Non-Analytic Logics

Abstract

We are now going to consider five logics, KB, DB, B, S5 and G that, for various reasons, do not fit the pattern of earlier chapters. The most eccentric of these is the logic G which, as defined by its model theory (Chapter One #2) is not compact. The other four logics, KB, DB, B and S5 are compact, but have other pecularities that set them apart from the logics of earlier chapters. Now KB, DB, B and S5, in many ways, resemble each other rather closely, and differ markedly from G. We have chosen to treat the five in a single chapter largely as a matter of convenience. The main theoretical reason for grouping them together is a negative one: none of them (apparently) have tableau systems of the sort considered in earlier chapters, in which each step must involve a simpler formula than any previously on the branch. We elaborate on this point.