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Interpretability logic
 Mathematical Logic, Proceedings of the 1988 Heyting Conference
, 1990
"... Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbertstyle programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength ..."
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Cited by 32 (9 self)
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Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbertstyle programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength of theories, or better to prove
The Interpretability Logic of all Reasonable Arithmetical Theories
 ERKENNTNIS
, 1999
"... This paper is a presentation of a status quaestionis, to wit of the problem of the interpretability logic of all reasonable arithmetical theories. We present both the arithmetical side and the modal side of the question. ..."
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Cited by 9 (5 self)
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This paper is a presentation of a status quaestionis, to wit of the problem of the interpretability logic of all reasonable arithmetical theories. We present both the arithmetical side and the modal side of the question.
The Interpolation Theorem for IL and ILP
 Uppsala University
, 1998
"... In this article we establish interpolation for the minimal system of interpretability logic IL. We prove that arrow interpolation holds for IL and that turnstile interpolation and interpolation for the modality easily follow from this. Furthermore, these properties are extended to the system ILP. ..."
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Cited by 2 (0 self)
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In this article we establish interpolation for the minimal system of interpretability logic IL. We prove that arrow interpolation holds for IL and that turnstile interpolation and interpolation for the modality easily follow from this. Furthermore, these properties are extended to the system ILP. The related issue of Beth Definability is also addressed. As usual, the arrow interpolation property implies the Beth property. From the latter it follows via an argumentation which is standard in provability logic, that IL has the fixed point property. Finally we observe that a general result of Maksimova [11] for provability logics can be extended to interpretability logics, implying that all extensions of IL have the Beth property. Keywords Interpretability Logic, Interpolation Properties, Beth Property, Fixed Point Property. 1 Introduction 1.1 Some History Interpretability logics are extensions of provability logics introduced by Visser in [15]. There the modal logics IL, ILM and ILP a...
Interpretability over Peano arithmetic
"... We investigate the modal logic of interpretability over Peano arithmetic (PA). Our main result is an extension of the arithmetical completeness theorem for the interpretability logic ILM ! . This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than sin ..."
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We investigate the modal logic of interpretability over Peano arithmetic (PA). Our main result is an extension of the arithmetical completeness theorem for the interpretability logic ILM ! . This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a theorem answering a question of Orey from 1961. All these results also hold for ZermeloFraenkel set theory (ZF). 1 Introduction Provability logic Provability logic is concerned with the investigation of various metamathematical relations with the aid of modal logic. For instance, provability logic has been used to investigate different notions of provability, interpretability, conservativity, and tolerance. The fundamental connection between modal logic and metamathematics is provided by certain functions called realizations. These are primitive recursive functions ...
A simple proof of arithmetical completeness for ...conservativity logic
, 1996
"... H'ajek and Montagna proved that the modal propositional logic ILM is the logic of \Pi 1 conservativity over sound theories containing I \Sigma 1 (PA with induction restricted to \Sigma 1 formulas). I give a simpler proof of the same fact. ..."
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H'ajek and Montagna proved that the modal propositional logic ILM is the logic of \Pi 1 conservativity over sound theories containing I \Sigma 1 (PA with induction restricted to \Sigma 1 formulas). I give a simpler proof of the same fact.