Provability interpretations of modal logic
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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.
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- 1.George Boolos,Friedman’s 35th problem has an affirmative solution, Abstract*75T-E66, Notices Amer. Math. Soc.22 (1975), A-646.Google Scholar