivity in certain cases and to a degree is in the mind of the beholder. But is that always the case? Are hot and cold contrapositives only because we stipulate that they are? Or is there something about the relation of certain temperatures and their respective effects on the human nervous system that should lead us to believe that there is something “natural” about their contrapositivity?

The following concepts could have any number of contrapositives: daydreaming, thinking, dreaming, preoccupied, gifted or passionate. Preoccupied, for example, could have alert, distracted, ready, focused, at attention or obsessed. Interestingly enough focused could be a synonym of preoccupied just as easily as it could be a contrapositive. Obviously the difference depends on two different meanings of “preoccupied,” but it does serve to further illustrate the aforementioned fluidity in determining contrapositives.

The trio of vaginal sex, oral sex and anal sex can function as counterparts in three possible contrapositive pairs. The prominence of pussy fucking as the only procreative activity in the group, however, reduces somewhat the evident contrapositivity of the pair butt fucking and skull fucking.

Literature abounds in examples. As You Like It lists contrapositives at III ii ll. 29 ff. Note also the references to “philosophy.” On the other hand, another Elizabethan poet wrote “What else is hell but loss of blissful heaven?” fully aware that some apparent contrapositives are really complements. The entire verse reads:


What else is hell but loss of blissful heaven?

What darkness but lacks of lightsome day?

What else is death but things of life beriv’n?

What winter else but pleasant spring’s decay?


And Demetrius (On Style Ch 2 37) refers to contrasting incompatible styles as ἐναντία.

There are lots of discussions of opposition in Plato himself that are unrelated to the cosmological and social issues dear to the Presocratics. In the Lysis (215 E) Socrates rattles off a list of contrapositives that include not only dry/wet and cold/hot but also bitter/sweet, sharp/blunt and empty/full. In The Meno 70C the term ἐναντία is used conversationally  and without philosophical significance to mean opposite situations. Thessalians were willing to subject their views to public debate while the Athenians were not (an interesting contradiction of the Periclean view of the Athenian personality).

The apocryphal Alcibiades II distinguishes between complements and something like contrapositives (138C – 140E) without coming up with a clear concept of contrapositivity. It also shows how there could be different candidates for a contrapositive to a given concept (Cf. esp. 140C) and entertains the notion of a third term between a concept and its contrapositive (140A-B). However, it is asserted without justification that a given concept cannot have two opposites (contrapositives) (140B). The term used is ὑπεναντία. Longinus in On the Sublime III 5 uses ὑπεναντίον with a clear sense of contrapositivity. Also in XLIII 6 he speaks of lofty stylistic techniques and their ἐναντία.

In the Protagoras 332A ἀφροσύνη is identified as the opposite of σοφία presumably, despite the alpha privative, in the sense of contrapositivity. In 332 C-D high and deep voices are contrasted as opposites. Vlastos (Platonic Studies p. 274 ftn. 15) rattles off his own list of “contraries” in Plato’s dialogues. Strangely, he ignores the Lysis. Plato’s favored topic, namely the opposition between good, bad and neither good nor bad prompt our interest not so much as examples but because, like the beautiful, the relations between objects that can be characterized as good, bad or indifferent is of philosophical interest.

In a number of locations Aristotle also addresses issues in the context of the Presocratic framework of likes and opposites. For example De An. II 4 416a20 assesses current theories of nutrition which fall into two opposing camps, those who claim that likes (τὸ ὅμοιον) nourish likes and those who claim that nourishment comes from opposites (τὸ ἐναντίον). Lacking examples, it is not clear whether what he means by “opposite” in this context is the same as my contrapositive. However, the reference to intermediate states indicates that he has in mind the theory of opposition on a scale which he propounds in the Metaphysics (on which more below). His theory of sensation in terms of opppsites can be found in II 11 422b 17. And in Met. I v 986a he lists a series of contrapositive pairs he attributes to the Pythagoreans.

Complementarity depends on, informally speaking, a set theoretical domain or universe, a prior sectioning off of objects of interest into a generic group. Venn diagrams and Boolean notions capture nicely how a domain or universe provides the framework for specifying two complements. Contrapositivity does not fit neatly into that kind of framework. Consider hydrophilia and hydrophobia as chemical terms. Many hydrophobic molecules, given the definition of “hydrophobic” favored by biologists, are not true contrapositives to hydrophilic molecules since they are not repelled by water molecules. Rather they are complements within a domain (roughly the domain of molecules). Likewise, for any larger domain that includes the domain of molecules, they belong to the complement of hydrophilic molecules. Entities that actually repel water molecules, on the other hand, are contrapositives to hydrophilic molecules without regard to whatever domain they may both belong to. This is one important “formal” property of contrapositivity, so to speak.

Another property is that there is no effective method for specifying the contrapositive to a concept as there is for specifying its complement. It depends on our imagination. But with a dose of ingenuity we can come up with a contrapositive to almost anything. I am right now looking at a light bulb and no obvious contrapositive comes to mind for the concept of a light bulb. A dark bulb? I’m not sure whether there are such things as dark bulbs but no one said contrapositives had to exist. The complement of the concept of a light bulb, on the other hand, is the concept of everything in a domain that is not a light bulb. The complement of a bulb would be anything that is not bulbous whether or not it had an actual shape or not. Perhaps the contrapositive of a bulb would be something anti-bulbous. More on this below.

Along these lines Gérard Genette wrote an interesting article (pp. 171-183) about a 17th century popish priest named Étienne Binet. The latter composed a book of rhetorical tropes containing almost entirely what appear to be rather arbitrarily paired contrapositives. At first glance they don’t seem to have the kind of relation that Genette describes as contrarieties and antinomies. But, like the indigenous informants Lévi-Strauss alludes to, Binet in the course of his exposition provides us something of a basis for seeing how these pairs could be set in opposition to each other. For example, flowers and stars are related because flowers are earthly and stars are heavenly. I suppose we can intuit that earth and the heavens are contraries. Hence Binet calls flowers earthly stars and stars flowers of the sky. Unlike candidates for logical relations, the fact of these contrapositives cannot be gainsaid.

Being directly opposite something in space or spatial opposition, which is the original sense of the word ἐναντίος, seems to be particularly easy to specify. Indeed the possibilities for spatial opposition appear effectively unlimited. What, after all, could count as a spatiotemporal particular that did not admit of being directly across from something else? A featureless sphere suspended in infinite empty space? There’s no thing opposite it. But could a place or position facing the sphere count as a satisfactory contrapositive? Perhaps. Space itself may be better viewed as the sphere’s physical complement, except that space is both inside and outside the sphere. Most likely the space outside the sphere counts as the spatial complement to both the sphere and the space occupied by the sphere.

Our descriptions of electromagnetic phenomena demonstrate that what we designate as like or opposite can be based on observed attraction and repulsion rather than the other way around. The fallacies of arbitrarily determining what are like or opposite and then trying to create a theory of stipulated oppposites on the basis of such an arbitrary determination are thereby avoided. Some opposites including some contrapositives are or may be natural. Vlastos (Ibid.) identifies a sub-class of contrapositives he calls “polar opposites.”  Attraction and repulsion qualify as polar opposites as do black/white and his own example of hot/cold. Vlastos identifies polar opposites as those contrapositives which admit of intermediate cases. A thing can be lukewarm or some shade of gray. There is another sense in which polar opposites, or at least those examples under examination, have an aura of naturalness of inevitability unshared by examples such as Yankees/Dodgers whose contrapositivity is mostly a matter of perspective. In this sense polar opposites are like complements in that part of the meaning of terms such as hot and cold is that they are opposites. But they are not complements by the very fact that they admit of intermediate cases. Vlastos confuses matters by conflating polar opposites, which are contrapositives, and complements. He further muddies the waters by his misuse of the logical term “antisymmetric.” The relation between opposites can very well be antisymmetric in case they are true complements. What he meant to say is that the class of things specified by one of two opposite terms is a subset but not necessarily a proper subset of the complement of the class specified by the term’s “contrary.” He should also have added that this is the case only when the classes specified by the contrary terms are mutually exclusive and that they may or may not be complementary.

It is sometimes difficult to talk intelligently about a phenomenon or a concept without conceptualizing a contrapositive to that concept. And the contrapositive needed is rarely just the complement. For example, in discussions of solipsism and the reality of the external world, we need a contrapositive concept like the internal world or the world of the mind in order to make sense. Indeed if we start from the spatio-temporal world (whose definition I will, for the purposes of this essay, leave aside), we have our choice of contrapositives. We can contrast the spatio-temporal world to mind or we can contrast it to a world of Platonic entities.

An amusing example of what often passed for philosophy in the Renaissance can be found in Firenzuola’s comments on Aristotle on opposites (p. 40). He applies the by now discredited (at least if the gentle reader has been paying attention to what I have written thus far) doctrine that if you know one of a pair of opposites, you also know the other, to a criterion for detecting moral qualities in women.

Contrapositive pairs such as sadism/masochism and male/female sexuality figure importantly in psychoanalytic theory. Also T.B. Bottomore apparently has a theory of the composition of society via opposites. (Cf. Trivedi (p. 11) in Chattopadhyahya and Kumar eds. Lang. logic and Science in India.)

Analysis in terms of contrapositive opposition has become an important tool for the social sciences and humanities in the last half century or so. In La Pensée sauvage (p. 190) Lévi-Strauss introduces the concept of a principle of opposition. His analysis is a classic in what we have come to recognize as structural analysis via opposition. This sort of approach has become particularly familiar in literary scholarship. Gérard Genette (Figures I p. 35) looks at Baroque poetry in terms of pairs of oppositions and a very Saussurean analysis of their non-referential and purely systemic nature. In Lévi-Strauss’ case the contrapositive opposition is not part of his own conceptual apparatus, a lens he uses to understand the phenomena he studies. Rather, he attributes oppositional thinking to the isolated societies he studies. It is the lens they use to organize and make some sense of physical nature and the organization of society, i.e. their own societies. The idea arises when field anthropologists, in the process of gathering data about alternate conceptual frameworks, are faced with claims of opposition whose basis is not immediately clear to the scientist approaching the informants’ information from the outside. For example cold skin is opposed to feathers and from this opposition is derived oppositions of water, lizards and frogs, on the one hand, and emus, ducks and other birds, on the other. Apparently certain species of trees are also conceptually opposed by belonging to one or another of these two groups. Another mysterious opposition is between frogs and opossums. The solution lies in discovering the basis of similarity that unites opposed groups such that an oppositional differentiation can take place within the larger group constituted by the basis. We could hardly expect that the bases used by Western biologists would be naturally those employed by the alternate conceptual structures of so-called primitive tribes. Accordingly frogs and opossums can be conceptually opposed because they both leap. Some species of trees and birds are classed together because the birds make their nests in those trees. Other species of trees are classed together with fish and with water itself because they grow in the proximity of ponds and swamps. Another principle of similarity is “working together.” An intriguing line of study would be to relate these independently developed classification systems (some of which are much finer grained than their equivalents in Western biology, (Cf. Lévi-Strauss ibid. p.15)) to attempts by Western biologists to find appropriate morphological characteristics whose groupings would accurately reflect a history of descent, even though the history itself might be vague. The idea of a principle of opposition sets some limits on what we might call a purely nominalist contrapositivity. We almost always if not always require of anyone who claims two items of a pair are contrapositively characterized that he explain himself. We prefer to understand how the items are opposed and that means specifying some wider concept that connects them. This is not unlike the set theoretical universes of which two complementaries are subclasses. The important difference, of course, is that complementary classes are Boolean and that there is an effective method for specifying the universe containing them. Everything that is a member of one of the two complementary classes and nothing else is a member of the universe. As I have been at pains to argue, there is no such effective method for specifying the members of a universe containing contrapositive classes. Short of enumeration, i.e. purely extensional specification,  in which case the principle of opposition would be unnecessary, an attempt to define the universe is likely to include members outside the conjunction of the contrapositives and exclude items that belong in the conjunction. If you define the universe containing the classes of frogs and opossums as the class of things that leap, then you include too much. If you define it by enumeration as consisting of the class of frogs and the class of opossums, then you lose the meaning of the principle “things that leap.” You end up turning the classes of frogs and opossums into complementary classes. As far as I can tell there is no mechanism within standard first order logic for defining principles of opposition, just as there is no such mechanism for defining contrapositives. This is only to be expected.

We don’t as of now have an effective method for generating contrapositives. Unlike complements, there is nothing in the structure of formal systems, i.e. nothing in our understanding of the meaning of syncategorematic terms, that will yield by a rule of mechanical computation the contrapositive of a given concept, or, speaking in loose set theoretical terms, yirld a set that is also a portion of the universe of a given set but has no members in common with the given set and is not its complement. There is nothing like linguistic negation that will automatically specify a contrapositive concept. Indeed, as we have just seen, there appears to be no single contrapositive to a given concept. This situation restores some of the value of contrapositive thinking despite Plato’s criticisms.

Formal Properties of Contrapositivity

Contrapositives are not contextually defined or capable of formal specification. But they are a phenomenon worth philosophical investigation. The examples of sports rivalries should bring out the fact that there is nothing natural about the way things are identified that makes them or the concepts they fall under contrapositives. There is nothing about being-a-Yankee that makes whatever falls under that concept an essential contrapositive to whatever falls under being-a-Dodger. Of course, some contrapositives do seem to occur naturally like the electromagnetic poles or color wheel relata. It seems more appropriate to say that we discover them than that we devise them. Others, like the Guelphs and the Ghibellines or sports leagues or political parties are social creations even though they appear “natural” to anyone who participates in the societies of which they are a part. Others like cold and hot, good and evil or agile and clumsy seem to depend a great deal on the perception of the beholder. It is noteworthy that these examples are akin to Lockean secondary qualities. What is interesting about these types of contrapositives is that someone may very be able to understand the meaning one pole of the contrapositive pair without having the other pole in his conceptual repertoire. That is, someone might have the concept cold but not the concept hot even though he understands the concept not-cold. As long as someone has the concept of complementarity or negation on his mental repertoire then he cannot understand a concept unless he also understands the concept’s complement. As far as contrapositives are concerned someone may in fact have both concepts but not consider them to be opposite to each other. Finally two men might both consider an object to be cold but disagree whether another object is hot by comparison. Perhaps we can extend the caution about conventional contrapositives provisionally and with less certainty to all qualifications that lead us to regard particulars as contrapositive. I shall assume that there is nothing natural about being hot and being cold that forces us to consider hot things and cold things as contrapositives. Accordingly whatever formal properties I specify with respect to contrapositive concepts are conditional. If we regard them as contrapositives, then this is what we mean.

Concepts can fall under higher order concepts that in their turn can be regarded as contrapositive concepts. The correct analysis of a situation where one contrapositive concept can be predicated of another is precisely one where the concepts in question are meant to characterize properties of concepts themselves. Take the contrapositive pair intriguing and boring. It would follow only from the vagueness of these concepts that we could regard something as simultaneously intriguing and boring. So let’s just agree, for the sake of this illustrative example, that if something is intriguing it cannot be boring and vice versa. These are mutually exclusive contrapositives but not complements. Now, there is nothing to prevent us from thinking that the concept of being-boring is itself something intriguing even though whatever falls under the concept of being-boring could in no way be intriguing. Perhaps La Rochefoucauld would agree that there is something vastly intriguing about being boring. This sheds light on why non-existence (or even nothingness – not quite the same concept but what is said here about non-existence applies equally to nothingness) could exist. There is nothing on the face of it that prevents us from assuming that concepts may exist (setting aside the thorny problem of the meaning of existence). Since non-existence is a concept, it could without contradiction be said to exist even though anything that falls under the concept of non-existence could not without self contradiction be said to exist. This is the way Plato solved the Parmenidean paradox of existence in The Sophist.

This example suggests a further qualification of any formal properties we propose for contrapositive concepts. Participants in a psychological experiment conducted by Hollins, Faldowski, Rao and Young identified two pairs of touch dimension and one stand alone dimension: roughness/smoothness, hardness/softness, sponginess. The fact that sponginess does not appear in practice to have a contrapositive (Cf. Cognitive Psychology, Robinson-Riegler and Robinson-Riegler , p. 81.)  indicates that contrapositives are not automatically generated from our familiarity with a given concept although with minimal ingenuity we might claim that firmness and watery are both candidates for contrapositives to sponginess. Incideentally, this is a good example of how there may be more than one contrapositive to a given concept. Along the same lines we might regard a contrapositive to roughness as either slickness or smoothness. If we take the set of rough things and then take its complement within an appropriate domain, obviously the slick and the smooth are (potentially overlapping) subsets of the complement; they function as contrapositives to roughness. There is nothing in the nature of the complement that dictates which of its subsets may be regarded as contrapositives.

Along these lines Nic. Eth. V I ll. 4 ff. gives examples of ἕξεις that are ἐναντία. Note that Aristotle says that we can know about one opposite notion from what we know about the other. This is not a logical or even strictly speaking a semantic relation. It comes in part from the fact that we construct these ideas.

When Wittgenstein (Phil. Unt. Nos. 596 ff.) speaks of different feelings and their opposites, he is in effect distinguishing between complements and contrapositives along with the appropriate scope distinctions. For example, the feeling of strangeness is different from the lack of a feeling of familiarity. If we don’t actively recognize something, it is not necessarily the case that we actively regard it as unfamiliar.

An amusing example of complementarity at play is Nietzsche’s malapropism schwermütige Gefühllosigkeit (Vom Nutzen… p. 300)

The binary codes that form the basis of modern computing are related to Boolean structures and the notion of complementary sets. In fact the historical source of computer coding is the Turing machine which itself arose from True/False models for the propositional calculus. Nevertheless Seth Lloyd (Programming the Universe, p. 18) mistakenly presents contrapositives such as hot/cold to illustrate his definition of binary bits. Hot/cold is not binary; hot/not-hot is.

With Boolean structures in mind let us turn to see what we can make of the different sorts of opposition from a formal standpoint. But nota bene: The use of set theoretical language in this essay is largely informal as a way of expressing an intuitive notion of complementarity. Technically, the issues raised particularly with respect to The Sophist cannot even be formulated in the Zermelo-Fraenkel version of axiomatic set theory and only with some difficulty in the von Neumann version. The following discussion may help illustrate the affinity between Russell’s Paradox and the Parmenidean/Platonic paradox of being. (Cf. Suppes, Axiomatic Set Theory, pp. 29-30).

Contrapositivity like complementarity is best understood as a second order, symmetrical, non-reflexive, non-transitive relation. It is a relation between concepts. There is no more than an inkling of logical levels in classical philosophy. But every case of individuals being called opposites comes down to the fact that they fall into opposing (either complementary or contrapositive) classes (partake of opposing Forms, to use Plato’s later terminology). A good man is the opposite of an evil man because good and evil are contrapositive concepts. Likewise coldness is contrapositive to hotness. But is a cold thing contrapositive to a hot thing? No, for, if it were, then Jenna at the beach and Jenna in air conditioning would be contrapositive to herself (although, if you introduce the appropriate qualifications you could make her opposite to herself). The term is often mistakenly used to mean a relation between the particulars that fall under those concepts. Indeed Plato commits this error. Regarding symmetry, in the Lysis Socrates argued that contrapositivity is not symmetrical in that, given two individuals, one may be the enemy of the other, but the other is the friend of the first. I shall assume his argument is invalid since it rests on a vagueness in Socrates’ understanding of the contrapositive terms themselves. He does not specify a respect in which one party can be a friend or an enemy of another. So, taking into account the second order nature of contrapositivity, for two concepts, F and G, F is contrapositive to G if and only if G is contrapositive to F. Good is contrapositive to evil if and only if evil is contrapositive to good. But the resulting relation between items that fall under these concepts is not necessarily symmetrical. It is not the case that Socrates is good if and only if Thrasymachus is evil. Socrates’ point is that friendliness is not obviously symmetrical between individuals. I add that there is a need for further qualification. Nevertheless, depending on the circumstances, “…is a friend of…” and “…is an enemy of…” could express symmetrical relations between individuals. But such circumstances do not involve contrapositivity between individuals. Those relations need not be paired with a contrapositive relation in the description of the situation. Furthermore, contrapositives F and G themselves, understood independently of each other, need not stand for symmetrical relations. In fact, just as hotness, goodness, Guelphness etc., F and G need not be relations at all. This is not the case for F and not-F, assuming our conceptual apparatus includes negation. The situation is confused in the Lysis because liking or φιλία is a first order relation that, Socrates argues, is not necessarily symmetrical. But this first order relation is grafted on to a characterization of individuals such that they fall under concepts that are related by the second order concept of contrapositivity. As a result, Socrates arrives at numerous self-contradictory conclusions partly because the concepts he uses belong to different logical orders, but mostly because it is a matter of indifference whether individuals who are characterized in one way should also be characterized in another way, which latter characterization happens to fall under a high order characterization.

The particulars that are characterized by contrapositive properties are also characterized by a complex first order property to the effect that they fall under some contrapositive concept (and related first order qualities such as “qualified by the contrapositive property…”). For example, hot and cold are contrapositive concepts. A hot particular is qualified by the contrapositive property hot. It is also qualified by the property of falling under a contrapositive concept. This derivation is, of course a formal maneuver, but its validity depends on the actual contrapositivity of the first property, being-hot. The idea of second order concepts may appear troubling, but I believe it is intuitive (with the proviso that intuitivity is largely culturally and historically conditioned). I believe it can also be defined rigorously within a logical metatheory that includes particulars and concepts, even though a satisfactory definition of particulars and concepts may itself be lacking.

As a second order predicate “contrapositive” is always a symmetrical relation (excluding for the moment what I call the Cherubino exception). The first order predicates that fall under it are not (or need not be). We might be able to invent artificial relations like “contrapositively cold to…” but these relations do not mean the same thing as “cold” and are clearly derivatives from the more natural second order relation.

The formal behavior of contrapositivity is different from that of likeness. Likeness is a first order relation that holds between two particulars that fall under the same concept. For example, two items are alike if they are both hot. But likeness does not thereby qualify as a second order concept in the way contrapositivity does. The fact that two particulars may be alike because they are both hot does not of itself imply that hotness is like any other concept except in the very broad sense that it is like any other concept that qualifies more than one particular. The likeness of two concepts is unrelated to the likeness of particulars that may be qualified by those concepts. In addition, likeness is reflexive and transitive for both particulars and concepts whereas, as we saw, contrapositivity is not necessarily either.

Contrapositivity is a second-order sometimes symmetrical relation, but one should not make the mistake of confusing contrapositive concepts with pairs of converse relations or symmetrical relations in general. They are not at all the same though quite often they are thrown into the same bag. Father/son expresses a converse relation and not a pair of contrapositives. The relation is expressed in the 2-place predicates “… is the father of…” and “…is the son of….” Hot/cold and good/evil, on the other hand, are pairs of contrapositive concepts and do not stand for pairs of converse relations. “…is good” in the relevant sense is a predicate that can be satisfied by a single argument. “…is good for…” or “…is good to…” are indeed relations and are perhaps required in analyses of “…is good” but they are not relations in any way that is similar to “…is father of….” You cannot say “for all x and y, x is good for y if and only if y is evil for x”; but you can say “for all x and y, x is the parent of y if and only if y is the child of x”. Likewise Roman/Carthaginian could be regarded as a pair of contrapositives, but “…is Roman” is no more a relation than “…is Scipio.” While hot and cold are contrapositives, “hotter than…” and “colder than…” are distinct two place symmetrical predicates (“hotter than…” is not the same concept as “hot”). Left/right is a mixed case. The combined concept is a symmetrical relation, but left and right as independent concepts can also be regarded as contrapositives. For example, left-handedness and right-handedness are contrapositive one-place predicates and not relations.  Master/slave (or better master/mastered) of Hegelian fame is mixed like left/right. “Master of…” is can stand for a two place predicate, but master and slave could also be regarded as contrapositive concepts. My use of the phrases “can be regarded” and “could be regarded” indicates how fluid the notion of contrapositivity is compared to logically precise terms like “symmetrical relation.” Contrapositives cannot be specified by an effective method in the same way as complementarity, for, if that were so, then two items would be contrapositive to each other no matter how we regard them. Roman and Etruscan might have to be contrapositives even if we in fact considered Romans and Etruscans as no more than neighboring Italian peoples. For the same reason contrapositivity is not something we can logically deduce from an analysis of the concepts. We always put something into the mix. There is, for example, nothing inherent in the converse relations “to the left of” that forces us to regard “to the right of” as somehow totally opposed in the same way that we tend to regard good and evil as totally opposed. An analysis of the meaning of those relations might strongly incline us to regard them as both contrapositive and as converse relations. But, unlike converseness, contrapositivity cannot be expressed by a biconditional or any similar formula. The question for Plato's discussion in the Lysis is: Is contrapositive a contrapositive to like? In one sense they cannot be compared because like is a first and second order symmetrical relation and the only way contrapositive can be a relation is as a second order relation that may or may not be symmetrical. Like is a two place predicate that can apply to particulars. Contrapositive, on the other hand, only applies to concepts and not particulars. If somebody is a father there is a particular that is a child. The two are asymmetrically related. But if something is hot, there is no particular that is cold by virtue of it’s being hot. “…is hot” is not a relation, but “…is father of…” is a relation. This illustrates the difference between contrapositivity and converse relations.

There might be a way to logically specify natural and even conventional contrapositives as complements by limiting a universe (i.e. a class in which other classes are included) in a purely extensional way using the apparatus of elementary set theory. Given a properly limited universe, sets of particulars whose members are qualified by contrapositive concepts are simply complementary. Define the universe strictly enough and the set of cold things is the complement of the set of hot things. In the universe of Florentine political parties at a certain date (assuming there are no more than two parties), the set of Guelphs is the complement of the set of Ghibellines (“You’re either for us or against us.”). Does this mean that, given a concept or a set, its contrapositive can be specified in a purely logical way just like its complement? Not really. The restriction of the universe remains an extra-logical operation since it depends on simple enumeration or a very precisely specified concept of set membership that does not involve simple mechanism like negation. There is also the lingering question of whether an individual can be qualified contrapositively to some other individual and yet not belong to a complementary class that may be specified in any other way than the purely extensional one just outlined. A traitor or a ringer could be a Ghibelline, i.e. contrapositively qualified to the Guelph qualification and yet not belong to any set specified simply as not-Guelph. He is both Guelph and Ghibelline. In a similar manner an individual can be contrapositively qualified with regard to other individuals but not belong to a set complementary to those individuals depending on how the universe of complementary sets is specified. A cup of hot coffee can be regarded as contrapositively qualified with respect to cold diamonds even though it is not a member of either of the complementary sets composing the domain of hot and cold jewels.

Complementarity and negation are expressions of what some people call a Boolean structure. That is, a proposition that predicates something of a subject and the negation of that proposition (likewise a proposition that predicates the complement of the original proposition) adhere to the Law of Excluded Middle. If one of the pair of propositions is true, then the other one is false. Contrapositivity does not involve a Boolean structure of paired propositions. For that reason there exist no known truth functional rules that define a relation between pairs of contrapositive propositions. Yet there is a relation between them that is something like negation. Right now finding or inventing contrapositive concepts is something of a creative procedure. That is to say there is no known effective method or mechanical procedure or algorithm for determining a contrapositive.

The preceding comments lead naturally to a loose formal structure of contrapositivity (disregarding the Cherubino exception for which see below): If predicates F and G are contrapositive predicate terms, then for a individual term a such that Fa and Ga are well formed: (1) Fa and Ga may both be true but usually are not. (2) ¬Fa and ¬ (Ga v Fa) may both be true or both not true or (Ga & ¬Fa) may be true. (3) We can extensionally specify a universe U (where a universe is just a set of which the sets specified by F and G are subsets) such that, for any member x of U, (x)((Fx v Gx) & (Fx ≡ ¬Gx) & (Gx ≡ ¬Fx)), assuming (x)((Fx v Gx) & (Fx ≡ ¬Gx) & (Gx ≡ ¬Fx)), is well formed.  Conditions (1) and (3) define complementarity between F and G in the universe U. For any x such that ¬Fx and ¬Gx are both true x will be a member of a (depending on the universe) complement of the union of F and G. (1) (2) and (3) imperfectly capture the rather meager logical skeleton of contrapositivity. (1) and (2) say much the same thing but (2) emphasizes that a set complement may contain members that are not contrapositives of members of its complement. Unlike the definition of complementarity these conditions do not capture what makes specific contrapositives, e.g. hot and cold, contrapositive. That is a matter of decisions we make about the non-formal semantic content of the concepts in question. But those conditions do express what propositions about contrapositives must state or presuppose if we were to eventually decide that the concepts in question are contrapositive. (1) captures limited incompatibility. (2) captures the fact that contrapositivity need not but may apply to the members of a complement of a concept. (3) adds to (2) the fact that the union of the sets specified by some contrapositive concepts is composed of two complementary sets. When the set of members of the contrapositive of the set specified by a given concept is a subset of the complement of that set, the members of the contrapositive set have special properties that do not belong to other members of the complement. If we think that contrapositivity reduces to this baggy loose logical form, then (1)-(3) are necessary and sufficient conditions for contrapositivity without the prior assumption that F and G stand for contrapositive concepts. That is a highly doubtful assumption, since it would make hot and warm, for example, contrapositives. If we think, on the other hand, that something essential to specific cases of contrapositivity has not been captured, then, barring the Cherubino Exception, (1)-(3) are simply necessary conditions for contrapositivity. As I have been arguing, a contrapositive to a concept cannot be specified by a formal rule in the sense that a complement is specified by way of negation or set complementarity. The contrapositive is a counter-positive, an independent concept that has a relation of complete opposition to its counterpart. In many cases it is hard to even imagine a concept’s contrapositive. Finally, the three conditions informally formalized above are based on the assumption that some universe has been specified. Accordingly they only characterize the hot coffee cold diamonds example by way of something approaching extensional specification. Accordingly we should also recognize the possibility of contrapositives outside a concept’s complement in a given universe. Complete clarity requires that the universe including both complements has to be specified and at least some contrapositives may fall outside that universe.

There appears to be no apparatus in standard first order logic with set theory for capturing contrapositivity. What about directional vectors from elementary algebra perhaps combined with Gödel numbering to mark the contrapositive concepts? Some vector pairs are inverse relations. Some contrapositives are inverse relations like East/West. But not all. This is because not all contrapositives, e.g. good/bad, are relations of any kind. Equally some inverse relations, e.g. parent/child, are not contrapositives. The sets of inverse relations and contrapositives intersect. The fact that contrapositive classes are non-intersecting subclasses of a given domain can be represented by Venn diagrams, but what makes them opposite cannot be represented. On the other hand we may be able to devise a mathematical model of contrapositivity using something like vector analysis (This would capture the oppositeness as a formal property without providing any information of what makes a given pair opposites). And, if algebraic geometry can ultimately reduce to first order logic with set theory (a very big if), then there can be a logical representation of contrapositivity. But as things stand there is no effective method on the order of negation or set complementarity to generate contrapositives. They are untamed. Aristotle (Met. X iv) outlines of theory of maximal contrariety which is a notion similar to but not the same as the idea of contrapositivity discussed here. On Aristotle’s theory see more below.

I have referred above to what I call the Cherubino exception to the mutual exclusivity of contrapositive concepts, i.e. to the idea that a particular cannot be both F and G if F and G are contrapositive to each other. The Ionian Presocratics may have maintained that opposite qualities (more accurately opposite somethings) such as hot and cold could simultaneously inhere in objects of perception. Unfortunately precisely because the issue they addressed involves perception it does admit of an alternative explanation in terms of secondary qualities, which is no more than adding an adverb of respect to an incomplete predicate. In the Philebus 36B Socrates, while engaging in a mind numbingly spurious argument for the existence of the soul, raises and ultimately rejects the possibility of someone’s having unqualified simultaneous feelings of pleasure and pain. Even if valid, the feelings Socrates considers are simultaneous only because they also are adverbially qualified either by their location or by their intensity (to which we might add other respects such as the physical pain of hot coals on a torture victim simulataneous with his sensed pleasure at the nobility of his sacrifice). 46 C-E adds another exception where apparent simultaneity of feelings of the contrapositives pain and pleasure is really a feeling of one and the increase of the other (admittedly a phenomenologically suspect line of reasoning). In The Republic Bk. VII 523D ff. he addresses the possibility of simultaneous sensation of opposites though his actual concern is most likely difference of standpoint as a source of perceptual error (He may also solve the problem to his satisfaction by invoking intelligibility as a necessary concomitant to unadulterated sensation). In his discussion of motion (Physics III i 20 ff.) Aristotle seems to take it as a given that outright contrapositives such as hot and cold cannot simultaneously characterize a particular when he observes that when properly qualified contrapositives such as potential and actual may do so. On the other hand, he argues in IV xiii that a point of time can be both a beginning and an end citing (222b) by way of analogy the fact that a circumference can be both concave and convex. Obviously these are also cases where the simultaneous presence of contrapositive qualities in a thing is due to a difference in the respect with which these qualities apply to the thing. What I have in mind with the Cherubino exception is a situation where something could be described as hot and cold without the alternative explanation of hot and cold really meaning being perceived by someone as hot or cold since the perceived object and the perceiver are the same thing.

Chaucer as well as Mozart knew what the feeling was like:

Allas! what is this wonder maladye?

For hete of cold, for cold of hete, I deye.'


Empson (p. 214) includes the simultaneous hot and cold under his seventh type of ambiguity. This passage is worth reading from the point of view of a philosophy of contrapositives not only because of the wealth of examples of opposites and simultaneous opposites Empson finds, particularly in Keats and Crashaw, but also because of the sexual element (p. 217) he discerns in the ambiguity due to simultaneously instantiated opposites. Returning to classical antiquity, we find Longinus suggesting that the “super contrapositives,” hot and cold, could be experienced simultaneously though his evidence for the claim is based on a misreading of Sappho’s φαίνεταί μοι (Cf. p. 81 ftn. 3); and  Aristotle (Nic. Eth. V ix) raises a similar case about an action being simultaneously voluntary and involuntary in a quote of a line from a lost Euripides play.

A situation of the Longinus kind arises in cases where the opposite qualities characterize the sensations of the perceiver himself.  If someone or something can feel hot and cold and the same time and if hot and cold are true contrapositives, then the class specified by one contrapositive may intersect the class specified by the other. They are not mutually exclusive even though they are opposed. In other words, Fx and Gx from (1) above are both true. If, however, saying that something is hot and cold at the same time is a vague or metaphorical statement that needs qualification of respect, then it need not follow that the contrapositive classes in question intersect. I cannot be sure whether Cherubino’s state is reliably recounted or not in his famous aria, that is, whether it is accurate to describe Cherubino as being simultaneously hot and cold (Cherubino’s actual words indicate a succession of hot and cold feelings; no matter, there are other examples that imply simultaneity such as Imogene’s gelo ed ardo in Romani and Bellini’s Il Pirata, Act I scene 3 27. Since there is more than one example of this poetic figure in opera librettos, it may very well have been a standard trope in Italian poetry or even a conversational figure of speech; that would only strengthen my example. But before the heyday of Italian opera we have Spenser’s “He gan to burne in rage, and friese in feare,/ Doubting sad end of principle vnsound” (Bk V Canto xi 2) which is distinctive in not being immediate motivated by feelings of love.). It is clear that it cannot be analyzed by adding qualifiers of respect in the same way that Count Almaviva can be described as simultaneously good in one respect and evil in another. It is also clear that Cherubino’s feeling hot or cold is not equivalent to a Lockean secondary quality. That is, it is not a quality Cherubino perceived in an object external to himself. A Lockean secondary quality would be in play if the Countess and Susanna disagreed as to whether Cherubino’s complexion was light red or dark pink. Hot/cold is what Cherubino feels about himself, and who are we to question the sincerity of his report? It is something like a secondary quality to the extent that it is something Cherubino feels about himself as an apperceived object, so to speak. Nevertheless what Cherubino perceives about himself is that he can be characterized by the first order contrapositive qualities feeling-hot and feeling-cold. Any accusation of logical inconsistency that I can ascertain would presuppose the incompatibility of contrapositives that Cherubino’s report denies. There is actually more to the poetic moment than meets the eye. Cherubino’s aria makes us feel at one remove what it is to be hot and cold simultaneously. In fact, we cannot understand the aria emotionally unless we in some sense know what it is to feel simultaneously hot and cold. Despite all this I find it nearly impossible to call up introspectively on my own what the feeling of being simultaneously hot and cold would be like (I suppose I should wait for some occasion when I fall hopelessly in love and reinspect my feelings under those circumstances). Is it shivering and sweating from excitement and fear? Does Cherubino mean he feels heat inside his body but the surface of his skin is cold? The former alternative would seem to admit the compatibility of contrapositives and the latter would explain it away. Nevertheless, the beauty of Da Ponte’s poetry and the suggestiveness of Mozart’s music, not to mention all our memories of first shy love, bring us ever so close to understanding how Cherubino feels. Such an understanding is not applicable to amorous feelings in general. Moreover, it is a physical state we seem to sense and remember. Nevertheless the phenomenology of the situation is uncertain. Lacking a strong intuition, I would simply repeat that simultaneous instantiation of contrapositive qualities may be possible.

A different take on a similar situation comes from Edward De Vere who cites contrapositively paired emotions (He calls them contraries), which serve as an outward appearance to mask what the speaker really feels: “Thus contraries be us’d I find / Of wise to cloak the covert mind.” DeVere explains the apparent simultaneity of incompatibly contrapositive feelings by describing one as feigned and the other as the true feeling of a “covert mind.”

