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In this paper, we introduce a proof system NQGL for a Kripke complete predicate extension of the logic GL of provability. While GL is defined by K and the Lβb formula ✷(✷p ⊃ p) ⊃ ✷p, NQGL does not have the Lβb formula as its axiom, but has a non-compact rule, that is, a derivation rule with countably many premises, instead. We show that NQGL enjoys cut admissibility and is complete with respect to the class of Kripke frames such that for each world, the supremum of the length of the paths from the world is finite. [ABSTRACT FROM AUTHOR]

We analyze semantically the logical inference rules in cut-free sequent calculi for the modal logics which are obtained from the least normal logic K by adding axioms from T, 4, 5, D and B. This implies Kripke completeness, as well as the cut- elimination property or the subformula property of the calculi. By slightly modifying the arguments, the finite model property of the logics also follows. [ABSTRACT FROM AUTHOR]