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Free and Projective Bimodal Symmetric Gödel Algebras

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Abstract

Gödel logic (alias Dummett logic) is the extension of intuitionistic logic by the linearity axiom. Symmetric Gödel logic is a logical system, the language of which is an enrichment of the language of Gödel logic with their dual logical connectives. Symmetric Gödel logic is the extension of symmetric intuitionistic logic (L. Esakia, C. Rauszer). The proof-intuitionistic calculus, the language of which is an enrichment of the language of intuitionistic logic by modal operator was investigated by Kuznetsov and Muravitsky. Bimodal symmetric Gödel logic is a logical system, the language of which is an enrichment of the language of Gödel logic with their dual logical connectives and two modal operators. Bimodal symmetric Gödel logic is embedded into an extension of (bimodal) Gödel–Löb logic (provability logic), the language of which contains disjunction, conjunction, negation and two (conjugate) modal operators. The variety of bimodal symmetric Gödel algebras, that represent the algebraic counterparts of bimodal symmetric Gödel logic, is investigated. Description of free algebras and characterization of projective algebras in the variety of bimodal symmetric Gödel algebras is given. All finitely generated projective bimodal symmetric Gödel algebras are infinite, while finitely generated projective symmetric Gödel algebras are finite.

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Authors and Affiliations

  1. Department of Mathematical Logic and Discrete Structures, I. Javakhishvili Tbilisi State University, University str., 13, 0147, Tbilisi, Georgia

    Revaz Grigolia, Tatiana Kiseliova & Vladimer Odisharia

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  1. Revaz Grigolia
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  2. Tatiana Kiseliova
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  3. Vladimer Odisharia
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Grigolia, R., Kiseliova, T. & Odisharia, V. Free and Projective Bimodal Symmetric Gödel Algebras. Stud Logica 104, 115–143 (2016). https://doi.org/10.1007/s11225-015-9630-3

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