Another example of a possible co-instantiation of contrapositive phenomena comes from Nietzsche’s concept pair Apollonian/Dionysian. Assuming these are genuine contrapositives, Nietzsche’s historical argument is that they are present simultaneously in the tragedies of Aeschylus and, arguably, Sophocles. He also implies that the pleasure in make believe and the ecstasy of inebriation, contrapositive affects of the Apollonian and Dionysian, can be felt simultaneously by the the spectators in a tragic performance.

Wittgenstein’s lapidary comments on religious belief in Lectures and Conversations (pp. 53-59) provide a good example of differences in the way negation can be used in what we say and how we conceptualize complementarity. Wittgenstein brings up someone who believes in the Last Judgment and goes on to say that he himself doesn’t contradict the believer, he doesn’t believe the opposite. Contrapositivity is not what he has in mind. A candidate for a contrapositive to what the believer believes would be belief in something like a Last Judgment where the judge was not Jesus but Satan or Baal or somebody (Unlike complementarity, which is Boolean, there can be any number of contrapositives to a given statement). It becomes clearer in the course of the notes that what Wittgenstein does have in mind is a scope distinction in the negation of the believer’s statement. What Wittgenstein denies is that he himself believes that there will be no Last Judgment. When he says he doesn’t think about it, he means that it is not the case that he believes there will be a last Judgment. In technical terminology he is distinguishing between the negation operator falling inside or outside the scope of the belief operator. This example points up an interesting fact about complementarity. Not believing that there will be a Last Judgment and not thinking about it both fall under the complement to believing that there will be a last Judgment. Agnosticism would also be part of the complement. Of course, being an agnostic and not thinking about the topic are two different things. Finally, as membership in the universe (i.e., the range of possible members of the class) containing a class and its opposite is defined in different ways, so what counts as a complement to the class can change. The complement of the class of Last Judgment believers may or may not include rocks or numbers depending on how you limit the containing class.

Strawson (Intr. To Logical Theory pp. 6-8) comments on predicate compatibility in a way that touches on what I am saying about complementarity vs. contrapositivity. His goal is to elucidate different kinds of inconsistency and different sorts of contradiction. The terminological framework he chooses is the scholastic Square of Oppositions as derived from Aristotle’s logic. Strawson’s observations lead ultimately to complex issues so I will do no more than make a few lapidary comments of my own such as are relevant to contrapositivity as discussed here: (1) His description of compatibility ranges does not apply to logical negation without a restriction on the domain of quantification. Strawson is not alone in committing this sin of omission. (2) He does not directly address contrapositives in my sense. (3) I am not here concerned with Strawson’s issue which is that some types of statements are not covered (explained?) adequately by theories of linguistic meaning that do not account for circumstances of utterance (not just time/place/speaker indexing; although a more fleshed out notion of speaker indexing might go a long way to providing such an explanation). Also (pp. 16-19) he defines logical contraries in a way that is similar to but not the same as my notion of contrapositivity. On p. 18 he explains his example of a non-complementary predicate in terms of his framework of circumstances and statements. He does not use the elementary apparatus of sets and domains that I use.  It is interesting to note his need to use background terms like “antithetical” and “opposite” which function something like genera including contraries and contradictories as species. Strawson’s understanding of incompatible predicates on p. 54 ftn. 2 comes down to saying that incompatible predicates apply only to subsets of complementary sets. This is to say that some or all contrapositives are incompatible but not all incompatible predicates are contrapositive. There are broader issues as to whether contrapositives must belong respectively to the opposing sets of a complementary pair as long as both belong to a single domain, and also whether some contrapositives may reside altogether outside a domain consisting of two complementary subsets. I address these issues below in a discussion of Aristotle’s theory of opposition.

On p. 106 Strawson seems guilty once again of the omission of not recognizing the importance of clearly specified domains when specifying complementary classes. His expression ‘-people who play bridge’ does not mean the same thing as the ordinary language phrase ‘people who do not play bridge.’ The domain of the latter is people while the domain of the former is unspecified. Accordingly the class referred to by ‘-people who play bridge’ includes or can include mice or mountains while the class referred to by ‘people who do not play bridge’ does not. (Any number of alternatives such as Russell’s Theory of Descriptions or scope distinctions could also be used to analyze this situation.) This ambiguity with regard to complementary classes leads to a similar ambiguity in his understanding of negation. But cf. p. 112 where Strawson accounts for precisely this in defining a universe of discourse in terms of a range of possible values. However, this raises more questions: (1) Is the universe of discourse coextensional with the range of possible values? (2) Is the union of the complementary sets determined by the universe of discourse coextensional with the universe? (3) Does the idea of possible values introduce a modality into the interpretation of first order logic?

Let me caution against reading too much into what I have to say about opposition and contrapositivity. Some might see it as trying to discovera “logic” of contrapositivity, or to impose one on it. Others might look upon contrapositivity as an example of something that cannot be subject to the effective method of conceptual analysis but rather requires “free variation” in the human imagination. I do not advocate either view and I don’t think my observations necessarily lead in either of those two directions. However, elements of both views have merit. In a broader sense I have no doubt that at least one reader should by now be wondering what I'm getting at with this account. Quite honestly I’m not getting at anything and had no end in sight when I began this project. I started with noticing something and then on that basis began accumulating observations and let them lead me where they may.

But what is the source of and what validates everything that I, Aristotle (See below) and Plato (obviously in order of descending merit) have said about the concepts of otherness, contrapositivity and opposition? Clearly noticing contrapositivity doesn’t come from the application of any logical or set theoretical laws. Rather it is a kind of observation whose object is language and what we would say about things. It is not observation like empirical canvassing by way of questionnaire because statistical results to the contrary might not cause us to retract what we have said as invalid. It is not phenomenological in the Husserlian sense since it is hardly more introspective than a physical observation (What after all is the difference between “The earth is round” and “We can’t call coldness hot”? They can be phenomenologically reduced in parallel to “It seems to me that the earth is round” and “It seems to me that we can’t call coldness hot.”(Note the psychologistic implications of the latter reduction in that it deals with apparently logical or at least semantic concepts.)). Yet, if you think about it, observations such as this compose about 95% of philosophizing (or thereabouts!). They are part of what I like to call extraspection, a concept I may explore elsewhere.

Asymmetrical relations as discussed by Russell in Principles of Mathematics Ch.XXVII pp.227 ff. are related to but not the same as contrapositives. Instances such as good/bad and Russell’s own examples taken from Kant such as pleasure/pain are not in fact asymmetrical relations but contrapositive one-place predicates although the comparatives derived from them such as better or more pleasant can be regarded as expressing transitive asymmetrical relations. Another of Russell’s examples is right handedness/left handedness, important in geometry, which can be analyzed into a group of second order relations between converse first order relations. But these contrapositives by themselves do not constitute series. Russell also glosses some of these contrapositives, to which he adds color opposites like red/blue, as synthetic incompatibilities which he goes on to define, almost certainly too narrowly, as “cannot exist in the same spatio-temporal place” (p. 233). Incompatibility is partially captured by the condition I outlined above of non-intersecting but not necessarily complementary classes, i.e. condition (2). Still, Russell’s translation into incompatibility doesn’t capture the special sort of oppositeness involved in contrapositivity. It doesn’t capture what makes good/evil contrapositive and blue/red incompatible but not contrapositive.

In Categories VII 7b ll. 15 ff. Aristotle makes comments about existence and the master/slave opposition which may be mistakenly understood to belie my analysis of contrapositivity and my application (link to Mackie) of it to specific philosophical issues. Anyone who has read with understanding the above passages will immediately see that it does not. But, in order to avoid misunderstandings, I will try to spell out why Aristotle’s passage is compatible with my own analysis. Aristotle’s conclusion is that that a master (δεσπότης) must exist if and only if a slave (δοῦλος) exists. That is to say an individual one of whose qualities is to be a master must exist if and only if another individual one of whose qualities is to be a slave also exists. This is consistent with my analysis to the effect that contrapositivity (among which we may count the master/slave opposition) is a relation between concepts which implies certain first order relations between the particulars which fall under those concepts. In more precise language, Aristotle’s point is that if a slave were to cease to exist, then the former master would cease to be a master. If all Jefferson’s slaves were to die, Jefferson would no longer be a master, but, everything else being equal, he would not cease to exist. He would continue to exist as Jefferson the revolutionary, Jefferson the President or Jefferson the former master. A couple of ambiguities in the Greek interfere with a properly fine-grained analysis. In the first place, the notorious ambiguity of εἶναι as standing for both the copula and the verb “to exist” stands in the way of Aristotle clearly distinguishing between existing as disappearing from the universe and existing-as. Of course, short of some indiscriminate plague, Aristotle was not making the physical observation that the deaths of slaves entail the deaths of masters. Also the vagueness of the English word “master” inclines us to mistakenly regard δεσπότης as standing for a simple meaning that ranges over a non-relationally defined class of individuals. If we translate δεσπότης as “slaveholder” instead we can see more clearly that it ranges over a class of individuals defined as members of certain n-tuples, viz. n-tuples of slaveholders and slaves and not just particular masterful persons. If we fiddle with the translation of δεσπότης a bit further to come up with “slaveholder of…”, then we can see its true nature as an incomplete one-place predicate contrapositive to another one-place predicate, “slave of….” Indeed Aristotle may have assumed just such an analysis by the placement of his comments in a discussion of πρός τι and not of ἐναντίος. The difference between the contrapositive pairs master/slave and good/evil (important in theological debates) becomes clear. Master/slave involves an opposition of concepts that are contrapositive and also relational (πρός τι). Good/evil involves an opposition of concepts that are contrapositive but not relational (Better/worse is contrapositive and relational like master/slave). Accordingly masters could not exist as masters (i.e. slaveholders) if there were no slaves for them to hold. But good things could very well exist as good things even if there were no evil things or even a concept of evil things. If we understand “good” as a term of approbation, there is some relation to other value judgments we make. Delicious food may exist and be judged by us to be delicious even if there were no disgusting or nauseating food, simply bland food by way of contrast. But even if we regard goodness as a property of an object or action and not only as attributed by a value judgment such as if “good” were equivalent to “judged-to-be-good,” the existence of evil things or actions would still not constitute a necessary condition for the existence of good things or actions. Feeding the hungry would still be good even if no mass murderers ever existed.

The intriguing issue Plato raises in The Sophist is whether a contrapositive has to be other, in other words whether a contrapositive must belong to, or, to use Platonizing language, partake of or participate in, the complement of that to which it is contrapositive. It is, for example, possible and even likely, according to the Stranger, that non-existence or nothingness belongs to the class of existing things. This leads to our next topic.

We have come a long way from our initial puzzlement over the significance of the arguments in the Lysis. But in a sense not much has really been accomplished yet. What we have found is that Plato most unsuccessfully tried to find logical flaws in Empedocles’ cosmology (and by extension elements of the cosmologies of Parmenides and Anaximander as well). The question now is with what did Plato replace the supposedly discredited explanatory system of opposition and what role if any opposition plays in the finished Platonic system.

Opposition in Plato’s Mature System

“Logical” Opposition. Even though Plato had no conception of anything approaching logic in Aristotle’s strict sense there are some passages in his dialogues that prefigure Aristotle’s purely logical definition of contrapositivity (a definition which in turn has close to no relation to the contrapositivity I have been discussing). In the Laches 196B and 197A ἐναντία are completely incompatible assertions in an argument. The second passage uses the term ἐναντιούμενος which means opposing another person’s argument or contradicting some statement he made. The first refers to self-contradiction (ἐναντία ἡμῖν αὐτοῖς λέγειν; the same phrase occurs, applied self-referentially, in the Meno 82A), the intended conclusion of the sort of peritropic argument favored, though so rarely pursued successfully, by Socrates himself. This usage is not philosophical; it comes instead from the lexicon of more conversational Greek as used in Athenian litigation. In the Protagoras 339B-340D the phrase ἐναντία λέγει αὐτὸς αὑτῷ also means self contradiction in argument (There is some significance, which I will not explore here, in the fact that the phrase occurs in the middle of a discussion of the distinction between being and becoming and yet another Platonic attack on poets, in the person of Simonides.) None of these passages relate specifically to the issues discussed in the Lysis where contrapositivity is addressed as a property of properties, but they are worth pointing out because they refer to linguistic or disputational or logical contrapositivity as contradiction and touches an important element of the Socratic method, viz. the rooting out of self-contradiction by way of analysis of concepts and statements.

The Meno 74D floats an idea related to the examples in The Lysis when Socrates states some geometrical figures (σχήματα) can be ἐναντία…ἀλλήλοις. His example of round and straight figures seems to imply a kind of contrapositivity, but an earlier comment mentions only difference or distinction (ἄλλα ἔστι, 74C).

The Protagoras 331A-333B includes a classic statement of Plato’s recognition of higher order entities and the almost inevitable category error in the discussion of whether justice itself is just (Specifically the issue is whether the divinely sanctioned is just and vice versa. The self-reference enters if one considers, as Hippocrates is prepared to do, the divinely sanctioned and justice to be the same thing). The explicit manner in which the self application of Forms like justice, or at least the application of Forms to other Forms, is discussed supports, I believe, the view that the same process is at work in The Sophist where the question is whether non-existence itself exists. In trying to make Plato’s ontology more commonsensical Cornford completely misses this thread. Likeness is also treated but only with respect to the parts of virtue and not as an opposite to ἐναντίος. In fact Hippocrates asserts that even the most opposed things have “a certain small likeness.”

The Phaedo (104B ff.) deals with a kind of relation between opposites that differs from the relations of liking and disliking or attraction and repulsion at issue in the Lysis. This is the sort of relation that we might call exclusivity or complementarity properly speaking and it is broadly logical in nature.  Taking odd and even numbers (i.e. positive integers) as an example, Socrates argues that the “possession” of one of the opposed pair of properties by a number excludes its possession by the other. That is, if a number is odd it cannot also be even and vice versa. Socrates’ ultimate conclusion - namely that the soul possesses (falls under) the concept alive, which is incompatible with the concept dead in the same way that evenness and oddness are incompatible, and that, therefore, the soul is immortal - is, of course, utterly specious. Socrates, for example, never examines his assumption that the soul is what causes a body to be alive. Nor does he consider the possibility that the soul might be a non-living thing that causes life in the body. Or that there may be a soul specific type of going-out-of-existence that is not death. But let that pass. How Socrates reaches the ridiculous conclusion illuminates some of Plato’s difficulties with the notion of opposition. Socrates’ argument does not account for change or what the Greeks would call “becoming.” If you add one peach to the three peaches in the basket, then what had been odd, namely the number of peaches in the basket, becomes even and an opposite admits its opposite. Plato did not distinguish between things that can fall under one concept and then another from things that cannot, i.e. things for whom the property expressed by a concept such as threeness is essential. The number of peaches in the basket is in a loose way an example of the first sort and the number of peaches in the basket before I put in the fourth peach, as well as any set that is equipollent to the cardinal number3 itself, are loose examples of the second sort. This is not just due to the fact the number of peaches in the basket is a general concept that is instantiated while the number 3 could be used as a referring term. For, in the same way Jenna can be hot while she is sitting by the pool and grow cold when she enters air conditioning. Assuming “Jenna,” “3” and “Socrates’ soul” are all referring terms, Socrates needs to prove his assumption that being alive is essential to our being able to pick out and identify someone’s soul (After all what did Odysseus see in Hades but dead souls?) in the same way that the assumption of being odd could be essential to our being able to pick out and identify the number 3 (irrespective of whether oddness is simply a conventional property of every other positive integer following the number 1 or oddness is built into a more formal set theoretical definition of the natural numbers).There is also the error, which I have focused upon throughout this essay, of confusing contrapositivity and complementarity. Just as the number of peaches in the basket may be either even or odd, the value of y in the equation x2 = y may be even or odd. In case it is even, and our universe of discourse is restricted to the positive integers, it belongs to the complement of the set of odd numbers. So does the number 2. We can affirm that any member of the universe is either odd or even but not both. Evenness is also contrapositive to the concept oddness. But if our universe of discourse widens to include the real numbers, the value of y may obviously also be odd. Indeed the values of the general concept expressed by the equation x2 = y may be odd or even or neither in a wider universe. Since the rationals are a subset of the reals, then the complement of the set off odd things is no longer coextensive with its contrapositive set, i.e. with the set of those objects that fall under the contrapositive concept evenness. ¼ as a value for (1/2)2 is neither odd nor even (alternatively it could be even but only at the cost of blurring the definition of odd and even). Following Socrates’ analogy, the soul may just be a rational number (The Pythagoreans would approve of that conclusion; of course, Raskolnikov’s soul would be an irrational number). It is this situation where a number can belong to the complement of the set of odd integers but not to its contrapositive (the set of even integers) that Socrates fails to recognize. Nevertheless the passage brings out these ideas that Plato would further develop in the Theory of Forms and that would later come to be recognized as involving conceptual, logical and set theoretical issues: (1) Having (ἔχει) a quality such as oddness (περιττός) as distinct from being identical with that quality (τὸ περιττόν; ἐστι is used for both identity and predication) (2) Abstract concepts (ἰδέαν and μορφῇ ) (3) Essential (ἀνάγκη)  properties (4) Higher order concepts and the subsumption (δέχομαι and ἔλθοι) of concepts under other concepts as well as the notions of mutual subsumption and self subsumption (5) The idea of contrapositivity (still confused with complementarity in the notion that opposites don’t admit opposites) (6) The distinction between a physical cause and logical or syllogistic subsumption (Fire physically causes the charcoal’s heat, but the subsumption of fiery under hot “causes” the object set on fire to be hot). Many of these notions were germinated by the Sophists, but still it’s not bad for only a few paragraphs.

In the Theaetetus 157 A-B, Socrates throws out the idea that being and becoming are mutually exclusive but the word ἐναντίοι is not used. Of course, Plato’s ultimate doctrine is directly contrary to the one entertained in this passage to the effect that all things are constantly becoming and can be apprehended only in context. It seems to me that this passage is not particularly relevant to the issue of contrapositivity.

Cosmological and Epistemological Opposition. Plato did not abandon the project of structuring ontological or cosmological categories by the concepts of opposition and likeness. In the Lysis he concerned himself only with criticizing the view that liking and dislike or attraction and repulsion are valid relations between likes and unlikes. In the later dialogues he proposes alternative theories that are not dependent on the concepts of attraction and repulsion or even likeness and unlikeness in quite the same way as the theories of the Presocratics. Plato’s cosmological alternative is in the Timaeus. His proposals as far as lovers are concerned are scattered throughout the dialogues but they appear in greatest detail in the Republic, Phaedrus, and the Symposium.

The theory of ἐναντίαν αἴσθησιν in the Republic  Bk. VII 523C ff. evokes the notion of opposite perceptions. This significant passage makes clear Plato’s recognition that properties that can be opposed to other properties, such as large and small, apply to an object in only a qualified manner. We cannot simply say that a finger (Socrates’ example) or a building is large. We need to say in what respect it is large. This involves two notions. First, a finger or a building is large only by way of comparison with other fingers or buildings. Secondly, that sort of comparison according to Plato is not the work of our visual or tactile or other sensations alone. Human understanding or reason (λόγος) must be involved in order to make the comparison and to apply the comparative quality properly to an object. We do well to separate these notions. For it isn’t necessary to accept a theory involving separate “faculties” of sensation and understanding in order to recognize the validity of the logical (more properly speaking, semantic) point that some qualities, such as those involving comparison, cannot be meaningfully asserted of an object unless certain blanks are filled in on the occasion we make the assertion. The fact that Plato knew this indicates that Socrates’ attacks on the comprehensibility of some ἐναντίοι in the Lysis was a kind of feint. Plato knew (or he came to know) that there was a correct analysis of the apparent contradictions. Assuming he knew it as early as the Lysis he simply did not engage in that analysis perhaps in order to dramatize the confusion of Socrates’ companions and to intensify the burlesque of Empedocles’ cosmology.

In the Republic Bk. VII 524C ff. the word ἕτερος appears as equivalent to δύο κεχωρισμένα, viz. “distinct.” Distinctness and complementarity are obviously not the same (although otherness could cover both) and this passage is another example of Plato’s inconsistency of usage (A similar confusion occurs in The Sophist). Nevertheless ἕτερος is clearly a philosophically significant term in this passage for reasons independent of the distinction between complementarity and contrapositivity because distinctness is something intelligible (νοήσει) and not necessarily perceptible without reflection.  Intelligence and sensation are ἐναντίοι in the sense of apprehending objects contrapositively, so to speak. But the means of apprehension are not the only contrapositives in question. Socrates argues for a different sort of contrapositivity when he distinguishes between objects that appear to us along with their ἐναντίοι and those that don’t. The former stimulate νοήσις or philosophical reflection and the latter don’t. I suspect that the theory of standards and measures as it appears in the Timaeus is meant as a refinement of the rough schema of contrapositive objects in this passage.

Further along (Bk. X 602 C ff.), in the midst of an attack on mimetic art, ἐναντίος is used to designate contradictory opinions (δόξας). In 603 D in particular ἐναντιομάτος refers to a kind of operational self-contradiction. I find this interesting because exposing operational self-contradiction in his interlocutors, viz. peritropics, lies at the heart of the Socratic maieusis. It indicates how central the notion of ἐναντιότης remains in Plato to understanding our mental structure and the interplay of concepts; it also leads us to suspect that the ridicule of opposition in the Lysis should not be taken at face value; it is part of a larger and more complex picture. However, the first sketch of Plato’s alternative to a simple theory of opposition also appears in this passage. Objective measuring of quantities is a superior approach to gaining actual knowledge of the qualities of an object because it is not subject to apparent contradiction based on comparison in the same way as the purely qualitative assertion of contradictories. While it is not necessarily contradictory to maintain that an object a is both hot and cold because hot and cold require adverbial modification of respect in order to be meaningful or else as comparatives require the specification of another item to compare with a, nevertheless the alternative of stating that a is so many degrees using a metric of temperature does not invite the sort of confusion that results from using unmeasured pairs of opposing qualities to make statements about objects. Representational arts like painting (specifically scene painting) and poetry are inferior because they remain bound to purely qualitative appearances and to the “lower” parts of the soul that apprehend those appearances. Plato expresses the superiority of the calculating part of the soul by way of a political metaphor that I’m sure the likes of a Derrida and some Marxists would go to town over. If we submit to purely oppositional or comparative qualities, he says, we let ourselves be ruled (ἄρχειν) by the world. If, however, we use objective scales of measurement, which are the products of our understanding and souls, then we rule the world. This is a sketch as I mentioned and a more complete exposition of the theory of measure will appear in the Timaeus, of which more below.

The Statesman 284 B-E contains a perfunctory statement of what Plato proposes as a replacement for the theory of oppositions. In his doctrine of μετρητική Plato distinguishes between two ways of regarding measurable opposites (e.g. big/small). They can be compared to each other as true ἐναντίοι or they can be measured (μετροῦσι) separately relative to their mean (μετρίον, 283 E ff. esp. 284 D).


Plato should be given credit. The concept of measurement by a standard was not contemplated by the Presocratics he attacked (although it may be present in ovo in Pythagorean number theory). And in many respects it is decidedly more useful than the qualitative assessment represented by the concepts of likeness and opposition. But the standard Plato lights upon is not fully independent. Rather, it is determined anew in each situation by the qualities he proposes to measure. Those qualities, which are still regarded as essentially opposites, determine a mean or mid-point. Distance from this mid-point is the quantity of measurement. Opposition remains essential to Plato’s concept of measurement. The fact that measurement depends on a mean in Plato’s doctrine may have been suggested by the affinity between the words μετρίον (mean; this is clearly not the same as our arithmetical concept of a mean between two quantities) and μετρέω (measure).  An advance as far as physical measurement is concerned would be to introduce the notion of a standard (inch, meter etc.) independent of and not determined by the qualities in a specific occasion of measurement. This approach would replace contrapositive concepts (big/small) and their mean (middle sized) with non-contrapositive qualities expressed by one-place predicates (e.g. 2 inches long). Clearly these standards are still specified in terms of what is to be measured. There is a further relation between the very idea and existence of a standard and a quantity to be measured, but considering that philosophical issue would take us too far afield as would a consideration of how this Platonic doctrine might have prefigured Aristotle’s concept of the mean.


Nevertheless, this further advance has decidedly untoward consequences for the Plato’s theory of the political personality and political morality which he not only wishes to include under the same umbrella as physical measure, but also constitutes the ultimate purpose of his arguments regarding opposites and measurement. The Stranger’s concluding analysis of opposite personalities such as the meek and the brave relies on the overall interpretation of measurement by way of a μετρίον. The μετρίον of the properly ruled state would be to not allow either the meek or the brave to dominate but to interweave the two. There is no equivalent in political psychology, at least not so far, for the centimeter (Can we speak of 3 units of meekness?). Nor does Plato want one. The essence of his doctrine of the political mean (and of its adoption and extension by Aristotle) is that moderation and the interweaving of extremes is good policy. And the notion of an independent standard by its very autonomy forfeits the context dependent element of a mean and thereby the notion of moderation as well. If we choose to take the step beyond Plato and accept independent standards of measure solely for physical quanties (a choice that is very nearly unavoidable, although Heidegger might consider it a further degradation into the slough of metaphysics), our decision reinforces a distinction between an ἐπιστήμη of the inanimate and an ἐπιστήμη of the animate. So Plato’s somewhat dubious parallel between the mean as a way of measuring quantities and the mean as a kind of personality type raises as many problems as the doctrines he proposes to replace.


Indeed affects and psychological states are notoriously resistant to quantification. Benjamin Péret’s perhaps gnomish line “la joie comme la peine se mesurent au centigramme Je connais la balance” (Le Grand jeu, p. 74) is not entirely irrelevant in this respect.


The Philebus 23 C ff. clarifies and elaborates the theory of measure sketched out in The Statesman. In the first place, it distinguishes between units of measurement and the mean by the simple device of ignoring the mean at least initially. What replaces conceptual or qualitative opposition is a metric system based on autonomous units. But before the metric can be applied some adjustments have to be made in our understanding of opposite qualities. The emphasis shifts away from the higher order concepts themselves such as heat and cold and away from the relations between these higher order concepts. Instead Plato focuses his attention on first order particulars - hot or cold objects – and considers their being hot and being cold as necessarily comparative. It is not enough to say that a particular of some sort is hot. We must specify that it is hotter than something else. And that the something else is colder than the first. This shift in perspective has the decided advantage of avoiding some of the paradoxes of likeness and opposition that were evident in the Lysis. Of course, Plato would not drop altogether the hard won notion of second order objects such as qualities. After all they constitute much of the subject matter of the Theory of Forms. And the Parmenides and The Sophist provide evidence that many other paradoxes of opposition are not solved by recourse to comparatives. Nevertheless, in the Philebus Plato’s shift in perspective gives him the tool whereby he can introduce autonomous units of measure into our understanding of opposite concepts and the particulars qualified by opposite concepts. He can elaborate on this to the effect that an item such as a cooking stone is (1) not just hot, and (2) not just hotter than a pebble in a brook, but (3) hotter than the pebble by x units of measurement. The cooking stone is, for example 5o C hotter than the pebble. There is a quantum of vagueness between what Socrates calls his second and third group among the groups (εἴδη) that comprise all things. I surmise that by the second group he means numbers in themselves and by the third group he means numbers applied as units for measuring qualities in particulars. But, even if there is no clear distinction between the two groups, I believe the point stands to the effect that Plato has devised a significant alternative to unmeasured qualitative opposition as a logical device (Since Reinhardt attributes a logic to Parmenides, I see nothing wrong with an understanding of something like logic to Plato as long as the proper caveats are kept in mind) and as a principle of cosmological explanation.


The meaning of ἄπειρος is also not entirely clear. This reflects no doubt a vagueness in conversational Greek usage of the term and also and especially Anaximander’s doctrine of indefiniteness. In the present case I once again surmise that Plato doesn’t mean infinite in the sense of an infinite sequence such as the natural numbers or in the way that the physical universe could be considered as spatially infinite. Rather, his understanding is a combination of Anaximandrean conceptual indefiniteness and an infinite progression similar to the density of the rational numbers. The idea seems to be that, if you just say the cooking stone is hotter than the brook pebble, your statement could mean any of a potentially uncountable number of states of hotness on the part of the cooking stone. But, if you assign a definite metrical value to the difference in temperature between the two, then you fix a definite single meaning to the statement that one is hotter than the other. The idea of the mean does not disappear in this discussion; indeed ethical maxims and conclusions are meant to be the end result of Socrates’ digression. However, at this stage moderation in human behavior bears almost no demonstrable relation to physical metrics of measurement. Socrates does not even attempt to deduce one from the other; nor does he propose any metrics for human behavior or psychology. The best thing he can come up with is a kind of analogy between the two. Or, more properly put, deliberate obfuscation. For Socrates mentions indefiniteness of quantity and excess in the same breath. Protarchus suggests, and Socrates readily agrees, that introducing measure or limit into opposites produces results desirable in ways other than simple accuracy. In fact Socrates doubles down on Protarchus’ suggestion by producing examples like health and illness where, by means of the introduction of limits, medical cures result. Likewise an easing of the discomfort of excessively hot or cold weather as well as different kinds of beauty such as physical and musical beauty and the “beauty” of the soul come about because of the imposition of limits glossed as ἔμμετρον. His suggestion contains two errors. The first is the introduction of quantities of measurement into our way of observing opposite qualities does not by itself produce the kind of desirable effect Socrates predicts in the physical world. He hints that there might be some sort of relation by devising a fourth group of things he calls causes, but he does not elaborate aside from saying that everything (apparently including qualities and limits) needs a cause. The second error is a confusion that appears egregious between the concept of limit (πέρας) as a unit of measurement and limit as the opposite of excess. Socrates’ reasoning is that just as limits or numerical measurement clear up the indefiniteness in comparative attributions of physical qualities, so limits on wild and unrestrained human behavior will solve analogous undesirable consequences. Unmasking the ambiguity of the concept of limit, of course, destroys the analogy.


The Timaeus takes elements from many of the cosmologists who preceded Plato - Parmenides, Democritus and the Pythagoreans in particular - and combines them into a single theory. The philosophical question is whether his combination is a synthesis or a pastiche. The Timaeus is syncretistic, to say the least. Some would call it a mish mash. Plato’s cosmology is a bookish sort of system. It is a an attempt on the part of someone who never seemed to show the slightest interest in original empirical physical observation to fuse together elements from the written and oral cosmological tradition into a systematic whole while ignoring those elements he rejects.  Plato proposes a cosmogonical myth of a demiurgic creator, a father and θεός, and the creation of the universe as a single event to replace the cyclical model of Empedocles and other Presocratics. He also constructed a cosmology of a spherical cosmos (derived from Parmenides) articulated by proportions (an idea derived from Pythagoras) and composed of soul and body (Anaxagoran in spirit) which latter is itself composed of the four elements: earth, fire, water, air (Empedoclean in spirit). The Same or ταὐτόν is identified with the soul and the sphere of τὸ πᾶν. Θάτερος or the other is identified with the body and the perceptible. This distinction also has Parmenidean roots. The attraction (Plato does not say ϕιλότης; but he does say ὁμόφυλον in 81A-B.) of like to like (ὅμοιον πρὸς ὅμοιον) remains an explanatory principle of physical phenomena such as seeing (44C). The apocryphal dialogue Epinomis (988 D) even includes a claim that the doctrine that like causes like is part of the cosmology in the Timaeus. Apparently Parmenides’ eternal and uncreated One is turned into the original creator by Critias (who ignores the paradoxes of plurality) and merged with Anaximander’s νοῦς (Reinhardt traced the origin of this maneuver to Xenophanes’ theological cosmology - an event (Ereignis?) that Heidegger would argue was one of the inaugural mistakes of Western metaphysics). Τὸ πᾶν, whose definition was the overriding concern of the Ionians, is rid of its logical meaning and viciously reified. In the Timaeus τὸ πᾶν is a copy of the original ταὐτόν or creator. It has a ψυχή. The gods are copies of this copy. The visible world is a creation of the Olympian gods with material furnished by the original creator. Note this cosmogony seems to be Egyptian in origin with a dash of Hesiodic theogony. But over and above that it represents a compromise between anthropomorphic religion and the burgeoning abstract concepts in the philosophical tradition and in that respect represents a net regression from Xenophanes. Gods and abstract entities are interchangeable; or perhaps the words that designate them are interchangeable. Or perhaps the anthropomorphic gods are the creatures of an original abstract and partially anthropomorphic creative force. Personally speaking, one of the most disturbing things about Plato’s cosmology is that the perfect sphere created by the demiurge eats its own shit (33 D;  Heli Rekula’s immortal performance piece Hyperventilation appears in this context to be a blow for sexual equality - earth mothers can eat their own shit too), but, hey, I’m squeamish about things like that.

The fact that Plato’s mature cosmology borrows elements from Empedocles sheds some ambiguous light on Socrates’ refutations in the Lysis. For the concept of likeness that gets thrown out with the bath water of opposition in Socrates’ demonstrations of semantic inconsistency reappears doctrinally in the Timaeus. Was Plato just trying to show the ineffectiveness of argument, or at the very least arguments in favor of Empedocles’ doctrines, in the Lysis? Or did he change his mind? Neither alternative is particularly satisfactory.

Several passages in Laws give some context and point to what may be the motives for the reductio that Socrates performs in the Lysis.  Bk. X 889B-D refers to a mechanistic (τύχῃ) cosmology described in terms of the interaction of opposites without the intervention of external volition (ψυχή or νοῦς). 891C adds a presumably Presocratic theory (most likely Archelaus’) that the elements, fire, water, earth and air are the constituents of nature and that something like ψυχή comes along later. 896E – 897B proposes instead the genesis of the ἐναντίοι from the original impulse of the ψυχή the self-mover and prime mover. Taken together these passages indicate what might be the purpose of the Lysis. Plato wanted to show that opposites cannot interact simply by virtue of being opposite. The argument is proto-logical or semantic. There is nothing in the definitions of individual pairs of opposites that can compel us to conclude that they like or hate each other or that particulars characterized by opposites like or hate each other. The unstated conclusion is that volition or νοῦς is necessary to achieve at least some of these results. The target may partly be Empedocles but it includes mechanistic thinkers as well, since Empedocles did not employ the concept of τύχη attributing instead what may be interpreted as a type of volition or ἐπιθυμία to the opposites themselves, viz. ϕιλότης. The context of these passages also makes explicit the reason why Plato thought it so important to refute the idea of interaction between contrapositives as an explanatory principle, in particular a cosmological explanatory principle. He associated that idea with Presocratic materialism and he rejected materialism on moral grounds. He believed that materialism led to atheism which led to murder, rape, sedition and listening to the wrong kinds of music. In a way Plato’s rejection of the contrapositive theory was not the result of a dispassionate objective evaluation (Although the Athenian Stranger says that in our attempts to convince the κακοί σοφοί we must try to repress our natural rage or θυμός). The desired conclusion dictates the initial position. The νοῦς theory is not so much true as what Plato wants to be true. Of course the screwball self-mover proof is produced to show the objective validity of the νοῦς or ψυχή theory, but even this bit of specious reasoning proudly displays its confirmation bias.

The elaborate theories adumbrated in The Republic (See below) and the Timaeus constitute models in the modern sense used by physical scientists. They involve a picture of how the world could be structured and a conceptual thesaurus for understanding that structure. There is very little argument for their validity outside the arguments that make his core concepts such as the Forms and the cosmos-creating mind plausible. Plato’s other arguments are directed against the then available competing systems. Some of those, like the ones proposed by the atomists and the Pythagoreans, are assimilated into Plato’s own model. Others like Empedocles’ system and the system implicit in Heraclitus’ writings are shown to be untenable. None of this adds up to the Platonic system being inevitable. In particular it would not be able to overcome and would eventually mostly yield to competing models in the form of Aristotle’s physics and, of course, modern empirical science, as adumbrated in the ironically mechanistic De rerum natura. Scientists would provide the arguments missing in Lucretius by way of accumulated observations. Nevertheless a type of the Platonic model persists. It informed Galileo’s thinking and continues, in a much altered version, as the reigning view among many mathematicians. One may regard as a commonplace (although I do not see this commonplace as sufficiently heeded by many philosophers) that philosophy does not consist in arguments alone. The devising of models for the cosmos, for society and for the human mind is where the real creative work of philosophy lies. It is such visions that serve as the basis for philosophical argument. Without this creative work philosophy does not become a neutral arbiter between “world views.” It would be difficult if not impossible to pursue that rather anodyne activity without installing some of the presuppositions of the world views under scrutiny in the judges’ seat. What philosophers in fact do is fill the vision vacuum with Aristotle’s “what men say” or the everyday English of the ordinary language philosophers, a version of Moore’s befuddled rider of the Clapham omnibus. Our ordinary way of looking at things, while it may not be ignored, is not a necessarily a touchstone for assessing the validity of competing visions. It may just be a heap of unconnected idées reçues.

