Weak interpretability
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Assume T and S are formal theories. Slightly simplified, T is said to be weakly interpretable in S iff the language of T can be translated into the language of S in such a way that the translation of every theorem of T is consistent with S. Of course, there are some natural conditions on admissible translations here, such as the necessity for a translation to preserve the logical structure of formulas.
This concept, together with interpretability, was introduced by Alfred Tarski in 1953. Weak interpretability is a special case of the concept of tolerance introduced by Giorgi Japaridze in 1992. See also Interpretability logic.
References
- A.Tarski, A.Mostovski and R.M.Robinson, Undecidable Theories. North-Holland, Amsterdam, 1953.
- G.Japaridze (http://www.csc.villanova.edu/~japaridz/), A generalized notion of weak interpretability and the corresponding logic. Annals of Pure and Applied Logic 61 (1993), pp. 113-160.
- G.Japaridze (http://www.csc.villanova.edu/~japaridz/), The logic of linear tolerance. Studia Logica 51 (1992), pp. 249-277.
- G.Japaridze (http://www.csc.villanova.edu/~japaridz/) and D. de Jongh, The logic of provability. Handbook of Proof Theory. S.Buss, ed. Elsevier, 1998, pp. 476-546.