Abstract

We prove here a series of original incompleteness results for propositional modal logic for both the relational semantics and the neighborhood semantics with an emphasis on finitely axiomatizable cases. We prove general such results for both semantics and also find simple logics complete for the N-semantics but not the R-semantics. ;In Ch. 1, we define the operator , which for a given semantics yields the smallest complete logic to contain L, and we show how its basic properties can be used, in conjunction with the results of later chapters, to show that incompleteness is widespread throughout the modal logic lattice. ;In Ch. 2 we show that all consistent normal extensions L of KT and of K4 have degree of incompleteness 2) ) R- and N-incomplete "semantically equivalent" logics which are valid in exactly the same semantic structures, or frames). We also show that there are large "pockets of incompleteness," stretches of the modal logic lattice consisting only of incomplete logics. Although some of our results were proved independently and earlier by Blok for the R-semantics , our method is model-theoretic, applies also to the N-semantics, and is more syntactically constructive. Where others have dealt with logics which are not fully axiomatized we deal with explicitly axiomatized logics: e.g., for each f.a. L extending KT or K4 we show that infinitely many of the L' s.e. to L are f.a. ;In Ch. 3, we establish in detail--so far as we know, for the first time--that there are f.a. normal modal logics which are "differrentially complete" . In Ch. 4, we produce another simple d.c. logic L. it is almost surely f.a.; if so, it refutes a conjecture of Gerson, since L extends KT. ;In Ch. 5 we produce logics L and L, each f.a. by means of very simple axioms and each complete for a class of simple finite R-frames, yet whose "logical join" is R-incomplete. In Ch. 6 we find an L S4 with 2) semantically equivalent logics L' all extending S4; of these L', infinitely many are f.a