Hyperintensions

Standard possible worlds semantics has been known from the start to have a problem with granularity: for a wide range of naturallanguage (NL) entailment patterns, not enough meaning distinctions are available to make predictions consistent with robust intuitions. Though numerous solutions have been proposed, often of great ingenuity and technical sophistication, none of these has gained widespread acceptance. As a result, most semanticists have made a practical decision to work in a framework known to have dubious foundations and leave the foundational problems to mathematical logicians. Here a new approach is proposed which may be simple enough and conservative enough to be practical for working empirical and computational semanticists. More specifically, I show how the use of a higher-order logic with definable subtypes leads to a novel and surprisingly straightforward solution of the granularity problem. I also call attention to a hitherto unnoticed problem in standard approaches to NL semantics having to do with nonprincipal ultrafilters and show why it does not arise under my proposal. The two main technical innovations that drive the proposal are (1) axiomatizing NL entailment as a preorder (as opposed to an order) on the set of (primitive) propositions, and (2) defining worlds as certain sets of propositions (viz. ultrafilters). These innovations provide just the tools we need to develop a formally explicit theory of hyperintensions 2, mathematical models of Fregean senses of a finer granularity than the familiar intensions (functions to extensions from worlds, where the worlds in turn are theoretical primitives) of mainstream Kripke/Montague-inspired NL semantics. 0