Algebraic characterization of completeness properties for core fuzzy logics II: predicate logics

We deal with general results on (several notions of) completeness of a wide class of predi-cate fuzzy logics. In particular, we characterize strong completeness by model-theoretic and algebraic means and give a constructive procedure for disproving finite strong completeness. We assume that the reader is familiar with the syntax and semantics of both propositional and predicate fuzzy logics (see e.g. [3, 5]) The weakest logic we consider here is the logic MTL intro-duced in [2]. Recall (see e.g. [4, 5]) that we define core fuzzy logics1 as axiomatic expansions of MTL obeying the following ‘substitution ’ law: ϕ ≡ ψ ` χ(ϕ) ≡ χ(ψ). Like in the propositional case we introduce several notions of completeness. Definition 1 Let L be a (∆-)core fuzzy logic and K a class of L-algebras. We say that L ∀ has the SKC if for each predicate language Γ, theory T, and formula ϕ the following are equivalent: • T ` ϕ. • (M,A) | = ϕ for each A ∈ K and each model (M,A) of the theory T. We say that L ∀ has the FSKC if the above condition holds for finite theories. Finally, we say that L ∀ has the KC if the above condition holds for the empty theory. Recall, that an embedding of two algebras with lattice subreduct is called σ-embedding if it preserves infinite suprema and infima. The following is the main theorem characterizing the notion strong K completeness. Theorem 1 Let L be a (∆-)core fuzzy logic in a propositional language P. Then the following are equivalent