Would the Presocratics have recognized Socrates’ counterexamples as valid? Would they have conceded that the attraction and repulsion of likes and opposites appropriately applies to all his examples? Would they have characterized Socrates’ examples as genuine likes and opposites? Or would they have demurred? Whatever the validity of Socrates’ counterexamples it could very well be argued that, as physics and perhaps as sociology also, the theories of Empedocles and Heraclitus do more justice to actual observable phenomena than any alternative Plato may have had to offer. Nevertheless we can see in dialogues like the Lysis the replacement of one conceptual thesaurus (what Kuhn might call one paradigm) by another. The genetics or cosmogony of likes and opposites gives way to a genetics of forms. It is not so much that one theory gives way to another (After all Heraclitus’, Anaximander’s and Empedocles’ views are competing theories within a single conceptual thesaurus). Rather one way of conceiving what questions are to be asked gives way to another. For example, the question of how opposites relate is no longer relevant (This is different from a conclusion that it is wrong to say that things are generated by the interaction of opposites). Plato replaces that problematic and the conceptual grab bag it involves with issues such as how the apparent world is generated by ideas or how the perception (more properly apprehension) of an εἶδος is possible. The moment of transition, however, does take the form of an argument, namely Socrates’ argument that no theory of the attraction and repulsion of likes and opposites can apply to all cases of ὁμοιότης and ἐναντιότης. Socrates’ arguments do not prove that some theory of likes and opposites can or cannot serve as a viable cosmogony. Rather, it discourages the search for such a theory.

Plato turns to the Same and the Other in a passage (Timaeus 43 D ff.) concerned with the origin of the mind’s discriminatory capacities and openness to error. The discussion is mixed with a great deal of talk about spheres and proportions – ideas that arrive in a direct line from Parmenides and the Pythagoreans – and does not address the sorts of issues raised in the Lysis. It is interesting that Plato seems to reify the Same and the Other as second order entities although English could just as easily express this with the slightly less mystical terms sameness and otherness. Τἀναντία appears in the passage to mean the opposite of truth (44 A), but in a way that is unrelated to cognitive sameness and otherness. The trinity of the Same, the Other and essence in the Timaeus (35B and 74A) reappear as the highest categories in The Sophist 244-245. However, the word for “other” in these discussions is ἕτερος not ἐναντίος. That important dialogue will receive a separate treatment below. The theory of perception in the Timaeus employs the concepts of likeness or similarity but does not refer to, indeed seems to allow no room for ἐναντίοι. Presumably whatever might be ἐναντίος to a given perceived similar would not be perceived, but that would also be the case for anything else unperceived contrapositive or not.

Plato’s best answer to Empedoclean contrapositivity is in the Philebus 23C ff. esp. 25C where the contrapositives are construed as relations (Instead of hot/cold we have hotter/colder) and understood as indefinite (ἄπειρος), which could mean either that something is both hotter and colder at the same time (such as in 12E ff.), or that there is an infinite series of hotter-thans. The solution to both is to propose an objective measure (πρὸς ἀριθμὸν ἀριθμὸς ἢ μέτρον) in the form of number. This could be a measurement like a thermometric degree. It is explicitly an objective relational term. Instead of object a simply being “hotter” than object b, we now say it is twice as hot (διπλασίου). The latter understanding also requires an objective standard to identify an objective non-relational property of an object (e.g. 80 degrees F) so we can answer the question: twice as hot as what? In 25E Socrates proclaims his triumph over Empedoclean contrapositivity – the so-called limiting of the unlimited (i.e. making precise what had been indefinite) which comes down to assigning a numerical value to a property that had appeared as one counterpart in a qualitative contrapositive relation. It may be that the theory proposed in the Philebus is derivative from discoveries already made by Presocratic philosophers, most likely from the Pythagorean school. Nevertheless, it answers neatly problems left over from Plato’s earlier purely negative evaluation of the concept of contrapositivity and suits the purposes of the Platonic system. It works for hot and cold and for musical qualities. Does it work for contrapositives like lover/enemy (Lysis) or cautious/courageous (Statesman)? I suppose this is where the vaguer some what metaphorical solution of weaving from The Statesman comes into play. It is also an issue that causes social science to our day to stumble. Rigorous definition of contrapositive terms and the specification of a reliable numerical measure by which they can be compared can be surprisingly elusive.

The  Philebus 43A alludes to Heraclitus in the course of a discussion of human awareness of pain and pleasure. This should remind us that throughout Plato mistook Heraclitus’ ontological doctrine as an epistemological issue (Reinhardt (p.29) made a similar observation about misunderstandings of Parmenides, particularly with regard to Aristotle) or as answerable by way of purported facts about what can and cannot be known. He misleadingly diverted a theory as to what is in the world in the direction of a theory of what could be known (and on this basis proceeded to construct his own theory of what there is in the world based on mere appearances and formal realities). More sympathetically stated, he assumed without proof that the only road to an ontological theory was via epistemological fact.  The source of this methodological misstep may have been Protagoras and the general skepticism attributed to the sophists. Whatever may be the case historically, while Plato may have provided an answer to sophist skepticism with his own theory that the Forms could be known and only the Forms could be known, he did not satisfactorily “refute” or provide a valid alternative to Heraclitus. Rather, he doubled down on Parmenides’ putative unwarranted assumption to the effect that existing things are just things that can be known by the human mind (or represented in language, depending on your interpretation of Parmenides).

In the Republic IV 437B ff. Plato gives a series of examples of opposites, but this passage does not relate to the theoretical discussion in the Lysis  (Cf. esp. 438 A where hot and cold are improperly assimilated to relations such as taller-than). For one thing ἐναντίοι are not contrasted with ὁμοίοι. For another they are not related to each other except as being mutually exclusive in the respects Plato mentions. One can only speculate that Plato does not need in this passage to address relations between opposites other than the most fundamental. Indeed he does not even mention the Presocratics because he may have felt that he had previously cleared away the Presocratic problematic of likeness and opposition in the Lysis.

Likewise Bk V 475E ff. refers to opposites such as good/bad, just/unjust and κάλος/αἰσχρός but the context is one in which Socrates is trying to show that each term of an opposition is an individual unit although the two appear grouped together. The point of this passage seems to be to argue that lovers of beauty and knowers of beauty, i.e. true philosophers, maintain the same kind of duality, individuality in actuality and duality in appearance so to speak. For the purposes of the present discussion this passage seems to imply a recognition on Plato’s part that contrapositives, or at least those contrapositives he gives as examples, are not related by the sort of links of meaning that we would recognize as logical. That is, they differ significantly from complements. This seems to be at least part of the meaning of calling each pole of the dualities he cites ἓν ἕκαστον. In The Sophist 247 A justice and presumably injustice are labeled ἐναντίοι but only by way of example. The nature of contrapositivity is not at issue in this passage.

Much more important is the appearance (Republic Bk. VII 524B – 526) of Plato’s own theory of ἐναντιότης to replace the Empedoclean physicalist theory. Ἐναντίοι are νοητά (and presumably ἰδέα) that can only be apprehended by νοήσις (Cf. also Theaetetus 186 B where Socrates speaks of τὴν οὐσίαν… τῆς ἐναντιότητος although it is typically unclear whether this phrase means the existence of opposition or its specific essence or both). The mere αἴσθησις or sensation of opposites like large and small does not distinguish between them. Only by the addition of νοήσις (reflection or understanding) and the apprehension of the νοητά bigness and smallness can sensed or perhaps perceived opposites be truly apprehended. Νοήσις also accounts for the apprehension of distinct single entities (ὄντα) and the one itself (τὸ ἕν). Plato’s distinction between thought and αἴσθησις may have come from and is certainly prefigured by Alcmaeon (Cf. Theophrastus De sensu 25 ff. Fragment 24 A5. Theophrastus’ term is φρονεῖν). It contradicts the McDowellish view that Theophrastus attributes to Empedocles to the effect that thought and perception are the same thing. An implication of Plato’s model could be that contrapositivity is not a property in the world; it is instead imposed on our take on the world by whatever conceptual scheme we possess that includes concepts involving contrapositivity. Of course, that is not the implication that Plato himself draws. His conclusion is that the physical world and the sense impressions we have of the world are two sides of an illusion. To the extent that large and small, as νοητά, are also ἰδέα, large and small belong to the realm of true reality. The ambiguity of how the apprehension of a distinct second order entity (including individuality itself - τὸ ἕν) relates to the apprehension of individuals, and, over and above that ambiguity, the dubious acceptability of the very idea of the existence of Platonic Forms, would, of course, reappear in Husserl’s Logical Investigations. It is not my intent in this essay to adjudicate between the two.

Republic Bk. IX 583B ff. schematizes about as well as one could hope the conceptual relations between contrapositivity and complementarity. Plato’s purpose as usual is to denigrate sensual pleasure in favor of the pleasures of the mind. The ἐναντίοι in question are pleasure and pain. The phrase “something that is neither pleasure nor pain” specifies what we might call the complement of the union of the sets of pleasant and painful sensations (although, as we saw above, the universe in this scheme should be restricted to sensations). An anaesthetized lack of feeling is comprehensibly complementary to feelings of pain and pleasure lumped together. However, the terms μεταξύ and ἐν μέσῳ add unwarranted content to the notion of complementarity. There is nothing to suggest that what is neither pleasurable nor painful should be “between” pleasure and pain. It is also unwarranted to describe the complement as ἡσυχία since anesthesia is hardly equivalent to a peaceful feeling which is itself a positive and very possibly contra to both pleasure and pain. From a strictly logical standpoint there is nothing wrong with Socrates’ argument that cessation of pain is not a pleasure and vice versa, since it invokes the distinction between contrapositives and complements. However, empirically, and for lack of a more detailed phenomenology of pleasure and pain, Socrates’ argument does not show that the experience of a release from pain could not also be an experience of pleasure and vice versa. Incidentally, the beginning of a modern discussion of pleasure and pain can be found in Burke.

Opposition in Psychology and Social Theory:  The notion that there is a sort of parallel between the physical and the social world is a thread that weaves through much of Presocratic philosophy. (An idea that has seen many lives; cf. e.g. Edgeworth quoted by Rawls, p. 164.) It can be argued that Plato shattered that parallel by devising two different solutions to contrapositivity in the cosmos and contrapositivity in the polis. The first as proposed in The Statesman and the Philebus is the concept of a standard of measure and is applicable to the cosmos. The second as it appears in the same dialogues is the art of weaving together opposed personalities and applies to the polis. Let’s take a look at Plato’s treatment of contrapositivity in the realm of psychological and social concepts throughout his dialogues in order to see how Plato arrived at what appears to be his ultimate position.

Eryximachus’ speech in The Symposium raises issues of opposition with respect to human interactions, and, in the context of the dialogue, most pointedly sexual relations. Depending on my mood I can regard what he says as either a repudiation of the arguments in the Lysis or a refinement of Socrates’ implicit conclusions. Eryximachus appears to reference Empedocles’ theory of universal attraction or ϕιλότης from the outset (186A). He returns to cosmology in 188A where, in what can only be described as a leading elision, he substitutes σώφρονα for Ἔρως οὐράνιος. The opposites (which he refers to indifferently as ἕτερα, ἀνόμοια and ἐναντιώτατα) he mentions are hot/cold, bitter/sweet and dry/moist to which he adds an extended discussion of the opposition of sickness and health. One effect of Eryximachus’ speech is to add a dimension of moral approval and disapproval to the principles Empedocles called ϕιλότης and νεἶκος, a dimension which at the very least is not explicit in Empedocles. Those principles are assimilated to or renamed as Ἀφροδίτη or Ἔρως Οὐράνιος and Πάνδημος. The effect of the former is to produce a kind of reconciliation or, using music as a relevant example, harmony between opposites. The latter creates discord. The consequence of Eryximachus’ theory is to prescribe treating illness and also human relations in general by the agency of σώφρονα, obviously a normative and not a descriptive conclusion. The transition to the normative comes by way of the example of medicine whose evident purpose is to produce a better physical state. Moral prescriptions are conceptualized along that model. Eryximachus gets to this conclusion (186 B-C) by invoking the theory rejected in the Lysis to the effect that unlikes love and desire unlikes.

The Republic Bk. I 349D ff. gets into the nuances of meaning involved in likeness-and-unlikeness opposition (ἀνομοίου τε καὶ ἐναντίου 350B). Socrates’ examples are meant to show that like or similar persons strive to outdo their peers and not their inferiors and thereby concludes, after a bit of shifty reasoning and some not so subtle word play, that unjust rulers are bad in that they try to take advantage over both their peers and their inferiors. The relation between likeness and πλεονεκτεῖν was not brought up in the Lysis.

The Gorgias starting at 510B raises an issue of whether a member of a state should try to be as like (ὅμοιος) the ruler of that state as possible in order to secure personal safety and power even if the ruler is an evil person or group and performs unjust acts. Despite the fact that the same sort of relation is at issue using the same terminology as in the Lysis, this dialogue veers in a different direction. The issue as it appeared in the Lysis of whether two like persons should like or hate each other does not arise. Rather, it is assumed that, in order to be φίλος to a ruler the subject should try to be as much like that person as possible. By way of various compounds based on ὅμοιος the argument shifts ground from likeness in general to sharing a viewpoint or ethos with a ruler or fitting in with the majority in a democracy. Socrates, as might be expected, argues that this strategy could lead to very bad things if the ruler or the majority are evil. The fact that the argument in the Gorgias is so different from that of the Lysis, indeed that it assumes what the Lysis set out to question, indicates that the aims of the two dialogues are very different. The Lysis is more pointedly directed against Empedocles while the Gorgias is concerned with setting the stage for Plato’s political theory and his theory of the Form of the Good.

In the Theaetetus 176 A Socrates argues that there has to be a ὑπεναντίον to the good, i.e. τὰ κακά. This recalls the superlatives which appeared to function as contrapositive terms in the Lysis. However, it is unclear in this short passage whether Socrates is just making a wise observation about the way life is or whether he is arguing logically to the effect that the existence of the good necessarily entails the existence of the evil. If the latter he is obviously committing the goddist error evident in some defenses of theism with respect to the so-called problem of evil.


The concept of ἐναντία enters with a vengeance in The Statesman 305E ff. The passage, whose ultimate goal is to explain the political task and craft of social συμπλοκή, begins with the Stranger isolating a pair of qualities that, he says, are simultaneously ἐναντία and φίλια. These are ἀνδρεία and σωφροσύνη (or κοσμιοτής 307B). The opposition between contrapositive and liking hearkens back strikingly if not explicitly to the Lysis. The Stranger and the young Socrates take as a given that those qualities like each other (a more comfortable if not entirely literal English version that would obscure the reference to the Lysis might be that they are more than compatible with each other, they go together). Accordingly the Stranger takes up the task of showing that they are also paradoxically at odds with each other (ἔχον διαϕοράν) and therefore contrapositive.


It is interesting to note that Plato here introduces a couple of ideas that are significant for the ultimate shape of his overall logic, so to speak. Aristotle would eventually exploit these ideas in constructing his alternative to the Platonic Theory of Forms. The first appears in the phrases ἐναντία… εἴδη and ὑπενατίοις γένεσι. These formulations are practically Boolean in nature but nothing much is made of them from a logical standpoint since Plato’s focus is political theory. The other idea is to qualify opposition by way of articulating in what respect a pair of items might be opposed or contrapositive. It is expressed in the curious phrase ἔς τι (306C, so curious that some editors emend it to ἐστι). Plato’s meaning seems to be the same as that expressed by Aristotle with the word κατά. Once again Plato doesn’t make much of this idea logically; he in fact pays little notice to the implicit resolution the notion of “in a respect” provides to some of his paradoxes of contrapositivity (Levinson in Anton & Kustas Essays in Ancient Greek Philosophy I, p. 271 notices this). However, as we saw, it doesn’t undermine the general tendency of the Lysis where Socrates proved to his satisfaction that contrapositives could not love, hate, both love and hate or neither love nor hate each other. The Stranger sets out to show that the concept pair under examination in The Statesman, viz. ἀνδρεία and σωφροσύνη, can be both naturally allied and incompatible (literally like and dislike each other), but the broad sweep of his argument does not easily reduce to a simple formula of the respect in which they are more than compatible vs. the respect in which they are incompatible.


The Stranger pursues his argument in terms of multi level contrapositivity. The concepts of ἀνδρεία and σωφροσύνη can also fall under a high order contrapositivity, that of earning our praise or earning our blame. The fact that praise and blame are themselves contrapositives is used to show how the two aspects of ἀρετή are contrapositive and incompatible.


Plato describes the statesman’s τέχνη as one of weaving together contrapositive personalities to produce a well managed state in such a way as to reconcile the incompatibility of these personality traits and neutralize potential conflict between opposing personality types. Of course his benign social interweaving can occur only after what can be described as a proto-Fascist social engineering: a program of eugenics for those who pass muster and enslavement, exile or execution for anyone so depraved as to be irreconcileable.  A specific application of the eugenic interweaving proposal occurs in The Statesman 310E and is taken up again in Laws 773 B-C. Plato speaks of encouraging (as opposed to legislating) marriage between opposite rather than similar social and personality types. He also speaks of two different types of ruling expertise, one over inanimate objects and the other over animate objects. Are the two solutions, i.e. measurement and interweaving, meant to address each of the two types of ruled object exclusive of the other? That seems to be the case. In any event, along with the cosmology of the Timaeus where opposition as a tool for the description of the natural world was replaced by the concept of μετρίον or objective standards, the integration of opposites in The Statesman may regarded as another alternative to Empedocles’ system. The question remains why does contrapositivity pose a problem? The problem with respect to animate objects is the dissolution of the state when one or the other of the two contrapositive personality types dominates. But, aside from all the bad arguments in the Lysis, Plato does not seem to provide a clear statement of what the problem of inanimate contrapositivity (e.g. hot/cold) may be, although a good case could be made for the superior usefulness of an objective standard.


In Philebus 34E ff. Socrates speaks of feelings of thirst and hunger, which he characterizes as examples of “emptiness,” i.e. thirst or hunger, and contrasts emptiness with its opposite (ἐναντίος) “fullness,” i.e. satiety. This is supposed to be an argument to prove the existence of the soul since he goes on to argue that a (presumably newborn) body cannot grasp (ἐφάπτοιτο) a fullness it has never experienced. Seeing through this argument will be left as an exercise. Nevertheless Socrates introduces a number of (in this context) second order contrapositives which relate to παθήματα like hunger and satiety. These include pain/pleasure and hope/hopelessness. 36C introduces another contrapositive pair, true/false (Cf. also 37D on rightness and its opposite (ὀρθότη ἢ τοὐνατίον)), and goes on to discuss how these second order qualifications can apply to sensations and emotions, most specifically the contrapositive pair pleasure/pain. It is worth noting in passing that Socrates argues by analogy (38D-40E) to a doctrine that pleasures and pains can legitimately be described as true or false (Cf. also 42B for a different analogy based on observational distance). This procedure does not get as much scholarly ink as its more flashy cousin, peritropics, but argument by analogy appears more frequently in the Socratic dialogues than we may sometimes assume. Socrates tends to use analogy when he is speaking with sympathetic interlocutors and peritropics or other ways of demonstrating self-contradiction or conflict of beliefs when dealing with hostile debating partners. Socrates does not speculate about an objective unit of measure or μετρίον for the contrapositive feelings under discussion although he introduces terms like σφοδρά in acknowledgement of the fact that these feelings can have greater or lesser intensity.

Opposition is misapplied in the possibly apocryphal Epistle VIII 354E-355 in the contrast between law and pleasure. These are not real complements and obedience to one does not exclude “obedience” to the other. They are not real contrapositives either, and even if they were a person could still “obey” both. My reference to this passage might be regarded as somewhat of a stretch since Plato never uses the word ἐναντία in the context and he clearly is not addressing the sort of logical issues that drive the Lysis. Freedom and slavery, moreover, could be construed as real contrapositives. I only mention the passage here because the attack on ἡδονη is a persistent theme in Platonic philosophy and central to his ethics.

Deleuze’s distinction between types of difference in Différence et Répétition pp. 84 ff. includes contrapositives among his examples of non-representational difference. Other examples appear to constitute something like a purely extensional specification of a group (not defined by division of a more inclusive group). The mysterious treatment of what appears to be an ontological problem in the middle of a social situation in the Lysis is mirrored throughout by Deleuze’s shifting from what appear to be logical issues (in his discussion of Leibniz) to ethical issues (in his discussion of Nietzsche) (Cf. also p. 184 on opposites in The Republic and p. 187 on intensities). If there is a relation between Deleuze and Plato, the question naturally arises of how does what Deleuze says relate to Plato’s eventual solution, i.e. the replacement of opposition by degree in the Timaeus?

Opposition in the Parmenides: Let me start right off by saying that 148 D-E could very well be my favorite passage in all Plato, indeed paradoxically one of the seminal moments in the history of Western philosophy. It consists in Parmenides (obviously on a tip from DiVinyls) wondering whether The One masturbates or joins in orgies. Thrilling.  Incidentally both options court contradiction. Just like the group sex loving One, the wanker One is no longer one in that it belongs to a many that includes one of its parts, namely its penis.

The Parmenides appears to take up the discussion of likes where the Lysis left off. But a new element is introduced in the form of a writing by Zeno containing a kind of back door proof of Parmenides’ famous dictum that all is one.

(I should like to make in passing the following observations that are worth notice though not relevant to the subject of this essay: (1) The rumored homosexual relationship between Parmenides and Zeno, provides further evidence as if any were needed that homosexuality in Greek philosophy was not just a Platonic obsession. What Plato did was thematize it. (2) Zeno’s argument is presented in the form of a private reading of γραμμάτα. This scene sheds interesting historical light on how philosophical ideas were broadcast in 5th century Greece. It also, in my opinion, casts serious doubt on Derrida’s contention in La Pharmacie de Platon that whatever is objectionable in Plato’s philosophy is somehow intertwined with historical phonocentric prejudices and a rejection of the tool of writing. (3) The term λόγος makes one of its many appearances here to mean “argument” (128 C) but it is carefully balanced against the much less heralded τεκμήρια which means “proof.”)

As summarized by Socrates, Zeno’s argument, which kicks off the discussion, is that entities cannot be many (…πολλά ἐστι τὰ ὄντα. I see no reason why this can’t be rendered into more readable English such as, “There can’t be many things,” or “There can’t be several things,” or even “There can’t be more than one thing.”) Otherwise entities would be both like and unlike, which is impossible (ἀδύνατον), i.e. the assertion that entities are both like and unlike is self-contradictory. “Like” is represented here by the reappearance of ὅμοια of Lysis fame. Its counterpart is ἀνόμοια or “unlike,” which raises the possibility of a distinction between complementarity and contrapositivity. The word for “other” (ἕτερος) does not appear here (Later in the dialogue (143 B and 145 E - 146 A) Plato would use ἕτερος in the course of a similar sort of argument; I will make some comments on 146A ff. and 147 C ff. below) and ἐναντίος qualifies concepts (εἴδη) and not non-conceptual particulars (τὰ ὄντα). Socrates omits the proof of Zeno’s (and Parmenides’) contention, but presumably the proof is Parmenidean: If there were several things they would be distinct and therefore unlike, but because each one could be characterized as “one of several” (Note: it is easy to avoid the category error of characterizing each particular as “several” or “many”) they are alike. Given this second premise, the proof of the negation of the major premise follows by modus tollens.

Socrates’ first response comes as a passing, almost sarcastic, aside. You need to read into his words to tease it out, and it may be that Socrates’ very indirection (the fact that he doesn’t state his point outright) is why Pythodorus worried that Zeno might take offense. Once understood, however, Socrates’ hints raise a kind of Gödelian grin. For Socrates plugs Zeno’s and Parmenides’ separate proofs themselves into the variable position “particular” (τὰ ὄντα) and asserts that their two proofs are both different and the same, both like and unlike. Socrates doesn’t spell that out; instead he turns to a more straightforward response. This introduces two crucial ideas. The first is a distinction between qualities of particulars and qualities of the concepts under which particulars fall. (I translate τὰ ὄντα as “entities” or “particulars” as legibility dictates. “Entities” is obviously more literal. Nothing should be assumed as to modern philosophical meanings of “particular.”) The second is the idea of “participation” as an alternative to “is-ness,” so to speak, with its unwelcome implications of identity and existence. “Concept” is my rendering of εἶδος and it is quite clear to the parties of the dialogue (130 B-C) that Socrates regards likeness and unlikeness themselves as εἴδη very much on a par with their more famous, counterparts, the good, the beautiful etc. What is important about the concept of likeness is that it is introduced not simply as a hypothetical independent entity (αὐτὰ καθ’ αὑτὰ), but as related to other particulars in a specific way. Particulars are not identical (εἶναι) to likeness or unlikeness. Rather they can be qualified as like or unlike each other.

“Qualified” is my thoroughly un-Platonic and largely Aristotelian rendition of the idea of participation (expressed as μεταλαμβάνειν and μετέχειν). This way I can set aside both the ontological gusto and the ontological confusion of participation without sacrificing what is essential to the argument at hand. Today we would call likeness and unlikeness second order entities and whatever can be qualified as like or unlike first order entities (or, more accurately, since there are respects in which second order entities can be like or unlike, higher and lower order entities). Socrates concludes that, while it is a contradiction to assert that a particular is identical to both likeness and unlikeness (assuming, as Zeno does, that likeness and unlikeness are mutually exclusive), there is no contradiction in asserting that a particular can be qualified as both like and unlike (i.e. participates in or partakes of both likeness and unlikeness) as long as we understand in what respect two things are alike and in what respect two things are unlike (“In what respect (they can be qualified)” is my rendition of the somewhat vague phrase τοσοῦτον ὅσον ἂν μεταλαμβάνῃ (129 A)). It is easy to see that likeness or unlikeness in respect of… or in virtue of… resolves the aporia raised in the Lysis. But, having made this important distinction, Socrates does not recognize that the concepts likeness (αὐτὰ τὰ ὁμοιά) and unlikeness can themselves without contradiction be qualified as like or unlike other concepts (129 B). The recognition of that would have to await The Sophist. In fact, The Sophist 255 E ff. recapitulates Socrates’ argument on the more solid ground of Plato’s theory of Forms and the interaction of Forms. Nevertheless, packed into this short passage is the distinction between existence, identity and qualification (predication, participation) most notably as a solution to paradoxes that arise when the distinction is not made. (To repeat, I freely use terms such as “concept” and “predication” even though they are not part of the Platonic lexicon. Cornford’s strictures (pp. 273 ff.) against interpreting Platonic Forms in terms of Aristotelian logic are well taken. However, that does not exclude our understanding Plato’s solutions to the issues he addressed as a nascent and not entirely self aware distinction between logical and ontological theories.) We also see the beginning of a distinction between types of entities and how certain things can be said about entities of one type that cannot be said about entities of another type (Cf. also Theaetetus 182 A-B where a distinction is made between an abstract quality or ποιότης and a thing being qualified in a certain way). Socrates says that mutually exclusive εἴδη are ἐναντία. He is clearly not using ἐναντίος in this passage in my sense of “contrapositive.” Rather it appears to be the equivalent to ἀνόμοια (which applies to non-concept particulars) as applied to concepts. But no counterpart is mentioned to ἐναντίος such as ὅμοιος is the counterpart of ἀνόμοιος. So it is hard to give a precise meaning to ἐναντίος in this context.

Parmenides’ response to Socrates’ analysis doesn’t really hinge on the like/unlike distinction so I won’t go into detail about it. Let me just mention a couple of defects. First of all, Parmenides assumes, without really defending it, that μεταλαμβάνειν requires a sort of real life and indeed spatial existence for an εἶδος to partake of. And, of course Socrates agrees with him. After all, the theory of ideas is more Platonic than Parmenidean. But that conclusion doesn’t necessarily follow from some given understanding of μεταλαμβάνειν such as would be sufficient to make Socrates’ point. Understood as predication there is no reason to require some sort of ontological status to what is predicated of lower order particulars that is equivalent or analogous to the status we attribute to the lower order particulars themselves. But Parmenides’ actual argument requires just that. For he attributes all sorts of spatial qualities to εἴδη, qualities that may obtain for certain kinds of first order entities but need not obtain for higher order entities such as ideas. Parmenides introduces the image of a sail covering several sailors as an illustrative metaphor, but the conclusion he draws indicates that he takes this image literally. Yet there is no reason why predication is the sort of thing to which spatial or even non-spatial part/whole qualities need apply.

Parmenides follows up with a distinct argument to the effect that, if particulars can be said to have a certain quality, then that quality itself must have the same quality (viz. itself?) or a version of the same quality. So, if Jenna is blonde by virtue of “participating in” blondness, then blondness must also be what it is because it participates in blondness or some higher order version of blondness. The second alternative introduces an infinite regress, which, for reasons unstated, is a bad thing (Infinite regress is bad if you’re looking for an explanation, but explanation is not the purpose at hand. Socrates is trying to avoid paradox). In a variation on this argument Parmenides opines that particulars which can be characterized by the same quality must also “participate in” likeness and so must the quality that characterizes them. The participation relation (understood as resemblance) implies a sort of third man likeness relation between particulars and ideas. Jenna and Nikki both participate in blondness and both participate in likeness.  But blondness also participates in likeness since, applying to Jenna and Nikki, it is like Jenna and Nikki by virtue of ideational resemblance (ἐοικέναι, ὁμοιώματα) to particulars. But this also leads to infinite regress. Needless to say, Parmenides makes a glaring assumption to the effect that the way blondness is “like” Jenna and Nikki in virtue of their blondness is in some way comparable to the way they are “like” each other in virtue of their blondness. To be fair, Parmenides seems to make exactly this point in 133 C ff.

Parmenides’ specious argument (139 E to 140 D) to the effect that τὸ ἕν cannot be like or unlike itself or another thing (This is just one of the many things τὸ ἕν cannot be in Parmenides’ litany) is based on his hypothetical partial definition of τὸ ἕν as something that cannot have parts. Incidentally Parmenides further assumes throughout this passage that things can be distinct and singular only in virtue of some entity he calls The One. (A source of the confusion can be traced to 159 D where in one sentence Parmenides treats being singular (ἕν… ἔστιν) and “having one” (ἔχει…ἕν, presumably “having The One” since in 159 C Parmenides uses the definite article, ἔχειν…τὸ…ἕν) as synonymous. The confusion is between having a property and being identical with that property. In 159 E Parmenides commits a similar error with respect to likeness and unlikeness.) That is not an assumption required by Zeno’s initial exposition. The bulk of the dialogue deals with Parmenides’ attempt - riddled as it is with what we would consider to be category mistakes - to show that participation in qualifications or μετέχειν courts the same paradoxicality as identity. He also argues that what we would call higher order concepts themselves such as oneness or singularity, existence, likeness, wholeness, number and identity, regarded as equivalent to or indicating entities, are not immune from paradox. Other renditions of the family of category mistakes to which Parmenides’ arguments belong are confusing concrete and abstract objects, type fallacies of the Russell sort and confusion of object language and metalanguage. Each of these renditions has its own nuances but they are all related. And Parmenides’ errors could be unraveled using any rendition.

In 146 D Parmenides qualifies sameness itself (αὐτό τε ταὐτὸν) and otherness (τὸ ἕτερον) as ἐναντία. The only consequence of this instance of apparent contrapositivity is that sameness and otherness cannot be “in” each other. The rest of the argument derives from that consequence. The argument itself is clearly another instance of category error and a vagueness about the difference between sameness and identity. However, it is interesting that Parmenides (or Plato) does draw a distinction between the relation of otherness (ἕτερος) and the relation of contrapositivity (ἐναντίος). Since not much effort is expended on defining ἐναντία, however, it is difficult to tell whether in this passage Parmenides means contrapositivity in the sense that I have defined it. There is an implication of complementarity about ἕτερος, however, since Parmenides asserts that if things are different (other), they cannot be the same – the ensuing dialectical, i.e. peritropic, portion of the argument asserts further that they are the same in virtue of being different, i.e. they are both different (NB the peritropic argument is based on an understanding of sameness and difference as qualitative and not numerical). Suffice it to say that Parmenides does see some sort of distinction between otherness or difference and contrapositivity, but he does not explain what the distinction consists in. Rather he rushes to paradoxical conclusions based on bits and pieces of assumptions about the meanings of those terms.

In 147 C to 148 C Plato uses all the terms for otherness along with likeness and self identity in a single context. Ταὐτόν and ὅμοιον are both (allowing for some vagueness of meaning) symmetrical, reflexive and transitive relations, but they do not mean the same thing. Anything that is self-identical (basically anything) is like itself. But two things that are like each other need not (or cannot depending on how we understand “two”) be identical. It seems that ἕτερος and ἀνόμοιος mean the same thing in this passage, but that is not spelled out (another synonym, the adverb χωρίς appears later in 159 B). Specifically it is not clear whether either term or both signifies a complement when applied to a concept or group of things. It is equally possible that both terms could signify no more than numerical difference among particulars (presumably τὰ ὄντα), but that is not spelled out either. It is most likely that Plato did not distinguish between having different properties and being numerically different. At least he did not sufficiently differentiate the two sorts of relation to even deny that such a distinction obtained. Self identity is treated throughout as just one πάθος (affection, state, quality) among others. Parmenides also introduces another term (ἄλλοις and ἀλλοῖον) to signify numerical difference between particulars and between concepts. But it is also unclear in what way, if any, this new term differs in meaning from its sister terms, ἕτερος and ἀνόμοιος (In 164 B Parmenides states that τὸ ἄλλο and τὸ ἕτερον are synonymous). For whatever it’s worth, two new substantives, ἕτεροῖα and ἀλλοῖα are introduced in 161 A-B.

In 158 E Plato uses ἐναντίος as qualifying the properties (Plato’s term is πάθη) ὅμοια and ἀνόμοια themselves and not the particulars that are alike or unlike. This corresponds to what we would call a logically higher order. He also (159 A) introduces the superlatives for both terms expressing opposition: ἐνατιώτατα and ἀνόμοιότατα and later in 160 C he uses the apparently synonymous πᾶν τοὐναντίον. This may be just a rhetorical flourish. However, it is clear that if two items or properties are most unlike or most contrapositive they cannot share certain other properties and that an assertion that they do would be self-contradictory (Cf. also 160 A). Those items or properties are contrapositive and also belong to complementary sets.

It is also uncertain how Parmenides regards his apparently paradoxical conclusions. He utters them with a straight face and his interlocutor, Aristoteles, mostly agrees though with varying degrees of discomfort. The Parmenides comes to an abrupt halt, but there are hints of at least a partial solution in The Sophist. Still, it should be noted that the conclusions are paradoxical only if we make two assumptions. The first is that otherness signifies logical (or Boolean) complementarity in Parmenides’ mind. If not, there is no paradox. Jenna and Nikki are both other and alike and an assertion to that effect is not self-contradictory. The second assumption is that all the concepts in question are completely meaningful in themselves in the same way that less (but obviously still) controversial first order concepts such as shape or spatiotemporal location (setting aside the fact that Parmenides tries to drag these concepts into his dialectic; he succeeds only if the initial steps succeed), are completely meaningful without further explanation. If not, i.e. if you cannot say that two things are “other” or unlike in Plato’s sense of the terms without specifying a sense or respect in which they are other or unlike, then Parmenides’ conclusions are not self contradictory. (This is largely the import of phrases like ὁμοίως εἰρήκαμεν (The Sophist 256 A-B) and πῃ (256 C and 259 C-D) which Cornford and Fowler both translate as referring to the sense in which terms are used. Most of the Parmenidean paradox dissolves if you keep this qualification in mind. As applied to the concept of everything (τὸ πᾶν), however, it won’t go away if you assume there is an unqualified use of “everything” as in “just everything.”)They are just vague. Jenna is like Nikki in that she is blonde. She is unlike Nikki in that she is the author of How To Make Love to a Porn Star. So, with these qualifications in mind, we can assert without contradiction that Jenna and Nikki are alike and unlike at the same time. Parmenides reserves the term ἐναντίος in this passage to qualify opposing concepts such as likeness and unlikeness. But if those concepts are simply contrapositive in the same way that good and evil are simply contrapositive, then there is no contradiction in the assertion that both qualify a given particular. Contradiction comes about only if ἐναντιότης entails mutual exclusivity or logical complementarity, i.e. whether it is equivalent to ἕτεροιότης (160 D) in this sense. What is confusing about the Parmenides is that none of the characters seem to recognize these two assumptions. (In 158 E Parmenides comes oh so close to recognizing the need to qualify likeness and unlikeness in terms of the respect in which things are like or unlike. The term he uses is ταύτῃ which is a relative pronoun in the declensional form of an instrumental dative. It means “in which respect,” referring back to an adjective appearing earlier in the sentence (viz. ἄπειρα) (Cf. Theaetetus 158 E for a similar use of τῇ as a dative of respect).

In his earlier treatment of Zeno’s aporia of like and unlike (described as ἐναντίοι) and its assimilation to what Parmenides says about the one and many (Parmenides 127 E ff.) Socrates mixes up two distinct ideas: (a) second order concepts (the Forms αὐτὰ καθ’ αὑτὰ) and (b) the need for adverbial qualification of some qualities (what, following Frege, we might call unsaturated predication). He is right to say there is no paradox in the assertion that non-conceptual objects are both like and unlike, both one and many (He undoes both paradoxes simultaneously). But he is wrong in saying that it would be paradoxical to call the Form Oneness itself one among many, for it is distinct from the forms of likeness and rest etc. Likewise he is wrong in finding a paradox in saying that the Form of likeness is both like and unlike. It is like other Forms in that they are all Forms and unlike other Forms in that they are different from each other and, as far as anyone who does not believe that the Forms are all that exist, from whatever is not a Form. There is no reason that the argument about unsaturated predication that applies to first order individuals should not also apply to the Forms. N.B. Socrates interprets the paradox of the One strictly in terms of the singularity of non-conceptual and, by our lights, second order objects. He does not address the more radical assertion that there is only one thing in the world, although taking out the paradoxes of singularity could very easily be regarded as undermining the only proof of that more radical assertion.

