I. Aristotle's exposition and defence of the principle of non-contradiction

One prominent feature of Aristotle's defence of these principles is that while Hegel's contemporaries usually characterised them as Denkgesetze, laws of thought or thinking, Aristotle himself does not. He regards them rather as claims about what is or being. He formulates the principle of non-contradiction in two main ways: ‘It is impossible for the same thing to belong and not to belong simultaneously to the same thing and in the same respect’ (Metaphysics G.iii.9).Footnote ⁴ Sometimes he states it in terms of ‘being’: ‘It is impossible for the same thing to be and not to be at one and the same time’ (K, v. 1; G.iii). Aristotle is here on the same terrain as Hegel. Whatever Hegel believes about contradiction he agrees with Aristotle that it is a matter of being or of essence, not simply a law about how we do, or ought to, think. If it were only law of our thinking, that would seem to leave open the possibility that even if we cannot, or should not, think in contradictory ways, things might nevertheless be self-contradictory, even if we cannot or should not think about it.Footnote ⁵ But Aristotle insists that things really are not self-contradictory—and Hegel seems to say that they really are contradictory. Another interesting reason for not treating the principle of non-contradiction as a law of thought, either in the descriptive or the prescriptive sense, is that I inevitably incur the risk of contradiction if I confess my fallibility. In giving a lecture or writing a book, I make many claims, each of which I believe to be true. But at the beginning I confess that I am bound to have made some mistakes. Then I have contradicted myself. I assert that p, that q, that r, and so on, but also that one or more of these propositions is false. But the things themselves are not self-contradictory. If I am right in saying that I have made some mistakes, then one or more of my claims must be false.Footnote ⁶

Understandably enough, the principle of non-contradiction is so fundamental that Aristotle finds it difficult to justify it. One relevant claim he makes is that no one disbelieves it. He admits, of course, that some philosophers seem to reject it—Heraclitus, for example, and Protagoras, and later he could have added Hegel. But they don't really disbelieve it.Footnote ⁷ Of course, people often believe things that are implicitly contradictory, but when the contradiction is pointed out to them and made explicit they almost invariably modify their claim, however reluctantly. But the universal, or near universal, acceptance of the principle of non-contradiction makes it hard to argue for it. There seems to be no more fundamental principle from which we could derive it. So now Aristotle tries his second defence. The details of this are complex and obscure. But the rough idea is this. You get your opponent to say something definite, such as ‘Socrates is a man’. But then he has to say ‘Socrates is not a dog’. Now the trouble begins. If he believes that everything is contradictory, he also has to add ‘Socrates is not a man’ and ‘Socrates is a dog’. The upshot of this is that the same applies to all predicates. Socrates is all of the following: a man, a dog, a mountain, a ship, and also none of them. And the same will be true of everything else. Everything ends up being the same single thing, a conglomerate of all possible predicates and of their negations. We might remember that in English the word ‘man’ is quite ambiguous, depending on whether it contrasts with ‘beast’—its central use—or with ‘woman’, ‘boy’, ‘officer’, or, in a metaphorical use, with ‘mouse’. Socrates may well be a man in one of these senses, but not a man in some other sense. Still, Aristotle is alert to such ambiguities and he explicitly rejects the exploitation of ambiguity to defend contradiction.

This argument is effective only against someone who claims that everything is contradictory. It does not work against someone who claims only that some things are contradictory, that, for example, there are contradictory states of affairs in Sparta, but not in Athens, or that Socrates is unique in being both a man and not a man.Footnote ⁸ But Aristotle's opponents do seem to have claimed that everything is contradictory. And so too does Hegel in his account of contradiction in the second part of his Logic, the doctrine of essence. ‘All things are in themselves contradictory’, he says (Alle Dinge sind an sich selbst widersprechend) (WL: 381).Footnote ⁹ There are occasions when it looks as if the principle of non-contradiction is too rigid, when we want to answer a question with ‘Yes and no’ rather than with a straight ‘yes’ or a straight ‘no’. Is someone standing in the doorway in the room or not? Yes and no. But he is definitely standing in the doorway. We cannot say that he both is and is not standing in the doorway. We cannot respond to every question with ‘Yes and no’. If we replace concepts that are too vague to describe the situation, such as ‘in the room’ and ‘out of the room’, with concepts that are for our purpose more definite, we can give a straight answer. In fact, universal contradictoriness would seem to undermine Hegel's own claim. Is everything contradictory? Yes, it is and no, it is not. Does Hegel believe that everything is contradictory? Yes, he does. But no, he does not.

