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2016•Logic Journal of IGPLOpen access

Weak arithmetical interpretations for the Logic of Proofs

Roman Kuznets, Thomas Studer

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Abstract

Artemov established an arithmetical interpretation for the Logics of Proofs LPCS, which yields a classical provability semantics for the modal logic S4. These Logics of Proofs are parameterized by so-called constant specifications CS that state which axioms can be used in the reasoning process, and the arithmetical interpretation relies on the constant specifications being finite. In this paper, we remove this restriction by introducing weak arithmetical interpretations that are sound and complete for a wide class of constant specifications, including infinite ones. In particular, they interpret the full Logic of Proofs LP. Keywords: Logic of Proofs, arithmetical interpretation, BHK semantics

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What this paper is about

Artemov established an arithmetical interpretation for the Logics of Proofs LPCS, which yields a classical provability semantics for the modal logic S4. These Logics of Proofs are parameterized by so-called constant specifications CS that state which axioms can be used in the reasoning process, and the arithmetical interpretation relies on the constant specifications being finite. In this paper, we remove this restriction by introducing weak arithmetical interpretations that are sound and complete for a wide class of constant specifications, including infinite ones. In particular, they interpret the full Logic of Proofs LP. Keywords: Logic of Proofs, arithmetical interpretation, BHK semantics

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Available abstract

Artemov established an arithmetical interpretation for the Logics of Proofs LPCS, which yields a classical provability semantics for the modal logic S4. These Logics of Proofs are parameterized by so-called constant specifications CS that state which axioms can be used in the reasoning process, and the arithmetical interpretation relies on the constant specifications being finite. In this paper, we remove this restriction by introducing weak arithmetical interpretations that are sound and complete for a wide class of constant specifications, including infinite ones. In particular, they interpret the full Logic of Proofs LP. Keywords: Logic of Proofs, arithmetical interpretation, BHK semantics

Key concepts: Arithmetic function, Mathematical proof, Arithmetic, Computer science, Mathematics, Philosophy, Discrete mathematics, Geometry

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