Abstract
In this paper the PA-completeness of modal logic is studied by syntactical and constructive methods. The main results are theorems on the structure of the PA-proofs of suitable arithmetical interpretationsS ϕ of a modal sequentS, which allow the transformation of PA-proofs ofS ϕ into proof-trees similar to modal proof-trees. As an application of such theorems, a proof of Solovay's theorem on arithmetical completeness of the modal system G is presented for the class of modal sequents of Boolean combinations of formulas of the form □p i,m i=0, 1, 2, ... The paper is the preliminary step for a forthcoming global syntactical resolution of the PA-completeness problem for modal logic.
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Gentilini, P. Syntactical results on the arithmetical completeness of modal logic. Stud Logica 52, 549–564 (1993). https://doi.org/10.1007/BF01053259
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DOI: https://doi.org/10.1007/BF01053259