Soft Computing

, Volume 22, Issue 3, pp 839–844 | Cite as

Kripke semantics for fuzzy logics

Methodologies and Application


Kripke frames (and models) provide a suitable semantics for sub-classical logics; for example, intuitionistic logic (of Brouwer and Heyting) axiomatizes the reflexive and transitive Kripke frames (with persistent satisfaction relations), and the basic logic (of Visser) axiomatizes transitive Kripke frames (with persistent satisfaction relations). Here, we investigate whether Kripke frames/models could provide a semantics for fuzzy logics. For each axiom of the basic fuzzy logic, necessary and sufficient conditions are sought for Kripke frames/models which satisfy them. It turns out that the only fuzzy logics (logics containing the basic fuzzy logic) which are sound and complete with respect to a class of Kripke frames/models are the extensions of the Gödel logic (or the super-intuitionistic logic of Dummett); indeed this logic is sound and strongly complete with respect to reflexive, transitive and connected (linear) Kripke frames (with persistent satisfaction relations). This provides a semantic characterization for the Gödel logic among (propositional) fuzzy logics.


Fuzzy logics The basic fuzzy logic Gödel logic Dummett logic Kripke frames Soundness Completeness Semantics 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Human and animal rights statement

This article does not contains any studies with human participants or animals performed by any of the authors.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TabrizTabrizIran

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