In mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, 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 theory

Formal theory can refer to:* Another name for a theory which is expressed in formal language.* An axiomatic system, something representable by symbols and its operators...

. Slightly simplified, T is said to be interpretable in S if and only if the language of T can be translate

Translation

Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. Whereas interpreting undoubtedly antedates writing, translation began only after the appearance of written literature; there exist partial translations of the Sumerian Epic of...

d into the language

Language

Language may refer either to the specifically human capacity for acquiring and using complex systems of communication, or to a specific instance of such a system of complex communication...

 of S in such a way that S proves the translation of every theorem

Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

 of T. Of course, there are some natural conditions on admissible translations here, such as the necessity for a translation to preserve the logic

Logic

In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

al structure of formula

Formula

In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....

s.

This concept, together with weak interpretability

Weak interpretability

Weak interpretability is a special case of the concept of tolerance introduced by Giorgi Japaridze in 1992.Assume T and S are 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...

, was introduced by Alfred Tarski

Alfred Tarski

Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...

 in 1953. Three other related concepts are cointerpretability

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 proves every formula whose translation is a theorem of T...

, logical tolerance, and cotolerance, introduced by Giorgi Japaridze

Giorgi Japaridze

Giorgi Japaridze is a logician, at Villanova University in Villanova, Pennsylvania. In the past his contributions were primarily into the areas of provability logic and interpretability logic...

 in 1992-1993.

See also

• Interpretation (model theory)
  Interpretation (model theory)
  In model theory, interpretation of a structure M in another structure N is a technical notion that approximates the idea of representing M inside N. For example every reduct or definitional expansion of a structure N has an interpretation in N.Many model-theoretic properties are preserved under...
  
  
• Interpretation (logic)
  Interpretation (logic)
  An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation...
  
  
• Interpretability logic
  Interpretability logic
  Interpretability logics comprise a family of modal logics that extend provability logic to describe interpretability and/or various related metamathematical properties and relations such as weak interpretability, Π1-conservativity, cointerpretability, tolerance, cotolerance and arithmetic...