Proof Polynomials: A Unified Semantics for Modality and lambda-terms (original) (raw)

Unified Semantics for Modality and lambda-terms via Proof Polynomials

Algebras, Diagrams and Decisions in Language, Logic and Computation, 2002

It is shown that the modal logic S4, simple-calculus and modal-calculus admit arealization in a very simple propositional logical system LP, which has an exact provabilitysemantics. In LP both modality and-terms become objects of the same nature, namely, proof polynomials. The provability interpretation of modal-terms presented here maybe regarded as a system-independent generalization of the Curry-Howard isomorphism ofproofs and-terms. 1 IntroductionThe Logic of Proofs (LP, see Section 2) is a system in the propositional ...

Proof Polynomials vs. lambda-terms

1998

Abstract: The Logic of Proofs provides a basic framework for the formalization of reasoning about proofs. It incorporates proof terms into the propositional language, using labeled logical operators" t:" with the intended reading of t: F being" T is a proof of F". In the current paper we demonstrate how the typed lambda calculus and the modal lambda-calculus can be realized in the Logic of Proofs.

Polynomial Ring Calculus for Modal Logics: A New Semantics and Proof Method for Modalities

The Review of Symbolic Logic, 2011

A new (sound and complete) proof style adequate for modal logics is defined from the polynomial ring calculus (PRC). The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra–Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S5, and can be easily extended to other modal logics.

Proof Theory and Algebra in Logic

Short Textbooks in Logic, 2019

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Proof Theory of Modal Logic

Applied logic series, 1996

SCOPE OF THE SERIES Logic is applied in an increasingly wide variety of disciplines, from the traditional subjects of philosophy and mathematics to the more recent disciplines of cognitive science, computer science, artificial intelligence, and linguistics, leading to new vigor in this ancient subject. Kluwer, through its Applied Logic Series, seeks to provide a home for outstanding books and research monographs in applied logic, and in doing so demonstrates the underlying unity and applicability of logic.

A note on some explicit modal logics

2006

Artemov introduced the Logic of Proofs (LP) as a logic of explicit proofs. We can also offer an epistemic reading of this formula: "t is a possible justification of φ". Motivated, in part, by this epistemic reading, Fitting introduced a Kripke style semantics for LP in . In this note, we prove soundness and completeness of some axiom systems which are not covered in [8].