Well-behaved modal logics

Abstract

Much of the literature on the model theory of modal logics suffers from two weaknesses. Firstly, there is a lack of generality; theorems are proved piecemeal about this or that modal logic, or at best small classes of logics. Much of the literature, e.g. [1], [2], and [3], confines attention to structures with the expanding domain property (i.e., if wRu then Ā(w) ⊆ Ā(u)); the syntactic counterpart of this restriction is assumption of the converse Barcan scheme, a move which offers (in Russell's phrase) “all the advantages of theft over honest toil”. Secondly, I think there has been a failure to hit on the best ways of extending classical model theoretic notions to modal contexts. This weakness makes the literature boring, since a large part of the interest of modal model theory resides in the way in which classical model theoretic notions extend, and in some cases divide, in the modal setting. (The relation between α-recursion theory and classical recursion theory is analogous to that between modal model theory and classical model theory. Much of the work in α-recursion theory involved finding the right definitions (e.g., of recursive-in) and separating concepts which collapse in the classical case (e.g. of finiteness and boundedness).)The notion of a well-behaved modal logic is introduced in §3 to make possible rather general results; of course our attention will not be restricted to structures with the expanding domain property. Rather than prove piecemeal that familiar modal logics are well-behaved, in §4 we shall consider a class of “special” modal logics, which obviously includes many familiar logics and which is included in the class of well-behaved modal logics.