Israel Journal of Mathematics

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

Provability interpretations of modal logic

  • Robert M. Solovay
Article

Abstract

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    S. C. Kleene,Introduction to Metamathematics, Van Nostrand, New York, 1952.MATHGoogle Scholar
  4. 4.
    Saul Kripke,Semantical analysis of modal logic I, Z. Math. Logik Grundlagen Math.9 (1963), 67–96.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    M. H. Löb,Solution of a problem of Leon Henkin, J. Symbolic Logic20 (1955), 115–118.MATHCrossRefGoogle Scholar

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

Personalised recommendations