Notre Dame Journal of Formal Logic

Provability and Interpretability Logics with Restricted Realizations

Thomas F. Icard and Joost J. Joosten

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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.

Article information

Source
Notre Dame J. Formal Logic Volume 53, Number 2 (2012), 133-154.

Dates
First available in Project Euclid: 9 May 2012

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1336588246

Digital Object Identifier
doi:10.1215/00294527-1715653

Mathematical Reviews number (MathSciNet)
MR2925273

Zentralblatt MATH identifier
06054421

Subjects
Primary: 03F45: Provability logics and related algebras (e.g., diagonalizable algebras) [See also 03B45, 03G25, 06E25] 03B45: Modal logic (including the logic of norms) {For knowledge and belief, see 03B42; for temporal logic, see 03B44; for provability logic, see also 03F45}

Keywords
provability logic interpretability logic restricted substitutions GLP ISigma1 PRA

Citation

Icard, Thomas F.; Joosten, Joost J. Provability and Interpretability Logics with Restricted Realizations. Notre Dame Journal of Formal Logic 53 (2012), no. 2, 133--154. doi:10.1215/00294527-1715653. http://projecteuclid.org/euclid.ndjfl/1336588246.


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