Israel Journal of Mathematics

, Volume 25, Issue 3–4, pp 287–304 | Cite as

Provability interpretations of modal logic

  • Robert M. Solovay


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.


Modal Logic Binary Relation Propositional Variable Truth Assignment Kripke Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    George Boolos,Friedman’s 35th problem has an affirmative solution, Abstract*75T-E66, Notices Amer. Math. Soc.22 (1975), A-646.Google Scholar
  2. 2.
    Harvey Friedman,One hundred and two problems in mathematical logic, J. Symbolic Logic40 (1975), 113–129.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    S. C. Kleene,Introduction to Metamathematics, Van Nostrand, New York, 1952.zbMATHGoogle Scholar
  4. 4.
    Saul Kripke,Semantical analysis of modal logic I, Z. Math. Logik Grundlagen Math.9 (1963), 67–96.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    M. H. Löb,Solution of a problem of Leon Henkin, J. Symbolic Logic20 (1955), 115–118.zbMATHCrossRefGoogle Scholar

Copyright information

© Hebrew University 1976

Authors and Affiliations

  • Robert M. Solovay
    • 1
  1. 1.IBM Thomas J. Watson Research CenterYorktown HeightsUSA

Personalised recommendations