# Cointerpretability

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In [mathematical logic](/source/mathematical_logic), '''cointerpretability''' is a [binary relation](/source/binary_relation) on [formal theories](/source/theory_(mathematical_logic)): 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'' proves every [formula](/source/formula_(logic)) whose translation is a [theorem](/source/theorem) of ''T''. The "translation" here is required to preserve the logical structure of formulas. 

This concept, in a sense dual to [interpretability](/source/interpretability), was introduced by {{harvtxt|Japaridze|1993}}, who also proved that, for theories of [Peano arithmetic](/source/Peano_arithmetic) and any stronger theories with [computable](/source/computable_set) [axiomatization](/source/axiomatization)s, cointerpretability is equivalent to <math>\Sigma_1</math>-conservativity.

==See also==
* [Cotolerance](/source/Cotolerance)
* [Interpretability logic](/source/Interpretability_logic)
* [Tolerance (in logic)](/source/Tolerance_(in_logic))

==References==
*{{citation
 | last = Japaridze| first = Giorgi | authorlink = Giorgi Japaridze
 | doi = 10.1016/0168-0072(93)90201-N
 | issue = 1–2
 | journal = [Annals of Pure and Applied Logic](/source/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
 | 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 = Studies in Logic and the Foundations of Mathematics
 | title = Handbook of Proof Theory
 | volume = 137
 | year = 1998| doi-access = free
 }}.

Category:Mathematical relations
Category:Mathematical logic

{{logic-stub}}

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