Unfortunately (assuming that Plato regards the consequences Parmenides draws from his hypotheses as indeed self-contradictory – a conclusion never explicitly stated in the dialogue) Plato does not seem to explicitly recognize that two particulars can be alike in one respect and unlike in another respect without contradiction. He had the concept but didn’t see that it undermined Parmenides’ argument. Yet we are not let in on what Parmenides chooses to conclude from his “consequential” analysis of the hypothesis that there is such a thing as τὸ ἕν. It could be that Parmenides is devising a reductio argument to the effect that there is no such thing (though he argues that the same paradoxical conclusions are supposed to arise if The One does not exist (160 B ff.); but that argument rests on the further fallacies of assuming that a negative existential proposition entails a positive existential proposition and the classic Parmenidean error of confusing predication and existence). Alternatively his reductio could be that the new concepts Socrates introduced of εἴδη, μεταλαμβάνειν and μετέχειν lead to the same paradoxes as his own concepts of being and The One and so are not satisfactory substitutes (I see no basis in the text for Cornford’s remark to the effect that “...Parmenides himself acknowledges that the Forms are a necessity of thought….”(p.11)) This is the interpretation I prefer, even though, in order to maintain his dialectic of the existence of non-existence (162 A-B) even in the conceptual framework of partaking of (a converse twin to predicating of) from descending into utter linguistic porridge, he has to substitute a new term, μετεῖναι for Socrates’ original μεταλαμβάνειν (Indeed he showed a marked preference for μετέχειν as his argument progressed). Or else he may be unaware of paradox. The abrupt end of the Parmenides leaves this last alternative open as a possibility.

There is an extremely important passage at the end of the dialogue, however, that points to a quite different conclusion. The concept of otherness is much used the argument in favor of this conclusion, although its role is the by this time a dreary repetition of the point that otherness is a symmetrical relation and requires the existence of both relata if one of them exists and is properly qualified as different. Since that point is applied at this juncture to the specific case of The One and The Many, the important conclusion is a proof of the existence of The One, or, stated in terms more acceptable to modern metaphysics, the necessary presence in the world of singular, distinct, specifiable particulars. Parmenides’ conclusion is that there could not be such a thing as plurality (alternatively we could not conceive of such a thing as plurality) unless there is also such a thing as singularity (unless we also understand singularity). I do not regard this as the overarching conclusion of the dialogue since the bulk of the earlier arguments are not designed to support it. But it is important because it raises an ontological issue that is often not clearly understood and accordingly missed. This is the need for the defense of the notion of determinate, independent, specifiable particulars. Placing it at the end of the dialogue indicates that Plato had a sense of its importance. Parmenides' principal argument in this passage seems to start from the premise that the only alternative to a concept of singularity is the concept of plurality. But we cannot understand plurality unless we understand singularity and so there must be such things as particulars; there must be oneness. Interestingly enough in 164 D he brushes against the idea of an infinitesimal (τὸ σμικρότατον and κερματιζόμενα are the terms he uses), but dismisses it with the independent argument that no matter how far you divide objects or however small a particle you identify, it is always large with respect to something else. And it also possesses the property of singularity. Therefore, for the purposes of his argument there are no such things as infinitesimals. The conclusion of the dialogue might be viewed as a counter to a supposed Heraclitean position that singularity is a fiction generated by certain arbitrary features of the language we use, that it is impossible to adequately describe or denote an independent particular such as a river because of our extra-linguistic intuitions as to the constancy of its change. Notably, stated in this way, Parmenides’ doctrine is no more than the assertion of logical complementarity and in that way it is akin to tautology. Presumably the logical Heraclitean would reject both singularity and plurality in favor of some sort of continuum. In addition, skepticism about the necessity of singularity is much more far reaching than Heraclitean physical change or the notions of a continuum and infinitesimals allow. These latter notions admit logical or grammatical singularity simply by virtue of using terms like “river” and “continuum” which purport to denote singular albeit ever changing particulars. The skeptical attack on singularity could oblige us to jettison subject terms from our language and what they designate from our conceptual schemes altogether (assuming, of course, that that is possible). Nevertheless, Plato’s dialogue in effect concludes with what might be regarded as the strongest possible statement of at least one version of the fundamental Parmenidean doctrine – a doctrine Plato accepts and further elaborates with his own theory of forms: ἕν εἰ μὴ ἔστιν, οὐδέν ἔστιν.

Opposition in The Sophist: The consequences of Plato’s rejection of the concept of contrapositivity and later partial retraction of that rejection rippled through his cosmology, social theory and ontology. Yet those seem like small stakes games next to the action in The Sophist, a veritable private room for metaphysical high rollers. To switch metaphors, The Sophist has long been a kind of rite of initiation, a pons asinorum an understanding of which separates the philosophical men from the boys. What follows is complicated, so buckle your seat belts.

In addition, because The Sophist is so dense I will refer to secondary literature more than I would normally feel necessary. Part of the problem with understanding The Sophist is that Plato makes claims about very different things, but throughout uses the same term (more properly terms, sharing one root morpheme) to refer to those things and so clouds the differences between them. These different things include: (a) entities (by which he mostly means first order entities, particulars, individuals etc.) which may or may not exist, (b) Forms or second order entities which in turn may or may not exist and among which we may count the Forms existence and non-existence themselves (although, as we shall see, Plato fails to distinguish between existence and self-identity, on the one hand, and between non-existence and difference, on the other), (c) participation in a Form (which an Aristotelian linguistic turn would redescribe as predication), (d) facts, i.e. whatever is referred to by a true statement and paradoxically referred to by a false statement, and (e) true and false statements themselves (It should be noted that the Stranger’s introduction of statements into the mix sets the stage for Aristotle; once part of the problem had been identified as linguistic, it became easier to sort out the confusions). The terms that Plato uses to designate (a) and (d) and, for the most part, (b) are τὸ (ὄντως) ὄν, τὸ μὴ ὄν and (significantly) μὴ ὄν. The term used to express (c) and by extension (e) is εἶναι (and its conjugations) mostly as predication but this term is also used to express the ideal entity or Form existence and so, in a tricky way, falls under (b) often simultaneously so that the copular “is” is merged with the predicative “is” of existence and the referring term is merged with the verb. (I find it amusing that the great Quine should fall prey to the trap of confusing the copula with the “is” of existence (“On What There Is” pp. 1 ff.), but to be fair he may have merely feigned the confusion to set up a criticism of one of his Platonist opponents. In any event he proposes his own untangling of the confusion – his own sartorial combing of Plato’s beard by distinguishing between naming and meaning.)

The confusion occurs right out of the gate (236 E) as soon as the Stranger has finished his slapstick classification of sophists and tantalizes Theaetetus with the aporetic or perplexing nature of describing illusions or falsehoods or lies. The phrase he uses to introduce the issue is ψευδῆ… ὄντως εἶναι, literally “a lie really is.” Both Fowler and Cornford (Cf. also pp. 312-313 where Cornford dallies with the very logical positivist notion of a statement being meaningless – an issue which appears only tangentially in The Sophist if indeed it arises at all in the commentators’ sense; for the Stranger λόγοι either are or are not, which we can interpret either in terms of existence/non-existence or in terms or truth/falsehood and never once is it even hinted that meaningful/meaningless is at issue) place all their chips on εἶναι as existence. But they lose big time. If we understand ψευδῆ along with the Stranger as a false assertion (λέγειν ἢ δοξάζειν) and εἶναι as “exists,” there is no contradiction (ἐναντιολογίᾳ). To say that a falsehood does not exist is not to say, imply or presuppose that it does exist. It exists as soon as it is uttered; if someone were to utter a falsehood and then say that his first statement never existed, his second statement would be false but not self-contradictory. To get to anything like an aporia we have to change the meaning of one of the terms. We could replace ψευδῆ with something like “what the ψευδῆ refers to or stands for or means.” (Aristotle seems to make precisely this point in reverse in Categories X 12a 5 ff. and 13a 37 ff. Cf. also Russell’s discussion of Lotze’s notion of propositional being in Principles of Mathematics p. 450.) Then, if we continue to understand εἶναι as “exists,” we get something like a contradiction in that we are asserting the existence of something that, since it is the referent of a falsehood and would exist only if the falsehood were true, doesn’t exist (Quine analyzed meaning and naming in a similar way in “On What There Is,” pp. 1 ff.). I say “something like a contradiction” because, if we were to just designate the referent of the falsehood and not simultaneously assert that it doesn’t exist we are not in contradiction. We may be wrong in our assertion, but we don’t contradict ourselves. Alternatively we could choose to understand ὄντως εἶναι as a second order predicate about statements, to the effect that what the falsehood asserts is not true, i.e. that its content really obtains, the fact it suggests is really part of the world. Then, as long as we assert and don’t simply designate the falsehood, we do contradict ourselves but the contradiction is harmless since one of our assertions - either the first order falsehood or the second order assertion that what the false asserts to be false is really true - is simply false. We are in effect asserting, e.g. “‘Jenna is a particle physicist’ (This is the ψευδῆ) is false (The real world where she is not a particle physicist is ὄντως εἶναι) for Jenna is not a particle physicist.”  The grand peritropic argument of being and non-being that would rattle noisily through the corridors of the history of philosophy is introduced with a verbal Doppelsinn that crumbles under analysis. Aristotle’s distinction (Met. V xxix) between false things and false statements seems to be consciously directed at this passage.

The confusion between things that don’t exist or non-existent entities or facts that don’t obtain, on the one hand, and non-existence as a Form, on the other, occurs in the Stranger’s next speech (237 A) where τὸ μὴ ὄν εἶναι is self-contradictory only in some version of the first meaning, a version that provides its own take on the sentiment that nothing is but (what) is not. (In 237C the Stranger introduces the term τὸ τί as a more abstract version of τὸ ὄν if indeed that were possible. Nevertheless for the sake of his argument they are equivalent.) This speech introduces a passage that is more or less a recitation of an Eleatic cum sophist style refutation of any attempt to characterize sophistic assertions as falsehoods and illusions. The Stranger would ultimately walk back this refutation but for the nonce he tries to present it in as convincing a light as possible. Moreover, in his ultimate answer to the claim he attributes to the sophists to the effect that false statements don’t exist (263 E ff. and 264 C-D), the Stranger continues to confuse the non-existence and the supposed falsehood of a false statement. He purports to prove that false statements exist in the mind as a kind of silent discourse. Presumably the sophists would argue that they do not deny that kind of existence. Even out loud and verbally anyone can make a statement that he believes to be false and that statement would exist in some sense. The sophist is saying that, because of the Parmenidean paradox, the supposedly false statement is really true. We have to keep in mind that the sophist’s goal is to justify his willingness to successfully argue both sides of an issue.

In his next speech the Stranger doubles down by introducing a new term, τὸ μηδαμῶς ὄν (237 B) and later the apparently synonymous τὸ μὴ ὂν αὐτὸ καθ’ αὑτό (238 C). Since he does not at this point define either new term or even distinguish either term from the simple τὸ μὴ ὄν, nothing in the immediate text provides any evidence that they are different concepts from τὸ μὴ ὄν. Cornford suggests some sort of distinction (pp. 203 ftn 1 and 213 ftn. 1) but he does not explain what the difference between “things which are not the fact” and “sheer nonentities” is supposed to be or what the phrases “totally unreal” and “absolute nonentity” are supposed to mean. The point of Plato scholarship is not to add to the confusion.

There may be room to interpret τὸ μηδαμῶς ὄν as a contrapositive to τὸ ὄν understood as existence or perhaps entityhood and to consider τὸ μὴ ὂν αὐτὸ καθ’ αὑτό as somewhat ambiguous. Plato does use the term ἐναντίος in 240 B and D in order to signify an opposition with τὸ ἀληθινόν a kind of stand-in for τὸ ὄν. This occurs in the context of a discussion of images in which Theaetetus attempts to differentiate an image from what it is an image of.  He makes an important distinction that, I feel, foreshadows Aristotle’s focus on the adverbial “in a respect” as a way of qualifying predicates that are otherwise incomplete. He distinguishes between existence πως and existence ἀληθῶς or ὄντως. Images exist πως with respect to what they depict or resemble and presumably they exist ἀληθῶς as images. What is depicted in an image also exists ἀληθῶς as itself but in most cases it doesn’t depict anything. The Stranger tramples on this distinction, however, by just restating the Parmenidean paradox, and an opportunity appears to be lost. One could also question how convincing the parallel between images and assertions may be. Does the referential function that causes the peritropic problems in language really inhere in an image? Does an image really assert anything? And if it doesn’t it certainly could not assert a falsehood. It would seem to be that the existence or non-existence of an object we recognize in an image is exclusively related to statements about the image and not of anything essential to the image itself. (Aristotle’s comments about σκιαγραφία (Met. V xxix ll. 23-24) are relevant here.)  Furthermore, and to return to my topic, ἐναντίος in 240 B seems to mean no more than complementarity between τὸ μὴ ἀληθινόν and ἀληθοῦς if indeed it means anything that specific at all. 240 E attempts to distinguish between τὸ μηδαμῶς ὄν and τὸ μὴ ὂν, but it is not clear whether the distinction is between a contrapositive and a complement to τὸ ὄν. It is worth remembering, however, that the Parmenidean paradox - quoted in 237 A (The most succinct statement of the paradox occurs at 254 D in the last line of the Stranger’s speech that begins at C) and here attributed to unnamed sophists, to which Plato subscribes - only works with a concept that is strictly complementary to τὸ ὄν and an understanding of the copular “is” as meaning at least partially “exists”. To put things into the context of the dialogue, the Stranger turns to Parmenides because unnamed sophists use the Parmenidean paradox to defend themselves against charges that their doctrines are false. They cannot preach false doctrines, or so they affirm, because if you take the paradox seriously there is no such thing as falsehood. (This way of stating the sophists’ self defense probably accounts for Plato’s own confusion between the existence of a false statement and the assertion that an existing statement is false - a confusion cleared up by Aristotle in Met. VI iv and IX x. Cf. also XIV ii 1089a ll. 25 ff. If we understand them as saying that supposedly false statements are really true since the assertion that those statements are false assumes that they are true, then we can avoid the confusing assertion that false statements don’t exist. Of course, stated this way, the sophists’ defense does run afoul of the Law of Non-Contradiction.) So, in order to support the very possibility of accusing the sophists of teaching falsehoods, the Stranger has to undo the Parmenidean paradox. The implications of his attack are wider than a parochial squabble in 5th to 4th century BCE Athenian intellectual circles. It touches at the possibility of observations about the world that are not confined within the prison house of language, so to speak. And the reasons for the invalidity of the paradox help support the validity of Plato’s own theory of several Forms. That is one reason why so much space is devoted to dismantling it.

However, on 241D the Stranger begins his own ritual of killing the father by returning to the notion Theaetetus had essayed of existence in a respect. He uses Theaetetus’ own term (ἐστι πῃ) and, anticipating Aristotle, introduces one of his own: ἐστι κατὰ τι.

The Stranger begins by assimilating the Eleatic doctrine of the One to other more overtly mythical accounts of the nature of entities. It is important to keep in mind that the Stranger uses the term τὸ ὄν throughout this passage. Both Cornford and Fowler pile on the confusion in this critical passage by translating the occurrences of τὸ ὄν (242 C) as “real things” or “realities” introducing a lexical distinction that implies an unjustified semantic distinction. The Stranger’s term is indeed saddled with the ambiguities I noted above, but Cornford and Fowler simply make matters worse. The translator needs to choose one translation (the most generally accepted, however unsatisfactory, option is “being”) and stick to it throughout this passage and footnote occasions where Plato seems to slide from one meaning to another of his term.

In 242 B ff. Plato steps back from the immediate task of his refutation of Parmenides, focuses his attention on apparently mythical ontologies/cosmologies and reprises the critique of Empedocles that he had adumbrated in the Lysis. This time, however, Heraclitus and the Milesians are added to the mix. The arguments are also different. The Eleatic Stranger does not mention the second order logical difficulties with the notion of likes and opposites that had concerned Socrates previously. In fact he does not refer to likeness or oppositeness on the part of things (in the universe – a faute de mieux and potentially misleading turn of phrase since Plato does not single out physical objects as being the only kind of first order objects) or the second order entities that qualify things (in the physical universe) at all. His actual criticisms are twofold. The first is that the mythical nature of the Empedoclean/Heraclitean cosmology renders it unconvincing. The universe does not develop because of anthropomorphic behavior on the part of its elements. It is not a little ironic that Plato introduced his own anthropomorphic cosmological myth of the demiurge on the Timaeus. Perhaps he did not actually subscribe to the Stranger’s criticisms. Perhaps he incredibly missed the inconsistency. Or perhaps he felt that the demiurge was not a mythical personage on the order of Strife and Harmony.

Cornford confounds the apparent inconsistency in his gloss to the effect that things in the universe, according to Heraclitus and Empedocles, are “derived” from “primary” things “by processes they can only describe in mythical terms (p. 218)”. The fact that Heraclitus and Empedocles used mythical language does not mean that they could only use mythical language. Their use could very well have been metaphorical in the same way that the atheist atomist Lucretius spoke metaphorically of Venus Genetrix. Cornford also confounds Milesian philosophy, whose claims are basically ontological and descriptive, and Empedocles who proposes something more akin to a natural scientific explanation. Change occurs in the universe, according to Empedocles, because its components, whatever their ontological status, are related by love and strife. The Milesians asserted that, despite appearances, the elements of the universe are, i.e. are identical with, one of its components such as water, air etc. Heraclitus, as we know, included elements of both ontology and physical explanation in his philosophy. Cornford’s confusion derives from the Stranger’s error. He calls hot and cold as well as wet and dry τὸ ὄντα as if these qualities were equivalent to reductive entities such as Thales’ water or Heraclitus’ fire. He also fails to recognize Empedocles’ distinction between ἀρχαί and στοιχεῖα. In the end, whatever merits the Stranger’s “mythological” criticism may have, it does not apply to the Milesians or to the ontological tenets of Heraclitus’ philosophy.

The second criticism which includes a passage about “being” made famous by Heidegger, is what we might call logical or semantic in nature to the extent that the Stranger’s main concern is with definitions. He does not, as I noted, use terms such as ἀρχαί, στοιχεῖα or ῥιζώματα. Rather he enumerates by name quality pairs such as hot/cold and wet/dry as examples of what various Presocratics had claimed were the basic components of the material universe, ignoring the contrapositivity that relates each pair and failing to specify that a given philosopher chose and could only choose one member of a pair as his reductive entity. The logical error, as argued by the Stranger, lies in claiming that one member of the aforementioned pairs is all there is; by way of such a claim the Presocratic must hold or assume either or both: (a) that being is a third thing that the elements partake of, or (b) that one of the elements is all that is. The Stranger does not really criticize the first alternative (although he digresses to paradoxes involving being, the one and the notion of everything); however, interpreted in one way, he argues that the second is logically contradictory since it takes two things to be one or, interpreted another way, he says that the conclusion that there is only one thing, hot or cold, contradicts the premise that both things are. This strange argument seems to attribute a Parmenidean assumption to the Milesians, Empedocles et al., namely that only one thing exists, and the Stranger seems to subscribe to that assumption. But it is precisely that assumption and what it means that is at issue. It is one thing to say à la Parmenides that only one thing exists, on the one hand, and a very different thing to say that many things exist but they all fall under a single heading, such as water, on the other. Had the Milesians been around to defend themselves, they very likely would have found no problem with more than one thing or one kind of thing existing although a clear articulation of their side would be much facilitated by Aristotle’s (no friend of Milesian reductivism) insights into the logical nature of genus/species subsumption. And it is hard to see what Empedocles is doing in this discussion since his physics does not involve monistic reduction. The Stranger may have a valid line of attack against Parmenides, but the digression through the other Presocratics throws it off track.

The linchpin (Heidegger might call it an Ereignis) of the Stranger’s assimilation of the Presocratic cosmologies to Parmenidean paradoxes occurs in 243 D-E. This speech condenses into a few words the two objections I noted above, viz. (1) That a doctrine that there exist only two things presupposes the existence of a third thing, existence or being (NB this is a version of the Third Man Argument in the Parmenides), and (2) that a Milesian (or ostensibly Empedoclean) style reduction of everything that is to some one thing incurs contradictory assertions in the style of “Everything is two and everything is one.”

Both Cornford and Fowler turn this speech and what follows into gobbledygook by mistranslation. In a sort of comic paso doble each one mistranslates one term and gets the other one right. Fowler mistranslates τὸ πᾶν as “the universe.” This captures the fact that Milesian concerns were primarily cosmological and had to do with the physical universe. But it obliterates the Parmenidean point which is more what we would recognize as logical in nature. There is no contradiction unless everything with a capital “E” is said to exist or be two or one in quantity. Otherwise existence and non-aqueous stuff, for example, could exist, just not in the physical universe.  Fowler is mostly right in translating εἶναι as “being.” The confusion between all entities, on the one hand, and existence, on the other, is Plato’s. Cornford reverses the error. He correctly translates τὸ πᾶν as “all things,” but turns around and translates εἶναι as “real” or “reality” which also obliterates the Parmenidean argument to contradiction. Existence and the losing side of a Milesian oppositional pair could consistently be said to exist even though they are not real in some unspecified (by Cornford) sense of “real.” Moravcsik (“Being and Meaning in The Sophist” pp. 26-29) sees that there are two distinct ideas in play, viz. being and all or everything (as well as the One and the Whole although he does not apparently understand the absurdity of calling these two concepts all-inclusive and topic-neutral (Cf. esp. p. 37)). However, he goes on to opine that Plato’s identification of “existence” and everything is some sort of valid theory instead of a confusion and error on Plato’s part. Moravcsik also fails to recognize that the paradoxes arise from the concept of everything and infect the Forms only to the degree that they are identified or associated with (the concept or class of) everything. Since Moravcsik treats existence as an indefinable simple he can say nothing for or against that identification. Note the tepid restatement in predicational terms of Parmenides’ isomorphism between being and thinking on p. 37. Moravcsik tries to fix the mess he’s made by introducing his own distinction between “exists” and “is real” (p. 41). But in the same vein he does not realize that the paradox problems arise from universality (or “topic neutrality”) and not from existence. So, if “existence” is essentially equivalent to “everything,” we need only to explore the notion of everything and drop artificial constructs like reality.

The following passage addresses the Parmenidean doctrine of the One (as well as the Whole; it is noteworthy that ambiguities in the notion of τὸ ὅλον recall those of the One since the meaning of that term shifts back and forth between “everything as a whole” and “a thing considered as a whole and not the sum of its parts”) and covers some of the same ground as the Parmenides. The purpose is to show that what the Eleatics say about the One involves the same contradictions that they argued for regarding τὸ ὄν. They are caught in their own trap. If you say something like τὸ πᾶν or a name or a part is equivalent to the One, then you are also saying that the One is really two things. You are saying both that there are two things and there is only one thing (assuming that the One is the only thing). Cornford and Fowler suffer some of the same problems as they did with τὸ ὄν in translating τὸ ἕν sometimes as “the One” and sometimes as “unitary” but the confusion here is more properly laid at Plato’s own doorstep.

The famous passage about gods and giants that follows is basically a swipe at what appear to be contemporaneous representatives of the Milesian tradition whom Plato characterizes in strikingly modern language as materialists or body (σῶμα) only ontologists. The Stranger’s refutation of materialism is typically shoddy and involves an incredible concession that souls have to exist. The reasoning is that, if souls exist, then why shouldn’t other non-material objects such as wisdom and justice also exist in some invisible and non-tangible way. The Stranger defines “being” on behalf of both the materialists and their counterparts, the idealists, as a capacity or power of acting on other things or being acted upon by other things. Theaetetus accepts this definition (ὁρίζειν) with nary a murmur and without noticing that it is prone to precisely the same self-contradiction as the reductive definitions of the Milesians to the effect that anything said about things that don’t have that capacity implicitly states that such things exist and accordingly attributes to them that capacity etc. In his digression into the history of philosophy the Stranger seems not so much to have refuted Parmenides as to have forgotten about him. He does in fact switch terminology in this passage. Instead of τὸ ὄν he mostly starts using οὐσία (anticipating thereby some of Aristotle’s terminological juggling) with its apparently more distinct flavor of material substance or physical objecthood as a limitation of the broad concept of anything and everything. If it is physical objects only that the Stranger refers to when he defines being as a kind of causal capacity, he doesn’t contradict himself. But his definition is hardly ἱκανόν. All he really does is engage in a shell game (the first of many that set up shop in the history of philosophy whenever philosophers get in a pickle – Unpack that metaphorical structure, literary critics!). He appears to escape the Parmenidean paradox of being by surreptitiously talking about something else. And he hides this sleight of hand by way of the shared lexical root of τὸ ὄν and οὐσία. Another problem is that the Stranger also defines the being of ideal objects as a capacity or power of acting on other things or being acted upon by other things. And, of course, occurrences of τὸ ὄν, εἶναι and ἔστι are sprinkled throughout the passage. So it appears he really doesn’t mean just physical objects at all. But in that case he tips back onto the edge of paradox.

The Stranger recognizes the relevance of his earlier arguments at 250B ff. to at least one version of the definition of being that he has just put on the table. He focuses on the tacitly Heraclitean doctrine that everything is in constant motion and the contrasting tacitly Parmenidean doctrine that everything is in permanent stasis. He applies the ontological Third Man argument to the ἐνατώτατα motion and stasis to the effect that both exist (or alternatively don’t exist) and that therefore existence must be some third existing thing, existence itself. If valid at all (Its validity depends on the difficult to formulate presupposition that motion and stasis are all that exist) this argument is much weaker than the self-contradiction argument derived from Parmenides. In addition, the Stranger does not mention capacity (δύναμις), choosing instead to concentrate on the presumably related notions of motion/stasis without showing how they are related to capacity. Nevertheless, the Stranger uses his ontological Third Man argument to upend some fairly weird doctrines, although it is likely that the Stranger is reducing what the apparent targets of his criticisms actually taught to caricature.

In another sudden shift of topic (248E ff) Plato seems to inaugurate his own reduction of being to an entity (in alternative language, conversion of the mega-concept of entity into that of a causally interactive object), a reduction that Reinhardt (roundly misinterpreted by Heidegger who made that misinterpretation the centerpiece of his much flawed history of philosophy) detected in Anaxagoras as one of the original sins of philosophy. The unexpected and possibly reifying phrase τῷ παντελῶς ὄντι signals this shift and the tantalizing though misguided reductive reification that would bait scholastics into suggesting a basis for theology in universal ontology.

How does opposition enter into all of this? In 240 D the phrase τοὐναντία οὖσι can only refer to the complement of (real) entities, since the Stranger throws it out in the context of the Parmenidean Paradox. Likewise the phrase παντάπασιν ἕτερον (255 E) seems to mean non-intersecting complementarity (I realize this phrase is redundant; I use it for emphasis). The passage beginning at 254 E is less clear. The Stranger introduces sameness and otherness (ταὐτὸν καὶ θάτερον) as γένη with which the other γένη - being, non-being, motion and stasis - can “intermingle (ξυμμιγνυμένω)” presumably by way or ideational participation or, in alternative language, generic subsumption or being-predicated-of. His immediate purpose is to show that sameness and otherness must be distinct from being, motion and stasis and so must be counted as fourth and fifth γένη. His argument is that if either sameness or otherness were (The only possible sense to attribute to “were” is identity, but of course that raises new difficulties) either motion or stasis, then, since both motion and stasis partake of sameness and otherness, if motion were either sameness or otherness, rest would partake of what we might call the specific characterizing traits of motion. That is rest would be in motion. A symmetrical consequence would obtain if rest were either sameness or otherness. The faults of this argument with its rampant category and order errors and ambiguities in the meaning of ἔστιν should be obvious by now. The interesting point is that θάτερον and τοὐναντίον) occur together in a single speech (255 A-B). The Stranger may understand ἐναντίος in this passage as synonymous with the word θάτερος. Indeed his argument only works, to the extent that it does work, if ἐναντίος is understood as “complementary” (If motion were not a member of the complement of rest but simply a concept contrapositive to rest, then the second order nature of the Form motion would be more evident as well as the consequence that there would be no contradiction in saying that motion is at rest) although, since no clear distinction is made, one is left with the impression that it also means “contrapositive.” But, of course, as it is stated, the Stranger’s argument fails no matter how we understand ἐναντίος.

Alfred Jarry seems to allude to this passage from The Sophist in Scene VII of his drama César-Antéchrist (pp. 290-292) where the characters profess a sort of limited identity of opposites and specifically reference the pair stasis/change. What they say doesn’t make a lot of sense, but there seems to be some ground to the belief, endorsed by the volume’s editor, that for Jarry words (not meanings) are all that count. …car le signe, says the templier, seul existe.

Owen’s phrase “…the other extreme” (in Plato I, Vlastos ed. pp. 234-235; cf. also pp. 261-262 and Moravcsik pp. 69-70) and the context show that Owen had an inkling of the contrast between complementarity and contrapositivity that pervades Plato’s examples. However, Owen immediately proceeds to thoroughly confuse the issue by his interpretation of the different meanings contributed by negation depending on its placement in the sentence, i.e. its logical scope. Owen draws his example, i.e. a distinction between (1) “not-is F”, where “F” stands for a predicate expression such as “white”, and (2) “is not-F” from Aristotle (Cf. Met. XIV ii 1089a ll. 15 ff. for the observation that if “is” has several meanings, so does “is not,” but Aristotle is not clear about how this affects the Parmenidean paradox as discussed in The Sophist), but the confusion comes in the way he applies it to Plato. He states that (2) does not designate (where “designate” could mean either “name” or “range over” or both) the contrapositive (Owen calls it the “contrary”) of a quality. “...is not-white” does not mean “… is black.” But he does not tell us what, in his opinion, it does designate. The obvious candidate is the complement of F but Owen’s failure to understand complementarity contributes to the hopeless garble in his understanding of (1). Drawing on the analogy to (2) Owen sets out of play the possibility that (1) could designate a contrapositive to “being” (I use scare quotes because Owen does not specify what meaning of “being” he has in mind), i.e. a second order entity such as non-existence. He says this option “is not open.” But he immediately proceeds to confuse contrapositivity with complementarity with his statement that (1) “cannot even be applied to anything in the contrary state.” Presumably things “in the contrary state” constitute a class complementary to the class of F’s. And since F in (1) is “being” tout court with no saturating (to use Frege’s phrase) predicate expression, the complement of F in (1) is the class of things that are not. Accordingly since Owen, following Plato, confuses the class of things that are (τὸ ὄν) with the universal class (τὸ πᾶν), it is self-contradictory to say that anything does not belong to that class. There is “no intelligible contrary (complement – WD) to being or to what is.” This is, of course the Parmenidean paradox. But, just as the paradox need not arise (as we have seen) if we distinguish between the copula and the predicate “exists,” so the paradox in this version also disappears if we carefully distinguish between the universal class and the predicate “exists.” Indeed if we do distinguish between the copula and the predicate “exists,” Parmenides’ dictum that being and thinking are the same collapses altogether (unless one retreats to the inane tautology that “is” means “is talked about”). Plato’s solution whereby non-being is assimilated to otherness or difference, along with the unacceptable consequences of that solution, becomes unnecessary. Owen’s introduction of the Platonic doctrine of intermediates as a modification of contrapositivity throws no light on what is at issue in The Sophist. The truly weird notion of “proportion of being” should be treated with the contempt it deserves. Of course, this passage occurs in the context of the general point of Owen’s essay to the effect that there is no concept of existence in the doctrine of being propounded in The Sophist. That general point is well taken as far as it goes but it suffers from (1) failing to recognize the notion of universality or the all in generating the paradoxes of being (He does come ever so close to cracking the puzzle on pp. 259 and 261 with phrases like “exhaustive dichotomy” and “leaving something out of the inventory”), and (2) failing to provide an adequate definition of “exists” to mark off that concept from identity and the unsaturated copula. This latter defect is not answered by pointing to ordinary language examples of our use of the term “exists” (e.g. A: “The copy of Montaigne I thought was in the library does not exist”) and treating “exists” as an indefinable simple. For Plato in The Sophist ultimately proposes to treat all senses of “exists” as precisely synonymous with “is identical to….” And Aristotle would expand upon this conclusion to interpret statements like A to mean “No object in the library can be described as a copy of Montaigne” or else “No object in the library is identical to a copy of Montaigne,” or , to phrase things in Aristotle’s technical terminology, “No matter in the library has the form copy-of-Montaigne.” Owen expresses the difference between “exists” and other meanings for the copula (p. 265) in a garbled way with his distinction between reference and reference with “the conditions for truth.” Among its myriad problems this approach continues to imply that any statement (as opposed to simple naming) about a non-existent object is false, the very consequence he wishes to avoid.

The same conclusions can be drawn if we turn to Cornford’s translation and commentary (p. 280): “Neither Motion (κίνησις which could also mean “change”) nor Rest can be (identical with) anything that we say of both of them in common.” Using this terminology the Stranger’s reasoning can be expressed as follows: Assume that Motion is identical with Difference (θάτερος). But Rest partakes of Difference so, by substitution of identicals, Rest would partake of Motion. Therefore Rest would be in motion (μετασχόν).  Etc.

Vlastos (Platonic Studies pp. 270 ff. and 302-305) makes an analogous point about The Sophist 252 D in his rather peculiar terminology of different types of predication (Cf. also Bostock (p. 110 - 116 and passim (in Oxford Studies in Ancient Phil. ed. Annas) for a distinction between abstract nouns used to name vs. used to generalize). The analogy is straightforward but needs some disentangling. Also there is some value in rescuing Plato interpretation from the overly Xtianizing impurities introduced by Vlastos. If the Form Motion is understood as the contrapositive of, or at least distinct or different from, the Form Rest, then we can without Parmenidean contradiction say it is at rest, i.e. it partakes of rest, by way of second order predication. Vlastos calls this “ordinary predication” but he fails to note that it must be second order in order to make sense. If, on the other hand, motion is understood as specifying a class that is complementary to a class specified by rest, then we cannot consistently say that whatever falls under motion also falls under rest. This is what Vlastos for thoroughly irrelevant reasons calls “Pauline predication.” Complementarity and the logical properties it entails can occur at any order of predication, though the issue here is first level predication. Vlastos recognizes complementarity (pp. 301 and 303) when he refers to “mutually exclusive classes” but he confuses complementarity with oppositeness in general. A theory of second order objects (Forms) is not the only supplement required by a theory of first order complementary classes. There are also, as I have been at pains to point out, first order contrapositive predicates. First order objects may fall under or be “embraced by (Vlastos)” contrapositive predicates in Vlastos’ “Pauline” sense without contradiction. This is especially clear if we regard the first order predicates “in motion” and “at rest” as incomplete and requiring qualification. To use (and probably abuse) an Aristotelian example, a rotating sphere could be described as simultaneously in motion and at rest. We might also think of Galileo’s notion of relativity to an observer as a measure of velocity. Speaking of Aristotle, one passage in the Prior Analytics (I xxiv 22-24) seems to state that distinct predicates such as apply to different particulars can also apply to each other, but the passage is unclear.  (Bostock (pp. 112-115) notes a distinction between oppositeness and otherness, but fails to see that the correct distinction is among three concepts: difference (typically a first order relation between individuals, but it can be n-order if higher order entities are nominalized), complementarity (a second order relation between classes) and contrapositivity (an ambiguous two-place predicate that characterizes concepts and the classes they define as well as the individuals that belong to those respective classes). Difference is covered by what Bostock calls "naming" (although the issue for Plato is the Forms themselves as second order individuals), while complementarity and contrapositivity characterize different aspects of what Bostock calls "generalizing.")

The next step in the dialogue is to investigate whether or not Sameness is an irreducible, i.e. independent γένος or identical with one of the other γένη such as τὸ ὄν. The Stranger’s negative conclusion derives from the assumption, supposedly just established, that Motion and Rest partake of being, i.e. they exist. He asserts that, if being and Sameness were the same, then the conclusion would impose itself that Motion and Rest are the same since they would of necessity partake of Sameness. It doesn’t take a great deal of reflection to see that the Stranger does not distinguish clearly between self-identity and being the same thing as something else (The passage is so short it does not lend itself to any further analysis as to what we mean when we say a thing is the same thing as something else). If Sameness is understood as self-identity, then the Stranger’s argument fails, since Motion and Rest are indeed self-identical just as they both exist (in some undefined sense of “exist”) even though they differ from each other. In fact it is quite appropriate and perhaps unavoidable to conclude that something is an entity if and only if it is self-identical (The left to right entailment is more evident than the right to left entailment as long as we haven’t specified what we mean by “entity”).

