A solution to the completeness problem for weakly aggregative modal logic

Abstract

We are accustomed to regarding K as the weakest modal logic admitting of a relational semantics in the style made popular by Kripke. However, in a series of papers which demonstrates a startling connection between modal logic and the theory of paraconsistent inference, Ray Jennings and Peter Schotch have developed a generalized relational frame theory which articulates an infinite hierarchy of sublogics of K , each expressing a species of “weakly aggregative necessity”. Recall that K is axiomatized, in the presence of N and RM , by the schema of “binary aggregation” For each n ≥ 1, the weakly aggregative modal logic K n is axiomatized by replacing K with the schema of “ n -ary aggregation” which is an n -ary relaxation, or weakening, of K . Note that K 1 = K . In [3], the authors claim without proof that K n is determined by the class of frames F = ( W, R ), where W is a nonempty set and R is an ( n + 1)-ary relation on W , under the generalization of Kriple's truth condition according to which □ α is true at a point w in W if and only if α is true at one of x 1 ,…, x n for all x 1 ,…, x n in W such that Rw, x 1 ,…, x n .