Journal of Symbolic Logic

Arithmetical Interpretations of Dynamic Logic

Petr Hajek

Abstract

An arithmetical interpretation of dynamic propositional logic (DPL) is a mapping $f$ satisfying the following: (1) $f$ associates with each formula $A$ of DPL a sentence $f(A)$ of Peano arithmetic (PA) and with each program $\alpha$ a formula $f(\alpha)$ of PA with one free variable describing formally a supertheory of PA; (2) $f$ commutes with logical connectives; (3) $f(\lbrack\alpha\rbrack A)$ is the sentence saying that $f(A)$ is provable in the theory $f(\alpha)$; (4) for each axiom $A$ of DPL, $f(A)$ is provable in PA (and consequently, for each $A$ provable in DPL, $f(A)$ is provable in PA). The arithmetical completeness theorem is proved saying that a formula $A$ of DPL is provable in DPL iff for each arithmetical interpretation $f, f(A)$ is provable in PA. Various modifications of this result are considered.

Article information

Source
J. Symbolic Logic, Volume 48, Issue 3 (1983), 704-713.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183741330

Mathematical Reviews number (MathSciNet)
MR716632

Zentralblatt MATH identifier
0546.03012

JSTOR