32 , 2017-07-31 , Japanese Association for the Contemporary and Applied Philosophy (JACAP)
掲載日: 2017.7.31 Proof-theoretic semantics (PTS) is a research program in logic and philosophy that tries to give semantics for various expressions, not by assigning denotations to them in a model-theoretic way, but by describing directly the "use" of them in a proof-theoretic way. The usual task that PTS purports to achieve is divided into two parts. First, it explains the meaning of sentences with the tools of proof theory, especially introduction- rules and elimination-rules in the natural deduction formalization. Second, drawing on the explanation, it justi es some system of logic, by showing the two kinds of rules are in "harmony". The early PTS by Dummett or Prawitz took the usual form of unilateral natural deduction as basis, and can justify only intuitionistic logic. However, according to Rumm t, if we move to a bilateral natural deduction, and broaden our attention to encompass denial, not only assertion, we can justify a logical system (we call it "system R") into which classical logic can be embedded. In this essay, I will criticize his argument, and present an alternative foundation for system R. In the rst part, I will summarize the basic ideas of PTS. In the second part, I will follow Rumm t's argument, which tries to justify system R by his bilateral PTS, and show one of his "coordination-rule" crashes with the meaning of sentences given by bilateral operational rules, so this rule cannot be justi ed. In the third part, we present a new system E, and justify this system with our bilateral and falsi cationistic PTS. System R can be embedded into system E, so my argument also gives justi cation to system R. This part also analyzes the reason why Rumm t's PTS cannot justify system R, and our PTS can.