II. Aristotelian contradiction versus Hegelian contradictions

I shall argue that the sense in which Hegel believes that everything is contradictory—in the account of essence I just referred to—does not conflict with the claim that nothing is contradictory, as Aristotle understands it. The reason for this is that whereas Aristotle is concerned with a predicate and the negation of that very same predicate, arguing that a predicate cannot both apply and not apply to the same thing, Hegel is concerned with the application of opposite predicates to one and the same thing. Opposition within a thing is often described as a contradiction.Footnote ¹⁰ A quotation from Lytton Strachey's Eminent Victorians exemplifies this use of the word. Speaking of General Gordon, Strachey says: ‘In the depths of Gordon's soul there were intertwining contradictions – intricate recesses where egoism and renunciation melted into one another, where the flesh lost itself in the spirit and the spirit in the flesh’ (Strachey Reference Strachey1918: 222f.). Strachey describes Gladstone in similar terms, though here he speaks misleadingly of ‘incompatibles’ rather than ‘contradictions’:

Strachey continues in a way that suggests Hegel's transition, in the Logic, from contradiction to ground (Grund):

I quoted these passages not only because of their rhetorical grandeur, but also to illustrate the vagaries of English usage. But it is better to look at a simpler case, Catullus's poem odi et amo, I hate and love Lesbia, presumably at the same time.Footnote ¹¹ Hate and love are opposites. To say you hate someone usually implies that you don't love them, and saying you love them usually implies that you don't hate them. But they are not logically incompatible. Loving someone does not entail not hating them. It is possible to love and hate at the same time. So Catullus is not saying that he both loves Lesbia and does not love her. Nor is he saying that his love or his hatred is lukewarm. He definitely loves her and definitely hates her. But because love and hate are not logically incompatible, his situation and his declaration of it is not contradictory in the way that Aristotle forbids. On the other hand, it does involve a contradiction in Strachey's sense. It is not entirely innocuous like some of Hegel's other exhibits—for example, that the way up is also the way down and that the road to the East is the way to the West: that's obviously so, it all depends on which way you are going. We can here introduce negation—as opposites, love and hate are negations of each other. In fact, ‘I hate her’ is ordinarily regarded as a stronger negation of ‘I love her’ than ‘No, we are just good friends.’ But opposition is Hegelian negation, not the negation of a proposition: ‘X loves Y’ does not entail ‘X does not hate Y’. And the upshot of it is Hegelian contradiction, or Stracheyan contradiction, not Aristotelian contradiction.

Does this entitle Hegel to say that everything is contradictory? Perhaps General Gordon, Gladstone, and Catullus are special cases. I will come on to that later. For the moment I'll take issue with Strachey's claim that there are no chimeras in the physical universe. One such chimera is the atom itself, which is held together by negatively charged electrons and positively charged positrons. Another, rather different example, is light. In some respects its movement resembles that of particles, and Newton held that light consists of particles. In other respects its movement resembles waves, so others believed it to consist of waves. These two models were regarded as logically incompatible with each other, and so they are on the macroscopic level, but they were kept in play by their success in dealing with the phenomena that respectively support them. Leaving aside the details, the resolution of the contradiction is that tiny particles moving at vast speeds behave like waves. So there is no logical contradiction—light is both a particle and a wave. We might regard that as a Stracheyan contradiction, and even as the grounding of the contradiction in quantum mechanics. Another such contradiction is that between, say, a human body and the surrounding atmosphere: the integrity of the body is sustained by the surrounding pressure and the body would explode if the atmosphere were removed.Footnote ¹² Such interdependencies as these play a part in Hegel's treatment of the principle of identity.Footnote ¹³ Of course, these Stracheyan contradictions differ from each other in important respects. General Gordon's contradiction brought him to a sad end, whereas Gladstone's did not. Catullus's contradiction may have tortured him, but it helped to confirm his status as a poet. Atoms and light are blissfully unaware of their contradictory nature.

