mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

, weak interpretability is a notion of translation of logical theories, introduced together with

interpretability In mathematical logic, interpretability is a relation between formal theories that expresses the possibility of interpreting or translating one into the other. Informal definition Assume ''T'' and ''S'' are formal theories. Slightly simplified, '' ...

Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...

in 1953. Let ''T'' and ''S'' be

formal theories Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements ( forms, in Ancient Greek). They may refer to: Dress code and events * Formal wear, attire for formal events * Semi-formal atti ...

. 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 In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...

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 In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...

. A generalization of weak interpretability, tolerance, was introduced by

Giorgi Japaridze Giorgi Japaridze (also spelled Giorgie Dzhaparidze) is a Georgian-American researcher in logic and theoretical computer science. He currently holds the title of Full Professor at the Computing Sciences Department of Villanova University. Japaridze i ...

in 1992.

References

*. Written in collaboration with Andrzej Mostowski and Raphael M. Robinson. *. * *{{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 Proof theory

See also

References