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Towards applied theories based on computability logic

Published online by Cambridge University Press:  12 March 2014

Giorgi Japaridze
Giorgi Japaridze*
Affiliation:
Department of Computing Sciences, Villanova University, 800 Lancaster Avenue, Villanova, Pa 19085, USA. E-mail: giorgi.japaridze@villanova.edu, URL: http://www.csc.villanova.edu/~japaridz/
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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-logicbased 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 2 , June 2010 , pp. 565 - 601
Copyright
Copyright © Association for Symbolic Logic 2010

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References

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