III. Being and nothing

We are not finished yet, however, with Aristotelian contradiction. In his account of contradiction, Aristotle presupposes that there are determinate entities, such as Socrates, with determinate natures, such as humanity, and does not explain, in these passages, how these come about.Footnote ¹⁴ Might contradiction be lurking at this deeper level? Hegel, in his Logic, attempts to explain such matters, and seemingly does so more thoroughly than Aristotle ever did. Moreover, Hegel's Logic opens with what looks like a blatant contradiction, that ‘Being and nothing is the same’ (EL: §88R). Hegel himself characterises this, and its upshot ‘becoming’, as a contradiction: ‘In becoming, being, as one with nothing, and nothing as one with being, are only vanishing [terms]; because of its contradiction becoming collapses inwardly, into the unity within which both are sublated; in this way its result is being-there’ (EL: §89).Footnote ¹⁵ Hegel then adds: ‘There is nothing at all anywhere, in which contradiction—i.e. opposed determinations—cannot and should not be exhibited’ (EL: §89R).Footnote ¹⁶ This addition gives the game away. ‘Being’ (Sein) and ‘nothing’ (Nichts) are, like love and hate, ‘opposed determinations’, not strict contradictories. Thus to say that each is the other does not involve an Aristotelian contradiction. It does not entail that Being is not Being, or that Nothing is not Nothing. (To make this more plausible we might interpret Hegel's dictum as implying that if something, such as God, is characterised as pure undiluted Being, with no other qualities, this amounts to saying that God is Nothing. Whatever the merits of this claim, it is not contradictory in Aristotle's sense.) Moreover, whatever contradiction there may be between Being and Nothing, the contradiction lies not within an existing entity, but in abstract concepts, whose interrelationships somehow give rise to particular entities. There is no paradox in the suggestion that concepts may embody contradictions. The problem arises only if concepts are supposed to transmit their contradictions to existing things.

IV. Kant's antinomies and Zeno's paradoxes

There are, however, other areas in which we might still find Hegel endorsing Aristotelian contradictions: firstly, Kant's antinomies, and secondly, Zeno's arguments against change and motion—though, since Hegel believed in change and motion, he would have to regard them as arguments for the view that change and motion, which undoubtedly occur, are nevertheless contradictory. That is a fairly common dilemma in philosophy. There are phenomena that seem to occur, but which also seem to be contradictory. Some familiar examples are: free will, weakness of will, motion, time, direct perception of objects, and knowledge of other minds. There are four possible types of response to such dilemmas:

1. 1. You can accept the contradictions and agree that the phenomenon in question does not occur. According to Plato's early dialogues this was Socrates's belief about akrasia or weakness of will. McTaggart (Reference McTaggart1908) similarly denied the existence of time owing to its supposed contradictoriness.

2. 2. You can accept the contradictions, but nevertheless insist that the contradictory phenomenon occurs. It is not easy to find actual cases of this approach. Perhaps Zeno himself is one of them. Some commentators, and possibly Hegel himself, believe Hegel to be such a case.

3. 3. You can accept the contradictions, but reinterpret the phenomenon so as to avoid them, for example by giving a behaviourist analysis of our mental life or a phenomenalist account of physical objects. It is often hard to distinguish this from the first approach, to decide, that is, whether the supposedly contradictory phenomenon has been acquitted of contradiction and reinstated in a sanitized form or, rather, analyzed out of existence.

4. 4. You can deny the contradictions and accept the occurrence of the phenomena, sometimes going on to unravel the contradictions and sometimes not. Peter Strawson (esp. Reference Strawson1985) was a practitioner of the first procedure, attempting to defend the existence of such phenomena as knowledge, free will, colour and mentality, phenomena accepted by common sense, and to defend them in the form in which common sense interprets them. By contrast, with no attempt at verbal argument Diogenes the Cynic purported to refute Zeno of Elea's denial of motion by simply walking up and down, and Dr Johnson similarly refuted Berkeley's denial of matter by kicking a rock. There is also a strain of Diogenes's and Johnson's brusque common sense in Strawson, as when, in response to the anti-realist analysis of the feelings of others and of past events, that is alternative 3, he proclaimed: ‘The seas of argument may wash for ever around these rocks of truth, but the rocks are not worn away’ (Strawson Reference Strawson, Strawson and Montague2011: 122).

