A Metalogical Theory of Natural Language Semantics

We develop a framework for natural language semantics which handles intensionality via metalogical constructions and deals with degree truth values, in an integrated way. We take an axiomatic set theory, ZF, as the foundation for semantic representations, but we make ZF a metalanguage for part of itself by embedding a language L within ZF which is basically a copy of the part of ZF consisting of set expressions. This metalogical set-up is used for handling propositional attitude verbs (limited to believe in this paper). We define a truth function which determines the truth value of an L-proposition p with respect to a theory T. Theories are sets of L-propositions with associated truth values, and can be viewed as a (much more well-defined) replacement for possible worlds. We develop a mechanism for defining belief worlds as theories. We simultaneously develop two different versions of our system -- a Boolean version where the set of truth values is {0, 1}, and a degree-truth version where the set of truth values is the interval [0, 1] of real numbers. We use degrees of truth in handling a broad class of semantic predicates that we call base-focus predicates, which include generalized quantifiers as well as many adverb and adjective senses, and certain discourse-level predicates.