Towards applied theories based on computability logic

TL;DR

This paper introduces CL12, a new deductive system for computability logic, which extends classical predicate calculus and offers a constructive, computationally meaningful alternative for developing applied theories like arithmetic.

Contribution

The paper develops CL12, a sound and complete deductive system for computability logic, and demonstrates its application by reconstructing Peano arithmetic within this framework.

Findings

CL12 extends classical predicate calculus with additional expressive power.

CL12 is proven sound and complete with respect to computability logic semantics.

A computability-logic-based version of Peano arithmetic is constructed.

Abstract

Computability logic (CL) (see http://www.cis.upenn.edu/~giorgi/cl.html) 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.