Interpretability logics comprise a family of modal logics that extend provability logic to describe interpretability or various related metamathematical properties and relations such as weak interpretability, Π1-conservativity, cointerpretability, tolerance, cotolerance, and arithmetic complexities. Main contributors to the field are Alessandro Berarducci, Petr Hájek, Konstantin Ignatiev, Giorgi Japaridze, Franco Montagna, Vladimir Shavrukov, Rineke Verbrugge, Albert Visser, and Domenico Zambella. The language of ILM extends that of classical propositional logic by adding the unary modal operator and the binary modal operator (as always, is defined as ). The arithmetical interpretation of is “ is provable in Peano arithmetic (PA)”, and is understood as “ is interpretable in ”. Axiom schemata:
Rules of inference:
Rules of inference: