Omitting types in fuzzy logic with evaluated syntax

Mathematical Logic Quarterly 52 (3):259-268 (2006)
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Abstract

This 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 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 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 constants for all truth values, or it has these constants for truth values from [0, 1] ∩ ℚ only

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References found in this work

On Fuzzy Logic I Many‐valued rules of inference.Jan Pavelka - 1979 - Mathematical Logic Quarterly 25 (3‐6):45-52.
On Fuzzy Logic I Many‐valued rules of inference.Jan Pavelka - 1979 - Mathematical Logic Quarterly 25 (3-6):45-52.
Joint Consistency of Fuzzy Theories.Vilém Novák - 2002 - Mathematical Logic Quarterly 48 (4):563-573.

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