A modal provability logic of explicit and implicit proofs

Annals of Pure and Applied Logic 161 (3):388-403 (2010)
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Abstract

We establish the bi-modal forgetful projection of the Logic of Proofs and Formal Provability GLA. That is to say, we present a normal bi-modal provability logic with modalities □ and whose theorems are precisely those formulas for which the implicit provability assertions represented by the modality can be realized by explicit proof terms

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Explicit provability and constructive semantics.Sergei N. Artemov - 2001 - Bulletin of Symbolic Logic 7 (1):1-36.
The logic of proofs, semantically.Melvin Fitting - 2005 - Annals of Pure and Applied Logic 132 (1):1-25.
On logic of proofs and provability.Elena Nogina - 2006 - Bulletin of Symbolic Logic 12 (2):356.

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