A generalized notion of weak interpretability and the corresponding modal logic

Annals of Pure and Applied Logic 61 (1-2):113-160 (1993)
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Abstract

Dzhaparidze, G., A generalized notion of weak interpretability and the corresponding modal logic, Annals of Pure and Applied Logic 61 113-160. A tree Tr 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. We consider a propositional language with the following modal formation rule: if Tr is a 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 yields exactly the schemata of PA-provable sentences, if Tr is understood as “Tr 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

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References found in this work

Arithmetization of Metamathematics in a General Setting.Solomon Feferman - 1960 - Journal of Symbolic Logic 31 (2):269-270.
The interpretability logic of peano arithmetic.Alessandro Berarducci - 1990 - Journal of Symbolic Logic 55 (3):1059-1089.
Relative Interpretations.Steven Orey - 1961 - Mathematical Logic Quarterly 7 (7‐10):146-153.
Relative Interpretations.Steven Orey - 1961 - Mathematical Logic Quarterly 7 (7-10):146-153.

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