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.
KeywordsModal Logic Binary Relation Propositional Variable Truth Assignment Kripke Model
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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