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Algebraic Kripke Sheaf Semantics for Non-Classical Predicate Logics

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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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Suzuki, NY. Algebraic Kripke Sheaf Semantics for Non-Classical Predicate Logics. Studia Logica 63, 387–416 (1999). https://doi.org/10.1023/A:1005207121661

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  • DOI: https://doi.org/10.1023/A:1005207121661

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