Self provers and Σ1 sentences

Logic Journal of the IGPL 20 (1):1-21 (2012)
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Abstract

This paper is the second in a series of three papers. All three papers deal with interpretability logics and related matters. In the first paper a construction method was exposed to obtain models of these logics. Using this method, we obtained some completeness results, some already known, and some new. In this paper, we will set the construction method to work to obtain more results. First, the modal completeness of the logic ILM is proved using the construction method. This is not a new result, but by using our new proof we can obtain new results. Among these new results are some admissible rules for ILM and GL. Moreover, the new proof will be used to classify all the essentially Δ1 and also all the essentially Σ1 formulas of ILM. Closely related to essentially Σ1 sentences are the so-called self provers. A self-prover is a formula φ which implies its own provability, that is φ → □φ. Each formula φ will generate a self prover φ ^ □φ. We will use the construction method to characterize those sentences of GL that generate a self prover that is trivial in the sense that it is Σ1

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Author's Profile

Joost Joosten
Universitat de Barcelona