Weak interpretability

In mathematical logic, weak interpretability is a notion of translation of logical theories, introduced together with interpretability by Alfred Tarski in 1953.

Let T and S be formal theories. Slightly simplified, T is said to be weakly interpretable in S if, and only if, 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.

A generalization of weak interpretability, tolerance, was introduced by Giorgi Japaridze in 1992.

See also

• Interpretability logic

• Tarski, Alfred (1953), Undecidable theories, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland Publishing Company, MR. Written in collaboration with Andrzej Mostowski and Raphael M. Robinson.
• Dzhaparidze, Giorgie (1993), "A generalized notion of weak interpretability and the corresponding modal logic", Annals of Pure and Applied Logic, 61 (1–2): 113–160, doi:, MR.
• Dzhaparidze, Giorgie (1992), "The logic of linear tolerance", Studia Logica, 51 (2): 249–277, doi:, MR
• Japaridze, Giorgi; de Jongh, Dick (1998), "The logic of provability", in Buss, Samuel R. (ed.), Handbook of Proof Theory, Stud. Logic Found. Math., vol.137, Amsterdam: North-Holland, pp.475–546, doi:, MR