Simplified Kripke style semantics for modal logics K45, KB4 and KD45

Abstract

In this paper, we show that logics K45, KB4 (= KB5) and KD45 are determined by some classes of simplified Kripke frames without binary accessibility relations between possible worlds. These frames are ordered pairs of sets <W,A>, where W is a non-empty set of worlds and A\subseteq W (a~set of common alternatives to all worlds in W). From a frame <W,A> we can construct models of the form <W,A,V>, where V is a standard valuation which to formulae and words assigns truth-values with respect to the set A. For K45 we use the class of all simplified frames; for KB4 we have the case that A=0 or A=W; and for KD45 we use frames with A\ne 0.

Moreover, to each of these logics we also assign a suitable class of finite Euclidean relational frames which satisfy conditions for normal extensions of K5 presented by Nagle.