Infinitary Classical Logic: Recursive Equations and Interactive Semantics

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 >>.