Omitting types in fuzzy logic with evaluated syntax

Abstract

AbstractThis paper is a contribution to the development of model theory of fuzzy logic in narrow sense. We consider a formal system EvŁ 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 EvŁ. The truth values form an MV‐algebra that is either finite or Łukasiewicz algebra on [0, 1].The classical omitting types theorem states that given a formal theory T and a set Σ(x 1, … , 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 EvŁ, that is, we prove that if T is a fuzzy theory and Σ(x 1, … , xn ) forms a fuzzy set , then a model omitting Σ also exists. We will prove this theorem for two essential cases of EvŁ: either EvŁ has logical (truth) constants for all truth values, or it has these constants for truth values from [0, 1] ∩ ℚ only. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)