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August 2022 Structural Completeness in Many-Valued Logics with Rational Constants
Joan Gispert, Zuzana Haniková, Tommaso Moraschini, Michał Stronkowski
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Notre Dame J. Formal Logic 63(3): 261-299 (August 2022). DOI: 10.1215/00294527-2022-0021

Abstract

The logics RŁ, RP, and RG have been obtained by expanding Łukasiewicz logic Ł, product logic P, and Gödel–Dummett logic G with rational constants. We study the lattices of extensions and structural completeness of these three expansions, obtaining results that stand in contrast to the known situation in Ł, P, and G. Namely, RŁ is hereditarily structurally complete. RP is algebraized by the variety of rational product algebras that we show to be Q-universal. We provide a base of admissible rules in RP, show their decidability, and characterize passive structural completeness for extensions of RP. Furthermore, structural completeness, hereditary structural completeness, and active structural completeness coincide for extensions of RP, and this is also the case for extensions of RG, where in turn passive structural completeness is characterized by the equivalent algebraic semantics having the joint embedding property. For nontrivial axiomatic extensions of RG, we provide a base of admissible rules. We leave the problem open whether the variety of rational Gödel algebras is Q-universal.

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Joan Gispert. Zuzana Haniková. Tommaso Moraschini. Michał Stronkowski. "Structural Completeness in Many-Valued Logics with Rational Constants." Notre Dame J. Formal Logic 63 (3) 261 - 299, August 2022. https://doi.org/10.1215/00294527-2022-0021

Information

Received: 3 August 2021; Accepted: 5 March 2022; Published: August 2022
First available in Project Euclid: 25 September 2022

Digital Object Identifier: 10.1215/00294527-2022-0021

Subjects:
Primary: 03B52
Secondary: 03B22 , 03B50 , 08C15

Keywords: admissible rule , fuzzy logic , Gödel logic , Łukasiewicz logic , product logic , quasivariety , rational Pavelka logic , structural completeness

Rights: Copyright © 2022 University of Notre Dame

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Vol.63 • No. 3 • August 2022
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