Abstract
AbstractThis paper is dedicated to extending and adapting to modal logic the approach of fractional semantics to classical logic. This is a multi-valued semantics governed by pure proof-theoretic considerations, whose truth-values are the rational numbers in the closed interval $[0,1]$ . Focusing on the modal logic K, the proposed methodology relies on three key components: bilateral sequent calculus, invertibility of the logical rules, and stability (proof-invariance). We show that our semantic analysis of K affords an informational refinement with respect to the standard Kripkean semantics (a new proof of Dugundji’s theorem is a case in point) and it raises the prospect of a proof-theoretic semantics for modal logic.
Highlights
In a previous paper [24], two of the current authors introduced an informational refinement of the standard Boolean semantics for classical propositional logic
We showed how to use HK in order to fractionally interpret modal formulas: any modal formula A is evaluated in terms of the ratio between axiomatic top-hypersequents and the totality of the top-hypersequents figuring in any derivation of A
We argued that fractional semantics conveys a different way of thinking about proof-theoretic semantics due to its commitment to the primacy of the notion of proof
Summary
In a previous paper [24], two of the current authors introduced an informational refinement of the standard Boolean semantics for classical propositional logic. Any proof in which no application of the complementary axiom occurs will necessarily end with a hypersequent signed with “ .” by Theorem 2.4, the formula A is valid and, by the rule of denecessitation (from A to A), we get the claim of the theorem.
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