Structural Completeness in Many-Valued Logics with Rational Constants

0559061 - ÚI 2023 RIV US eng J - Journal Article
Gispert, J. - Haniková, Zuzana - Moraschini, T. - Stronkowski, Michał
Structural Completeness in Many-Valued Logics with Rational Constants.
Notre Dame Journal of Formal Logic. Roč. 63, č. 3 (2022), s. 261-299. ISSN 0029-4527. E-ISSN 1939-0726
R&D Projects: GA ČR(CZ) GA18-00113S; GA MŠMT(CZ) EF17_050/0008361
Institutional support: RVO:67985807
Keywords : admissible rule * fuzzy logic * Gödel logic * Łukasiewicz logic * product logic * quasivariety * rational Pavelka logic * structural completeness
OECD category: Pure mathematics
Impact factor: 0.7, year: 2022
Method of publishing: Limited access
https://dx.doi.org/10.1215/00294527-2022-0021

The logics R Ł, 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, R Ł 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.
Permanent Link: https://hdl.handle.net/11104/0332481