A generalized notion of weak interpretability and the corresponding modal logic
The paper introduces two interpretability-style relations between theories: tolerance (generalized weak interpretability) and cointerpretability. Intuitively, \(S\) is tolerant with \(T\) if \(T\) can be translated into \(S\) in such a way that the translation of any theorem of \(T\) is consistent with \(S\); and \(S\) cointerprets \(T\), if \(S\) can be translated into \(T\) in such a way that \(S\) proves every formula the translation of which is a theorem of \(T\). The set of all valid principles of tolerance is modal-logically axiomatized and its decidability is proven. As for cointerpretability, it turns out to be a characterization of sigma-one conservativity (over strong theories) in terms of translations.
scientific article; zbMATH DE number 416884
scientific article