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Cirquent calculus is a new proof-theoretic and semantic approach introducedby G.Japaridze for the needs of his theory of computability logic. The earlierarticle "From formulas to cirquents in computability logic" by Japaridzegeneralized the concept of cirquents to the version with what are termedclusterng and ranking, and showed that, through cirquents with clustering andranking, one can capture, refine and generalize the so called extended IFlogic. Japaridze's treatment of extended IF logic, however, was purelysemantical, and no deductive system was proposed. The present papersyntactically constructs a cirquent calculus system with clustering andranking, sound and complete w.r.t. the propositional fragment of cirquent-basedsemantics. Such a system can be considered not only a conservative extension ofclassical propositional logic but also, when limited to cirquents with no morethan 2 ranks, an axiomatization of purely propositional extended IF logic inits full generality.

A cirquent calculus system with clustering and ranking