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
==References== *{{citation | last = Tarski | first = Alfred | authorlink = Alfred Tarski | location = Amsterdam | mr = 0058532 | publisher = North-Holland Publishing Company | series = Studies in Logic and the Foundations of Mathematics | title = Undecidable theories | year = 1953}}. Written in collaboration with Andrzej Mostowski and Raphael M. Robinson. *{{citation | last = Dzhaparidze | first = Giorgie | authorlink = Giorgi Japaridze | doi = 10.1016/0168-0072(93)90201-N | issue = 1-2 | journal = Annals of Pure and Applied Logic | mr = 1218658 | pages = 113–160 | title = A generalized notion of weak interpretability and the corresponding modal logic | volume = 61 | year = 1993| doi-access = }}. *{{citation | last = Dzhaparidze | first = Giorgie | authorlink = Giorgi Japaridze | doi = 10.1007/BF00370116 | issue = 2 | journal = Studia Logica | mr = 1185914 | pages = 249–277 | title = The logic of linear tolerance | volume = 51 | year = 1992}} *{{citation | last1 = Japaridze | first1 = Giorgi | author1-link = Giorgi Japaridze | last2 = de Jongh | first2 = Dick | author2-link = Dick de Jongh | editor-last = Buss | editor-first = Samuel R. | editor-link = Samuel Buss | contribution = The logic of provability | doi = 10.1016/S0049-237X(98)80022-0 | location = Amsterdam | mr = 1640331 | pages = 475–546 | publisher = North-Holland | series = Stud. Logic Found. Math. | title = Handbook of Proof Theory | volume = 137 | year = 1998| doi-access = free }}
Category:Proof theory