The Stranger then chooses Sameness and Entityhood (τὸ ὄν) to pursue his argument, concluding that if Motion were (sic) Sameness just as it is an entity (it having already been assumed that Motion as well as Rest partake of being, i.e. are entities), then Motion and Rest would be the same since Rest also partakes of Sameness and Entityhood. Either way the argument commits the error of failing to recognize logical (or in Plato’s case “ontological” would be the more proper term) orders. The conclusion does not follow even if we were to grant the unlikely assumption that Motion is identified with either Sameness or Difference. The higher order entity (viz. Platonic Form) could very well itself be in motion (or at rest) even though it is not identical with Motion (Of course, even applying the concept pair moving/at-rest to a non-spatiotemporal entity is a category mistake, but that is over and above the logical order mistake). The error is clearest with respect to the qualification of sameness. A thing can be self-identical without thereby being Identity or being identical with everything else that is self-identical although it is like everything else in that respect. So, even if Motion were identical with Sameness, Rest could partake of Sameness (self-identity) and indeed be in motion without being the same as either Sameness or Motion. (Ackrill in Plato I, ed. Vlastos, p. 208 fails to recognize a distinction that Plato apparently made between self-identity as a one-place predicate and identical-to as a two-place relation. This allows him to endorse the bland conclusion that if being an entity were the same thing as being identical, then all things would be identical presumably to each other. That conclusion depends on understanding identity strictly in terms of identical-to. Since Plato did make that distinction he could go on to say that non-existence is equivalent to being identical to something in the universe, i.e. it is self-identical, but not that it is identical to everything in the universe.)

It is unlikely that either Plato or the Stranger actually subscribed to this argument (Cf. Vlastos, pp. 23-24 for discussion of the issue of the self-predication of the Forms). Moreover, the Stranger pretty much overturns the reasoning in his summary starting at 255 E and also at the end of the dialogue when he states that a Form can partake of its opposite without thereby becoming identical with its opposite and that a Form doesn’t have to partake of itself. There is a better version of the argument which runs as follows. Assume a thing is in motion if and only if it is self identical. A thing at rest is not in motion. Therefore nothing at rest is self-identical. This version avoids the logical order error. It is also tautological as long as we accept the obviously faulty major premise. But, even if we clean up the errors of logical and ontological order in the actual version of the Stranger’s argument, it would still run afoul of what I have been calling the incompleteness of certain predicates (assuming for the sake of argument that identity, i.e. sameness, and difference are real predicates). That is, you cannot meaningfully say of two things that they are the same or that they are different unless you specify a respect in which they are the same or different. In a broader sense, the relation between self identity and the assertion that two things with different names could in fact be identical is obviously a continuing philosophical issue.

The Stranger’s short speech at 256 B to the effect that the Form Motion can be at rest makes many of these points (Astoundingly he does not summarize his previous arguments so much as overturn some of them). Quite appositely he distinguishes between participating in a Form (predication involving in some languages a grammatical copula) and being identical with a Form (Vlastos gets this right in ibid. pp.286-287). This is a major contribution on Plato’s part that allows us to get over the Parmenidean paradox (although mistaking the essence of the paradox as having to do with being or entities and not with the concept of everything or all entities would keep philosophizing based on the paradox alive for many centuries. While making a different point Vlastos (p. 306) alludes to the association of the key concepts of Platonic philosophy and indeed of all philosophy - concepts such as being, identity and difference to which we may add oneness or singularity - with the notion of everything or all. These concepts “must characterize everything whatever;” they are “adjectives of universal application.” Vlastos doesn’t seem to recognize, or he doesn’t recognize in this passage, that the paradoxicality these concepts exhibit stems from their universality and not from any other meanings that characterize them specifically and distinguish them from each other). Plato shows how a Form can participate in another Form (How Motion can exist or be at rest) and not be identical to the Form it participates in. The speech at 256 A-B also corrects the confusion between self-identity and relational identity to another thing that had invalidated his argument that Sameness and Otherness must be distinct Forms. (Owen pp. 256-258 (Cf. also Wiggins pp. 288-289 ftn. 13 and p. 291) denies that Plato’s reduction of “is not” to “is different from” involves corresponding adjustments in our understanding of “is”  in such a way as to turn it, in at least some contexts, into the “is” of identity and not the “is” of one-place predication. This leads to absurdities such as denying that a statement such as “It is not the case that a is different from b” entails “a is identical to b” (assuming both statements are abbreviations and that they are subject to Leibniz’ Law).)

The Stranger’s argument for the generic nature of Otherness is supposed to follow the same path, but it contains a curious error missing in the previous argument about sameness. What he must mean to say is that, if τὸ ὄν and θάτερον are the same thing, then things that are, i.e. that partake of being, must also partake of Otherness. If they exist, they must be other than something. But there are things that exist but are καθ’ αὑτά, i.e. they are not other than anything. So τὸ ὄν and θάτερον cannot be the same (i.e. the same Form). What he in fact says is that if τὸ ὄν and θάτερον each partook of both καθ’ αὑτά and relatedness, then etc. It is possible that this is some sort of textual corruption. It is also possible that the meaning of μετεῖχε has shifted away from “participate” to something closer to “encompasses.” The use of the dative in this occurrence of the expression instead of the more common genitive (jn Plato and elsewhere) has nothing to do with a shift of meaning. It merely reflects the fact that the Stranger is considering the possibility of concurrent participation in two things, viz. καθ’ αὑτά and relatedness. Cornford (p. 281) notices the problem, but his contrived solution (ftn. 2) to the effect that “partaking” must be a symmetrical relation between Forms suffers from four faults: (a) it assumes καθ’ αὑτά and relatedness are Forms which they are not at least in this discussion which limits the generic Forms to the five under discussion, (b) it does violence to Greek grammar, (c) it turns into mush the oppositeness of Forms such as existence and non-existence which as complementary had made the Parmenidean paradox comprehensible and as contrapositive points the way to its resolution, and (d) it seems to imply that participation between Forms must be symmetrical which is absurd since it would mean being doesn’t exist and justice is unjust. (Ackrill punts on this passage in Plato I, Vlastos. ed. P. 221 ftn. 9 but the above comments should fill in his argument.) Nevertheless, despite the scrambled grammar, we can reconstruct the argument. Indeed 256 C-E reiterates and reconfirms the distinction between predication or partaking and identity for all five of the genera under discussion.

At 257 B the Stranger makes the distinction between ἐναντίος and ἕτερος that has been my theme throughout this essay. He appears to distinguish these concepts precisely in terms of complementarity and contrapositivity as I understand complementarity and contrapositivity. Moreover, he associates complementarity or conceptual otherness with the negation sign in language and provides an almost perfectly phrased Boolean explanation of otherness or complementarity as the class of things that are not qualified by the predicate that lies in the scope of a negation sign. This critical distinction occurs apparently for the sole purpose of showing how each of the five γένη can be different from (other than) the other four and yet not be its contrapositive or its opposite in the way that hot and cold are opposites, or, to use his own example, in the way that great and small are opposites but not complements since the complement of either quality includes middle-sized (and technically things to which size measurement doesn’t apply). His sense of otherness, however, is ambiguous. It can mean non-identity in general and not complementarity, for at least part of his argument is that the Forms are different from each other (without necessarily being complementary) and that the things that partake of different Forms are different from each other (i.e. the Forms determine non-intersecting, but not necessarily complementary classes). I suspect this ambiguity is at least partly due to the fact that Plato did not have the concept of a universe of discourse.

Wiggins pp. 299-300 asserts, that if Plato understood negation in terms of class complementarity, then he runs into difficulties. Or else, if that is not what Plato meant, then those who take him to mean it must be mistaken. Wiggins’ reason is that any complement equated with negation must be an “unrestricted” complement and so presumably fall afoul of paradoxes like Russell’s. This is unrestricted nonsense. In the first place Wiggins seems to have no idea of a universe of discourse or of a restricted domain – ideas on which Boolean complementarity is based. In addition, there is no reason why the equivalence of negation and complementarity cannot be understood in the context of Zermelo’s revision of set theory. Finally - another point that Wiggins misses – set theoretical paradoxicality is the twin of the Parmenidean paradox that for Plato seemed to disrupt any coherent discourse. Thus Plato’s answer to Parmenides would also disqualify any objections to an understanding of the links between negation and set complementarity based on the threat of unrestricted set concepts. On p. 302 Wiggins seems not to understand the difference between Other as complementarity and Other as something-different either numerically or numerically and definitionally. As a result he confuses otherness as a relation holding between the five Forms and otherness as a relation holding between things that participate in the Forms. Plato’s primary concern is with the first although admittedly he applies his conclusion to first order particulars as well. In any event Wiggins’ unrestrictedness objection applies only - if it applies at all, which it does not – to complementarity and not to numerical difference since the latter is not tied to a class concept (Of course, you can construct a class out of anything, but that has no impact on the present distinction). The fact that two things are numerically different does not entail that one of the two things is in fact “everything else.”

The Stranger’s first speech at 258 B is fairly garbled but, engaging in a bit of wishful thinking, I would like it to mean that, just as certain items (particulars or Forms) can be other and/or contrapositive and still exist, in the same way a Form such as non-being can be contrapositive to being while those things that fall under non-being can be other than (complementary to) those things that fall under being. At the same time the Form non-being can partake of being and so be other to the non-existing things that partake of non-being. If this isn’t what he said, it is what he should have said if only because it provides a logical lead-in to the following passage at 258 C which returns to Parmenides and supports our conclusion. The complement/contrapositive distinction combined with the distinction between participation (predication) and identity, produces a resolution of the Eleatic paradoxes which is in some ways superior to Aristotle’s version.

However, if Plato means, as later passages suggest, that there is no such thing as non-existence and that apparently non-existing things are really existing things which are merely different from some other existing things, then we have a problem. Plato’s replacement of negation by otherness or difference leaves no room for negative existential statements. In that case 258C contains an error and the arguments he uses are certainly invalid. Not only does Plato miss the distinction between difference or otherness and existence, he appears to deliberately obscure that distinction. Entertain for the moment the idea that τὸ ὄν means existence in the sense that the earth’s second moon doesn’t exist or Charlus doesn’t exist. In that case to say that Charlus or earth’s second moon is each respectively a τὸ μὴ ὄν is to say something more than that they are just different from something else (Russell appears to advocate just such a view, at least with respect to physical objects, in Ch. LIV of The Principles of Mathematics). The moon is different from the planet Mars and both exist; equally earth’s second moon is different from the planet Mars, but in addition it doesn’t exist. Marcel Proust is different from Montesquieu and both existed at a time, whereas Charlus is different from Montesquieu but Charlus never existed (Setting aside refinements such as Meinongian or Husserlian unreal objects).

Ackrill in “Plato and the Copula: Sophist 251-259” in Vlastos ed. Plato I p. 211 and 222 interprets the phrase (κίνησις) ἔστι δέ γε διὰ τὸ μετέχειν τοῦ ὄντος as “existential” presumably as opposed to predicative or asserting identity. To the extent that his comment makes sense, it is mistaken. Plato, i.e. the Stranger, is asserting that Motion is not an entity. That is, it does not participate in the all-encompassing genus, being. He does not mean that motion exists in the sense that the ninth planet from the sun exists while the tenth planet, although an entity, does not. His perhaps deliberate obscuring of this distinction allows him to “solve” the Parmenidean paradox of non-being (which relies on an understanding of being as the all-encompassing genus) by concluding that non-being is just difference from other things. That conclusion amounts to a rejection of an understanding of existence as a net addition to the stuff of the universe, so to speak, in the same spirit as Aristotle’s rejection of creation ex nihilo. Whichever interpretation of “exists” in Plato we opt for we run into problems, although the problems differ. If we choose a proto-Aristotelian view of existence as participating in some Form or other, then Plato’s solution of the paradox of non-being (construed as non-existence) doesn’t pose a problem. We then, however, have the problem of why being should be a Form, indeed one of the Five highlighted Forms, since it is synonymous with identity. If, on the other hand, we construe existence as involving a net addition to the universe and non-existence as a net subtraction, then it appears that Plato ignores this meaning of existence. In addition there is the risk that the Parmenidean paradox may not be answered.

At pp. 236-241 op. cit. Owen argues that the Platonic approach to dissolving the Parmenidean paradox by way of negation can be understood in terms of the scope of the negation operator, and that we do not need to invoke an existential meaning of the verb “to be” in addition. There is validity to this argument if we add that it incorporates an imperfectly understood (imperfectly understood by both Plato and Owen) distinction between complementarity and contrapositivity to parallel the scope distinctions of negation. The valid point is that Plato’s ultimate theory of being does not incorporate being as existence (or nothingness as non-existence, p. 248). However Owen does not recognize or does not give sufficient credence to the fact that the notion of existence (as signified by a one-place predicate) is not just missing from the theory of The Sophist, Plato actually undertakes to replace (to “reduce” in modern parlance) non-existence with difference and – by extension though less explicitly and with no acknowledgement of the problems this generates – to replace existence with identity. It is not clear whether Plato had a concept of existence in The Sophist and proposed to explain it away or if he simply did not entertain such a notion and proceeded straight away to treat all cases of “is not” as disguised examples of unsaturated one-place predicates or of negated identity statements. However that may be there is a significant resemblance between the ultimate viewpoint expressed in The Sophist and Aristotle’s Form/Matter theory of being. Wiggins pp. 288-289 ftn. 13 makes a doubly converse claim. He states that “to be” for Plato means “to be something” which latter he distinguishes from both predication and identity - appropriate if you go beyond Plato and distinguish between “Socrates is a man” and “Socrates is white”. Wiggins then goes on to point out the problems this raises for the negation of “to be.” Problems of negation aside, this Aristotelianizing view may be more satisfying in itself, but it does not correspond to what Plato actually said. In The Sophist Plato was not concerned with individuals such as Theaeteus but with the intermingling of the five Forms. Nevertheless it is a valid point that, if non-being reduces to otherness, then being is indistinguishable from self-identity and its independence as Form is belied. Given the three options for interpreting being, viz. (a) existing (b) self-identity and (c) instantiating a non-accidental species, Plato does not seem to even recognize (a), while (b) and (c) lead to problems with negation and the non-independence of the Form being.

Certainly there is no clear reason why this kind of non-existence as a Form should not exist if the Form existence exists (The Stranger makes this point in 258 D-E but only after he has doubled down on his definition of τὸ μὴ ὄν; he regards non-being as some fact about existing but distinct particulars not being the case instead of some particular not existing). There is nothing self-contradictory about the assertion that the Form non-existence exists. The arguments the Stranger uses to support the conclusion referred to in 258E (that to say something about non-existence is to paradoxically say it exists) are derived directly from the Parmenidean paradoxes (that to say something about non-existent objects is to paradoxically say they exist) that he had already disposed of with respect to first order individuals by distinguishing between the “is” of predication (to use an Aristotelian formulation in lieu of “participation in a Form”) and the “is” of existing in the sense that Proust exists. “Charlus does not exist (is a non-being)” is not self contradictory because the first “is” in that statement is a copula and the second “is (a (non-)being)” means “exists.” (I choose to ignore arguments against the predicability of existence proposed by later philosophers because they would take us too far afield; I will argue elsewhere that the doctrine of the non-predicability of existence arises from a confusion of being with the highest genus or the notion of “all”.) Of course, any number of things can be said about the individual Charlus even though he doesn’t exist. He is, for example, a French peer. He is homosexual, etc. But Charlus’ non-existence does not reduce to the fact that he is different from Montesquieu. Even if in some way Charlus and Montesquieu shared all the same properties, Charlus would still not exist. Something like the tenth planet Eris is an interesting case.  Suppose we decide, as the experts have, that it isn’t really a planet according to our criteria for classifying heavenly bodies as planets of our sun. Then it is true to say that Eris (the existing Eris) is not a planet (Apparently it is a dwarf planet). This is covered by Plato’s copular sense of “to be” as “to participate in a Form, viz. the Form planet.” But it would be equally true to say that the planet Eris does not exist; if you list the planets of our sun there will be no planet Eris among them. A heavenly body we chose to name Eris does exist but that is not the planet Eris because the planet Eris does not exist. The heavenly body Eris and the planet Eris have all the same properties (NB, just like the heavenly body Eris, the planet Eris does not have the property planethood becaue it isn’t a planet) and both are different from the other nine planets, but the heavenly body Eris exists and the planet Eris does not.

The scholarly material that includes Ackrill, Vlastos, Moravscik and Malcolm succeeded in showing how three different concepts - viz. predication via the copula, identity and existence - were in play when Plato theorized about being in The Sophist. Malcolm makes the most telling point, endorsed by Vlastos, to the effect that Plato does not clearly distinguish between existence, on the one hand, and predication or identity, on the other. But no one in this discussion, as far as I can tell, realizes that Plato perhaps intentionally assimilates non-existence to difference or otherness. For Plato, to say that a Form is not is to say that it is not something. When we say that the Form being is we mean that it is distinct from the other Forms. Plato either had no notion of existence as a net addition to the universe, or he chose to reject it like Aristotle rejected creation ex nihilo. The unpleasant consequence of Plato’s assimilation of non-existence to otherness is that it turns existence into identity. Indeed a Parmenidean might use this consequence to defend his thesis that only one thing exists since all existing things are identical. More apposite is the objection that, lacking some sort of theoretical bandage, non-existing things are construed as not self-identical. It is unclear whether Plato meant to apply this analysis to particulars as well (Owen in Plato I, ed. Vlastos, p. 233 ftn. 20 asserts that he does). If he did then the conclusion would be: To say that a particular is not is to say that it is not F where “F” stands for a first order one-place predicate whose meaning does not involve identity. That is he would completely assimilate “is” to predication or subsumption of a particular to a class. The result would be to make the statement “Charlus is a French peer” false. That is, it would fail to distinguish between “Charlus is a French peer” and “Charlus is a real or existing French peer.” Malcolm in particular argues, probably successfully, that Plato does not have a notion of existence which is not reducible to predication or identity. However, Malcolm fails to make clear how he himself understands “to exist” as a meaning for εἶναι. Specifically he does not distinguish between an Aristotelian-style analysis of existence as some sort of form/matter combination and existence such as would result from ex nihilo genesis. Understood in the former sense a Platonist could argue that existence is adequately accounted for in terms of the identity/difference analysis. It may be that Owen and Bostock propose just such a defense. More fatal is that Malcolm himself in his “reconstruction in modern terms” of what Plato says assumes that the predicative copula and the predicate “exists” are compatible meanings of a single word instead of distinct notions ambiguously signified by a single word in Greek and other natural languages. That assumption is unfounded. Moreover, Malcolm falsely claims that “is” is implied or presupposed in natural language verbs that appear in the predicate position of a proposition in natural language. One other weakness in Malcolm’s interpretation is that he feels compelled to regard the Form being, notably as it appears in phrases such as μετέχειν τοῦ ὄντος, as a grammatical connective and so different in kind from the other Forms which are simple (complete) predicables. Schipper (“The Meaning of Existence in Plato’s Sophist” in Phronesis Discuss points in Schipper in my bibliographical entry; check Runciman first whom she relies on) and Berger (“Rest and Motion in the Sophist” in Phronesis) distinguish between predication and existence but neglect to provide an adequate account of what is meant by existence.

Following the Stranger’s lead we can also distinguish between second order predication involving the copular “is” about the Forms themselves and their existence or non-existence. That is, we could without self-contradiction say lots of things about the Forms even if some specific Form or indeed all of the Forms didn’t really exist. It may be the case that the ontological standpoint of Form participation projects an illusion of contradiction (How can something participate in something else if it doesn’t exist?) that disappears with a reformulation in terms of the standpoint of how we use our language, i.e. from the standpoint of predication. The background for Plato’s blindness to the equal applicability of his distinction between predication and existence regarding first order objects to second order entities such as Forms lies in his rejection of the very idea of contrapositivity that informs Socrates’ dialectic in the Lysis (Remember contrapositivity in the Lysis is a relation between concepts (what would later be called Forms) and is therefore second order).  (Indeed as we saw in the Timaeus, he believed he found a sufficient replacement for contrapositivity in the concepts of relative measure and objective measure. Of course, it is difficult to imagine what Timaeus-style relative or objective measures of existence might be. Proust is more existing than Charlus? Proust measures ten units of existence and Charlus only three? At a superficial glance existence cannot even be measured by way of intensive magnitudes in the Russell/Meinong sense.) That rejection underlies his use of the loaded term ἐναντίος in 258 B and E. Because the notion of contrapositivity was not clear in Plato’s mind, he did not see how non-existence could be a Form, so to speak, contrapositive to existence and yet not be a member of the complementary class to the class of existing things, i.e. not participate in itself. Plato’s slide from one meaning of ὄν to the other is clearest in 238C. In his third speech in this passage the Stranger uses the phrase μὴ ὄντι which means “a thing that doesn’t exist.” But in his next speech he shifts to τὸ μὴ ὄν where the addition of the definite article could make the phrase mean either “the non-existing thing (i.e. non-existing things)” or non-existence itself (the Form non-existence). The Stranger derives his paradox about the existence of non-existence by assimilating the two different meanings of τὸ μὴ ὄν. As I mentioned, the Stranger reverses his position on the existence of non-existence in 258E ff. only after he has changed his definition of non-being (νῦν εἰρήκαμεν) to the ontological equivalent of false statements. His quite reasonable argument is that, if there were no such thing as false statements argumentative discourse and perhaps language as a whole would be impossible. But the fact that the meaningfulness (he says “existence”) of false statements gets embroiled with the existence of negative facts points up the limitations of trying to describe linguistic phenomena in the ontological terms of Form participation.

However, in Plato’s defense, non-existence thus understood is the complement of existence, not its contrapositive. And in the passage in question the Stranger uses the term “contrapositive” and not the term “complement.” He says ἐναντίος and not ἕτερος. This leads to the strange quirk I mentioned above. We do not appear to have a concept strictly contrapositive to existence in the sense that hot is contrapositive to cold or good to evil. At least English and Greek do not have terms for such a concept. The appearance of negation in the grammar of non-existence is sufficient indication that that term designates a complement. The Stranger would have been better served to say we have no concept (alternatively there is no Form) contrapositive to existence, but that does not mean that we cannot say things about the concept of what is complementary to existence and even say that such a concept or Form exists if the concept or Form existence exists. It also doesn’t mean that one day we may not devise a contrapositive to existence. Parmenidean style contradiction does not exclude that possibility.

Assume, on the other hand, that τὸ ὄν in this passage simply means “is F,” i.e. “being” in the sense of predication or participating in a Form. Nothing changes. Everything I said about τὸ ὄν as existence applies in this case as well. Plato had started by dealing with complementarity as suggested by the negation term μὴ. But, because the relation between being-F and not-being-F is similar to the relation between identity and difference, Plato, as I noted above, slides from complementarity to otherness. He avoids the Parmenidean paradox, and so the purported sophist claim that everything is true, by simply changing the meaning of his terms. “Not hot” no longer means participating in the complement of hot, which meaning is the motor behind the Parmenidean paradox (This is clearer if we read “is not hot” as “not is hot” in such a way as to place the copula within the scope of the negation and give the impression that being hot is assumed in a statement that something is not hot – “Ice is hot – Not!”); now it means being different from some hot thing. Likewise coldness is no longer understood as the contrapositive of heat; it is just a different and completely independent unrelated concept. The inadequacy of Plato’s device should be self-evident. (Moravcsik (p. 74) recognized the inadequacies of this doctrine or a similar doctrine of the identity and non-identity of predicates, but he failed to provide any convincing evidence against the fact that this doctrine is exactly what Plato concluded.)

The Stranger uses these conclusions (258 B ff.) to establish that it is not self-contradictory to state that an assertion is false and that some state of affairs doesn’t really obtain. He returns to the argument he attributed to the sophists to the effect that you cannot call some assertions that they make false because in doing so you would contradict yourself. There is an apparent grandeur in the structure of his refutation, but in fact all he needed in order to make his point was the distinction between the copula and existential predication which he made at the beginning. The other elements of his theory - the notion of higher order objects, the distinction between complementarity and contrapositivity, and the distinction between identity and predicative subsumption - are only tangentially related to the consistency of calling an assertion a falsehood. The Stranger is casting a wider net. That is, he wants to provide a theoretical basis for the notion of a Platonic Form by showing how the Forms can be subsumed under each other. (Lest you think that the Theory of Forms is not at issue here, note that εἶδος and γένος are used interchangeably throughout this passage.) This takes care of the objection that the Form non-existence cannot consistently be said to exist. He also provides a conceptual basis for opposing the universalizing and reductive ontologies of some of the Presocratics by rejecting the idea of a single highest genus in favor of a schema involving five genera (This version, if the five genera are meant to be comprehensive, contradicts versions of the Theory of Forms Plato proposes in other dialogues, most notably The Republic). Unfortunately the Stranger confuses the distinction between contrapositivity and complementarity, on the one hand, and predication and identity, on the other. He seems to merge the two when he asserts that non-existence (τὸ μὴ ὂν) can exist. He does not clearly differentiate between the conclusion that the Form non-existence can be subsumed under the Form existence as an existing second order thing, on the one hand, and the answer to the sophist to the effect that we can talk about things that don’t exist without attributing existence to them, on the other. The existence or non existence of the Form non-existence is irrelevant to the issue of whether we can speak consistently of things that are subsumed under that form, that partake of non-existence. Greek grammar lends itself to this confusion in that τὸ μὴ ὂν can be translated as both “non-existence” and “something that doesn’t exist.” (Latin lacks a definite article altogether and as a result much later classical and scholastic philosophy exhibited that corruption in the (mis-)understanding of the issues that so bothered Heidegger).

Usually when he speaks of an opposite (including what I have chosen to call otherness in the passage that deals with the five genera) Plato is not entirely clear whether he means some second order thing such as a Form which is the referent of the term “contrapositive” or some lower order thing such as we might call a particular that is qualified by a concept that is contrapositive to another concept. His confusion is not entirely unexpected since he pioneered the theoretical recognition of abstractions and second order terms (Bostock (p. 91 “Plato on ‘Is Not’” in Oxford Studies in Ancient Philosophy ed. Annas) completely misses the first order/second order distinction which drives the argument about the interaction of the five Forms). There is, for example, a difference between coldness and a cold thing. But he often focuses on the abstract singular term, to use Quine’s phrase, and not on particulars qualified in a certain way. Accordingly the Stranger’s solution to a Parmenidean paradox is that, while a non-existing thing cannot without contradiction be said to exist, there is a sense in which non-existence itself can be said to exist (in some as yet undefined sense of “exist”). As class terms existence and non-existence are complements. As names of abstract entities Plato’ treats them as contrapositive (or at least as distinct) terms.

Furthermore, much of Plato’s argument has to do not so much with the abstract notion of contrapositivity in isolation, as a real logical or even ontological argument probably would, but with ambiguities in the meaning of specific contrapositive pairs of terms. Whether or not coldness itself can be said to be hot depends very much on how we understand coldness and how we understand what it means to be hot. If coldness is understood as a concept or a second order abstraction, we would think that hot or cold are not qualities that can be applied to it since we do not think of any second order entities as having a temperature. We may choose to conclude that coldness belongs to the extended complement to hot things (that is, the set of everything that isn’t a hot thing); or else we can take the apparent inapplicability to mean that any attempt to regard coldness as itself hot or cold is a category mistake. Likewise, if we can say that nonexistence exists, we must take “exists” to mean something like “can be talked about” and not “located in spatiotemporal reality.”

One solution to the problem would be to accept as a criterion that one or both of the two contrapositive concepts, existence and non-existence, must be of the sort that can be subsumed other concepts. This would allow us to understand the relationship in genus/species terms. Presumably “exists” is a concept that can apply to the abstract entity non-existence; this solves the paradox, according to the Stranger. Likewise “does not exist” can apply to existence; the assertion that it does would be false for Plato and true for some nominalists. However, since “is hot” is not, according to any non-metaphorical understanding of the term familiar to me, a concept that can apply to other concepts, we can make no sense of saying “Coldness is hot.” Nevertheless, an appropriate candidate for a hot-like concept might correspond to a jerry rigged genus that includes both hot things and concepts-of-temperature-sensation (One of the benefits of the idea of the extensional specification of set membership is that it allows us to think of entirely artificial or at the very least hitherto undiscovered sets like this one). Name this concept hot-plus and it would make sense to say that cold (i.e. the concept cold, coldness, cold itself, the second order entity cold) is hot-plus.

The Stranger’s reference to Parmenides by name and his detailed answer to the paradox begins at 258E. In the two speeches that straddle 258 E to 259 B he clearly means something by the terms ἐναντίος and ἕτερος that differs from the meaning I have attributed to those terms. The meaning of ἐναντίος in these speeches is complementarity not contrapositivity. It is only by saying that non-existence is not a complement of existence that he can claim that non-existing things actually exist. I suppose what he means is that just as something cold can also be swift since cold and swift are also not complements, an existing thing can also not exist in case those qualifications are also not complementary. By ἕτερος he seems to mean “something else” in the sense that that coldness is not the contrapositive of speed, just something else, i.e. just different. This meaning of ἕτερος comes down to “is not identical.” Being is ἕτερος from the other Forms, i.e. none of the five Forms is reducible to any of the others. None are identical.  It is possible that this is what Plato means by ἐναντίος and ἕτερος throughout the dialogue, and it is only ambiguity that prevents us from either affirming or denying that that is the case. I can only point out that, if this is what Plato means, then he has not really solved the Parmenidean paradox, because the paradox addresses the complement of being or all entities, whatever you choose to call it. If non-being is not the complement of being, then the paradox reappears for whatever the complement of being turns out to be. And that is precisely Parmenides’ point. You can call the complement to being whatever you want; as long as there is a complement to being, there is paradox. Understood this way, Plato may have established that the Form non-being could exist since it participates in the Form being, but he has not established that things which fall under the complement of the γένος being exist and that we can without contradiction assert that they exist. In order to do that he needs the other distinction between “exists” as a predicate and the copula. If we grant the latter, then we can move on from the paradox and assume that ἐναντίος or ἕτερος need not mean either “contrapositive” or “complementary.” Either term can simply mean “other” or “different from,” “not identical with.” And this latter meaning is sufficient to establish how it is conceivable that the Forms can be subsumed under each other. Nevertheless it is clear that being and non-being (understood as either nothingness or non-existence but not as a complement of being), just like hot and cold and fast and slow, are indeed opposites in a special way that I have been calling contrapositivity. And in The Sophist, just as in the Lysis, Plato fails to account for this special sort of relation. Moreover in the Stranger’s speeches in 258 D to 259 B he excludes from consideration complementarity as well. He makes clear that he is only concerned with otherness or difference and not complementarity. He also implies that what we thought was complementarity is in fact just otherness (non-identity). (Frede (p. 409) proposes an unexceptionable interpretation of otherness or “difference” in terms of class complementarity. The problem is that this unexceptionable interpretation is not Plato’s; it smacks of Aristotle.) This thoroughly unsatisfying argument fails to account for the fact that there are indeed complements, that, for example, the class of things that are is a complement to the class of things that are not, or that the class of cold things is, assuming a properly chosen universe, a complement to the class of not-cold things. Furthermore, it is kind of a shell game, as I observed above. Changing definitions on Parmenides does not answer his paradoxical argument. It merely changes the topic.

The speech at 260 D-E compounds the error. Plato once again asserts that he proved that non-existing things can exist (on the face of things, a contradiction), when what he really proved is that non-existence can exist in that the Form non-existence can partake of the Form existence even if what falls under non-existence cannot partake of existence. However, proving the latter does not answer the sophists claim that there are no falsehoods. In order to do that, the Stranger would have had to have shown that there is a genuine complement to the class of existing things (which he believes he had denied), and that the reference of falsehoods is to things that fall under that complement. Moreover, if the sophists’ self-defense is stated in terms of the truth and falsehood of statements and not in terms of the existence or non-existence of what those statements refer to, then the distinction between the copula and the predicative assertion of existence is not enough. For the sophist could claim that the copular “is” in a purportedly false statement affirms (its own) truth and so is presupposed in the assertion that it is false. Something must be added to the copula/predicate distinction. That something is the notion of higher order statements or so-called metalinguistic statements about an object language. Using that apparatus it becomes clear that the affirmation that a statement is false does not assume the truth of the statement since the object language statement is mentioned and not used. Likewise negation can be understood as an operator in the formal logical sense such as modifies the meaning of the copula. In this case also the truth of what falls within the scope of is-not is not assumed. The negated statement is a kind of object language translation of the metalinguistic assertion that a statement is false. In addition, if you pair this passage with the Stranger’s later proof (263 D ff.) that falsehoods exist in the mind, then you arrive at the uncomfortable conclusion that false statements both exist and don’t exist. The resulting violation of the law of Non-Contradiction (not the Law of Excluded Middle) doesn’t seem to disturb the Stranger or Plato.

One could argue, as some neo-Platonists did, that the Five Forms Theory adumbrated in The Sophist is a rejection of the highest genus problem, to use Aristotle’s formulation, and thus simply a rejection of the highest genus paradox in that Parmenides’ stricture could be understood in those terms. But Plato doesn’t argue for the acceptability of such a rejection. He merely shows how Forms can partake of other Forms that appear to be complementary partly by arguing that a Form need not partake of itself, and partly be asserting that there is no such thing as complementarity, only otherness or difference. The first argument is acceptable if you are willing to accept an ontology of Forms. The second is like arguing that objects can be both black and white because black and white are not complementary, just different.

Cornford’s interpretation of this notoriously difficult passage (pp. 289 ff.) emphasizes the copula/predicate distinction to argue that it accounts for true negative statements (Something that is-not-F is not something that is-not). Non-existing things (τὸ μὴ ὄν) do not belong to the set of existing things simply because the word ὄν appears in the phrase that designates them. They belong to the complement of the set of existing things. However, the actual text of The Sophist indicates that Plato surmised that the opposites being and non-being were not complements. And, as I just pointed out, this defangs his answer to Parmenides. Plato never addresses the problem of what to say about the genus, so to speak, that contains both existing and non-existing things, or alternatively the complement of existing things (a challenge Hegel and Heidegger, with truly romantic disdain for the dangers such a project might pose, would take up). Indeed Cornford misses the fact that Plato’s findings can be extended to the Form (or “predicate”) existence or being itself. He did not seem to recognize that the Stranger is at least partially concerned with the existence of the Form non-being. He really should have noticed this because he is careful to point out (p. 292) that τὸ ὄν is ambiguous, meaning alternatively: the entity, entities, being (the mass term) or existence. Moreover, he completely misinterprets 257 B when he takes τὸ μὴ ὄν to refer to particulars and asserts that the negation in this phrase does not negate the existence of a particular but rather asserts its difference or otherness from some other particular.  That is to say, τὸ μὴ ὄν denies that the particular has a property that another particular does have. Cornford’s interpretation would lead to the conclusion that Plato does not deal at all with existence at all but rather shifts the topic from existence (which was in fact Parmenides’ concern) to difference and predication or what Meinong would call Sosein. Plato’s actual point was that the Forms being and non-being were not true complements but simply different from each other, not that whatever partakes of these Forms may or may not be distinct. This point is just as ineffective against Parmenides as the conclusion of Cornford’s interpretation, but it says something quite different. And it has the further advantage of not putting out of play altogether any talk of the existence or non-existence of particulars (although in his general assimilation of non-being to otherness, Plato may well have subliminally agreed with Cornford’s interpretation; the non-being of particulars is just not an issue in the discussion of the five Forms that concludes The Sophist).

In any event, those theologians who thought they found a quick answer to the problem of evil in terms of the logical necessity of evil existing if good exists would have done well to read this passage in Plato. Plotinus of all people saw through the logical error in the theologians’ argument (Enn. 1.7.3 and 1.8 passim, although there is quite a bit of ambiguity in I.8.5). Ironically the very same Plotinus managed to turn the valid arguments such as appeared in The Sophist into the worst metaphysical drivel and in Enn. I.8.3 and in I.8.7, I.8.11  and I.8.15 ll. 4-6 ignored his previous insight and proactively sided with what some theologians say about “the problem of evil.” Those theologians were perhaps ill-served by Aristotle’s logical distinctions in the Prior Analytics where he confuses matters not so much by misunderstanding or misstating the point as by appropriating and then garbling Plato’s terminology – perhaps a case of poor transcription as with many other passages in that lecture series. Malcolm (pp. 136-137) claims to resolve the paradox of τὸ μὴ ὄν by distinguishing between that phrase’s function as naming an abstract entity and some broader sense (which presumably includes and may be coextensive with first order predication and identity), signified by εἶναι πως, in which it ranges over particulars that are not in one way or another. He does not recognize that the paradox may not obtain even if τὸ μὴ ὄν is interpreted as naming an abstract entity, non-existence, in the same way that it is not paradoxical to say that the concept color is not a color. He also fails to recognize that the paradox reappears whenever some broader sense of not being specifies a class complementary to some class specified as being in some broader sense.

Ackrill (in Plato I, ed. Vlastos, pp. 201-209) responds to Cornford on a different basis, but his interpretation leads to more disastrous results. Ackrill takes Cornford as denying that a statement such as “Theaeteus sits” (This example, of course, does not appear in The Sophist.) involves not so much an interweaving (συμπλοκή) of Forms as the participation of an individual in a single Form (ostensibly the Form Sitting). Ackrill argues that it does so involve an interweaving of Forms since it rules out the “concept” not-sitting (Note the egregious Aristotelianizing of Plato’s ontology). His conclusion implies that the negation of a statement involves participation in some sort of negative Form such as Not-Sitting. I don’t know where Ackrill got the idea of negative Forms but it certainly wasn’t from Plato. The only possible candidate for a negative Form Plato mentions is non-being and he does so because of his concern with the Parmenidean paradox. But there is no evidence that Plato would countenance the idea of a negative Form accompanying every concept and there is good reason to doubt whether he conceived even non-being as a Form. Even worse, Ackrill treats sitting as a Form whereas The Sophist strictly limits to Forms to the five under consideration. Ackrill ends up with an indefinitely large number of Forms each duplicated by its own negative shadow. If a statement such as “Theaeteus sits” can indeed be imported into the foreign context of The Sophist, then there is a simpler way of explaining how it involves a συμπλοκή εἰδῶν: The proper name “Theaetetus” designates an individual who participates in several Forms. Hence to assert that Theaetetus sits is to assert that that sitting (assuming sitting is a Form) interweaves with the other Forms in which Theaetus participates.

