Interpretability in

Annals of Pure and Applied Logic 161 (2):128-138 (2010)
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Abstract

In this paper, we study IL(), the interpretability logic of . As is neither an essentially reflexive theory nor finitely axiomatizable, the two known arithmetical completeness results do not apply to : IL() is not or . IL() does, of course, contain all the principles known to be part of IL, the interpretability logic of the principles common to all reasonable arithmetical theories. In this paper, we take two arithmetical properties of and see what their consequences in the modal logic IL() are. These properties are reflected in the so-called Beklemishev Principle , and Zambella’s Principle , neither of which is a part of IL. Both principles and their interrelation are submitted to a modal study. In particular, we prove a frame condition for . Moreover, we prove that follows from a restricted form of . Finally, we give an overview of the known relationships of IL() to important other interpretability principles

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Author Profiles

Joost Joosten
Universitat de Barcelona
Dick De De Jongh
University of Amsterdam

References found in this work

The interpretability logic of peano arithmetic.Alessandro Berarducci - 1990 - Journal of Symbolic Logic 55 (3):1059-1089.
Proof-theoretic analysis by iterated reflection.Lev D. Beklemishev - 2003 - Archive for Mathematical Logic 42 (6):515-552.
The logic of π1-conservativity.Petr Hajek & Franco Montagna - 1990 - Archive for Mathematical Logic 30 (2):113-123.
Modal completeness of ILW.Dick De Jongh & Frank Veltman - 1999 - In Jelle Gerbrandy, Maarten Marx, Maarten de Rijke & Yde Venema (eds.), Essays Dedicated to Johan van Benthem on the Occasion of His 50th Birthday. Amsterdam University Press.

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