Journal of Symbolic Logic
- J. Symbolic Logic
- Volume 75, Issue 2 (2010), 565-601.
Towards applied theories based on computability logic
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.
J. Symbolic Logic Volume 75, Issue 2 (2010), 565-601.
First available in Project Euclid: 18 March 2010
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 03F50: Metamathematics of constructive systems
Secondary: secondary: 03F30 03D75: Abstract and axiomatic computability and recursion theory 03F50: Metamathematics of constructive systems 68Q10: Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.) [See also 68Q85] 68T27: Logic in artificial intelligence 68T30: Knowledge representation
Japaridze, Giorgi. Towards applied theories based on computability logic. J. Symbolic Logic 75 (2010), no. 2, 565--601. doi:10.2178/jsl/1268917495. http://projecteuclid.org/euclid.jsl/1268917495.