Diviser et rassembler. Comment Aristote réforme la division platonicienne au profit de la déduction

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Existence, Meaning and the Law of Excluded Middle. A dialogical approach to Hermann Weyl’s philosophical considerations

International audienceIntuitionistic logic is often presented as a proof-based approach to logic, where truth is defined as having a proof. I shall stress another dimension which is also important: that of the constitution of meaning. This dimension of meaning does not reduce to proof, be it actual or potential, as standard presentations of intuitionistic logic put it. The law of excluded middle sits right at the junction between these two dimensions, proof and meaning: in intuitionistic logic, there is no proof for the law of excluded middle, but the law of excluded middle is a proposition that does have meaning. It is thus a problematic case. Better understanding how these two levels, meaning and proof, dissociate and interact is the purpose of this paper. I contend that the dialogical framework, a logical framework developing first and foremost intuitionistic logic (though it can also accommodate classical logic), allows to separate these two levels, meaning and proof, and show how the level of proofs rests on the level of meaning. In this respect, the law of excluded middle becomes a meaning-constitutive principle, even if it is neither proved nor refuted. The dialogical framework can thus integrate the philosophical considerations of Hermann Weyl's "intuitionistic episode" of the 1920s, which, I contend, already present a similar distinction between the level of meaning and the level of existence

DIALECTIC, THE DICTUM DE OMNI AND ECTHESIS

Given that demonstrations can be analysed into inferences, Aristotle is naturally understood as having constructed the theory of inference (or deduction) of Prior Analytics as a tool for his theory of demonstrative science in Posterior Analytics, and there is a long tradition of commentators, harking back at least to Pacius, according to which Aristotle’s belief that there is no demonstrative knowledge of singulars terms entails that inferences in Prior Analytics could not involve such terms. Thus, the typical syllogism ‘Humans are mortal, Socrates is human, therefore Socrates is mortal’ could not be truly Aristotelian. Still, there are a number of proofs within Aristotle’s own presentation of his theory of inference that appear at first sight to involve singular terms, among them, what has been called ‘proofs by ecthesis’, such as the proof of the convertibility of universal negatives or ‘e-conversion’:Now, if A belongs to none of the Bs, then neither will B belong to any of the As. For if it does belong to some (for instance to C), it will not be true that A belongs to none of the Bs, since C is one of the Bs. Although he does not use that word in this very passage, Aristotle calls the selection of a C ‘ecthesis’ (ἔκθεσις) – translated by Robin Smith and others before him as ‘setting out’. In this paper, we would like to address two problems engendered by the presence of proofs by ecthesis. (1) Do they involve singular or general terms? In the last century, Łukasiewicz notoriously argued that they involve only the latter. However, this issue remains unsettled, as we shall see in the next section. (2) Is ecthesis a separate procedure somewhat external to the theory of inference or is it constitutive of it? While ecthesis is usually treated in the secondary literature as an alternative mode of inference, i.e., as part of Aristotle’s inferential arsenal, so to speak, it is usually considered as not truly pertaining to his theory of inferences. In Robin Smith’s words, “it is virtually redundant”. If this were the case, one wonders why Aristotle did not simply do away with its occurrences, instead of marring his presentation with them. Hence this second problem.In order to answer both of these problems, we shall propose a new perspective on ecthesis, presenting it as a procedure such that (answering the second question) it will be seen as fully pertaining to Aristotle’s theory of inference, and (answering the first one) as involving both singular and general terms. These answers will be motivated in sections 2-3, with a critical review of alternatives in the secondary literature. But we should state at the outset that, according to our perspective, although it is part of the theory of inference, ecthesis is not at the same level, so to speak, as that of the rules of syllogisms and of conversion. With the dictum de omni one can recover the meaning explanation of the main building blocks of Aristotle’s theory of inference, the universal affirmative (AaB), universal negative (AeB), particular affirmative (AiB), and particular negative (AoB) propositions, and we see ecthesis as a procedure implementing the dictum, that allows one to prove the admissibility of the basic rules of his theory, i.e., rules of the first figure (Barbara, Celarent, Darii and Ferio), and the three conversion rules (for propositions a-e-i).While our perspective involves a bit of ‘formalism’, it is meant to be more historically sensitive than is usually the case in the secondary literature on logical aspects of Aristotle, as it relies on the claim that dialectic, far from being simply discarded by Aristotle when he wrote Prior Analytics, actually forms its historical context. In this we follow E. W. Beth, Kurt Ebbinghaus, Mathieu Marion & Helge Rückert, and claim that the dictum de omni at An. Pr.  2, 24b28-29 originates in a dialectical rule in Top.  2, 157a34-37, that involves one of the players, in their terminology (taken from Aristotle), Questioner getting the other player Answerer, to concede a few instances before she can introduce a universal affirmative proposition such ‘A belongs to all B’ (‘AaB’), and ask Answerer for a counterexample: if unable to provide one, Answerer must then concede it. To argue their point, Marion & Rückert followed a suggestion by Jan von Plato in using Martin-Löf’s Constructive Type Theory to read AaB as meaning that no c of type B – or no ‘c : B’ – can be found for which it is not the case that A(c). We shall here travel further along that path, using a dialogical take on CTT that yields an interactive logical framework called ‘immanent reasoning’, which we will adapt to Aristotle’s syllogistic. We motivate this approach to ecthesis in sections 2-3, provide rules for syllogistic reasoning within it in section 4 and proofs within this logical framework of Aristotle’s uses of ecthesis, and of rules of syllogism and conversion in section 5 and the Appendix. But, we begin with a brief overview of ἔκθεσις in the Prior Analytics, in order further to clarify the meaning of that expression and make our two problems more precise