Abstract
We introduce a first order extension of GL, called ML 3 , and develop its proof theory via a proxy cut-free sequent calculus GLTS. We prove the highly nontrivial result that cut is a derived rule in GLTS, a result that is unavailable in other known first-order extensions of GL. This leads to proofs of weak reflection and the related conservation result for ML 3 , as well as proofs for Craig’s interpolation theorem for GLTS. Turning to semantics we prove that ML 3 is sound with respect to arithmetical interpretations and that it is also sound and complete with respect to converse well-founded and transitive finite Kripke models. This leads us to expect that a Solovay-like proof of arithmetical completeness of ML 3 is possible.
Keywords provability logic  GL  Hilbert-style logic  a  modal first-order logic  reflection theorem  Gentzen-style logic  arithmetical interpretation  cut-elimination  sequent calculus  soundness
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DOI 10.12775/LLP.2013.030
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A Completeness Theorem in Modal Logic.Saul A. Kripke - 1959 - Journal of Symbolic Logic 24 (1):1-14.
The Modal Logic of Provability. The Sequential Approach.Giovanni Sambin & Silvio Valentini - 1982 - Journal of Philosophical Logic 11 (3):311 - 342.
On Modal Systems Having Arithmetical Interpretations.Arnon Avron - 1984 - Journal of Symbolic Logic 49 (3):935-942.
The Modal Logic of Provability: Cut-Elimination. [REVIEW]Silvio Valentini - 1983 - Journal of Philosophical Logic 12 (4):471 - 476.
The Predicate Modal Logic of Provability.Franco Montagna - 1984 - Notre Dame Journal of Formal Logic 25 (2):179-189.

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