Strong non-standard completeness for fuzzy logics

Authors:
Tommaso Flaminio
Affiliations:
Dipartimento di Matematica e Scienze Informatiche, Pian dei Matellini 44, 53100, Siena, Italy
Venue:
Soft Computing - A Fusion of Foundations, Methodologies and Applications - Special issue (pp 315-357) "Ordered structures in many-valued logic"
Year:
2007

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Abstract

In this paper we are going to introduce the notion of strongnon-standard completeness (SNSC) for fuzzy logics. This notionnaturally arises from the well known construction by ultraproduct.Roughly speaking, to say that a logic 𝒞 is strongnon-standard complete means that, for any countable theory Γover 𝒞 and any formula Φ such that Γ\nvdash 𝒞Φ, there exists anevaluation e of 𝒞-formulas into a𝒞-algebra 𝒜 such that theuniverse of 𝒜 is a non-Archimedeanextension [0,1]* of the real unit interval [0,1], e is amodel for Γ, but e(Φ)