Weak interpretability

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

Assume T and S are 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.

References

• Tarski, Alfred (1953), Undecidable theories, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland Publishing Company, MR 0058532. 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:10.1016/0168-0072(93)90201-N, MR 1218658.
• Dzhaparidze, Giorgie (1992), "The logic of linear tolerance", Studia Logica, 51 (2): 249–277, doi:10.1007/BF00370116, MR 1185914
• Japaridze, Giorgi; de Jongh, Dick (1998), "The logic of provability", in Buss, Samuel R., Handbook of Proof Theory, Stud. Logic Found. Math., 137, Amsterdam: North-Holland, pp. 475–546, doi:10.1016/S0049-237X(98)80022-0, MR 1640331