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「ANTI-REALISM, REASONS AND GAMES」
住所：東京都港区三田 3-1-7 三田東宝ビル 8F
e-mail：philosophy [at] abelard.flet.keio.ac.jp
When one wishes to introduce a new logic, the normal way to proceed is in three steps: first, one presents in an informal manner some semantic intuitions. Secondly, one presents a formal semantics that adequately captures these intuitions. Finally, one develops the corresponding syntax. However, new developments in logic hardly ever follow this sequence, as the second step is often skipped. One example is intuitionistic logic: on the basis of some philosophical theses, Brouwer made some suggestions of a syntactical nature in 1908, skipping altogether the second step. It is only later, in the 1930s, that Kolmogorov and Heyting proposed a formal semantics. More recently, Jean-Yves Girard introduced linear logic in a syntactical manner in 1987 and the search for a proper semantics is still going on today. As I see it, the role of philosophy is limited mainly to the first step, where a philosophical discussion can give good reasons to take seriously the intuitions that will be captured within the formal semantics in the second step. One should now notice that in both cases just mentioned, the least one can say is that the philosophical motivations are found lacking. Brouwer's mystical and mentalistic philosophy has been largely set aside as indefensible, while Girard's remarks concerning Engel's "dialectic of nature" have not attracted much attention.
We owe to Paul Lorenzen an extraordinarily rich intuitive idea, first presented in his 1958 paper 'Logik und Agon' (Lorenzen 1960), the fruitfulness of which we are barely encompassing today. This is the idea of defining logical particles in terms of rules of a two-persons (the proponent and the opponent), non-collaborative game and to define logical truth in terms of the existence of a winning strategy for the proponent. Lorenzen's original intention was to recover with his dialogical approach the rules for Beth's intuitionistic tableaux. But the equivalence theorem necessary between proofs in Gentzen's calculus LJ for intuitionistic logic and strategies for winning dialogues was obtained only in 1985 by Walter Felscher (Felscher 1985, 1986). This result came at the end of a long search, within Lorenzen's 'school', for the right set of restrictions on structural rules for dialogues needed to obtain intuitionistic provability. That this turns out to have been a complex, non-intuitive matter, just shows how perilous it is to jump from step one to step three and then try and fix step two.
In this paper, I wish merely to remain at the first stage and propose what seems to me the proper philosophical grounding for some semantic intuitions, which can be captured, at a later stage, by a formal semantics. I wish to propose two theses, concerning the nature of assertions and of meaning that should jointly provide a strong philosophical basis for game semantics. These two theses are a reworking of ideas first propounded by Michael Dummett. One of Dummett's most significant contribution has been to provide, in lieu of Brouwer's philosophy, a thorough philosophical argument in support of Heyting semantics, in the form of an argument for an antirealist theory of meaning for mathematical statements. However, in recent years his antirealist challenge has attracted less and less attention. The suggestions that I shall make imply a profound revision of this approach, aspects of which I cannot really discuss here. At least I should say this: Dummett calibrated his antirealist challenge in order to give support to the intuitionistic critique of the Law of Excluded Middle. In this sense, he already presupposed a logic at the third step in our rational sequence and tailored his discussion at the first step in order to lead us to it. In contrast, I shall not make any such presupposition here. My aim is to pick up some arguments, develop them and see were they lead us. Although my constructivist bias should be plain, I wish to remain uncommitted to any logic.