Infinitary Classical Logic: Recursive Equations and Interactive Semantics

TL;DR

This paper introduces an interactive semantics for infinitary classical logic, using debate-like interactions to characterize derivations and prove a completeness theorem for infinitary derivations.

Contribution

It develops a novel interactive semantics framework for infinitary classical logic and establishes a completeness theorem linking syntactic derivations to interactive correctness.

Findings

Unique solutions for recursive formula equations.

Interactive completeness theorem proven.

Semantic framework for infinitary logic established.

Abstract

In this paper, we present an interactive semantics for derivations in an infinitary extension of classical logic. The formulas of our language are possibly infinitary trees labeled by propositional variables and logical connectives. We show that in our setting every recursive formula equation has a unique solution. As for derivations, we use an infinitary variant of Tait-calculus to derive sequents. The interactive semantics for derivations that we introduce in this article is presented as a debate (interaction tree) between a test << T >> (derivation candidate, Proponent) and an environment << not S >> (negation of a sequent, Opponent). We show a completeness theorem for derivations that we call interactive completeness theorem: the interaction between << T >> (test) and << not S >> (environment) does not produce errors (i.e., Proponent wins) just in case << T >> comes from a syntactical derivation of << S >>.