We cannot escape the fact that 258 E – 259 A (Cf. also 238 C) explicitly denies that there is such a thing as the contrapositive (ἐναντίος) of τὸ ὄν, presumably non-existence as opposed to existence, or that there is a class complementary to the class of existing things (Frede also makes this point on pp. 409-410 “The Sophist on False Statements” Kraut ed.). In fact these passages seem to deny that we can say anything at all about either non-existence or non-existing things. The curious consequence would be that everything (or everything we can talk about) exists. And, as we also saw above, the only thing that Plato actually proved to be logically consistent is that the Form τὸ μὴ ὄν may exist, i.e. participate in the distinct Form τὸ ὄν (Cf. Russell’s reference to The Sophist, p. 73 (The Principles of Mathematics)). He did not show to be logically consistent the assertion that those things which participate in τὸ μὴ ὄν also participate in τὸ ὄν. To conclude that anything participates in both τὸ ὄν and τὸ μὴ ὄν, understood as complementary, would be logically contradictory. To conclude that nothing in fact participates in τὸ μὴ ὄν would be to conclude that everything exists, which is precisely what the sophists would say and which the Stranger spent a lot of breath denying. To say, as the Stranger does, that things we assumed didn’t exist are really just existing things that are distinct from other existing things does not solve this dilemma. In fact it reinforces the sophists’ claim. If everything exists and all things are just distinct from each other, then all statements are true and the sophist is justified in his claim to be able to successfully defend both sides of an argument.

Wiggins (pp. 286-287 ftn.12) swallows the Parmenidean bait and asserts with what one might call overweening self-confidence that nothing does not exist and cannot be said to exist on pain of self-contradiction. His argument for this view turns into a veritable cornucopia of schoolboy errors. In the first place, he fails to distinguish adequately between the term “nothing” as syncategorematic (i.e. combination of a negation sign and a quantifier) and the term “nothing” as designating an object (a Form or something like a Form in Plato’s case). In its syncategorematic usage talk about the existence of a purported referent of “nothing” is meaningless; it is grammatically excluded. You cannot say “There is nothing such that F does not exist” in any known language (although you can say “The phrase ‘There is nothing such that F’ does not exist” although you would be contradicting yourself and so uttering a falsehood). Wiggins seems to acknowledge this but turns it into a tautology and calls it “boring.” Plato’s conclusion is that the second order referring term “nothingness” is meaningful and can exist (i.e. can participate in the Form being) just as the related term “non-existence” is meaningful and can exist. Wiggins’ arguments involve one logical error after another. He fails to notice that a second order object nothing need not fall under itself and so does not not exist by definition (n.b. Plato’s un-Fregean belief that exists and its counterpart could function as both first and second order concepts, just like all the other Forms). He confuses identity with predication and further confuses first and second order predication in arguing that if nothing were to exist then something must be identical with nothing. And, in asserting the validity of the assumption that nothing is equal to nothing ((x) x≠*) where “*” denotes the object nothing, Wiggins makes an assertion that evidently falls under Russell’s paradox, would be excluded by Zermelo’s Axiom of Separation and so would be invalid in ZFS. Wiggins appears to dismiss this and sneers at the “model theorists” who invoke it. He fails to recognize that Plato solves the Parmenidean paradox by a move very similar to those embodied in Frege’s distinctions. Wiggins’ basic problem lies in confusing existence (or being) with the idea of all or everything (what set theoreticians call the unqualified universal set) and so attributes the genuine paradoxicality of unqualified universality to existence or to the class of existing things. It is amusing to note that Wiggins’ proposal that we cross out the term “nothing” has a precedent in the history of philosophy. Who else talked about crossing out? Let me see. Heidegger! Actually Heidegger wanted to cross out the term “being” but the two moves really amount to the same thing and both Wiggins and Heidegger base their inscriptions on the same faulty assimilation of being to everything.

Let me finish the discussion of The Sophist with a few random observations. It is worth pointing out that much of what I have to say about Plato is influenced not only by Aristotle but also by Frege and his distinction between concepts and objects and Meinong and his distinction between first and second order objects (Cf. further Russell, Principles of Mathematics p. 64 ftn. 2 who cites De Morgan for various meanings of “is,” and also Levinson “Language, Plato and Logic” in Anton & Kustas eds. Essays in Ancient Greek Philosophy I, p. 281 who notices “a logic of class inclusion” in The Sophist) . Let me refer the interested reader especially to Paragraph 94 of Die Grundlagen der Arithmetik (p. 123) where Frege observes that a concept can be valid even though nothing falls under it. If we add Frege’s definition of objectivity as (roughly) being the same for everyone, establish a sort of biconditional relation between objectivity and existence and note further that Frege seems to equate existing and falling under a concept (126), then we can conclude that the concept (Form) of non-existence exists or may exist even if no object falls under it. (Ackrill in Plato I, Vlastos ed. Pp. 213-214 and Wiggins in Plato I pp. 272-275 noticed the relevance of Frege to The Sophist. Of course, Frege explicitly excluded the Platonic idea that a second order object nothingness could correspond to the phrase “no thing” understood as a sentential operator. Wiggins (pp. 285-286) calls the relevance an “anachronism.” I prefer to think that much the same issue is discussed in both Plato and Frege. Wiggins also fails to notice that Frege’s model is actually much closer to Aristotle than Plato.)

Meinong’s terminology is a bit more tortured but it permits of an analogous conclusion. Findlay (pp. 109-110) suggests that Meinong found an intriguing solution to what is essentially the Parmenidean paradox of being. I won’t go into detail about this solution here but it essentially comes down to saying that references to non-existent objects need to be understood as occurring in what we would today call opaque contexts or indirect discourse.

Vlastos makes distinctions which are largely similar to my own in “An Ambiguity in The Sophist” (pp. 270 ff.). He does so using his own idiosyncratic terminology; barring a significant difference of meaning I see no reason to abandon the Frege-Meinong vocabulary which has by now become canonical. Furthermore, while Vlastos does clear up the ambiguities surrounding first order predication and identity especially as they appear in a particularly twisted way in the case of the five Forms and their interactions and self-participations, he misses an important problem which I discussed above in the Charlus and tenth planet examples. This is Plato’s sleight of hand in dropping from consideration any meaning of being (τὸ ὄν or εἶναι) as existence. Instead of treating non-being as simply non-existence, the complete (one might say ἁπλῶς) disappearance from the universe of some entity, he changes the meaning to difference. The upshot is that what “is not” in Plato’s solution is understood solely as “is different from” or “is other.” One unsatisfactory consequence is that both being and identity are turned into opposites (whether complements or contrapositives is not clear) of difference. This has the effect of losing the sense in which the Form Motion is contrapositive to the Form Rest; in Plato’s solution they are just different from each other. A more telling problem with regard to the being/non-being opposition is that the sense of being as sheer existence is dropped altogether (In fact this may have been embraced by both Plato and Aristotle; their ontologies very likely imply that coming to be and passing away are a mere change of shape on the part of something underlying; theirs is a non-additive ontology where nothing can simply appear or disappear without involving a modification in the ὑποκείιμενον and nothing can be added or taken away from the amount of ὑποκείιμενον in the universe. This is particularly unsatisfactory if we treat the underlying as physical matter; the situation with a more inclusive ontology is less clear. Incidentally, if anyone needs a proof that existence and identity are not the same thing, consider the fact that it is a contradiction to say that a particular is not self-identical, but it is not a contradiction to say that a particular doesn’t exist. Not to put too fine a point on things but the non-contradictoriness of the assertion of non-existence is brought out by the Charlus example. It is perfectly meaningful to say that Charlus doesn’t exist. As regards the Plato/Aristotle “solution,” the assertion of his non-existence does not mean that some object (or some underlying bit of matter) in the universe could have been Charlus but isn’t. Likewise the contradictoriness of the assertion that something is not self-identical is brought out by the fact that, by way of the substitution of identicals, an indefinitely large number of other contradictory statements would thereby be entailed. But there is another way of construing existence that doesn’t solve these problems so much as show that they arise because a definition of existence was sought in the wrong place. By way of teaser this other way consists in understanding “exists” as an umbrella term that collects together a number of distinct predicates (i.e. concepts) but which cannot be used in place of any of them without further explanation. However, an explanation of this idea will have to wait for another essay.

Anyone ever notice the strange structure of Plato’s five metaphysical forms: being, identity, otherness, motion and rest? The last four come in oppositional pairs but the first one does not. Yet there is a great deal of talk about an opposite to being, namely non-being. Motion and rest (depending on how we understand them) are contrapositives. Identity has to be reducible to self-identity while otherness is always relational. Being and identity (and otherness) are applicable to everything (i.e. ZFS everything to be absolutely philosophically correct). Owens calls them “ubiquitous predicates.”  Being and identity court the Parmenidean paradox only if they are identified with (are Fregean characteristics of) the unqualified universal class. Otherwise there is no paradox in talking about non-being or the non-self-identical (although there may be other problems in describing objects that are not self-identical). Even if we consider identity to be a ubiquitous predicate it remains a non-paradoxical opposite of sorts to otherness because the two are not complementary (Only the complement of a concept of the universal class invites paradox). One indication of this is that identity can be construed as a one place predicate and otherness is a two place predicate. Universality does not seem to apply to motion/rest unless you define rest as a logical complement to motion. The fact that Plato elevates these concepts to the logical generality of being and identity indicates that he did not have a purely physicalist, one might say Milesian, understanding of them. In place of motion and rest we are better advised to use the terms change and stasis. This broader understanding is confirmed by Aristotle’s treatment of the same terms. It is possible though hardly explicit that by speaking of an opposite to change Plato is challenging the Heraclitean doctrine of universal change. He makes way for a doctrine that in certain respects there is lack of change and certain particulars can be characterized as not changing. An implicit point might be that an understanding of change assumes an understanding of stasis and the existence of changing things can be expressed in language only by way of assuming (at least the potential)  existence of unchanging things.

Heidegger (Einführung in die Metaphysik p.85) inserted a parenthetical note that appears to be at odds with the conclusion I find to be implicit in The Sophist. He says that nothingness is not an entity although in a way it may belong to being. This is, of course, not Plato’s own conclusion. In addition, the Stranger resolved the Parmenidean paradox by showing that there is no contradiction in nothingness being an entity, i.e. a second order entity like the Forms, while those items that would fancifully fall under the concept of nothingness or partake of nothingness are not entities.

Deleuze Différence et Répétition pp. 88-89 (Cf. also p. 143) makes some comments on the dialectic of non-being in Plato’s Sophist which clearly state the issue but, other than that, in my opinion, they don’t add a whole lot. Moreover, if I understand him correctly, he has the meanings of ἐναντίος and ἕτερος in Plato exactly backwards. Cf. also p. 152 where the eternal return is described as excluding the negative apparently understood as complementarity.

In another passage Deleuze (Ibid. pp. 261-262) seems to find an alternative to Boolean complementarity and its propositional expression in negation - an alternative which is different from contrapositivity. He calls this alternative "being problematic" (l'être du problématique). Problematicness, so to speak, comes to the fore when origins as expressed by affirmative statements of fact are challenged, an occurrence Deleuze calls a universel effondement.

Problematicness is distorted, according to Deleuze, and misconstrued when statements are regarded as hypothetical, which seems to imply a kind of weakened version of problematicness. Hypothetical-ness seems to refer to the premises of a syllogism, but Deleuze doesn't state that explicitly. Something like non-contradiction and/or excluded middle arises from hypotheticalness, and so, Deleuze claims, the shadow of negation remains. Presumably negation would disappear if hypotheticalness were replaced by problematicalness. For example, says Deleuze, there are no hypotheses in nature, only problems (If you find this confusing, you're not alone). From there he concludes that it doesn't matter whether negation is a logical limitation or a real opposition, presumably class complementarity (If you find this really confusing, you're not alone).

Nothing in Deleuze's analysis so far has anything to do with the contrapositivity in my own discussion. However, Deleuze goes on to cite concepts which may be construed as contrapositive: the One and the many, order and disorder, being and nothingness. He calls these "les grandes notions négatives." In other words, they could, perhaps falsely, be interpreted as purely complementary, presumably because they do no more than specify complementary classes. He adds that it is fruitless to regard these opppositions as limits of a degradation (This may be a reference to Plato, although I am not sure why he calls the scale between opposites a degradation). It is equally fruitless to regard them as thesis and antithesis. The latter language is Hegelian, but it could be construed as expressing contrapositivity. Deleuze asserts that these interpretations do not achieve their goal (which seems to be to critique negation). That is because they both assume the primacy of the simple concept as presumably non-hypothetical.

Deleuze shares the goal of critiquing negation, but, he asserts, that critique has to be performed by way of a kind of external, non-negational thing. He calls that thing the ideational (ideel), differential and problematic element. For example variety can "denounce" the order/ disorder pair and something Deleuze calls ?-etre can denounce the being/non-being pair. The goal of the denunciation (disturbingly Marxist term that) would be a more profound link between problematicalness and difference. The differential elements of the Idea do not involve any kind of negation or negativity at all.

If you think I'm somewhat skeptical about all this, you're right. However, I do concede that Deleuze is circling around what may be some important social and ethical issues. The problem lies in his attempt to state these issues in the framework of pure logic and formal ontology. The garbled result just begs for derision. I guess the source for this cross fertilization comes from Hegel who used the same sort of peritropical reasoning in discussing logic cum ontology as well as ethics and politics. Presumably Deleuze wants to make room for self-contradictory presumably revolutionary behavior and Hegel believed that, if you could show someone his behavior was self-contradictory, he would change his mind and behave differently. Kierkegaard and, I guess, Marx were aiming at the same thing. But attacking formal logic in order to get at bourgeois rationality may be a fool's errand.

Having said all this, my goal here is to clear up a possible error in believing that this passage in Différence et répétition is in any way relevant to what I am saying about contrapositivity.

There may have been a third dialogue either lost or unwritten apparently titled The Philosopher that would have pursued further the Eleatic issues raised in Parmenides and The Sophist. I for one view The Sophist as not open-ended (unlike the Parmenides). It reaches a definite conclusion in the refutation of supposed sophist relativism and the defense of the Theory of Forms against Eleatic paradoxes.  As it stands, further development in this respect was left to Aristotle and his nominalist approach to Platonic Forms.

In Anal. Post. II vii 92b 14 Aristotle clearly the states that existence is not a genus, or, more accurately, when we talk about existing entities - τὸ ὄν – we are not talking about a genus or generic collection or class. This could be regarded as his own solution to the Parmenidean paradox which depends on τὸ ὄν meaning the collection of all things. However, things are not quite that simple. In Soph. Elen. X 170b 22-25 he responds to the Parmenidean doctrine to the effect that all things are one on the basis of linguistic ambiguity. Aristotle expressly assimilates πάντα and τὸ ὄν (thereby possibly endorsing or at least not engaging with the other Parmenidean doctrine that only things that can be thought qualify as entities). Furthermore τὸ ὄν could mean either “being” or “the entity” in Aristotle as well and in the latter sense (and probably in the first sense also depending on whether you take the gerund as signifying some sort of mass like the mass term “water” or signifying a collection of individuals as in “the working poor”) is grammatically distinct from ὄντα. In Phys.I iii 187a 1-11, part of a notoriously vague passage, Aristotle either himself confuses or accuses Plato of confusing non-existence and the (non-existing) things that are qualified by non-existence. But this is unclear because Aristotle is mainly concerned with refuting the Parmenidean doctrine that the all is one and in establishing that the term “being” encompasses a range of different meanings.

With these remarks we can segue to our next topic.

Opposition in Aristotle

Despite the tangible superiority, at least in my opinion, of Aristotle’s form/matter distinction over the Platonic theory of ideas and despite the fact that the new science of Aristotelian logic streamlined and made consistent whatever was valid in Plato’s views about how particulars could be qualified by general properties, nevertheless Aristotle was often much less disciplined than Plato in his use and understanding of certain crucial philosophical terms of art. His kaleidoscope of meanings for the various inflections of εἶναι is just the most notorious example. The same confusion reigns for the group of terms he uses to discuss the idea of opposition.


For our purposes the critical confusion lies in Aristotle’s use of the term ἐναντία. He proposes two distinct and unrelated theories about this notion. The first is what might be called the logical or syllogistic theory of contraposition which is set out in the logical treatises. The second is what might be called the conceptual theory of contrapositives (inclusive of contrapositivity of particulars) which is discussed throughout his work. Of course, concepts and predicates are tied at the hip to syllogistic deduction, but there is a gulf between the two sorts of opposition in Aristotle which is well nigh unbridgeable. In any event, it does not allow, given the conceptual apparatus at hand, the reduction or translation of one to the other. Aristotle’s views on opposition are woven throughout his writings, so, in order to make some sense of his theory, we need to engage in a mighty and detailed work of disentangling.

The logical theory of contraposition is incorporated in what we still call the Square of Oppositions and most appositely in the position occupied by ἐναντία in the Square. De Int. VII is the first introduction of the Square of Oppositions - the notion of opposition as applied solely to forms of propositions (ἀποφάνσεις) and not to concepts or particulars. Propositional opposition in general or ἀντικεῖσθαι is divided into ἐναντίως and ἀντιφατικῶς. The former is traditionally translated as “contrary” and the latter as “contradictory.” So the traditional translation as “contrary” is of the the same Greek word ἐναντία which I have throughout been translating as “contrapositive.” Clearly formal or logical ἐναντίαι are not the same thing as conceptual and particular ἐναντία such as are discussed in Plato and elsewhere in Aristotle. A few moments with modern logic (and the corresponding Venn diagrams) can help sort out how they are related.  In standard notion contraries (i.e. ἐναντία) can be represented by the pair (x)(Fx)/(x) ¬(Fx) and contradictories can be represented by (x)(Fx)/ ¬(x)(Fx).  That is to say, the difference lies in the scope of the negation operator. Now take the logical forms of syllogistic contrary and contradictory propositions and relate them to my elucidation of conceptual complements and contrapositives above and it should be clear that there is not a simple correspondence.

Of course, propositional contraposition in the modern sense (“p  q” and “¬q  ¬p” are propositionally contrapositive) is something different again. The pair of Theorems 45 and 46 in Chapters 4 and 5 of Suppes pp. 107 and 152, for example, are propositionally contrapositive. Nevertheless Aristotle’s use of ἐναντίως is reflected in the predicate calculus. His understanding of logical opposition as a feature of propositions is further developed in De Int. (X 20a 15 ff.) and Anal. Pr. (II viii 59b 6 ff., II xi 61b 33-62a 19 and II xv 63b 22-31). The propositional schemata “All x’s are F” and “No x is F” are again called ἐναντίοι. The difference from the contrapositivity in the Lysis is obvious. For example, “a is evil” and “a is good” could be considered propositions related to contrapositive concepts, but they are unrelated to the sorts of propositions that would fit into the Square of Oppositions, e.g. “Everything is good” and “Nothing is good.” As far as I can tell there is no equivalent to logical contrapositivity in Plato.

Aristotle’s conceptual theory of ἐναντία is distinct from the syllogistic theory. It can best be understood in the present context by incorporating it into my complement/contrapositive distinction. In this framework Aristotle’s specification of ἐναντία, which I will allow myself to call contrapositives, comes in the form of a theory that we already encountered in parts of Plato. Conceptual contrapositivity is introduced by way of a general discrimination of different types of opposition.

The general theory of opposition can be found in Categories X. The generic term is ἀντίκειμαι which is probably best rendered as indicating opposition in general. There are four types of opposition in general. The first is πρός τι which Aristotle understands to involve two place or relational predicates of comparison. If we say Socrates is better than Thrasymachus, we need to fill in this incomplete expression by expressing a πρός τι, e.g. Socrates is morally better than Thrasymachus but not a better athlete. I don’t find any special similarity between this type and the three other types of opposition. It finds its place in the context of this essay as what I referred to above as incomplete one place predicates or ellipses. It is opposition with respect to…. As noted above, recognizing that certain predicates are incomplete would help resolve some of the conundra in the Lysis (But not entirely. I.A. Richards throws out an interesting observation that comparison by respect is not the end of the story. It is a complex matter, even at the semantic level. Cf. “Toward a Theory of Translating” p. 250 in Wright ed. Studies in Chinese Thought). But, in addition, Aristotle cites two different kinds of πρός τι without recognizing how different they really are. The first is two place predication in general for which Aristotle gives the somewhat unfortunate example of knowledge and objects of knowledge. The other kind of πρός τι is the case of two extremes on a scale of values. His example is the opposition hot/cold. It is significant that Aristotle classifies this kind of opposition under πρός τι in the Categories because in the Timaeus Plato seemed to consider it as exemplifying ἐναντία in such a way that analysis by way of extremes on a graduated scale resolved paradoxes from dialogues such as the Parmenides and The Sophist. The same assimilation of scalar extremes to πρός τι seems to pervade Aristotle’s Physics and provides a way for Aristotle to assimilate conceptual contrapositivity to his genus/species model. Indeed this second kind of πρός τι opposition is that one of the two relational types that most directly reflects Aristotle’s conceptual contrapositivity. On the other hand, πρός τι opposition where the blank of “with respect to…” is filled in with the opposite pole of a contrapositive pair, is not at all the same as πρός τι opposition where the blank is filled in with a qualification that limits the domain of opposition.  Hot is not πρός τι cold in the same way that wine-loving is πρός τι with respect to things to drink.

The next type of opposition is ἕξις/στέρησις which corresponds generally to complementarity though Aristotle’s discussion in this passage does not show a clear recognition that it is necessary to specify a domain in order to deal with true Boolean complementarity. Rather, his idea that positives and privatives must be limited to certain domains (περὶ ταὐτόν τι, literally “the same kind of things”) which must be characterized one way or the other (πέφυκεν ἔχειν) captures a similar idea without the language of classes, or, to use more Aristotelian terms, the language of genus and species. To use terms from different philosophical periods we might call it the formal mode or de dicto analogue to complementarity.  A negated proposition corresponds roughly to a complementary class or species. In De Int. VI Aristotle, succumbing to a bout of πολλαχῶς, calls negation ἀντικεῖσθαι. Aristotle’s intuitions about negation and the scope of the negation operator (as in e.g. De Int. XII) are largely valid. They are just a bit garbled from a modern perspective.

Anal. Post. I x 77a 10-22 explicitly states the relation between negation and complementarity. One might say that the former is de dicto although the latter is not quite de re. Rather, it applies to a Boolean or Venn relation of classes or a characteristic of sets. Also Topics II iv 111a 14 ff. clearly uses ἐναντία to mean complements since his example involves negation (ὀρθῶς and μὴ ὀρθῶς). As an indication of how tenuous a grasp we have of the distinction, Tredennick renders this as the contrapositive opposition of correctness and error. Obviously a stone can be characterized by way of the complement or negation of being correct (It is not the case that this stone is correct about….) and not be characterized by its contrapositive (This stone is in error.)

Finally there are ἐναντία properly speaking. This type is also conceptual contrapositivity. Aristotle’s example of good/evil seems to indicate that, at least sometimes, he recognized the distinctiveness of what I have been describing as contrapositivity. It is interesting to note that, in order to discuss higher order opposition between the first order oppositions such as obtain between individuals and classes of individuals, Aristotle was obliged to introduce another term, ἀντίθεσις.

In Cat. X 11b ll. 35 ff. Aristotle makes a distinction between two kinds of ἐναντία, those like black/white which have intermediates and those like healthy/diseased which do not. His analysis of the latter kind introduces an idea which looks similar to that of the union of two complementaries being equal to their domain. Unfortunately the two ideas amount to the same thing. His idea (Cf. also X 12b 25 ff.) in this passage is that certain particulars and certain contrapositives are such that the former must be characterized by one of the two contrapositive terms. An organism must be (πέφυκεν) either healthy or diseased. Aristotle concludes that, as regards these kinds of ἐναντία there is no intermediate state or μέσον. Black or white alone, however, do not have to characterize an object and so there are intermediate shades of grayness between black and white (though presumably a qualifying object would in any case have to be colored; therefore even the scale of black/white/grey does not exhaustively characterize all objects). However, the obligatory inherence of opposites like healthy/diseased and odd/even in qualifying particulars is another way of stating that, if you limit your domain to the class of organisms, then all the members of the domain belong either to the sub-class of healthy or to the sub-class of diseased (unhealthy) things. Likewise, if you limit your domain to the natural numbers then all the members of the domain must be either odd or even. Apparently not everything Aristotle calls ἐναντία are in fact contrapositives in our sense. Some, as Aristotle views things, are just complementary. Yet his examples in this passage are indeed contrapositive; since they lack an intermediate gradation of values, both pairs, healthy/diseased and odd/even, are indeed contrapositive; the domain specification is purely extensional, i.e. artificial, and can be used to turn any pair of contrapositive concepts into complements. Likewise there is no way in English to limit the domain of good/evil in terms that are not simply redundant, i.e. extensional and conventional. Because he was a little vague on the notion of a domain Aristotle ignored the correct distinction.

The Topics builds on the theory of opposition outlined in the earlier books of the Organon. VIII xiii raises the issue of whether there is just one or many sciences of opposites. But it is not quite clear what Aristotle means. Does he mean that there is only one or that there are several actual scientific disciplines studying opposites in the sense that logic and medicine are distinct scientific disciplines? Or is he simply reiterating his point that there are different sorts of opposites? In any event his original distinction misses what I consider to be the crucial difference between complementarity and contrapositivity.

In Topics I xv Aristotle outlines a theory of semantic ambiguity that also involves a theory of semantic opposites. What he says clarifies some of the obscurities in the Prior Analytics. 105b 9 ff. discusses ἐναντίοι. He clearly means contrapositives in our sense. His point is that differences of contrapositive signals differences in meanings (εἴδει) on the part of the original term. He introduces examples of contrapositives that are worth mentioning, such as sharp/flat and sharp/dull or love (as an emotion)/hate and love (as a sexual act)/fill in the blank. He also explains ambiguity of meaning in terms of the incompleteness of the meaning of a word, an incompleteness that needs to be supplemented by specifying a respect (Cf. also II iii 110b 16 ff. which is basically a distinction of meaning by way of respects).  His example is clear sounds and clear colors, beautiful organisms and beautiful houses, different senses of “good” and sensual vs. intellectual pleasure. He also distinguishes ἐναντίοι from ἀντικειμένοι or complements (106b 13 ff.). He explains nested complements as Boolean style subclasses. For example, the complement of “To see” is “Not to see”; the complement of “To possess sight” is “Not to possess sight”; the complement of “To exercise sight” is “Not to exercise sight (presumably to keep your eyes closed).” Aristotle seems to think that each of the nested terms each has its own complement unrelated to the complements of the others. Thus to possess sight (to be able to see) and not to possess sight exhaustively constitute a universe of things that can see and things that can’t. Sleepers and people with their eyes closed are a member of one class of this universe, while stones and blind animals are members of the other. Exercising and not exercising sight constitute a universe where the line is drawn differently. Sleepers belong to the class of things that do not exercise sight. There is nothing wrong with this; you can make any valid distinctions you want. One thing that isn’t brought out by his clarification of the ambiguity of “to see” is that stones, for example, belong both to the class of things that don’t possess sight and to the class of things that don’t exercise sight unless the exercise/not exercise distinction is specifically limited to a universe of things that possess sight.  Finally the term ἕτερα or otherness (107a 18 ff.) is used to mean distinct genera of things that are not related by either complementarity or contrapositivity. Otherness is also distinguished from generic subsumption (ὑπ’ ἄλληλα). The latter is further divided into species and differentiation subsumption.

Topics V vi 135b 8 ff. returns to the general theory or science of opposition. Aristotle divides ἀντικειμένοι or opposites into several different types: (1) Apparent contrapositives (ἐναντίοι) (2) Opposites in a respect (πρός τι) (3) Complements (ἕξις and στέρησις) (4) Something called corresponding divisions. I call (1) a class of apparent contrapositives because, even though the examples Aristotle cites look like contrapositives, what he says about those examples does not fit all contrapositives. Assuming the Guelphs and the Ghibellines are contrapositive associations and assuming that papal political party and French political party are contrapositive properties, it does not follow that, if the Guelphs are a papal political party, then the Ghibellines must be a French political party. They could fall under the other contrapositive pair papal/imperial. Similarly Aristotle’s own example is subject to doubt despite the ambiguities of the terms themselves. Assuming advocated-by-Thrasymachus and advocated-by-Socrates are contrapositive properties, it does not follow that one must apply to evil if its contrapositive applies to good. What Aristotle says about (2), on the other hand, raises an interesting point. By distinguishing opposition of measure (double/half) from contrapositivity, Aristotle seems to leave room for rejecting Plato’s implied elimination (at least with respect to arithmetical and perhaps physical entities) of contrapositivity in the Timaeus by way of a standard of measure. This will become an important issue in the Metaphysics. Finally Aristotle’s treatment of true complements in this passage is in many respects genuinely Boolean. His point in 136a 36 ff. could be very succinctly put using the modern devices of a set theoretical universe or a domain of quantification.

Topics VI vii 146a 21-33 draws some of the consequences of the theory. On the one hand Aristotle shows how failure to specify a respect when qualifying something can lead to ambiguity and false conclusions. But his example mixes in the distinct issue of opposites. He also speaks of ἀντικείμενα as if they were the same thing as complements, although we know from other passages that negation specifies just one type of ἀντικείμενα. We would like to include contrapositives in our sense as another type, but Aristotle doesn’t do this in the logical treatises except by way of his adaptation of the theory of degrees from the Timaeus. The other type Aristotle in fact identifies reverts to the Square of Oppositions; it takes the form of “Everything is…” vs. “Nothing is…” Aristotle constantly uses contrapositives as examples of opposition but he does not make clear distinction of how they differ from complements.

Aristotle’s conceptual theory of ἐναντία actually appears before the deductive logical theory as embodied in the Square of Oppositions, at least if we take the logical treatises in their natural as opposed to their possibly chronological order. This understanding of ἐναντία first occurs in Categories  V 3b 25 ff. and 5b ff. The term has a clear sense of contrapositivity as I have defined it.

Categories VIII 10b ll. 122 ff. provides examples of ἐναντία κατὰ τὸ ποιόν. Aristotle mentions the important distinction between the quality itself and particulars which have that quality. The contrapositivity of the one implies the contrapositivity of the other. He also asserts, rightly, that some qualities specific to a given genus may have a contrapositive while others may not. Aristotle’s assertion very likely shows a recognition on his part often belied in other passages that contrapositivity, unlike complementarity, cannot be logically (or, to extend the point to the terms of modern philosophy, epistemologically) derived from a given concept. His color examples are not entirely apposite to this point, however, due to his lack of awareness of the color wheel.

Back in V 4a 10 ff. Aristotle uses examples of contrapositive quality pairs such as good/bad. There is much that is objectionable in this passage. Aristotle seems to be mistaken in his argument that only οὐσίαι can be qualified by ἐναντία. Ostensibly secondary οὐσίαι such as virtue can be both praised and dishonored. But even if his argument is limited to primary οὐσίαι (presumably physical particulars), which he nowhere concedes, counterexamples immediately suggest themselves. His own example is that a particular color, such as blue (a secondary οὐσία), cannot be characterized by the ἐναντία black and white. (Perhaps a black or a white person painted blue could be so characterized, but this challenge could be resolved by some sort of πρός τι.) However, any number of other ἐναντία suggest themselves. A particular shade of blue could, for example, be both appropriate and objectionable. The translators deal with the other obvious objection by picking up on later occurrences of ἅμα and adding the phrases “at the same time” or “simultaneously.”

Deleuze (Différence et répétition pp. 45 ff.) notes a conflict between logical and categorical opposition in the course of a discussion of what the mediaevals called transcendentals, but he doesn’t elaborate on the issues of the logical theory itself. Indeed Aristotle’s logical and conceptual doctrines of contrapositivity seem to deal with two different things. There is a relation between the two, but that does not mean that Aristotle doesn’t contradict himself by expounding the two doctrines separately. Are they in fact consistent or even compatible?

Passsages abound where ἐναντία clearly means contrapositive in my sense of the term and not complementarity. Categories VII xv ff. gives examples of ἐναντία which are in fact contrapositives. For example in l. 28 Aristotle cites the contrapositive pair master/slave and calls them ἀντιστρέφοντα. However, his use of πρός τι in this passage invites confusion. This qualifying phrase does not mean “in respect of…” as it does in other contaxts. Instead he means that two contrapositives are πρός τι to each other, viz. that they enter into a relation involving opposition. In other passages the issues that concern him are largely similar to those in the Timaeus, that is the type of contrapositivity at issue concerns extremes in the description and measurement of physical states. Categories VI 6a ll. 12-18 introduces this use of ἐναντία to signify two extremes which have a mean. Aristotle would reprise it especially in the Physics and De. An. II 11 424a. Note his interesting suggestion that we should use a spatial model, one might almost say metaphor, when we think about ἐναντία.

In Soph. Elen. XV 174b 1-8 Aristotle uses ἐναντίος in the clear sense of contrapositive. His somewhat devious advice involves obscuring the scope difference between actively disobeying (ἀπειθεῖν) and not obeying (perhaps because you didn’t hear or didn’t understand the command). The sophistical dilemma arises when the difference is neglected and the disputant assumes that the only alternative is active disobedience or the contrapositive of obedience. Another error is the false implication from the negation of one contrary as defined in the Square of Oppositions to the affirmation of its contrary. The same considerations apply to arguing about large and small. The term reappears in l. 37 but Aristotle’s meaning in that sentence is unclear.

In De Gen. et Corr. I 3 319a 31 Aristotle uses ἕτερος and ἐναντίος somewhat synonymously. But the examples throughout are clear contrapositives. Somewhat inconsistently from its use elsewhere, the term ἕτερος seems to be pressed into service as a way of identifying one member of a pair of contrapositives. It is important to remember that ἕτερος usually means at least two different things in Aristotle which he doesn’t clearly distinguish: specific difference and numerical difference. He may mean no more than numerical difference (of species) in this passage but that would make the appearance of ἐναντίος as an effective synonym surprising.

In De Gen et Corr. II 2 329b 19-21 Aristotle lists a series of haptic opposites: hot/cold, dry/moist, heavy/light, hard/soft, viscous/brittle, rough/smooth, coarse/fine. These are true contrapositives, not complements, and it is a matter of indifference to the physical theory he lays out whether they can apply to each other (Obviously they cannot) or whether they are complementary (They probably aren’t). In II 3 330a ff. Aristotle whittles these down to four. Their own ontological status is vague but they “attach” (ἠκολούθηκε) to “simple” (ἁπλοί) physical bodies in non-contrapositive pairs. This theory in summarized in 331a 14-16.

Physics VIII i 251a 29 ff. (Cf. also Met. X I 1052b l. 28.) uses ἐναντίας with a clear meaning of contrapositivity, i.e. opposite directions. In discussing motion and change (Phys. VIII vii 231b) Aristotle uses the important qualifier πως to argue that something can be ἐναντίος to two things simultaneously. His assertion could not apply to complements rigorously understood. It involves something like semantic contraposition.

Certain arithmetical notions invite a treatment that is somewhat akin to Aristotle’s conceptual doctrine of contrapositive opposition. In Topics IV iv 125a 25-33 (Cf. also VI ix 147a 29-32) the concepts of double and half designate inverse relations (In formal terms, if X and Y are sets and L   X x Y is a relation from X to Y then L-1 is the relation defined so that y L-1x if and only if xLy). Whether the predicates double and half are also contrapositives is, as the preceding discussion has taught us, largely a matter of choice. Aristotle’s point is that, given two relations inverse to each other, each relation will belong to a genus that is inverse to a genus including the other. Double and half belong to the genera multiple and fraction respectively. Aristotle does not mention that species could belong to any number of different genera, and so his methodological advice can only be effective if the appropriate genera are specified, a task that is not always straightforward.

In Topics I x Aristotle uses ἐναντίος in the Platonic sense to mean “contrapositive.” The equivalent to ἕτερος is ἀντίφασις (Note ἀντίφασις is about logic whereas ἕτερος in Plato mostly classifies things. Aristotle in this passage uses ἐναντίος in both ways.) His example is a slightly muddled since it actually contains two sets of contrapositives: friends/enemies and do good/do harm. The four possible combinations are (1) Do good to one’s friends. (2) Do harm to one’s friends. (3) Do good to one’s enemies. (4) Do harm to one’s enemies. Aristotle sets the muddle straight later on in II vii where he makes these distinctions and more. Note particularly 113a 14 ff. where he says the same “thing” can have more than one contrapositive. That should not be the case for a complement as I have defined complementarity.

