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June 2010 Towards applied theories based on computability logic
Giorgi Japaridze
J. Symbolic Logic 75(2): 565-601 (June 2010). DOI: 10.2178/jsl/1268917495

Abstract

Computability logic (CL) is a recently launched program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth that logic has more traditionally been. Formulas in it represent computational problems, “truth” means existence of an algorithmic solution, and proofs encode such solutions. Within the line of research devoted to finding axiomatizations for ever more expressive fragments of CL, the present paper introduces a new deductive system CL12 and proves its soundness and completeness with respect to the semantics of CL. Conservatively extending classical predicate calculus and offering considerable additional expressive and deductive power, CL12 presents a reasonable, computationally meaningful, constructive alternative to classical logic as a basis for applied theories. To obtain a model example of such theories, this paper rebuilds the traditional, classical-logic-based Peano arithmetic into a computability-logic-based counterpart. Among the purposes of the present contribution is to provide a starting point for what, as the author wishes to hope, might become a new line of research with a potential of interesting findings—an exploration of the presumably quite unusual metatheory of CL-based arithmetic and other CL-based applied systems.

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Giorgi Japaridze. "Towards applied theories based on computability logic." J. Symbolic Logic 75 (2) 565 - 601, June 2010. https://doi.org/10.2178/jsl/1268917495

Information

Published: June 2010
First available in Project Euclid: 18 March 2010

zbMATH: 1201.03055
MathSciNet: MR2648156
Digital Object Identifier: 10.2178/jsl/1268917495

Subjects:
Primary: 03F50
Secondary: 03D75 , 03F50 , 68Q10 , 68T27 , 68T30 , secondary: 03F30

Keywords: Computability logic , constructive logics , Game semantics , Peano Arithmetic

Rights: Copyright © 2010 Association for Symbolic Logic

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Vol.75 • No. 2 • June 2010
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