Abstract
We show that the modal prepositional logicILM (interpretability logic with Montagna's principle), which has been shown sound and complete as the interpretability logic of Peano arithmetic PA (by Berarducci and Savrukov), is sound and complete as the logic ofπ 1-conservativity over eachbE 1-sound axiomatized theory containingI⌆ 1 (PA with induction restricted tobE 1-formulas). Furthermore, we extend this result to a systemILMR obtained fromILM by adding witness comparisons in the style of Guaspari's and Solovay's logicR (this will be done in a separate continuation of the present paper).
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Hájek, P., Montagna, F. The logic of π1-conservativity. Arch Math Logic 30, 113–123 (1990). https://doi.org/10.1007/BF01634981
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DOI: https://doi.org/10.1007/BF01634981