The provability logic for Σ1-interpolability

Annals of Pure and Applied Logic 64 (1):1-25 (1993)
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Abstract

We say that two arithmetical formulas A, B have the Σ1-interpolation property if they have an ‘interpolant’ σ, i.e., a Σ1 formula such that the formulas A→σ and σ→B are provable in Peano Arithmetic PA. The Σ1-interpolability predicate is just a formalization of this property in the language of arithmetic.Using a standard idea of Gödel, we can associate with this predicate its provability logic, which is the set of all formulas that express arithmetically valid principles in the modal language with two modal operators, □ for provability in PA and for Σ1-interpolability. In this paper we determine this provability logic .It turns out that this problem appears to be quite similar to the problem of such modal description of relative interpretability, a problem, solved by Berarducci and Shavrukov. However, the present case is sometimes easier. In particular, we prove in this paper the fixed point property and the Craig interpolation property for our modal system ELH

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Citations of this work

Modal Matters for Interpretability Logics.Evan Goris & Joost Joosten - 2008 - Logic Journal of the IGPL 16 (4):371-412.
Self Provers and Σ1 Sentences.Evan Goris & Joost Joosten - 2012 - Logic Journal of the IGPL 20 (1):1-21.
Interpolation and the Interpretability Logic of PA.Evan Goris - 2006 - Notre Dame Journal of Formal Logic 47 (2):179-195.

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References found in this work

The Interpretability Logic of Peano Arithmetic.Alessandro Berarducci - 1990 - Journal of Symbolic Logic 55 (3):1059-1089.
Characters and Fixed-Points in Provability Logic.Zachary Gleit & Warren Goldfarb - 1989 - Notre Dame Journal of Formal Logic 31 (1):26-36.

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