In Topics VI ix Aristotle discusses what are actually three distinct phenomena under the rubrics of ἀντικειμένοι and ἐναντίοι. The first consists of the inverse relations already dealt with in Bk. IV. The second involves how ἐναντίοι are qualified and how the qualification or description of one of an oppositional pair has implications for the qualification of the other member of the pair. His example of the pair beneficial/harmful appears to be a true contrapositive and it appears further that Aristotle understands it to be contrapositive since he notes that something that is merely not beneficial will not necessarily be “productive of evil” or “destructive of good.” That Aristotle has a glimmering of the distinction between complementarity and contrapositivity is implied by his use of the word συμπλοκή but he doesn’t elaborate on the meaning of that term. Furthermore, the idea of descriptive consequences of ἐναντίοι could have implications for the issues raised in the Lysis – issues such as whether things qualified as opposites can like or dislike each other – but Aristotle doesn’t take his remarks further and does not allude to Plato. The third sort of opposition is στέρησις which obviously suggests complementarity. He doesn’t add much to what was said in Bk. IV but it is noteworthy that he uses a term μεταληφθῇ which as “substitution” would assume a great deal of importance for modern truth functional logic.

There are clear contrapositives in Aristotle’s moral theorizing as well. For example, the Nic. Eth. I xiii 16 and II iii 5-7 involves a use use of ἐναντίον with a clear sense of the contrapositivity of right and left. In fact the entirety of Bk. II viii involves a lengthy discussion of moral ἐναντία in the sense of contrapositives although his examples admit of degrees of opposition. In Topics VIII v 159b 30-34 Aristotle calls good and evil τἀναντία. Interestingly enough he does so in the course of a reference to Heraclitus.

On the other hand, a couple of instances of ἐναντία in the Politics seem to me to be just informal and conversational. For example, in III iv 1 1278b l. 15 τοὐναντίον just means “on the other hand” or “by way of contrast. Democracies and oligarchies are clearly not contrapositives in my sense. They are two instances of an array of possible political structures. Likewise the phrase “opposites create opposites” in VII 1307b 29-31 isn’t used in a rigorous theoretical sense even though the notion is borrowed from his physical theory. Aristotle does not elaborate on his observation since he clearly intended it as no more than a wise saw and an example of the way he thinks.

Contrapositives play a complex role in Aristotle’s physical and metaphysical doctrines. A discussion of Empedocles’ first principles in Met. I viii 989a l. 29 uses ἐναντία with a clear sense of the contrapositivity of hot and cold. It is interesting that qualitative contrapositivity in this passage interacts with Aristotle’s form/matter schema. There are  ἐναντίοι in De Gen. et Corr. I x 328a 32 which seem to be things that are capable of acting and those capable of being acted upon. In Phys. VIII iii 253b 20 ἐναντίος is used with a clear sense of contrapositivity (change from one contrapositive pole to another). Cf. also VIII viii 264b where ἐναντία in approximately the same sense are contrasted with ἀντικιμένη based apparently on a distinction between the end points of rectilinear and semi-circular segments.

Physics I iv resurrects the term ἐναντία in the context of a physical theory to mean opposing principles, though Aristotle in this passage uses the language of opposition specifically to describe the theories of the φυσικοί and other Presocratics. I v is as clear a statement as could be wished for of Aristotle’s attribution of a contrapositive based physical theory to the Presocratics. Met. I v 986a lists opposites and discusses opposition as a causal principle, the agency by which formed objects relate to matter or various material reductions; the opposites cause the things of the universe to be what they are.

However, even though in I vi Aristotle subscribes to physical theories involving ἐναντία, he goes on to argue that there can only be one pair and that the contrapositives involved need a third thing to “act upon (ποεῖν).” Clearly he intends to use his interpretation of physical principles, (both causal and ontologically reductive principles) the discovery of which constituted much of Presocratic method, to set up his more all-embracing ontology based on a Form/Matter schema. Notwithstanding Aristotle specifically mentions other contrapositive pairs in this chapter such as sweet/bitter, black/white, density/rarity and φιλία/νεἶκος. I vii 191a 6 raises the possibility of an ontology without opposition based simply on some unspecified underlying thing and the presence or absence in it of a single quality not defined via its opposition to something else. Nevertheless, in Met. XII x 1075a ll. 25 ff. Aristotle recognizes the importance of ἐναντία in Greek cosmological theories. He criticizes them to the effect that they don’t explain where ἐναντία come from. His own theory is that first principles like ὕλη and σοφία have no ἐναντία but are a kind of basis from which ἐναντία emerge (l. 34 and ll. 21-24).

Met. XIV i 1087a ff. summarizes his theory of the relation of ἐναντία to ἄρχαι. They both derive from something prior. This prior something is the ὑποκείμενον which Aristotle relates to the subject place in a sentence (1087b l. 1) - ὑπάρχειν (This parallel also involves the place of conceptual contrapositives in Aristotle’s logical theory on which more below). In De Gen. et Corr. II I 328b-329b esp 329a 24 ff. ἐναντίοι are identified as “accompaniments (μετ’) ” of a ὑποκείιμενον which is identified with ὕλη. Aristotle also makes the important point that the substratum or matter is inseparable (ἀχώριστον) from both the elements and from compounded objects.

Things get messy, however, when Aristotle tries to graft the theory of opposition not only onto his form/matter distinction but also onto the separate theory of polar opposition involving graded differences. In Met. X iv-v Aristotle proposes his theory of maximal contrariety which can be regarded as an alternative to conceptual contrapositivity. (Note on terminology: X v identifies ἀντίθεσις as the closest thing to contrapositivity in my sense and takes ἐναντία as roughly complements or extreme cases within a complement.)  He acknowledges in these sections that complementarity strictly speaking requires the specification of a genus exhaustively determined by the complementary species it subsumes. Examples of extra generic opposition would require the specification of a wider genus in order to be understood as complementary. (1056b ll. 1-3 is the closest approximation to our understanding that complements have to be defined in terms of a universe, broader class or wider set of which the complements are mutually exclusive subsets.) (Nic.Eth VII xiii 1 ff. applies the polar opposition model of ἐναντία as end points on a scale of moral values. With respect to the issues surrounding Mackie, maximal contrariety is no more logically implied by a concept than is contrapositivity.)

Polar opposition, however, requires considerable additional refinement of the basic genus/species model. For example, logically there need be no end point in either direction of a gradation although there may be physical limitations. In the same way neither maximally bad nor evil is logically implied by the concept of good, much less is the existence of bad things logically implied by the existence of good things. Incidentally Nietzsche (Menschliches II 2 67) tried to reduce all oppositions to gradations without polar extremes. His intention was probably to deny validity to hard and fast moral opposites like good and evil but he did not explain how opposites in general can be eliminated in favor of gradations.

The cornerstone of Aristotle’s assimilation is the theory that within a genus specified by its complements some pairs of opposing members are more opposed to each other than other pairs. Aristotle accounts for this by the notion of degrees of opposition between maximal contraries. The idea is that contrapositives are measurable extremes between which lies a scale of intermediate stages. However, some of his examples (e.g. odd and even) are not matters of degree at all and are better captured by the notion of conceptual or semantic contrapositivity as he appears to acknowledge in xv. Aristotle’s term μεταξύ is problematic in this regard. (Cf. also Met. V xxii 1023a l. 7.) Literally it indicates an intermediate, a middle status, and some translators have rendered it as “intermediate.” However, it is clear that in X iv Aristotle also takes it to mean “neither one nor the other of a pair of contrapositives.” 1055b 10 ff. confirms this. The distinction between not-equal and unequal expresses the difference between irrelevance and complementarity. Accordingly the actual meaning of μεταξύ in this passage is “irrelevant.” In that case, however, anything that is μεταξύ should not be included in the genus constituted by two complementary species. X v recognizes the difference between non-inclusion in a genus and being an intermediate in a scale. However, even there Aristotle fails to explain how something can fall under a genus and yet not be characterizable by either of the complementary species of the genus – he doesn’t make sense of the two different meanings he gives to μεταξύ. Take a genus G evenly divided into the species hot and not-hot. The hot species, according to Aristotle subsumes a sub-species of things that are maximally hot or Hot. The not-hot species subsumes a sub-species of things that are maximally not-hot or Cold. A continuum of sorts is supposed to stretch between the sub-species Hot and the sub-species Cold. Some things, like the number 3 belong neither to G nor to either of its species. They are irrelevant or μεταξύ in one of its senses. Others at some unspecified mid-point along the continuum do not, according to Aristotle, belong to either species, i.e. they are neither hot nor not-hot. These things are also μεταξύ. It is unclear whether Aristotle regards these as intra-generic to G or not. But in either case the way they are specified contradicts either the assumption that G is exhaustively specified by the species hot and not-hot or the assumption that G contains all things that can plausibly be hot or not-hot. This mid point is a weird entity that is neither hot nor not-hot and yet is less hot than members of the sub-species Hot and hotter than members of the sub-species Cold.

The upshot is that Aristotle tries to have his cake and eat it too by recognizing the need to limit a universe of discourse or genus in order to achieve the true complementarity of ἕξις and στέρησις, but at the same time effectively including irrelevant things within the genus in question by calling them μεταξύ along with true intermediates on a continuum. Both extra-generic or irrelevant things and intermediates along an intra-specific group of things that constitute a continuum between extremes are called μεταξύ. This seems to be a case of πολλαχῶς gone wild.

The same kind of problem attaches to his notion of primary contrapositivity (1055a 83-84) or πρώτη ἐναντίωσις which, as he defines it, would be a clear cut case of Boolean complementarity if taking it to mean contrapositivity didn’t contradict the definition of contrapositivity as maximal difference which opens iv. Understood as primary opposition πρώτη ἐναντίωσις is supposed to embrace maximal opposites which constitute a sub-species, so to speak of the complementary species. Aristotle tries to finesse his way out of the tangle by basically invoking his much used doctrine of ambiguity (πολλαχῶς), but, as should be clear by now, the Gordian knot can be cut more economically by distinguishing as three separate notions (1) Boolean complementarity, (2) conceptual or semantic contrapositivity and (3) measurable differences of degree along a scale. Not all contrapositives admit of intermediates although it is somewhat unclear whether all opposite poles of a measurable contiuuum are contrapositives.

X iv -v can be read as an attempt to capture the notion of conceptual contrapositivity (ἐναντία) within the logical framework of (a) the Square of Oppositions by associating conceptual contrapositives with propositional contraries, and (b) the adjunct notions of genus/species complementarity and privation (negation). The upshot would be to undo my claims of irreducibility. If successful, it would contravene my denial that each of a pair of contrapositive concepts such as hot/cold or good/evil logically implies its opppsite. His reasoning is as follows: When a genus or universe of discourse is limited to complementary species, examples such as the number 3, which are neither cold nor evil, do not belong to a species complementary to the species hot or good (For good/evil as a continuum with contrapositive extremes cf. below). The number 3 lies outside the genus in question altogether. But since true contrapositives are in fact a limiting case (effectively a subspecies) of specific complements, then true contrapositives are members of a complementary species and can indeed be logically deduced one from the other, presumably by some technique similar to negation. Accordingly Aristotle defines conceptuual contrapositivity such as hot and cold as a limit case within complementarity. It is maximal (μεγίστη) or extreme (ἐσχάτων) complementarity. In ll. 24 ff. it is also called πλεῖστον and τέλειος. As Aristotle himself asserts, the very idea of maximal difference (ultimately maximal complementarity) depends on the assumption that there is such a thing as degrees of difference and that contrapositivity can be understood as an example (albeit an extreme example) of a degree of difference. His “proof” relies on the troublesome technique of Aristotelian induction (ἐπαγωγῆς). The idea seems to be that specific opposites generate one another, but that within a given species there is a maximal opposite that is complete in itself (τέλειος) and need not be generated by anything. This maximal opposite is the contrapositive. There is reason to believe that the idea of generation is physical as in Bk. XIV and not logical (The text after all is the Metaphysics and not the Organon). In that case Aristotle’s argument is limited to physical objects and depends on a theory of generation or cause by opposites. The induction in question is presumably based on observations of generation by opposites such as the Presocratics may have performed. I will have more to say about the relations between deductive and conceptual opposition below.

Limited to physical occurrences, however, the argument is not strong enough to counter my understanding of contrapositivity which extends to all entities, physical or otherwise. For some of these entities it is irrelevant to ask whether they generate each other. If, however, Aristotle does, despite appearances, understand maximal opposition in a logical sense, then he begs the question.  He assumes that there are in fact members of a species that are complete in themselves and that these are the same things as can be characterized bycontrapositives. Indeed, as noted above, not all semantic contrapositives can plausibly be construed as end points of a continuum. While hot/cold could be construed as extremes on a continuum and there is an outside chance that the same could be said for good/evil, it is difficult to see how odd/even or non-quantitative contrapositives such as Sparta/Athens could be degrees on a scale. Aristotle acknowledges as much for examples such as unity/plurality, equal/not equal and dissimilarity/similarity. And if that is the case, the continuum theory does not constitute an effective method for identifying contrapositives within a genus, a point which in fact Aristotle seems to concede.

Another problem is that, while there are logical techniques, such as negation limited to propositions about members of a defined genus, by means of which we can logically deduce propositions about members of one complementary species of the defined genus from propositions about members of its complement, there is still no logical technique for picking out those complementary items which are in fact contrapositives. We can detect complements by way of (a) specifying the species hot of a genus of things that can be hot or not hot – a genus of which the number 3 is not a member - and (b) negation, which allows us to pass from a logically opposite proposition to a specification of the members of the species’ generic complement, its στέρησις, i.e. the members of the species not-hot. This is the sense of his discussion of the phrase τὸ πότερον in X v 1055b 30 ff. as well as πέφυκεν at 1056a 20-22. But there is no equivalent to negation for narrowing down the complementary class to those things that are actually cold and not simply, for example, warm. To do so requires the extralogical device of Aristotelian induction. Finally, even assuming he is correct in asserting there is no greater difference than extreme difference, no link is established with actual propositional contraries. He does not show us the relation between something being extremely not hot (presumably contrapositively cold) and the contrary propositions: (1) Everything is hot, and (2) Nothing is hot. In fact contrary propositions in a sense undo the effect of identifying the contrapositive subset of a complementary set. The same thing can be said about propositional contradictories. Aristotle does not show us the relation between something being extremely not hot (presumably contrapositively cold) and the contradictory propositions: (1) Everything is Hot, and (2) Something is not hot.If, however, Aristotle understands maximal opposition in a logical sense, then it begs the question. It assumes that It assumes that there are members of a species that are complete in themselves and that these are the same things as can be characterized by contrapositives.

Despite and in addition to all of the above, any number of passages in Aristotle seem either to outright contradict or at least indirectly deny the distinction I have drawn between complementarity and contrapositivity.

Post. Anal. I iv 73b 20-25 discusses a case of complementarity as if it were just contrapositivity (a natural number must be either odd or even; odd and even are contrapositive concepts but if you limit your domain to the natural numbers you produce complementary sets). Both examples he brings up involve “necessary” attribution precisely because they are complements. It is only under those circumstances that the Law of Excluded Middle can be invoked. Since the issue is attribution of qualities and not propositions, this discussion is not related to the classification of quantified propositions in the Square of Oppositions.

Topics I xiv 105b 34-38 gives examples of what Aristotle considers to be opposites: good/evil, black/white, hot/cold. But he doesn’t say whether he considers them to be ἔναντία or ἀντικειμένας or examples of each. He uses both terms in the same passage. In II vi 112a 24 ff. he courts the same ambiguity, treating health and illness as if they were complements although they could easily be understood as contrapositives. In 112b 10 ff. of the same section he appears to be guilty of outright error. He uses τὸ πολύ as if he meant “sometimes” (and not in the more instinctive sense of “usually,” as Tredennick erroneously translates). But the complement (ἔναντίος) of “sometimes” is not “always;” the complement is the union of “always” and “never” (I use the adverbs for the sake of readable English; they can easily be rendered into more correct if cumbersome paraphrases such as “the class of things that sometimes occur”)

In Topics IV iii Aristotle makes theoretical observations about opposites that, despite their broad validity, bring to a point the difficulties that come about from not distinguishing clearly between complements and contrapositives. The term he uses throughout is ἐναντίος. He mostly understands this term in the Boolean sense as a complement (following the logical convention from the Square of Oppositions, Tredennick translates it as “contrary”), which is appropriate since his goal is to position ἐναντίοι in his theory of genus and species. Problems pop up, however, on two fronts. The first is that his examples are most easily understood as contrapositives. Even Aristotle has an inkling of this and that is why he has to propose the concepts of intermediates (τὰ ἀνὰ μέσον) and extremes (τὰ ἄκρα). The other problem is that Aristotle does not deploy what set theoreticians would call a universe, that is, a class that includes the members of the two complementary classes (In Topics VIII iii 153a 34 – 153b 24 Aristotle states his version of a universe of quantification or discourse when he stipulates that ἐναντίοι or their differentia or both must fall under opposing (ἐναντία) genera. Also something like a universe of discourse appears in the Metaphysics). This leads him into difficulties when he tries to treat the contrapositives (Significantly he calls them independent genera) like good and evil as complements. He doesn’t or can’t deal with the fact that things we would not regard as appropriately described as either good or evil (Things like inanimate objects) nevertheless belong to a true complement of good and indeed to a true complement of evil. His genus/species model is unwieldy for dealing with this phenomenon since inanimate objects do not belong to a higher genus, such as, for example, things that we can call good or evil.  His mindset does not afford him the flexibility of specifying genera by stipulation, or, as we would say, extensionally. The idea of a negative intermediate (κατὰ ἀπόφασιν τὰ ἀνὰ μέσον) appears to be an unsuccessful attempt to fill the gap, for there is no reason why the contrapositive genus should not also be characterized as κατὰ ἀπόφασιν. Much of the second difficulty results from not recognizing the independent status of contrapositivity; his confusions are due to trying to cram contrapositives into the otherwise logically respectable model of complements. It is worth noting that Aristotle regards health and disease as true complements. Disease is the absence of health and vice versa. The only thing lacking in this instance is the specification of a universe. Depending on your universe, things like rocks and waves could also be treated as belonging to the complement of either healthy or diseased things or the union of those things.

Topics IV iv 124a 35 - 124b 14 deals with opposites (ἀντικειμένοι) that can be interpreted as complements or contrapositives. Blindness is a complement to sight if it is simply the lack of sight. It is the contrapositive if seeing and blind things belong to a universe that includes inanimate objects and plants, but where it is stipulated that the latter are neither seeing nor blind. Στέρησις or privation can be interpreted as designating complementarity (negation in de dicto terminology) only if the universe is limited to things that see and things that we choose to call blind. He reinforces this point in VI ix 147b 34 to 148a 3 using what I have been calling the device of respect. Blindness is lack of sight with respect to the eye. What Aristotle he calls ultimate (ἐσχάτῳ) genera are largely equivalent to universes in the logical sense, although the use Aristotle puts them to does not help him in distinguishing between complementarity and contrapositivity. Lines 7 ff. in the same passage probably only apply to complements already defined as excluding each other. His example of pleasure and good is unfortunate in that it does not pay attention to earlier strictures concerning qualification by respect. Murder may be pleasurable to a sadist, for example, and yet not be good. As we saw Aristotle tried with indifferent success to resolve these problems in the Metaphysics with a full blown theory of degrees of difference.

Despite his affinity to Plato and sporadic hostility to Aristotle, Plotinus occasionally commits the same ambiguity of talking about complements and calling them ἐναντία. In Enn. I.4.6 ll. 1-4 he calls εὐδαιμοεῖν the lack of some unpleasantness or other (He expresses this by the negation particle μὴ), but describes the relation as one of ἐναντίων.

Aristotle’s physical theory shuffles back and forth between confusing and distinguishing contrapositivity and complementarity. Phys. IV xii uses ἐναντίος (interchangeably with ἀντικείμενα) in a clearly contrapositive sense with regard to some notions from geometry. An example of geometrical incommensurability does exist, according to Aristotle, which means that its contrapositive commensurability does exist (outside time). That sort of biconditional relation would not obtain for many members of a class which is simply complementary to the geometrical incommensurability he cites. But in Phys. V I 225b he uses ἐναντίος explicitly to mean a complement. Throughout V i-ii he applies his logic of opposition to motion, discussing whether contrapositives and other kinds of opposition can be used to define motion or change. But in V iii 227a he reverts to classing ἐναντίος under ἀντικείμενος and distinguishing it from ἀντίφασις. As in his logical treatises, when he makes this distinction ἐναντίος is closer in meaning to ἕτερος or complementarity. ἀντίφασις does not necessarily mean contrapositivity in the sense of Plato’s examples of ἐναντία. Rather it is (rather loosely) tied to contradictories in the Square of Oppositions. These passages show that, although Aristotle opened the Physics by criticizing Presocratic theories of cosmological opposition, he does employ opposition in a unique way in his own physics. Indeed his analysis of motion or change is based on contrapositivities and he goes into exemplary detail in describing these contrapositivities. In V iv Aristotle uses ἐναντία to mean precisely contrapositives. Cf. esp. 229b for a list of additional examples that are all contrapositive. In V vi Aristotle describes a situation where contrapositivity is clearly distinguished from complementarity (He uses ἐναντία to designate both kinds of relations but the distinction is unmistakeable). Because he tries to explore all the logical possibilities involving the two relations and what they qualify things get a little complicated. Aristotle observes that there are two ἐναντία to κίνησις. The first is simply the opposite κίνησις, i.e. a motion or change in a direction or to a state opposite to the other κίνησις. This is what we would call a contrapositive. He calls the second a στέρησις by which terms he doesn’t mean void as he did in previous chapters of the Physics, but lack of motion. This is what we would call a complement assuming the appropriate Boolean understanding of “lack.” Like all complements, however, στέρησις is sensitive to qualifications. If you add the appropriate qualification a στέρησις switches from being a complement to a contrapositive. Remaining at a given point is in Aristotle’s view contrapositive to moving away from that point. Now the starting point and the point of arrival of a motion or a change are themselves contrapositives and so the lack of motion from the start is contrapositive to the lack of motion to the arrival (alternatively stasis at the start is contrapositive to stasis at the arrival). Aristotle observes that motion or change can also be qualified by a starting point or a point of arrival but he does not explicitly state what should be obvious, viz. that these two motions thus qualified are contrapositives, that in fact they are more precisely defined cases of contrapositive motions in general which he began the chapter by identifying. He does observe that the contrapositive pairs generated by qualified motion and stasis can be further classified in terms of a kind of chiasmatic contrapositivity. Stasis at the start is contrapositive to movement from the start to the finish (and apparently similar to movement from finish to start, but this is not clear). A term that is not a member of a contrapositive pair cannot be a party to motion or change to its opposite. The only opposition (the use of the privative indicates this opposition is one of complementarity) that can be applied to a particular designated by such a term would be between existence and non-existence. Our takeaway from the logical fine print should be that Aristotle had a grasp of the difference between contrapositivity and complementarity, but, since he used the same term for both relations, in this passage at least, his grasp was not as clearly stated as it might be. V vi 230a suggests more examples of contrapositivity and pursues his analysis of their relations: 230a ll. 19 ff. natural vs unnatural motion or change; 230b l. 8 easy vs. arduous destruction; 231a 5-7 stasis vs. forced motion or change; 230b ll. 13 ff. chiasmatic contrapositivity involving directions and motion vs. stasis. In 230b 11-12 he speaks of something like a general categorization for types of second order contrapositivity using the term τοιαδί. 231a introduces the notion of degree of contrapositivity: μᾶλλον…ἐναντίον. 230b 32 and 231a 14 introduces the notion of concurrent contrapositivities of a given term to two different contrapositive terms.

The uncertainties persist in the Metaphysics. In Met. VII vii 1032 b and 1033a  and VIII v 1044b ll. 30 ff. Aristotle seems to confuse contrapositivity and complementarity. Privation signifies complementarity. But the example of health and disease could be argued to be a case of contrapositivity. Inanimate objects are neither healthy nor diseased. Depending on one’s definition of healthy and diseased, a man could be called unhealthy but not diseased.  Cf. also IX I 1046a ll. 30-31 and IX ii 1046b ll. 14 ff. for a clear identification of ἐναντίον and στέρησις although it is unclear whether a στέρησις such as disease is straightforwardly a complement.

In Met. IV v 1009a ll. 23 ff. ἐναντία and ἀντιφάσεις are mostly identified as contrapositives but they are discussed in terms of being and non-being treated as complementaries. Contrapositivity is why, Aristotle explains, the same thing can both be (i.e. exist) and not be. Therefore, the doctrines of of potentiality and actuality (Met. IX ix refers to mutually exclusive contrapositives in the middle of a discussion of actuality and potentiality. But it is unclear whether this incompatibility is due to complementarity or to some other cause) and of opposition in a respect (κατὰ…) are invoked to adjust for the identification of this sort of opposition as complementariness. In Met. IV vi 1011b ll. 8-23 Aristotle makes clear the identification of both ἐναντία and ἀντιφάσεις with στέρησις or “lack,” i.e. the equivalent of complementarity. This is a return to the meaning of the Posterior Analytics and in conflict with the examples of ἐναντία in the Physics which are true contrapositives. Cf. also IV vii where the complementarity of ἀντιφάσεις is invoked to defend the Law of Excluded Middle.

In De An. I 5 411a Aristotle asserts that if we can discern (κρίνειν) one of a pair of opposites, we can also discern its mate, which, speaking strictly in the terms used in this essay, is untrue, assuming the opposites are contrapositives as they are in his example of straight and crooked. This is an example of the issue of the confusion of contrapositivity and complementarity that I raised with respect to Mackie. There is nothing that logically compels us to discern crookedness in the color blue (the color blue, not a blue line) given that we discern that it is not straight. Aristotle could appeal to some sort of semantical rule, but the net effect of that would simply be to restrict the domain of objects to which straightness applies. Thus we could accommodate the example by limiting our domain of reference to things that have to be straight or crooked, such as lines, and excluding things, such as the color blue, which can be non-straight without being crooked. Aristotle needlessly creates confusion by using ἐναντιώσεως, θάτερον and ἀντικείμενον to refer to just one kind of opposition in this passage.

There are points of contact between deductive contrapositivity and conceptual or what we might call semantic contrapositivity. However, as we shall see below, Aristotle is unsuccessful in reducing one to the other. Topics I x uses ἐναντίος in the Platonic sense to mean “contrapositive.” The equivalent to ἕτερος is ἀντίφασις (Note ἀντίφασις is about logic whereas ἕτερος in Plato mostly classifies things. Scholastics would say the former is de dicto and the latter de re. Aristotle in this passage uses ἐναντίος in both ways.) His example is slightly muddled since it actually contains two crossing sets of contrapositives: friends/enemies and do good/do harm. The four possible combinations are (1) Do good to one’s friends. (2) Do harm to one’s friends. (3) Do good to one’s enemies. (4) Do harm to one’s enemies. Aristotle sets the muddle straight later on in II vii where he makes these distinctions and more. Note particularly 113a 14 ff. where he says the same “thing” can have more than one contrapositive. That should not be the case for a complement as I have defined complementarity.

Still, the relation between conceptual contrapositivity and Aristotelian logic in general is tangled and tricky. In Categories XI 13b ff. he makes the gnomic comment that it can be proved by induction (ἐπαγωγῇ) that evil is the ἐναντίον of good, courage of cowardice and so on. He doesn’t explain what he means and his comment certainly contradicts my claim that there is no effective method (assuming Aristotelian induction is an effective method) for specifying the true contrapositive of a given concept as there is for specifying its complement. He may mean that the induction can occur over individual instances of good and bad behavior or over judgments of good and evil. Or he could have in mind a kind of survey of opinions about good/evil etc. But that is not at all clear.

A passage from Topics may throw some light on this comment. The fact, as I argued above in my comments on the formal properties of contrapositivity, that a contrapositive proposition such as ”Socrates is evil”cannot be logically generated (by “logically generated” I mean constructed by means of rules from another proposition, such as, for example, a contradictory proposition specifying a complementary class can be generated from a given proposition way of negation) from a given proposition such as “Socrates is good” is well illustrated by the discussion of ἐναντίοι (understood in this passage as contrapositives) in Topics II viii 113b ff. (Cf. also II ix 114b 7 ff.) Aristotle repeats the point from the Categoires that certain logical relations that may obtain between contrapositives have to be discovered “by induction” (ἐξ ἐπαγωγῆς). But now there appears to be more content to the notion of induction.Presumably what he means to say is that each case of contrapositivity is unique and more information is needed about the meaning of the individual contrapositive terms before the relations can be said to obtain. There is no effective method for detecting those relations, to use modern terminology. It is interesting that Aristotle’s position can only be applied to conceptual contrapositivity and not to the logical contrapositivity defined in the Square of Oppositions. I will address below Aristotelian induction and its relation to opposition in the Metaphysics.

A kind of informal backdoor analogy between logical and conceptual contrapositivity occurs at Met. XIII ii 1077a l. 15 where ἐναντίον is used to refer to opposite, presumably incompatible, conclusions. In addition, Met. XIII iv 1078b ll. 25 ff. relates dialectic specifically to the study of τἀναντία. Nic. Eth. IV 1 24-25 provides an interesting use of ἐναντία which bears some affinity with the Square of Oppositions. The fact is ἐναντία is such a common term in Greek that Aristotle has to use it frequently in contexts that are not easily analyzable into any of the more precise definitions.

Another complex issue is how contrapositive predicates relate to what we might call the structure of the proposition in Aristotelian logic. There is a striking passage where Aristotle’s views on conceptual opposition get entangled with both that aspect of his logical theory and with his physical theory. De Gen et Corr. I 1 314b 26 ff. and 315a 22 puts Empedocles’ ἐναντίοι in context in that the contrapositives (Note, the examples in 19-20 are genuine contrapositives) account for the generation of physical objects from an underlying ὕλη. Significantly Aristotle uses his logical term ὑποκείιμενον (e.g. Categories II 1a 20) in connection with the notion of a material substrate (ὕλη) (314b 3 and 27-28). This is problematic because a logical subject refers to a particular and it is the combination of the substrate with form (μορφή) (or variants thereof depending on the theory) that results in particulars or types of particulars. It is possible to argue that the subject of a proposition in Aristotle is already formed in some sense in virtue of being referred to by a name or a definite description. The predicate would add a form or highlight one aspect’s of the subject’s formation. Thus “Socrates” in “Socrates is brown” refers to some bit of matter or ὕλη that is formed by the μορφή Socrates. We might express that in the proposition “This is Socrates” where the indeterminacy of “this” indicates a subject of unspecified form. “Socrates is brown” adds a form or highlights an aspect of the form of the already formed ὑποκείιμενον Socrates. In stark terms, logically matter or ὕλη is already formed since the word “matter” can function as the subject of a sentence, but physically it is an unformed substrate. The contrapositive theory of predication as Aristotle sees it in Empedocles sits on top of this structure paralleling propositions and physical objects. Incidentally in the same book 315a 5 Aristotle uses ἐναντία λέγειν to mean “contradict oneself,” a linguistic usage with no obvious counterpart in physical theory.

Did Aristotle ever notice the stark difference between logical or deductive and conceptual or semantic contrapositivity and did he find some underlying relation between the two?

Chapter XIV of De Int. makes a number of important and up to that point undetected distinctions, but it does so in such a confused and condensed way that doubt has been raised as to the authenticity of Aristotle’s authorship. He raises the question as to whether the true ἐναντία to an assertion are its logical ἐναντία as previously defined or the assertion of something that appears to be a contrapositive quality. His example starts with “Every man is just.” The question he raises is whether the ἐναντία of this statement take the form of “No man is just” (the ἐναντία of the Square of Oppositions) or “Every man is unjust.” This latter proposition, or its schema, does not appear in the Square of Oppositions because it relies on the meaning of the predicate (in this case “unjust”) in addition to the purely logical device of quantification. Of course, “unjust” in this discussion could simply be the complement of “just.” In that case the opposition would be something like a logical contradictory. But since the opposite proposition is not a contradictory, we may presume unjustness, so to speak, is a contrapositive like evil. The ambiguity might be what Aristotle has in mind by distinguishing between spoken ἐναντία and ἐναντία in the mind but that is far from clear. Nevertheless XIV seems to me to indicate that Aristotle suspected that conceptual contrapositivity expressed a type of opposite to universal statements that is different from the logical ἐναντία he had defined in previous chapters. He does not cite the logical contradictory (ἀπόφασις) in 23a 30 ff. (“Some men are not just”) but instead brings up assertions in which the subject place is occupied not by a quantified expression but by a proper name (“Callias is just” etc.). The Square of Oppositions does not make use of proper names or singular referring expressions just as it does not make use of contrapositive concepts such as (potentially) unjust. Chapter XIV brings up logical and semantic issues that are outside the purview of the Square. In 23b 15 ff. he appears to try to assimilate negation and complementarity to the Timaeus model of contrapositivity as extremes at the ends of a scale of intermediate values (I shall ignore the confusing and largely irrelevant reference to accidental and inherent qualities) in a way that, as we saw, would be echoed in the Metaphysics. The not very satisfying and logically undefined idea is that logically contradictory assertions express contrapositive extremes. Nevertheless 33 ff. appear to express a clear distinction between complementarity and contrapositivity. In 24a Aristotle again tries to tie his expanded concept of ἐναντία to the oppositions as they appear in the Square of Oppositions. In his brief comments he seems to regard universalization as a substitution of subject terms in an assertion, and not, as modern logic would have it, an operation on a predicate function (Frege’s unsaturated predicate).

In Topics III vi 119a 32 to 119b 1 Aristotle promptly becomes thoroughly muddled. The problems in this passage must, of course, be regarded in the context of the apparent purpose of this book which is to provide successful arguments in practical contexts and not lay out steps toward deductively necessary conclusions. Aristotle writes almost as if he were providing a guide book for lawyers, particularly in his constant citation of “commonly accepted beliefs (ἔνδοξοι)” as the basis for both constructive and destructive arguments. And at times I sense a wink and a nod quality to the writing as if Aristotle was aware that his suggestions were just a way to dazzle the jurors. But that could just be my imagination. Nevertheless, the passage in question is very confused. That partly has to do with the fact that Aristotle deals with three distinct sets of opposites combined in single propositions. He is concerned, first of all, with what happens when one of a pair of opposites (The term he uses is ἀντικειμένοι) is asserted of (We would say “predicated of”) a subject which is one of a distinct pair of opposites. He overlays that schema with logical opposition derived from his theory of syllogism and the Square of Opposition. The general principles he lays down to the effect that (1) If all x’s are F, then some x is F, and (2) If no x’s are F, then it is not the case that some x is F (where identical variable letters stand for identical variables), are, of course, not to be disputed. Errors arise, however, when these logical schemata are applied to specific predicates with meaning in his example. (Since he calls this application an ἔνδοξον, he appears non-committal as to whether he endorses the validity of the example. But why cite the example at all, indeed use it as an illustration, if he didn’t believe it was just as valid as the schema?) The common beliefs he cites are (1) that if all pleasure is good, then all pain is evil, and (2) If some pleasure is good, then some pain is evil. Anyone who has been following my distinctions between complementarity and contrapositivity will immediately see that, on the face of it, the pairs pleasure/pain and good/evil are semantic contrapositives. Yet the inferences are not at all valid if those pairs are understood contrapositively. But, in fact, they are just as invalid if they are understood as complementaries, i.e. if pain is understood as lack of pleasure and evil is understood as lack of good. For, assuming all pleasure is good, some instances of its complement could still be good (Remember, we are talking about the instantiation of a logical schema) if there were good things that were not pleasurable. Indeed, depending on how you define “pleasure,” Socrates’ tribulations could be cited as a counterexample. The fallacy of Aristotle’s “some” example should be obvious.

The notion of existence plays a not unexpected complicated role in Aristotle’s handling of opposition. On the one hand, existence and non-existence themselves are opposites (it being an open question as to whether they are contrapositive or complementary). On the other, the question arises as to the relation between purely conceptual opposition and the existence of things qualified as opposite in one sense or another.

A good starting point is Categories XI 14a 7 ff. where Aristotle makes a point about particulars characterized by opposite qualities to the effect that the existence of one such particular does not imply the existence of its opposite. His examples are actually complements but his point is if anything more valid for contrapositives like good and evil. Relevant to Mackie essay. Put this in that essay also.

The Prior Analytics provides a specific application of the general existence issues regarding any pair of things whatsoever that fall under opposed concepts. At the same time Aristotle raises the more focused question of the existence of second order entities such as Platonic Forms for, as we saw above, the issue of whether Forms can be predicated of each other involves the predication of the Form being to itself and to its opposite and so, in a self-reflexive way, falls under the general problem of the existence of second order entities. Anal. Pr. I xxxiv esp. 48a 18 ff. states Aristotle’s version of what I have been calling the difference between the logical and ontological orders and of what Plato expressed as the problematic of the Forms applying to each other and to themselves as well as to particulars. By distinguishing a state or ἕξις from whatever is in that state, Aristotle provides a solution to the apparent paradoxes of ἐναντία applying to each other. What appeared as paradox in Plato (despite the resolution of the paradox as far as being is concerned at the end of The Sophist) is in Aristotle a simple logical misstep where, because of an ambiguity in the meaning of the terms involved, some true syllogisms appear to be false and vice versa. As is appropriate to the subject of a book on logic this solution is stated in terms of the meanings of words used in syllogisms although the distinction he draws does have ontological import. A ἕξις as Aristotle uses that term is nothing like a Platonic Form, although Aristotle refers to ἕξις with the meaning of both state and class to clear up the same sort of difficulty Plato found in, for example, the paradox of whether non-existence exists. Nevertheless, Aristotle does not address in this passage the ontological status of a ἕξις. Related passages include I xl which states clealy the difference between predication and identity with respect to an abstract object as well as Anal. Pr. II viii. and De Interp. 17 b 16 ff.

