A propositional system induced by Japaridze's approach to IF logic

Cirquent calculus is a new proof-theoretic and semantic approach introduced for the needs of computability logic by G.Japaridze, who also showed that, through cirquent calculus, one can capture, refine and generalize independence-friendly (IF) logic. Specifically, the approach allows us to account for independence from propositional connectives in the same spirit as the traditional IF logic accounts for independence from quantifiers. Japaridze's treatment of IF logic, however, was purely semantical, and no deductive system was proposed. The present article constructs a formal system sound and complete w.r.t. the propositional fragment of Japaridze's cirquent-based semantics for IF logic. Such a system can thus be considered an axiomatization of purely propositional IF logic in its full generality.