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An incomplete system of modal logic. (English) Zbl 0589.03005
The authors present a new system of modal propositional logic, called GH, that is ”incomplete in the following sense: there is a sentence that is not a theorem of GH even though it is valid in every frame in which all theorems of GH are valid”. This result is discussed with regard to the interpretation of ’\(\square '\) as provability. The characteristic axiom of GH may then be read as saying that the sentences which are provably ”equivalent to their own provability are in fact provable”. It is thus a formalized answer to L. Henkin’s question in ”A problem concerning provability” [J. Symb. Logic 17, 160 (1952)], and it is closely related to Löb’s theorem.
Reviewer: W.Lenzen

MSC:
03B45 Modal logic (including the logic of norms)
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[1] For terminology, nomenclature and notations not defined in this paper, see G. Boolos,The Unprovability of Consistency, Cambridge University Press, 1979, where the system GL is consideredin extenso, along with several other systems of modal propositional logic.
[2] R. Magari, ?Primi risultati sulla variet? di Boolos?,Bollentino U.M.I. (6)1-B (1982), 359-367. · Zbl 0487.03039
[3] David Lewis, ?Intensional logics without iterative axioms?,Journal of Philosophical Logic 3 (1974), 457-466. · Zbl 0296.02014 · doi:10.1007/BF00257488
[4] J. F. A. K. van Benthem, ?Two simple incomplete modal logics?,Theoria 44 (1978), 25-37, and ?Syntactic aspects of modal incompleteness theorems?,Theoria 44 (1979), 63-77.
[5] L. Henkin, ?A problem concerning provability?,Journal of Symbolic Logic 17 (1952), 160.
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