A solution to the completeness problem for weakly aggregative modal logic
Peter Apostoli, Bryson Brown · Journal of Symbolic Logic · 1995
We are accustomed to regardingKas 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 ofK, each expressing a species of “weakly aggregative necessity”. Recall thatKis axiomatized, in the presence ofNandRM, by the schema of “binary aggregation” For eachn≥ 1, the weakly aggregative modal logicKnis axiomatized by replacingKwith the schema of “n-ary aggregation” which is ann-ary relaxation, or weakening, ofK. Note thatK1=K. In [3], the authors claim without proof thatKnis determined by the class of framesF= (W, R), whereWis a nonempty set andRis an (n+ 1)-ary relation onW, under the generalization of Kriple's truth condition according to which □αis true at a pointwinWif and only ifαis true at one ofx1,…,xnfor allx1,…,xninWsuch thatRw, x1,…,xn.