Abstract
Algebraisation is a process of translating the syntax and deductive properties of given logic to an algebraic language. While propositional logics fit various algebraisation frameworks reasonably well, algebraisation of first-order logics has many difficulties. Recently, a class of quasicylindric algebras was introduced and investigated. It was proved that each superintuitionistic predicate logic is strongly complete with respect to a variety of quasicylindric algebras. In this paper, we prove that the semantics of pseudo-boolean models of Rasiowa and Sikorski and the Kripke semantics is subsumed by the semantics of quasicylindric algebras. We also expand results obtained by Larisa Maksimova on algebraisation of non-classical propositional logics to the case of first-order superintuitionistic logics. We consider such deductive properties of first-order superintuitionistic logics as the Beth property and the projective Beth property, the Craig interpolation property, the disjunctive and existential properties. We formulate algebraic equivalents which correspond to these properties in the language of varieties of quasicylindric algebras and establish equivalences of the logical properties and their algebraic counterparts.
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Tishkovsky, D. (2018). On Algebraisation of Superintuitionistic Predicate Logics. In: Odintsov, S. (eds) Larisa Maksimova on Implication, Interpolation, and Definability. Outstanding Contributions to Logic, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-69917-2_13
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