Interpretability logics comprise a family of modal logic

Modal logic

Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...

s that extend provability logic

Provability logic

Provability logic is a modal logic, in which the box operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich formal theory, such as Peano arithmetic....

 to describe interpretability

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...

 and/or various related metamathematical properties and relations such as 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...

, Π₁-conservativity, 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...

, tolerance, cotolerance and arithmetic complexities.

Main contributors to the field are Alessandro Berarducci, Petr Hájek, Konstantin Ignatiev, 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...

, Franco Montagna, Vladimir Shavrukov, Rineke Verbrugge, Albert Visser and Domenico Zambella.