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      Towards applied theories based on computability logic

      The Journal of Symbolic Logic

      Association for Symbolic Logic

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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.

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          Most cited references 12

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                Author and article information

                Journal
                The Journal of Symbolic Logic
                J. symb. log.
                Association for Symbolic Logic
                0022-4812
                1943-5886
                June 2010
                March 12 2014
                June 2010
                : 75
                : 2
                : 565-601
                Article
                10.2178/jsl/1268917495
                © 2010

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