When it comes to Kant's antinomies, it is not very clear to me which of these alternatives Hegel adopts. I think it is safe to say that he rejects the first alternative, that is, he does not accept Kant's conclusion that the phenomena in question are only the way in which things appear to us and not things in themselves. There is an element of the third approach. For example, he does not accept Kant's view that the opposition between freedom and necessity is irreconcilable, but this requires some reinterpretation of our concept of freedom (see, e.g., PM: §484). But does Hegel believe that Kant has exposed real contradictions, Aristotelian contradictions, in these phenomena? I think probably not. For one thing, in describing the antinomies, Hegel usually speaks of ‘opposition’ or ‘antithesis’, not of the affirmation and negation of one and the same proposition. That leaves open the possibility of reconciling the opposites in a way that Aristotelian contradictories do not. For another, Hegel implies that what is at fault in these antinomies is not the phenomena themselves, but the concepts in terms of which Kant describes them. If we apply common-or-garden concepts to these difficult questions, we end up in a contradictory muddle. We saw that earlier in the case of light. Let me take Kant's first antinomy, that there are insoluble problems in claiming that world is finite in space and also in the claim that it is infinite in space. Kant's argument against the infinity of space is somewhat murky, but there is an obvious problem with its finitude if we take it to imply that space has a boundary. As the Epicurean poet Lucretius asked: Where will a javelin go if you throw it from the boundary?

There are several possible answers to this besides Lucretius's conclusion that space is infinite in the ordinary sense. One answer might be that it requires more and more energy to proceed as the javelin gets closer to the boundary, so that infinite energy is needed to reach it. (This of course raises the problem of infinity again.) Another might be that if you hurl the javelin hard enough, it will come round and hit you in the neck, because space is curved and perhaps spherical.Footnote ¹⁷ It is surprising that Hegel did not consider this solution, since circular infinity, in effect finitude without a boundary, is one of his most fruitful ideas. Perhaps it is because, by parity of reasoning, it would have committed him to the circularity of time—which is harder to defend. So he left that for Nietzsche. Or again we might appeal to Aristotle's idea of potential infinity rather than actual infinity: you can always go further in space, but you must not ask about the whole of space. This is roughly Kant's own solution—and also Locke'sFootnote ¹⁸. Hegel may have favoured this solution, because in the section on Quantity in the Science of Logic, he defends Aristotle's idea of potential infinity against Pierre Bayle's dismissal of it, and regards Aristotle's handling of Zeno's paradoxes as exemplary.Footnote ¹⁹

V. Zeno of Elea and the moving arrow

I shall now leave Kant's antinomies and turn to Zeno's paradoxes, both because Hegel thought they were better than Kant's and because my goal is to talk about Greek philosophy. The paradox most relevant to Hegel is the arrow paradox. At a particular instant, t1, the arrow, or let us say the tip of the arrow, is at a particular point, p1. We can not, I think, appeal to Aristotle's belief that infinity is only potential, not actual. This forbids us from speaking of all the points on the arrow's path and all the instants of its flight, but it does not prevent us from picking out one point and one instant.Footnote ²⁰ So the problem is this. Pick any instant on the arrow's supposed flight. At that instant the arrow is stationary, since it cannot move any distance in an instant. So how can the arrow move at all?

