Omitting types in fuzzy logic with evaluated syntax

This paper is a contribution to the development of model theory of fuzzy logic in narrow sense. We consider a formal system Ev L of fuzzy logic that has evaluated syntax, i.e. axioms need not be fully convincing and so, they form a fuzzy set only. Consequently, formulas are provable in some general degree. A generalization of Gödel’s completeness theorem does hold in Ev L. The truth values form an MV-algebra that is either finite or Lukasiewicz algebra on [0, 1]. The classical omitting types theorem states that given a formal theory T and a set Σ(x1, . . . , xn) of formulas with the same free variables, we can construct a model of T which omits Σ, i.e. there is always a formula from Σ not true in it. In this paper, we generalize this theorem for Ev L, that is, we prove that if T is a fuzzy theory and Σ(x1, . . . , xn) forms a fuzzy set then a model omitting Σ also exists. We will prove this theorem for two essential cases of Ev L: either Ev L has logical (truth) constants for all truth values, or it has these constants for truth values from [0, 1] ∩Q only.