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2012 Provability and Interpretability Logics with Restricted Realizations
Thomas F. Icard, Joost J. Joosten
Notre Dame J. Formal Logic 53(2): 133-154 (2012). DOI: 10.1215/00294527-1715653

Abstract

The provability logic of a theory $T$ is the set of modal formulas, which under any arithmetical realization are provable in $T$. We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $\Gamma$. We make an analogous modification for interpretability logics. We first study provability logics with restricted realizations and show that for various natural candidates of $T$ and restriction set $\Gamma$, the result is the logic of linear frames. However, for the theory Primitive Recursive Arithmetic (PRA), we define a fragment that gives rise to a more interesting provability logic by capitalizing on the well-studied relationship between PRA and I$\Sigma_1$. We then study interpretability logics, obtaining upper bounds for IL(PRA), whose characterization remains a major open question in interpretability logic. Again this upper bound is closely related to linear frames. The technique is also applied to yield the nontrivial result that IL(PRA) $\subset$ ILM.

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Thomas F. Icard. Joost J. Joosten. "Provability and Interpretability Logics with Restricted Realizations." Notre Dame J. Formal Logic 53 (2) 133 - 154, 2012. https://doi.org/10.1215/00294527-1715653

Information

Published: 2012
First available in Project Euclid: 9 May 2012

zbMATH: 1255.03054
MathSciNet: MR2925273
Digital Object Identifier: 10.1215/00294527-1715653

Subjects:
Primary: 03B45 , 03F45

Keywords: GLP , interpretability logic , ISigma1 , PRA , provability logic , restricted substitutions

Rights: Copyright © 2012 University of Notre Dame

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Vol.53 • No. 2 • 2012
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