Hegel's solution is the nearest thing to an Aristotelian contradiction I have found in his writings. In his Science of Logic he says: ‘External, sensuous motion is itself contradiction's immediate existence. Something moves, not because now it is here and there at another now, but because in one and the same now it is here and not here; because in this here it is and is not at the same time’.Footnote ²¹ This looks like a real Aristotelian contradiction. The tip of an arrow is in a definite place, p1, at an instant of time, t1. It is also not in p1 at t1. The contradiction is, however, an instantaneous contradiction. The arrow is in a contradictory state at t1, but it is not in that state for any stretch of time. Not that particular state, at least. By parity of reasoning, at any other time it will be in the same type of contradictory state but a different one—being both at pn at tn and not at pn at tn. (Of course I can not continue using natural numbers here, speaking of the next instant as t2—there is no next instant: the number of instants in a stretch is not countable or denumerable.) But why should we agree that there is any contradiction in the first place? The arrow might hit its target at t1 and suddenly stop moving. Should we not say then that the arrow is at p1 at t1, and not not at p1? Moreover, even if the arrow continues its flight, saying that it is both at p1 and not at p1 does not look like an informative explanation of its onward movement, but just a fancy way of saying that it does move onward. Admittedly, the state of the arrow at t1 is quite paradoxical and can only be described in strange ways. It is moving at t1, but it does not at t1 move any distance. When does it start being at p1? Not at any earlier instant. So it must start being at p1 at t1, the very instant at which it is at p1. By parity of reasoning, it cannot stop being at p1 at any later instant. So it stops being at p1 at t1, the very same instant as that at which it starts being at p1 and is at p1. Aristotle himself raises an analogous problem about time alone. When does it stop being exactly 6 o'clock? It cannot stop being 6 o'clock after 6 o'clock, so it must stop being 6 o'clock at 6 o'clock—and again by parity of reasoning that is also when it starts being 6 o'clock.Footnote ²² This is all very paradoxical and possibly Stracheyan—things do not usually start and stop at exactly the same time, even if we say such things as ‘No sooner had the rain started than it stopped again’. But none of it need commit us to accepting an Aristotelian contradiction, the affirmation and negation of one and the same proposition.Footnote ²³ It probably has to be conceded, however, that Hegel himself believed that he had unearthed an Aristotelian contradiction, and one that infests everything that moves and changes.

VI. Aristotelian truth versus Hegelian truth

Now I propose to approach the topic from a different angle, that of truth. Aristotle does not mention truth in his discussion of contradiction, but he does give an account of truth elsewhere. In the Metaphysics itself he says: ‘To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true’ (Metaphysics 1011b25). Aristotle does not specify what happens if I say, of what is, that it both is and is not. It looks as if I have said something true and also something false. But obviously enough we have here another version of the principle of non-contradiction: to say of anything that it both is and is not, is always false. Hegel's view of truth, and his use of the word ‘true’, is diverse and complicated, but one thing that is clear is that he does not accept this Aristotelian account of truth. To say of what is that it is, may be ‘correct’, but it is not true. There are several reasons for this. One reason is that for Hegel truth is not a piecemeal matter: ‘The truth is the whole’ (PhG §20, 11). That is, a proper truth includes the process leading to it and its subsequent development. Another reason is that truth is not a matter of either–or, but of more-or-less (§39, 19). What we regard as truths, in science or philosophy, and perhaps in daily life, are decent approximations, true enough to be getting on with, but not absolute truth—whatever that might be. Both of these factors come into play if we consider a simple case: our successive views of the shape of the Earth. We tend to think that the belief that the Earth is flat is false, plain and simple, while the belief that it is round, i.e., spherical, is true. But that is not correct. Neither belief represents the exact truth, with no admixture of falsehood, but both give a more-or-less useful approximation to the truth. The Earth is obviously not completely flat. It has hills and dales, lumps and bumps, including ones that move about like ourselves. But in the medium range it is more-or-less flat, and the flat earth view works well for local surveys. The Pythagorean conception of a spherical Earth also works well for many purposes—it is a very big sphere, so naturally local areas can still be treated as flat. So this theory accommodates the truth in the flat Earth theory, but it works better for the long range. Later, however, it was found that the Earth is not a sphere, but an oblate spheroid or ellipsoid. That is, it is flattened at the poles and bulges at the equator. But even that is not exactly true—to represent all the local lumps and dips would be mathematically too complicated. So no theory in the sequence is exactly true, but none of them is wholly false. Each one deserves a response of Yes and No rather than a straight Yes or a straight No. If we call one theory, such as the spherical theory, true, this is because we are contrasting it with its main competitor, the flat Earth theory. Each one is an improvement on its predecessor, but accommodates the element of truth in its predecessor.

