Abstract
Kabakov has proved that for the finite validity (in Medvedev's sense) of intuitively unprovable propositional formulas it is necessary that an implication occur in the premise Β or else in the inference γ of some subformula of the type (Β → γ), and, consequently, that at least two implications be present. Here we prove that every finitely valid, intuitively unprovable formula contains the occurrence of an implication necessarily in the premise Β of some subformula of the form (Β → γ) and we also present an example of a similar formula containing exactly two implications.
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F. A. Kabakov, “Deducibility of certain realizable formulas of propositional calculus,” Z. für Math. Logik und Grundlagen der Math.,9, No. 2, 97–104 (1963).
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Yu. T. Medvedev, “On the interpretation of logical formulas by finite problems,” Dokl. Akad. Nauk SSSR,169, No. 1, 20–23 (1966).
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Translated from Matematicheskie Zametki, Vol. 20, No. 3, pp. 383–390, September, 1976.
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Skvortsov, D.P. The occurrence of an implication in finitely valid, intuitively improvable formulas of propositional logic. Mathematical Notes of the Academy of Sciences of the USSR 20, 771–775 (1976). https://doi.org/10.1007/BF01097248
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DOI: https://doi.org/10.1007/BF01097248