Israel Journal of Mathematics

, Volume 25, Issue 3, pp 287–304

Provability interpretations of modal logic

  • Robert M. Solovay

DOI: 10.1007/BF02757006

Cite this article as:
Solovay, R.M. Israel J. Math. (1976) 25: 287. doi:10.1007/BF02757006


We consider interpretations of modal logic in Peano arithmetic (P) determined by an assignment of a sentencev* ofP to each propositional variablev. We put (⊥)*=“0 = 1”, (χ → ψ)* = “χ* → ψ*” and let (□ψ)* be a formalization of “ψ)* is a theorem ofP”. We say that a modal formula, χ, isvalid if ψ* is a theorem ofP in each such interpretation. We provide an axiomitization of the class of valid formulae and prove that this class is recursive.

Copyright information

© Hebrew University 1976

Authors and Affiliations

  • Robert M. Solovay
    • 1
  1. 1.IBM Thomas J. Watson Research CenterYorktown HeightsUSA
  2. 2.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA