Article

Studia Logica

, Volume 63, Issue 2, pp 245-268

First online:

Proof-Theoretic Modal PA-Completeness II: The Syntactic Countermodel

  • Paolo GentiliniAffiliated withIstituto per la Matematica Applicata del Consiglio Nazionale delle Ricerche (IMA-CNR)

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Abstract

This paper is the second part of the syntactic demonstration of the Arithmetical Completeness of the modal system G, the first part of which is presented in [9]. Given a sequent S so that ⊢GL-LIN S, ⊬G S, and given its characteristic formula H = char(S), which expresses the non G-provability of S, we construct a canonical proof-tree T of ~ H in GL-LIN, the height of which is the distance d(S, G) of S from G. T is the syntactic countermodel of S with respect to Gand is a tool of general interest in Provability Logic, that allows some classification in the set of the arithmetical interpretations.

Proof-Theory Provability Logic countermodel of a sequent classification of arithmetical interpretations of modal logic