  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.
