A generalized notion of weak interpretability and the corresponding modal logic

AbstractDzhaparidze, G., A generalized notion of weak interpretability and the corresponding modal logic, Annals of Pure and Applied Logic 61 (1993) 113-160. A tree Tr(T1,...,Tn) of theories T1,...,Tn is called tolerant, if there are consistent extensions T+1,...,T+n of T1,...,Tn, where each T+i interprets its successors in the tree Tr(T+1,...,T+n). We consider a propositional language with the following modal formation rule: if Tr is a (finite) tree of formulas, then ♢Tr is a formula, and axiomatically define in this language the decidable logics TLR and TLRω. It is proved that TLR (resp. TLRω) yields exactly the schemata of PA-provable (resp. true) sentences, if ♢Tr(A1,...,An) is understood as (a formalization of) “Tr(PA + A1,...,PA + An) is tolerant”. In fact, TLR axiomatizes a considerable fragment of provability logic with quantifiers over ∑1-sentences, and many relations that have been studied in the literature can be expressed in terms of tolerance. We introduce and study two more relations between theories: cointerpretability and cotolerance which are, in a sense, dual to interpretability and tolerance. Cointerpretability is a characterization of ∑1-conservativity for essentially reflexive theories in terms of translations