Algebraic Kripke sheaf semantics for non-classical predicate logics

Studia Logica 63 (3):387-416 (1999)
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Abstract

In so-called Kripke-type models, each sentence is assigned either to true or to false at each possible world. In this setting, every possible world has the two-valued Boolean algebra as the set of truth values. Instead, we take a collection of algebras each of which is attached to a world as the set of truth values at the world, and obtain an extended semantics based on the traditional Kripke-type semantics, which we call here the algebraic Kripke semantics. We introduce algebraic Kripke sheaf semantics for super-intuitionistic and modal predicate logics, and discuss some basic properties. We can state the Gödel-McKinsey-Tarski translation theorem within this semantics. Further, we show new results on super-intuitionistic predicate logics. We prove that there exists a continuum of super-intuitionistic predicate logics each of which has both of the disjunction and existence properties and moreover the same propositional fragment as the intuitionistic logic.

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Citations of this work

First-order intensional logic.Melvin Fitting - 2004 - Annals of Pure and Applied Logic 127 (1-3):171-193.
Categorial modal realism.Tyler D. P. Brunet - 2023 - Synthese 201 (2):1-29.
On Harrop disjunction property in intermediate predicate logics.Katsumasa Ishii - 2023 - Archive for Mathematical Logic 63 (3):317-324.

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References found in this work

The mathematics of metamathematics.Helena Rasiowa - 1963 - Warszawa,: Państwowe Wydawn. Naukowe. Edited by Roman Sikorski.
Incompleteness results in Kripke semantics.Silvio Ghilardi - 1991 - Journal of Symbolic Logic 56 (2):517-538.

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