A modified subformula property for the modal logic KC4

Abstract

The modal logic KC4 is the smallest normal modal logic K with the additional axiom $\Box\Box A \mathbin{\supset} \Box A$ called C4, which claims density of the accessible relation. In this article, a modified subformula property for this logic is shown by devising an appropriate sequent calculus. The finite model property and decidability of this logic are easy corollaries.

It is peculiar that a modified subformula for this logic is defined for a set of formulas but not for a single formula.