This is how Heisenberg (Reference Heisenberg2000), in his lectures on quantum theory, treats his predecessors. Thales made a promising beginning by claiming that everything was water. He had the intuition that matter is the same stuff everywhere. But he did not account for its apparent diversity. So someone else comes along, Anaximander, who tries to account for that. And so it goes on until we come to Heisenberg, who expected later improvements to his own theory, but did not contemplate its complete overthrow. Hegel himself treated the history of philosophy in a similar way. He was not one of those revolutionary philosophers, like Descartes and Wittgenstein, who purport to forget about past philosophy and to start from scratch. In the second paragraph of the Preface to the Phenomenology of Spirit, for example, he compares the development of philosophy over the ages to the growth of a tree:

Thales, or whoever we suppose to have begun the development, made a promising beginning: his view is comparable to an acorn. It is obviously not the complete truth any more than an acorn is full-grown tree. So it needs to be supplemented, but not entirely rejected, by the next stage in the development. And so it goes on: Anaximander, Heraclitus, Plato, Aristotle, each add something to their predecessors producing branches, buds, leaves and so on until eventually we have a full-grown tree. At first sight, these philosophies seem incompatible with each other, but on closer inspection we can see that they form interlocking parts of an overall system which contradict each other, if at all, only in a relatively harmless Stracheyan sense, combining, say, empiricism and rationalism. Of course a philosophy may need some modification if it is to take its place in the overall system. Aristotle revised, but did not entirely abandon Plato's doctrine of forms, when he brought them down to earth, and Leibniz accepted Plato's belief in innate knowledge, while rejecting the doctrine Plato associated with it, the doctrine of the pre-natal existence of the soul. Some of Leibniz's own doctrines have undergone a similar conditional acceptance—his belief that the world consists of little souls by Heisenberg and, in a different way, by Louis Pasteur, and his belief in innate ideas by Noam Chomsky. In such cases the response to a predecessor's doctrine is not a straight No, but ‘Yes, but’.

Here we seem to have a sharp conflict between Hegel and Aristotle. On the one hand Aristotle has a bivalent or binary view of truth, holding that every sentence expressing a proposition has one and only one truth value—truth or falsity. In his logical works he presents scientific knowledge as taking a rigid syllogistic form. We start from a major premise, add a minor premise, and deduce a conclusion from it. For example:

If both premises are true, the conclusion must be true. If the conclusion is false, then at least one of the premises must be false.

Hegel was, of course, allergic to this procedure and does not apply it himself, at least in philosophical matters. In any case, it is of little use, as Aristotle acknowledges, when we are searching for the first premises of our arguments. In fact, Hegel was not very keen on propositions of this sort at all, and prefers the ‘speculative proposition’—a difficult matter that I will not enter into here.

VII. Aristotle and Hegel come together again

More important for my purpose is one striking feature of Aristotle's procedure—that by and large he does not use this deductive method when he is dealing with philosophical problems. What he does is this. He collects together what he calls ta phainomena, things that appear to be so, or endoxa, ‘reputable opinions’, and then tries to bring order and consistency into them.Footnote ²⁴ As he puts it in the Nichomachean Ethics:

In the Topics he says that dialectic starts from premises that are accepted ‘by all, or by most, or by the wise, and of those, either by all, or by most, or by those that are best known and most respectable’ (I.1,100b21–23; again following Frede's translation). In the Nicomachean Ethics he applies this procedure to a problem I mentioned earlier, akrasia or weakness of will. On the one hand, it seems obvious that there is such a phenomenon as weakness of will and most people believe that there is. On the other hand, a very wise man, Socrates, proposed a variety of ingenious arguments for his belief that no one does wrong willingly, that wrongdoing is always the result of ignorance, of not knowing the right thing to do. Aristotle's attempt to reconcile these two views is obscure and controversial. His central idea is to distinguish between two sorts of knowledge, proper knowledge and the type of knowledge one has when one is drunk or drowsy. It is only fully fledged knowledge that cannot be acted against. But the details of Aristotle's solution are of less interest here than his general strategy—which is to collect and sift reputable opinions on a topic and try to iron out their differences. More generally, Aristotle's works often begin with an account of the views of his predecessors, not only assessing them critically, but also scouring them for intimations of the causes or explanatory factors to be considered in the subject in question. The first book of the Metaphysics is primarily devoted to the views of previous philosophers from Thales to Plato, especially on the causes of things. Similarly, the first book of the Physics deals with the first principles of the subject as presented in previous theories (Physics I, esp. 1 and 5–6). The de Anima too begins by surveying previous philosophical accounts of the soul (De anima I, 2–5). So here Aristotle's methodology converges with Hegel's.