Some of the issues regarding the existence of members of a contrapositive pair persist among modern philosophers. Tanabe p. 67 seems to make the basic mistake of claiming that one term of a contrapositive pair cannot “exist” unless the other term also exists. This is not entirely clear, however, because he appears to be addressing alternative courses of action. He also does not explain exactly how freedom and contingency are contrapositives.



We should keep in mind that Aristotle uses the term ἐναντίος, in these passages at least, quite differently than does Plato and also in a manner different from his own conceptual definitions in other loci. And this is not due solely to the fact that Plato’s use is ontological and Aristotle’s logical. For Aristotle’s understanding of ἐναντίος is not quite equivalent to Plato’s ἕτερος either, although it is closer in spirit to that concept. Aristotle’s usage follows closely the lead of the Square of Oppositions and so ἐναντίος has something to do with “applying to none,” whereas for Plato ἕτερος sometimes signifies those things to which a Form, to speak Platonically, or a predicate, to use an Aristotelian derived concept, does not apply (It can also signify numerical non-identity). The difference lies in how narrowly the universe of discourse is specified. On the other hand, as we saw ἐναντίος in Plato signifies a Form completely opposed to but not complementary to another Form. In his logic Aristotle lays the emphasis on a single concept and what we would call the logical operations of negation and quantification. Plato looks at pairs of concepts and their relations to each other. Plato’s approach to ἕτερα as complementaries can also be expressed logically and even illustrated by Venn diagrams. His concept of ἐναντίος, on the other hand, while roughly illustratable by Venn diagrams cannot be expressed by the logical apparatus of Aristotelian syllogistic or the formalism of the first order predicate calculus. The standard translation for Aristotle’s ἐναντίος in the Square of Oppositions is “contrary.” As I pointed out, this is not equivalent to contrapositivity at all. The term he contrasts it with is ἀντικειμένος usually translated as “contradictory,” which expresses an idea related to but not the same as Platonic complementarity. In Anal. Pr. II viii Aristotle further relates his versions of propositional opposition to the ideas of not, some and all, i.e. what we would recognize as negation and universal and existential quantification.


In Met. IV ii 6 1003b ll. 23 ff. Aristotle takes a step beyond the logical treatises. In the course of relating the notion of opposites to the general science of being (Opposites, he says, are genera of the all encompassing notion of being), he endorses the view that existence (as well as unity and the other equivalents of existence) is not a predicate, to use modern terminology. Presumably this is because it is all encompassing and not a genus; only genera and species can predicate something of a subject. He recognizes the Parmenidean paradox of being as a sort of unrestricted totalizing predication although he does not really see the implications of that paradox for his general science of being. The upshot of the Parmenidean paradox is that nothing at all can be said (presumably predicated) about the all encompassing notion of being (although showing this requires an additional step of showing how an all-encompassing concept cannot itself be subsumed under some other more limited concept) nor about the things (i.e. everything) it subsumes solely as things subsumed by being. Many of the problems arise from not recognizing the difference between existence as a predicate and being as a sort of hyper-category whose principal role is to be all encompassing.

The other issue, obviously in sequel to The Sophist, is how the abstractions, concepts or forms existence and non-existence themselves are opposed. It is equally obvious that even addressing this issue assumes that existence is not equivalent to the all-encompassing notion of being. In Physics VI v esp. 235b Aristotle uses τὸ (μὴ) ὄν and (μὴ) εἶναι with a clear sense of existence and non-existence. There is nothing paradoxical in the assertion that everything must either exist or not exist since the context dictates that he only has physical existence in mind. Indeed the discussion of the existence of a statue in marble earlier in the Physics provides a framework for understanding the restricted domain in which he uses the concept of all or everything. To all appearances Aristotle simply dropped the logical notion of a highest genus when discussing physical and ideal (Platonic) existence in the rest of his philosophy. He did not, except perhaps in the Metaphysics, consider the relation between the two. We should note, however, his claim that the transition from existence to non-existence is κατ’ ἀντίφασιν. Existence and non-existence, even restricted to physical nature, seem to have to be complements since Aristotle insists that it is necessary that something either exists or doesn’t and also speaks as if existence and non-existence were absolutely opposed without an intermediate or irrelevant state. Of course, this sort of complementary opposition is not analogous to contradictory judgments from Aristotle’s logic. Nor is it compatible with regarding existence and non-existence as contrapositives in my sense (although a good case could be made that they are complements regardless of domain. Can anything neither exist nor not exist? Only Vedic philosophers can know for sure). It is, however, compatible with Plato’s view that the Form Non-Existence might itself exist, although that conclusion is not particularly Aristotelian.

Vlastos (Platonic Studies pp. 274-275) tries to untangle some of the confusion in The Sophist by using terms from Aristotle’s logic. The effect of his effort is to make matters more confusing. As we see, Aristotle’s logical theory of opposition answers Plato largely by changing the subject. His opposites include items that don’t appear in Plato and excludes other items that do. This means that in his logical theory of contraposition Aristotle misses the distinction between complementarity and contrapositivity altogether. Contrapositives in the Platonic sense would sneak back in as Aristotle’s theory of conceptual contrapositivity but overall he fails to make a connection between the logic and the more intuitive sense of conceptual contrapositive opposition such as he assumes in the Physics.

Aristotle, of course, rejected the very idea of a Platonic Idea and so raised the question of the existence of the abstraction existence (being) by effectively answering it in the negative. His reasoning, stated in modern terminology, was probably that the name “existence” fails of reference because there are no such things as Platonic second order entities. The proof of the latter is separate but the non-existence of the Form existence or being follows from it. The degree to which Aristotle rejects Plato’s two tiered ontology (Forms vs. other things) with a distinction between ontology and logic comes out forcefully in Anal. Post. I xxii 84a 34 ff. where he rejects the Forms altogether in his account of essential predication. In 83b 18 ff. he also rejects second order predication (Qualities cannot have qualities). In Topics VI vi 143b 11 ff. Aristotle proposes an argument against the existence of Platonic ideas on the basis of the fact that differentia cannot be predicated of a genus considered as a distinct (second order) entity (Note the use of ἕν on line 32 to signify distinctness); only whatever falls under the genus can be differentiated.

The complementarity of existence and non-existence naturally suggests Aristotle’s other conceptual understanding of ἐναντία as complements. One of the purposes of this essay has been to distinguish conceptual contrapositivity from complementarity. Because, as we have seen, the two notions are often confused we should expect to find, wherever a theory of conceptual contrapositivity appears, a theory of conceptual complementarity alongside it. And indeed Aristotle largely deals with complementarity hand in hand with conceptual contrapositivity, occasionally regarding them as distinct and sometimes confusing the two.

There are also textual inconsistencies throughout his treatment. If we pay attention to his examples Aristotle uses the word ἐναντία sometimes to mean “contrapositives” and sometimes to mean “complements.” But another term ἕτερος is sometimes invoked either to mean distinct or different as in numerical distinctness or to mean complementary on the analogy of specific difference. In the latter usage it overlaps with one sense of ἐναντίος. In his treatment the terms ἔναντία and ἕτερα often appear to mean the same thing or are otherwise confused. Ἕτερος , like the other terms, is effectively a meaning shifter in Aristotle’s texts. Categories VII uses ἕτερος in the sense of “other” or “different from,” i.e. “not identical” and not in the sense of complementarity. However, in Categories III 16-24, the first occurrence of ἕτεραι in Aristotle, the term has a clear sense of non-intersecting but not complementary genera. This relation of genera he distinguishes from what we may call nested classes (as in class inclusion). This recalls the shifting meanings of ἐναντίος. For example, cf. Met. 1018a25 and Cat. 6a18 where Aristotle appears to use ἐναντίος much the same way Plato does.

Two other terms which, as we have seen, come closer to the meaning of complementaries are ἕξις and στέρησις, i.e. a state and its lack or absence. We could interpret these terms logically in the sense of negation, but Aristotle mostly regarded them as applying to the world and not to propositions. In any event στέρησις in Aristotle’s usage conforms more closely to what I mean by complement than does ἕτερα. One way to interpret Aristotle would be to regard ἕτερα as distinct things, either numerically distinct particulars or distinct concepts. In the latter case the distinct concepts may specify complementary genera or species.

If we take account of the confusing terminology there remain numerous passages where Aristotle is talking about Boolean complementarity in the more or less strict sense. In De Gen et Corr. I 7 323b ff. ἐναντίος means complement (ἕτερος is used interchangeably) and is situated clearly within a genus/species framework. Note the change of terms when Aristotle speaks about Democritus. In Prior Analytics I XXVIII 44b 38 ff. Aristotle uses ἔναντία as equivalent to ἕτερα. That is, both designate concepts that cannot apply to the same subject (45a 5). Aristotle’s terminology is a little muddled because, as I noted above, he uses the same term ἀντικειμένας and its conjugational forms as both a generic term to cover all the types of opposition in the Square of Oppositions and to refer specifically to complementary opposition. This is particularly evident in Anal. Pr. II xv. However, the context makes Aristotle’s intentions and meanings somewhat clearer. Anal. Pr. II xxvi is also vague about whether ἀντικειμένας means the same thing as ἐναντίας or whether it is a more generic term including ἐναντίας and other types of opposition.

Met. V ix  1013a ll. 10 ff. and V x and XI vi 1063b ll. 16 ff. summarize Aristotle’s definitions of terms involving opposition. In these passages and in V xii 1019b ll. 22ff. ἐναντίον is explicitly linked to negation and so to complementarity. In X vii ἐναντία are distinguished as belonging to the same γένος and having μεταξύ (I discussed the issues regarding intermediates and irrelevants above). Cf. also X x as clear statement of ἐναντίον as complementarity and its relation to negation. Note especially his comment that universals and particulars, the top and the bottom of the genus-species pyramid, can have opposite properties.


Met. X viii defines of ἐναντίωσις as occurring within a genus. It could be either contrapositivity or complementarity but in 1058a l. 23 he says it is determined by negation or ἀποφάσει which argues strongly for complementarity. In l. 12 he also calls it διαφορὰ τελεία. In this passage ἐναντίωσις is distinguished from otherness or difference as belonging to different genera or being different genera. Note the use of the expression τινός τι to signify difference with respect to something. (An interesting sidelight is that 1058a l. 23 uses ὕλη in such as way as to indicate Aristotle means something more than physical matter. It means anything that underlies a differentiation. This seems to imply an extension of the schema of his physical theory of the universe to non-physical things as well, i.e. an extension of the form/matter structure to all entities. This should lead us to suspect that there is a certain conceptual relativism in so-called Aristotelian essentialism, an issue which will not be discussed here.)

Met. X iii seems to treat ἐναντία as complementaries since his example involves στέρησις. But this is not entirely clear since the passage does not explicitly distinguish complementaries from contrapositives. It does, however, notably distinguish otherness (presumably as pertaining to particulars) and difference (presumably specific difference), a distinction that is not as clear in other loci.

Anal. Pr. I xlvi is as good a statement of the distinction between complementarity and contrapositivity as anyone could hope for. It is just worth noting that Aristotle does not use the term ἕτερος. His terms are ἀπόφασις and ἀντικείμενα (in l. 15 ἀντικείμενα clearly means complementarity). He uses no term to designate contrapositivity at all. Instead he has to resort to tortured if largely accurate prose descriptions of the difference between complementarity and contrapositivity, a difference that can easily be illustrated by way of a Venn diagram. Aristotle’s analysis does make an important connection between the complementarity/contrapositivity distinction and what we today would call the scope of negation (or the negation operator, to use syntactical language; for more on scope differences cf my “Logical Counterparts”). Aristotle for the most part restricts himself to logic alone and the relations between propositions. In 33-34 he does say that truth is analogical to (ὁμοίως) to being but it is most like that by “being” he means no more than the grammatical copula, at least in this passage. In this way he appears to understand, and correctly so in my opinion, that what Plato had regarded as an ontological issue is no more than a matter of how we use words and construct sentences related to what our sentences are about.

Despite extensive and subtle investigations into the nature of opposition, Aristotle failed to make clear distinctions between complementarity, contrapositivity and otherness. His vagueness left the door open for fallacious arguments such as the theological argument that evil is logically implied by good and that the existence of evil things is logically implied by the existence of good things.

Opposition in Structuralism

Oppositions are not just a phenomenon of Greek philosophizing and Greek thought in general. Along with classification opposition provides a scaffolding for the way we all think when we undertake a discursive description and analysis of nearly any phenomon that interests us. If there is one result I would like to draw from the preceding it is that, when we do science or philosophy, it is not enough to just draw up lists of opposites. Opposition is a logically much ramified relation and under any circumstances we would do well to be precise as to what exactly we mean by opposites. However that may be, the so-called structuralist approach to the human sciences and literary studies explicitly reverts to oppositional concepts as a way of organizing its material. In an intriguing twist, Lévi-Strauss sees the structuralist approach as following upon a consistent tendency in human thought such as is exhibited even in so-called pre-literate societies.

In the course of summarizing some of the findings of the anthropologist Alfred Radcliffe-Brown (Le Totémisme aujourd’hui pp. 130-132), Lévi-Strauss presents a model of totemism and opposition as a particular approach to classificatory thinking and conceptualizing similarities, i.e. integrating disparate phenomena into a similarity class. Primitive oppositional structures, according to Radcliffe-Brown are constituted by a shared characteristic whose possessors are divided into sub-groups by opposing or contrasting sub-characteristics. The sub-characteristics provide a way of comparing and so describing the opposing members of the parent group. What is special about this way of thinking is that the oppositional pairs are not necessarily posterior; they in fact provide the means for conceptualizing the parent or similarity class. Thus the parent group of carnivorous birds derives from and is divided into predators and scavengers which are exemplified by falcons and crows respectively. Other parent classes not necessarily limited to primitive societies are nocturnal birds, birds nesting in tree trunks (the carnivorous members of which are divided into the child classes hunter birds/thieving birds), true species such as cockatoos (divided into the child classes black/white), tree dwellers (divided into the child classes diurnal/nocturnal) and the tao (divided into the child classes yin/yang). Other examples of opposing terms not necessarily limited to primitive societies are heaven/earth, war/peace, upstream/downstream, red/black, husband/wife and day/year. Lévi-Strauss calls this way of thinking a “union of opposing terms.” As regards their role as analogies to social institutions and behavior, these parent/oppositional-child classifications are simply means of “coding,” to use Lévi-Strauss’ term, the structure itself. Instead of thinking in terms of abstract concepts such as classes and oppositions, the societies in question think concretely solely in terms of the exemplifications such as falcons and hawks. Instead of thinking in terms of an abstract concept such as analogy, the societies in question assign a concrete pair as emblems of social groups (the affinity to the anti-Platonizing tendency of modern thought is probably not accidental). This is largely what Lévi-Strauss considers to be the truth behind totemism. Nevertheless, what is explanatory for psychology and sociology is the structure and not its contents.

La Pensée sauvage (pp. 168-170) throws out more examples of binary oppositions such as those used in the classificatory system of the Subanun in the Philippines: simple (single?)/multiple, open/closed, heavy/light, superficial/deep and furthest/closest. At pp 174-175 he notes opposing categories from the Osage and the Hawaiians that are similar to opposites found in the Presocratics: high/low, in front/behind, south/north and right/left.

On p. 130 he lists some opposites found in the The Laws of Manu: capable of motion/incapable of motion, with incisors/without incisors, with hands/without hands, timid/proud (or brave). Depending on the universe of reference all of these except the last are complements. Lévi-Strauss throughout speaks of the formal organization of similar and opposites as logical in nature. It is pretty clear that by “logical” he means the assignment of semantic values, usually dual or polar in nature, to a class of objects. The values are formal in the sense that they can be assigned to any empirical. Classical logic as we understand it from Aristotle onwards, however, includes no polarity except as exemplified in the Square of Oppositions all of which modern logic has effectively reduced to negation and quantification. Other types of conceptual opposition lie outside pure logic. Since many of the oppositions that Lévi-Strauss records are in fact contrapositives, they do not fit into the classical logic of opposition for reasons I have already discussed. Another distinction is that you cannot assign a truth value to any unit of Lévi-Strauss’ structures. And classical logic requires truth values. It is better to describe Lévi-Strauss’ structures as semantic or perhaps grammatical in Quine’s sense of “grammatical” (Philosophy of Logic passim) rather than logical.

A succinct statement of the theory of totemism in Le Totémisme aujourd’hui appears on p. 141 of La Pensée Sauvage. Totemic institutions invoke an analogical relation not so much between social groups and natural species themselves as between the differences that divide up those groups and species. (Where Lévi-Strauss uses the term homologie I substitute “analogy” in English to avoid confusion with the biological sense of “homology” which implies a derivation from a common ancestor.) He goes on to acknowledge the existence of direct one-to-one analogies between social groups natural species and emphasizes that he simply wants to focus on systematic or differential structure.

La Pensée sauvage provides more examples of classificatory opposition. On p. 73 we find intra-cultural oppositions (light/dark and characteristic-of-fresh/dried plants as color oppositions) as the genera of a system that permits the identification of colors in the world (as opposed to the Indo-European axes of color value and chroma). On p. 75 he cites a nested tree of oppositions: animate/inanimate, rational/irrational, male/female. He also brings up examples that are not dual oppositions. They are tripartite, 4-part and five-part. This raises problems since these multiple types of structure do not involve opposition in any Saussurean or otherwise traditional sense for opposition has to have a simple duality somewhere. Setting aside non-dual structures, the fact that the placing of a given species or phenomenon under one genus rather than another recalls the somewhat arbitrary nature of contrapositivity (arbitrary because it cannot be defined in terms of a Boolean algebra).There is an issue regarding the ultimate referential meanings of these analogies. Is Lévi-Strauss right to say that a totemic association derives its entire meaning from the system of oppositions into which it fits (Does he in fact say that?)? Nevertheless we should remember that these systems are defined by generic oppositional concepts that could very easily (or even have to) be regarded as referential which is in direct defiance to certain structures of post-modernist structuralism.

On pp. 100-101 Lévi-Strauss reveals a transformational system of oppositional pairs that is actually far more intricate and nuanced than anything found in the Presocratic writings. His broad conclusion is that what is important is the structure of opposition itself without the specific terms or objects referred to. In his system there appear to be at least three levels of generalization. The most abstract is a plus or minus value. Those values are filled in with populations (ensembles) of relative specific values such as life and death or group and individual. At the most concrete are the animals or plants and the individual humans or groups of humans affected, the individuals instigating the associations, and the specific taboos or ritual practices. Without presuming to draw ethnographic conclusions, from a layman’s point of view it appears possible that Lévi-Strauss may be wrong or misleading. He may have chosen his own more abstract classifications prejudicially in such a way that the group variations within a population may exhibit a predetermined goal of structuralist oppositional transformation. Moreover, his particular example may not include important concrete opposing concepts (e.g. fire/air or plant/animal) which the native informants are aware of and would acknowledge as opposites.

Another example of an overly rigid application of structuralist oppositional semantics can be found in Jakobson (p. 209) who nearly succeeds in ruining our enjoyment of Sonnet 129 with a heavy handed highlighting of the contrapositive “confounding contraries” in Shakespeare’s famous melancholic insight into desire and its satisfaction.

Opposition in Hegel

The three canonical reactions to contrapositivity come from Plato, Aristotle and Hegel. Plato rejected contrapositivity altogether. In its place he proposed objective standards of measurement for all objects and the concept of interweaving opposite personality types for humans in society (Timaeus). Aristotle reduced contrapositivity to the relation between the universal propositions “Everything is F” and “Nothing is F.” This response is the least satisfying because it defines something different from contrapositivity as conceived by the Presocratics. Aristotle did have a feeling for conceptual contrapositivity but he ran aground in trying to graft that notion onto syllogistic opposition and onto Plato’s idea of a scale of intermediate values between opposites. Hegel saw contrapositivity as the engine behind speculative reasoning. It may be argued, however, that Hegel’s actual examples of contrapositivity are either no more than instances of the paradoxes of “all,” i.e. of unrestricted universalizing statements, or else they do not really generate the contrapositive term from anything inherent in previously given concepts. This latter appears to be the case in the master-slave dialectic, for example, where nothing inherent in the concepts of the preceding stage of the dialectic necessarily suggest the concepts of master and slave. Hegel ended up with nice dramatic flourishes and perhaps some insight into human behavior, but much of what he argued did not surpass in philosophical rigor Socratic demonstrations that people may hold contradictory beliefs unawares.

We might say that what Hegel provides is a theory of human motivation and overall change based on the relation of opposing attitudes. In acting a certain way an oppositional pole produces an effect opposite from the one desired and so changes into its own opposite. Of course, this is an unjustly schematic treatment of Hegel’s philosophy as a whole which deserves at the very least a separate essay.

As far as logic is concerned there is no Platonic argument that can be used to justify Hegel's extensive use of opposition in his logic. That doesn't mean there isn't some sort of justification in Hegel himself.


Ethical Issues in the Lysis

Not much is written about the Lysis and most of that is not worth reading. One particularly bad article is by somebody named Robert G. Hoerber. After observing that most Lysis scholarship is concerned excluseively with issues of authenticity and dating, Hoerber announces his intention of reconstructing and assessing the actual philosophical content of the dialogue. His method will be a blend of something like literary hermeneutics and philosophical argumention, an approach popular among a certain school of Plato scholars. That is, he will balance the dramatic narrative and certain of its salient characteristics with the substance of Socrates’ reasoning to see what positive conclusions might be drawn - conclusions which are presumably over and above what Socrates states explicitly.


One problem I have with this sort of thing is that I have never yet run across anyone who practices it with Plato and manages to draw any conclusions that convincingly differ from what is explicitly stated as a conclusion to a given dialogue. Nevertheless, the Lysis seems to invite another try because, superficially at least, it is purely negative. It does not conclude with any doctrine explicitly affirmed by Socrates by way of myth or otherwise.


Hoerber begins his project, in a manner reminiscent of people who play Beatles’ songs backwards, by noting the omnipresence of triads in the dialogue. I’m just a country boy but to me these triads seem pretty much to be made up whole cloth by the commentator thereby proving the law that with a little ingenuity you can find anything almost anywhere. Perhaps he would have found even more triads had he tried reading the text upside down. He might even have discovered that Plato was a Chinese gangster.


Based on his discovery of latent triads Hoerber introduces a three part division of love with your standard academic faggot depreciation of fucking. Apparently being a tyrant and teaching a young man to be a good tyrant should replace physical κουροφιλία (to coin a phrase). Plato may have agreed (although Socrates’ comments on Archelaus in the Gorgias surely foreclose any purported superiority tyrant training may have over butt fucking; 510 A-E is obviously ironic if not sarcastic). There is a lot of evidence that he wanted moral education to be the major component of κουροφιλία and in his Fascist period he sympathized with legislating against homosexual sex acts (Clearly pedophilia can involve homosexuality as it did for the Greeks of Plato’s time; however, it is noteworthy that what Plato calls unnatural (Laws Bk. I 636c and Bk. VIII 838-839d) includes lesbianism and is not explicitly related to the age of the individuals involved. Moreover, the institutions the Athenian Stranger actually mentions are communal meals and exercise and not the sort of romantic kourophilic relation at issue in the Lysis as well as The Symposium and the Phaedrus) as well as incest, male/female anal and oral sex and masturbation.


You can draw your own conclusions about a contrast between Hippothales' romantic love and Lysis' parents' "sensible" love. But there is nothing in the dialogue that encourages you to critically compare the two except the dubious device of contiguity which could be read as a kind of Jakobsonian metonymy. Moreover, Socrates draws the most unacceptable consequence that parental lovers should allow the beloved to do whatever he wants. Much more telling against this reading is Socrates' demonstration later in the dialogue that parental love is self-interested and utilitarian after all. Socrates' actual transition to advising Lysis that he σοφὸς γένῃ comes by way of asserting that he will become useful and thereby beloved of others including his parents by way of his technical expertise. Become more σοφός and you increase the number of your lovers. All this was described in more detail in my outline of the argument of the Lysis above. Hoerber's attempt to turn the conclusions of the Lysis into a wise saying you can crochet on a throw pillow is academic faggotry at its most egeregious.


Hoerber's claim is that Plato provides "clues" that his hard and fast conclusion consists in some sort of hierarchy with butt fucking at the bottom and asexual virtuous hand holding at the top with something vague in between. I'm not interested in clues (pace Gadamer, Ricoeur, Leo Strauss and all the other hermeneutical clue hunters out there). Show me valid argument and explicit conclusions. Of course Hoerber could count himself among those who regard Plato not as a philosopher but as some sort of litterateur into whose texts the hermeneuticist can read whatever meaning strikes his fancy. In that case I don't see why the Lysis can't be viewed as a passionate defense of Krispy Kremes.


Hoerber is intellectually dishonest in his argument that Plato opts for mutual reciprocity as the only valid form of affection. His argument is that Socrates’ refutation of the reciprocity of φιλία is invalid because the φιλ- in wine loving is not the same as the φιλ- in human relations. Of course, this argument begs the question. Hoerber must first show that oinophilia and anthropophilia are relevantly different in order to disqualify Socrates’ use of the former (or something like it) as a counterexample. Human love or friendship after all might just be an exaggerated kind of fondness-for not unlike the feeling of someone who forms an unhealthy affection for a particular bottle of Nuits St. George. Barring philosophical validity Hoerber’s lit crit arguments do not pass the sniff test (a relevant metaphor as long as we’re talking about wine). A literal mind could say not without justification that, if Plato meant φιλία to be reciprocal, he would have said so. He was not one to leave such a sensational conclusion repressed. Accordingly Hoerber’s further conclusion that ἔρως is unrelated to φιλία fails by modus tollens.


Oinophilia and Hoerber's absurd and unsubstantiated claim of a "purely linguistic argument" on Socrates' part aside, Socrates' telling counterexamples to reciprocity are love for a newborn (who may not yet be capable of that kind of emotion) and love for a horse (Young Liz in National Velvet is both a horse fancier and the lover of a particular horse) who may not return the emotion. In his enthusiasm to reach his Sunday go to meeting conclusion Hoerber ignores these examples of what Socrates, ultimately agreeing with Solon, seems to regard as genuine unreciprocated love. Hoerber puts into Socrates’ mouth the faggoty opinion that ἔρως is somehow base and disgusting and that the "highest type" of φιλία is necessarily mutual. Why are these mutually exclusive? Obviously Hoerber never heard of a double dong.


In further support of his insipid view, Hoerber proposes a deliberate misinterpretation of the dramatic climax (222 a-b) where Socrates addresses the love triangle of Hippothales, Menexenus and Lysis directly. Hoerber adduces this passage as further support of his bland and misleading conclusion that true love is reciprocal (and by implication not physical since it focuses only on the so-called higher things). Actually Socrates' conclusion is not what Hoerber wants. Socrates is in fact making good on his promise to Hippothales by concluding that his argument has shown that Lysis must reciprocate Hippothales' love. Hoerber’s admission that with respect to “modesty” Lysis and Hippothales are οἰκεῖοι seems to undercut his view that Socrates wants to denigrate physical love (represented by Hippothales) in favor of spiritual companionship (ostensibly represented by Menexenus). The actual conclusion that Lysis must reciprocate Hippothales’ love is why Lysis and Menexenus became less than enthusiastic once it finally dawned on them what Socrates was driving at. Crucially Socrates uses the expressions γνησίῳ ἐραστῇ  and παιδικῶν (222 A-B) which refer almost unequivocally to the Greek institution of man-boy butt fucking.


Admittedly there are difficulties in interpreting this passage which seems to rely on knowledge of the actual historical situation involving these personages, information that we don't have. It also involves the difficult concept of οἰκεῖος. Hoerber seems to be unaware of the fact that at 222 B ff. Socrates explicitly denies that there is any difference in meaning between οἰκεῖος  and ὅμοιος (Admittedly he also denies that they are the same). Nevertheless, even if there was a difference between οἰκεῖος and ὅμοιος, the latter does not mean “identical” as Hoerber proclaims with breathtaking dishonesty and ignorance of freshman logic. Plato would have been talking sheer gibberish if by ὅμοιος he meant “identical.” Greek, in fact had a readily available term for “identical,” viz. αὐτός or τὸ αὐτό, if, for some bizarre reason, Socrates wanted to entertain as a plausible hypothesis that you could only be friends with yourself, a lesson Scarface learned the hard way. Hoerber’s passing comment that ἐναντίος is somehow an opposite to “identical” has been refuted by everything said at the bginning of this essay.


If Hoerber is right in his affirmation that the Lysis promotes a dogma about φιλία, then there should be an analogy of substance between this (in my opinion) clearly early dialogue and the Stranger's proposal for a sexual dictatorship in Bk. VIII of the Laws. In any event the Lysis is never referenced in the Laws. (Vlastos pp. 35-37 divorces the Lysis from later Platonic doctrine. His argument is relevant to the dating issue but I grant that positioning the Lysis as an early dialogue is not without controversy.)


Well, what about the notion that Plato’s ultimate doctrine in the Lysis, such as Hoerber purports to discover, prefigures Aristotle’s theory of friendship to such an extent that Aristotle may have derived his views from the Lysis, or at least used Plato’s dialogue as a starting point? Aristotle’s doctrine of friendship (an appropriate translation of φιλία in Aristotle’s case) is propounded in the Nicomachean Ethics Bk. VIII. The issue of the relation of similarity to friends seems to have been an ongoing conceit in Greek thought and poetry (VIII 1 6) and not derived solely from the Lysis. Nevertheless there is a striking parallelism between Aristotle and Plato as regards the types of φιλία.  Aristotle discusses similarity with respect to friendship at Nic.Eth.VIII iii 7. In iv 3 he mentions the issue of friendship between men who are both good, both bad and one good/one bad. In VIII viii 5-7 Aristotle reviews different issues which had been raised in the Lysis.


In the first two paragraphs of Bk. VIII Aristotle employs his usual method of observation and fact collection, the observation being in this case, as in other instances of social phenomena, about people’s opinions about things. Opinions about friendship are deduced from what people praise and how they behave. Aristotle makes quite clear, however, that he shares these opinions and he even deploys arguments based on goals (such as wealth preservation) and cites examples from animal behavior to support the essentiality or naturalness (φύσει) of the universal human praise of friendship. Unlike Plato, Aristotle was not one to oppose common opinion lightly and in the case of friendship he most definitely concurs.


The third paragraph appears to move in classic Aristotelian style from a statement of the “facts” to a review of prevailing theories that try to explain the facts concerning friendship. But he immediately dismisses physicalist explanations as irrelevant to his immediate concern with moral facts and so he returns to his analysis of what exactly φιλία may be. This analysis involves a rejection of what we remember as Socrates’ argument in the Lysis to the effect that unreciprocated love is still φιλία, because, in Aristotle’s view, true φιλία must be reciprocated and it is absurd to ask a bottle of wine, for example, to reciprocate the oinophiliac’s love. The phenomenon Socrates had identified as love for non-human objects is called by Aristotle a mere liking or φίλησις (VIII v 4), but making a nominal distinction does not establish a distinction of fact, which is what Socrates argued against. In the end Aristotle relies on our intuitions about what is and isn’t absurd, but it is clear Socrates had different intuitions, at least for the sake of argument. For Socrates an oinophiliac is a lover whatever opinions that delicious Margaux may have about him. But, even if we grant that Aristotle successfully set aside inanimate objects as valid candidates for φιλία, he would still need to deal with the unreciprocated love for newborn babies and horses. (Aristotle discusses maternal love in viii 3 but does not seem to recognize that this sort of φιλία violates his requirement that φιλία be reciprocal. In fact the entirety of 3-5 contradicts his earlier analysis. At xii 2 l. 25 – 3 l. 1 1161b he states outright that parents’ love for a newborn is not reciprocated.) Aristotle’s other argument is that true φιλία involves wishing well to the object of love for its own sake, whereas apparent φιλία for a bottle of wine is for the sake of another τέλος, i.e. the goal of our drinking and enjoying it. But Aristotle’s intiuitions in this case also are inadequate. Without shading over into actual perversion, the wine lover’s love for Margaux does not stop at wishing that it remain properly stored and carefully decanted. He loves the Margaux for its own sake and because of its qualities, the slightly acidic tang with overtones of cherry. The relation of these qualities to the goal of drinking is not obviously different from the relation of a human fiend’s qualities to the goal of pleasant conversation. You might even say that these qualities render the Margaux ἀγαθός.


Along the same lines it would seem strange to deny that Helena’s unrequited love for Bertram, or indeed Hippothales’ love for Lysis, is true φιλία. Aristotle might reply that those are not cases of friendship because friendship requires reciprocity (In iv 1-2 he qualifies this by distinguishing reciprocity in the broad sense from reciprocity which involves qualification by way of a respect; accordingly a lover-beloved relationship may be reciprocal but the respects in which each loves the other are different. This qualification does not cover a situation like that of Bertram who takes no pleasure in Helena’s attentions.). But that reply could equally well serve as a proof that Aristotle should not have used the term φιλία to name the moral phenomenon he wishes to analyze. And that may have been precisely Socrates’ point. That is to say, certain relations involve reciprocity. You can call these relations whatever you like. But, to stipulate that other relations that share certain characteristics with the reciprocity-involving relations must also involve reciprocity is to beg the question. And, by concluding that non-reciprocal instances of φιλία are thereby not φιλία at all or only by analogy to φιλία in his preferred sense (iv 4 31),  Aristotle becomes the worst mendicant of them all.


Aristotle does introduce an important new concept to handle cases of affection for animate objects who, for whatever reason, do not directly reciprocate. This is the concept of εὔνοια or good will (The parallels to Rousseau and Hume’s benevolence leap at once to the mind). Good will seems to account for cases of φιλία style feelings between adult persons who are completely unacquainted. But it is hard to call a parent’s love for a non-reciprocating baby mere good will. (For more on Aristotle’s discriminations of types of positive feelings cf. IX v 1166b 30 ff.)


At a deeper level we might try turning inside out the notorious Platonic/Aristotelian evaluative and moralizing hierarchy that rather arbitrarily and with no small amount of contempt places physical sexual activity, dispositions and feelings at the bottom, rises through self-interested friendliness and general good will to selfless φιλία based on certain essential but non-physical traits of the friends (This is where Aristotle stops), and ultimately to a kind of mystical love of the Good itself. Let us look at this hierarchy rather as an intuition on Socrates’ and even Plato’s part that there is a link that binds lust and physical sexual activity to some of those other more ethereal attitudes. In other words, Socrates understood that even the friendships among good people (and even the love of the Good if there is such a thing) are through and through sexual, that without the sexual element those “higher” types dissipate and disappear. By effectively re-defining φιλία as some sort of sterile bonhomie, Aristotle breaks the relation and represses the intuition that would require another two millennia to be regained in the work of Sade and eventually Freud.

Concluding Remarks

It shouldn’t need pointing out that nothing I say in this essay either supports or denies various mystical theories of opposition which emerged in antiquity, such as Xenophanes’ religious views or pronouncements about opposition in Orphism. Plato’s discussions of opposition, which are the focus of my attention here, tended to confine themselves to logical and metaphysical issues, although many other aspects of his philosophy were overtly religious in character. Plotinus did extend what Plato said about opposition to an entire mythology of good and evil, an extension which likewise does not interest me for the purposes of this essay.

Socrates did not invent argument. The sophists and some other Presocratics were using argument at exactly the same time Socrates flourished and many of his comments imply that argumentation, particularly legal argumentation, was a given when he came on the scene. Many of the passages involving reasoning in the Platonic dialogues are merely illustrative. Their conclusion is to show the inconclusiveness of argument, or at least argument of a certain sort. There is a point in many dialogues where argument is insufficient or hopelessly tangled and it has to be supplemented by myth. These are new myths invented by Plato usually to show how an individual was, is or could be acquainted with the Platonic Ideas. Some dialogues, like the Lysis¸ stop short of the mythopoeic stage. For Plato there are good and bad arguments like there are good and bad myths. When he disapproves of myths or art it is usually on moral or religious grounds. Good myths are the ones he devises himself and they usually function as allegories of moral behavior and, as I mentioned, icons of the way we come to know Platonic Ideas. Bad arguments can be refuted and thus shown to be bad. Some, like Lysias’ tract in the Phaedrus are that way because they are mostly concerned with elegance of expression and rhetorical techniques for inducing conviction rather than soundness of reasoning. Good argument depends on first knowing the truth and includes definition and discrimination.

There is now no necessary and sufficient specification of the properties of contrapositivity which allows us to draw conclusions with the same level of validity as, say, deductive logic. That does not mean that a logic of contrapositives may not be devised in the future. Never say never.

Opposition survives as something of a heuristic in empirical science in ideas such as electric and magnetic polarity or spin direction. But that is just a convenient way of organizing our concepts, like non-inheritance based morphological phylogenies in biology. Opposition usually just means that if one pole is manifested in a phenomenon, the other one can't be. Because of this Plato did not succeed in “refuting” Empedocles whose concepts of opposition and attraction survive as principles of explanation in the physical sciences even though some applications of these concepts are certainly not logically necessary.

Socrates as usual gets the final speech:

Οὐκοῦν ἐὰν ἀπολλύηται τὰ κακά, ἅ γε μὴ τυγχάνει ὄντα κακά, τί προσήκει τοῖς κακοῖς συναπόλλυσθαι; Οὐδέν. (Lysis 221B)