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Predicate Provability Logic with Non-Modalized Quantifiers

Giorgie Dzhaparidze
Studia Logica: An International Journal for Symbolic Logic
Vol. 50, No. 1, Provability Logic (Mar., 1991), pp. 149-160
Published by: Springer
Stable URL: http://www.jstor.org/stable/20015560
Page Count: 12
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Predicate Provability Logic with Non-Modalized Quantifiers
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Abstract

Predicate modal formulas with non-modalized quantifiers (call them $Q^{\prime}$-formulas) are considered as schemata of arithmetical formulas, where □ is interpreted as the provability predicate of some fixed correct extension T of arithmetic. A method of constructing 1) non-provable in T and 2) false arithmetical examples for $Q^{\prime}$-formulas by Kripke-like countermodels of certain type is given. Assuming the means of T to be strong enough to solve the (undecidable) problem of derivability in $Q^{\prime }GL$, the $Q^{\prime}$-fragment of the predicate version of the logic GL, we prove the recursive enumerability of the sets of $Q^{\prime}$-formulas all arithmetical examples of which are: 1) T-provable, 2) true. In particular, the first one is shown to be exactly $Q^{\prime }GL$ and the second one to be exactly the $Q^{\prime}$-fragment of the predicate version of Solovay's logic S.

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