Polytropy

The senior essay that I wrote at St John’s College was called something like ‘An account based in Aristotle of the Law of Contradiction’. I do not know now what the point was. I had read the Metaphysics in a preceptorial, so I decided to spend even more time with this book in writing my essay. I remember noting ultimately that humans could indeed be self-contradictory. Hector was an example. To Andromache he described two incompatible expectations: that their son would win renown, and that the boy would die as an infant when the Greeks took Troy.

The Metaphysics includes the following words about those who deny the Law of Contradiction,—in particular Heraclitus and his follower Cratylus (1010^a1–15, taken from Project Perseus):

—αἴτιον δὲ τῆς δόξης τούτοις ὅτι περὶ τῶν ὄντων μὲν τὴν ἀλήθειαν ἐσκόπουν, τὰ δ’ ὄντα ὑπέλαβον εἶναι τὰ αἰσθητὰ μόνον: ἐν δὲ τούτοις πολλὴ ἡ τοῦ ἀορίστου φύσις ἐνυπάρχει καὶ ἡ τοῦ ὄντος οὕτως ὥσπερ εἴπομεν: διὸ εἰκότως μὲν λέγουσιν, οὐκ ἀληθῆ δὲ λέγουσιν ̔οὕτω γὰρ ἁρμόττει μᾶλλον εἰπεῖν ἢ ὥσπερ Ἐπίχαρμος εἰς Ξενοφάνην̓. ἔτι δὲ πᾶσαν ὁρῶντες ταύτην κινουμένην τὴν φύσιν, κατὰ δὲ τοῦ μεταβάλλοντος οὐθὲν ἀληθευόμενον, περί γε τὸ πάντῃ πάντως μεταβάλλον οὐκ ἐνδέχεσθαι ἀληθεύειν. ἐκ γὰρ ταύτης τῆς ὑπολήψεως ἐξήνθησεν ἡ ἀκροτάτη δόξα τῶν εἰρημένων, ἡ τῶν φασκόντων ἡρακλειτίζειν καὶ οἵαν Κρατύλος εἶχεν, ὃς τὸ τελευταῖον οὐθὲν ᾤετο δεῖν λέγειν ἀλλὰ τὸν δάκτυλον ἐκίνει μόνον, καὶ Ἡρακλείτῳ ἐπετίμα εἰπόντι ὅτι δὶς τῷ αὐτῷ ποταμῷ οὐκ ἔστιν ἐμβῆναι: αὐτὸς γὰρ ᾤετο οὐδ’ ἅπαξ.

But the reason why these men hold this view is that although they studied the truth about reality, they supposed that reality is confined to sensible things, in which the nature of the Indeterminate, i.e. of Being in the sense which we have explained, is abundantly present. (Thus their statements, though plausible, are not true; this form of the criticism is more suitable than that which Epicharmus applied to Xenophanes.) And further, observing that all this indeterminate substance is in motion, and that no true predication can be made of that which changes, they supposed that it is impossible to make any true statement about that which is in all ways and entirely changeable. For it was from this supposition that there blossomed forth the most extreme view of those which we have mentioned, that of the professed followers of Heraclitus, and such as Cratylus held, who ended by thinking that one need not say anything, and only moved his finger; and who criticized Heraclitus for saying that one cannot enter the same river twice, for he himself held that it cannot be done even once.

The figure of the finger-wagging Cratylus was somehow appealing to me. I fancied there was a similarity between him and the Zen teacher Gutei described in the story ‘One finger Zen’ in 101 Zen Stories (part of Zen Flesh, Zen Bones). The translation that I quote here seems to be inconsequentially different from the Reps and Senzaki version that I am familiar with:

Gutei was a Zen teacher who had a habit of answering questions by simply raising a single finger. One day Gutei noticed a young boy imitating him. Someone had asked the boy what the master had taught that day, and the boy cheekily raised his finger. Gutei grabbed the boy suddenly and cut off his finger.

The boy yelped and ran away, but Gutei called out to the boy. The boy stopped and looked back. As he did so Gutei raised his finger. In that instant the boy was enlightened.

I used this story as an epigram for my essay. When asked by one of the examining tutors what the point was, I said something about wordless communication. Gutei’s finger enlightened the boy. But in fact I cannot say what this means. I still imagine that both Cratylus and Gutei had come to the understanding that words could not convey what needed to be conveyed. The finger-raising was a sign of this understanding. But then anybody can imitate the sign without having the understanding. Gutei punished a boy for this. We do not know if Cratylus was moved to do anything similar.

I ultimately became a mathematician specializing in logic. This means I studied logic as another kind of mathematics, not as an analysis of reasoning in general. Only after becoming a university teacher did I pay much attention to general accounts of logic as such. But one such account was that of Wilfrid Hodges, whose big red book Model Theory I had bought when it came out in 1993 (I was a student then). His Logic: An introduction to elementary logic (second edition, Penguin, 2001), I happened upon in an English-language bookshop in Amsterdam a dozen years later. (It may be a weakness of mine, but I am pleased to have in my possession a physical copy of the book, which I bought in an interesting place.)

I am impressed by this book’s admission that our view of logic may in future be revolutionized. And it refers back to Aristotle’s words on Cratylus. This is from the beginning of the chapter called ‘When is a sentence true?’:

Most contemporary logicians believe that there are two fundamental links between words and things. The first link is that declarative sentences are true in certain situations, and not true in other situations. The second link is that certain phrases refer to things in certain situations. Of course science advances, and it may be that some Einstein of logic will appear in A.D. 2050 and convince us that some quite different and hitherto unimagined notion is the key to the relations between words and things. But at present there is no sign of this.

Perhaps the first thinkers to take seriously the questions we now consider were the Heraclitean philosophers of ancient Greece, who maintained that ‘It is impossible to say anything true about things which change.’ One of them, Cratylus, found the whole matter so distressing that he thought it best to stop talking altogether, and simply waggle his finger.

And this is from the end of the same chapter:

Is there any difference between possible situations and the actual situation, apart from the fact that only the last one is actual? Do possible but not actual situations exist in the real world? Questions like these have made many logicians deeply unhappy about possible situations. The whole notion seemed much too speculative and metaphysical to them. They hoped that logic would guide us to greater certainties, not to perplexing questions about imaginary states of affairs. Alas, perplexing questions do not go away if we ignore them. There may be some better approach to the links between language and the world; but is it likely that Cratylus and his followers will lead us there?