Editing Interpretability

{{Short description|Concept in mathematics}}
{{for|the machine learning concept|Interpretability (machine learning)}}
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 [[Theory (mathematical logic)|formal theories]]. Slightly simplified, ''T'' is said to be ''interpretable'' in ''S'' if and only if the language of ''T'' can be translated into the [[language]] of ''S'' in such a way that ''S'' proves the translation of every [[theorem]] 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 [[Well-formed formula|formulas]].

This concept, together with [[weak interpretability]], was introduced by [[Alfred Tarski]] in 1953. Three other related concepts are [[cointerpretability]], [[tolerance (in logic)|logical tolerance]], and [[cotolerance]], introduced by [[Giorgi Japaridze]] in 1992–93.

==See also==
* [[Conservative extension]]
* [[Interpretation (logic)]]
* [[Interpretation (model theory)]]
* [[Interpretability logic]]

==References==
* Japaridze, G., and De Jongh, D. (1998) "The logic of provability" in Buss, S., ed., ''Handbook of Proof Theory''. North-Holland: 476–546.
* [[Alfred Tarski]], [[Andrzej Mostowski]], and [[Raphael Robinson]] (1953) ''Undecidable Theories''. North-Holland.

[[Category:Proof theory]]


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