Many-Valued Predicate Logics

Abstract

We are now ready to start our investigation of fuzzy predicate logics (or first-order logics, quantification logics). We shall develop logics broadly analogous to the classical predicate logic; in particular, we shall deal only with two quantifiers, ∀ and ∃ (universal and existential). Generalized quantifiers will be studied in later chapters. In Section 1 we shall develop the predicate counterpart BL∀ of our basic propositional logic BL; in Section 2 we prove a rather general completeness theorem for predicate logics (with respect to semantics over residuated lattices). Sections 3 and 4 are devoted to Gödel and Lukasiewicz predicate logics respectively; we show that Gödel predicate logic has a recursive axiomatization that is complete with respect to the semantics over [0,1], whereas for Łukasiewicz we only present a variant of Pavelka logic. (We show in the next chapter that Łukasiewicz does not have a recursive complete axiomatization.) We close Sec. 4 with some for remarks on the predicate product logic. Sec. 5 discusses many-sorted calculi and Sec. 6 introduces and studies similarity (fuzzy equality). This notion will be crucial for our analysis of fuzzy control in Chap. 7.