The Sophists' Challenge

Parmenides and Zeno both tried to prove the truth of Parmenides’ philosophy by means of (roughly) isomorphic logical arguments. Let us set aside for a moment the difficulties implied by the very notion that logical considerations may lead to absolute knowledge of the cosmos. We are then immediately taken by this incredible discovery. What is so marvelous and apparent is that Parmenides, Zeno and Plato alike share awareness to the isomorphism which today we know constitutes the heart of logic: sound inferences and refutations share a logical form. Sound inferences and refutations are distinct uses of the same inferential form, the same logical structure. The same valid inference is at once sound when taken to have true premises and is a refutation when taken to have a false conclusion.

The isomorphism was first formulated in a revolutionary, though somewhat inelegant, manner by Aristotle, in the opening to his Topics and again in the opening to his Prior Analytics. But it was already noticed and celebrated in the most epic philosophical episodes of all times: the meeting of Parmenides, Zeno and the young Socrates in the famous opening of Plato’s Parmenides. There, the young Socrates observes (somewhat sarcastically), that Zeno is at most an innovator of style: his claim for fame is merely his reformulating the Eleatic philosophy (giving it the form of a refutation rather than that of a sound inference, or of indirect proof rather than a straightforward one). Zeno, by the way, is quick to admit this (in the dialogue) in a manner that is perhaps humble and perhaps averse: he claims never to have pretended to offer more than mere structural variations over the Parmenidean theme.