mathematical logic Mathematical logic is the study of logic, 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 for ...

, 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 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 attire ...

. Slightly simplified, ''T'' is said to be ''interpretable'' in ''S'' if and only if the language of ''T'' can be

translate Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...

d into the

language Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...

of ''S'' in such a way that ''S'' proves 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 th ...

of ''T''. 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 betwee ...

. This concept, together with

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 weakl ...

, was introduced by

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. Three other related concepts are

cointerpretability In mathematical logic, cointerpretability is a binary relation on formal theories: a formal theory ''T'' is cointerpretable in another such theory ''S'', when the language of ''S'' can be translated into the language of ''T'' in such a way that ''S ...

, logical tolerance, and cotolerance, 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 1992–93.

References

* Japaridze, G., and De Jongh, D. (1998) "The logic of provability" in Buss, S., ed., ''Handbook of Proof Theory''. North-Holland: 476–546. *

, Andrzej Mostowski, and

Raphael Robinson Raphael Mitchel Robinson (November 2, 1911 – January 27, 1995) was an United States of America, American mathematician. Born in National City, California, National City, California, Robinson was the youngest of four children of a lawyer and a t ...

(1953) ''Undecidable Theories''. North-Holland. Proof theory {{logic-stub

Informal definition

See also

References