A cirquent calculus system with clustering and ranking

TL;DR

This paper introduces a new cirquent calculus system with clustering and ranking, providing a sound and complete proof system for propositional logic extensions related to computability logic and extended IF logic.

Contribution

It syntactically constructs a cirquent calculus system with clustering and ranking, extending classical logic and axiomatizing propositional extended IF logic.

Findings

System is sound and complete for propositional cirquent semantics

Extends classical propositional logic conservatively

Axiomatizes propositional extended IF logic with up to 2 ranks

Abstract

Cirquent calculus is a new proof-theoretic and semantic approach introduced by G.Japaridze for the needs of his theory of computability logic. The earlier article "From formulas to cirquents in computability logic" by Japaridze generalized the concept of cirquents to the version with what are termed clusterng and ranking, and showed that, through cirquents with clustering and ranking, one can capture, refine and generalize the so called extended IF logic. Japaridze's treatment of extended IF logic, however, was purely semantical, and no deductive system was proposed. The present paper syntactically constructs a cirquent calculus system with clustering and ranking, sound and complete w.r.t. the propositional fragment of cirquent-based semantics. Such a system can be considered not only a conservative extension of classical propositional logic but also, when limited to cirquents with no more than 2 ranks, an axiomatization of purely propositional extended IF logic in its full generality.