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Arithmetical Interpretations of Dynamic Logic

Petr Hájek
The Journal of Symbolic Logic
Vol. 48, No. 3 (Sep., 1983), pp. 704-713
DOI: 10.2307/2273463
Stable URL: http://www.jstor.org/stable/2273463
Page Count: 10
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Arithmetical Interpretations of Dynamic Logic
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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 α a formula f(α) of PA with one free variable describing formally a supertheory of PA; (2) f commutes with logical connectives; (3) f([α] A) is the sentence saying that f(A) is provable in the theory f(α); (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.

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