Axioms for a theory of semantic equivalence

Gödel-type semantic completeness theorems are established for a formal theory of semantic equivalence based on L.A. Zadeh's concept of a linguistic variable. The linguistics that is employed allows for the expression of propositions such as “it is not the case that ‘young’ is semantically equivalent with ‘not old’”, or, in symbols (young(x) ≅ ∼old(x)).The result is a two-leveled semantics which, at the lower level, is a multivalent interpretation of fuzzy assertions (e.g., ∼old(x)) in terms of fuzzy subsets of a given universe and, at the upper level, is a two-valued (classical) interpretation based on the fact that two closed fuzzy assertions either do or do not have the same truth value. The main proof is of the Henkin variety, employing adaptations of the algebraic methods found in Rasiowa 9] and Rasiowa and Sikorski 10].