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.

Examples

Logic ILM

The language of ILM extends that of classical propositional logic is provable in Peano arithmetic”.

Axiom schemata:

1. All classical tautologies

Rules of inference:

1. ”

2. ”.

The completeness of ILM with respect to its arithmetical interpretation was independently proven by Alessandro Berarducci and Vladimir Shavrukov.

Logic TOL

is a tolerant sequence of theories”.

identified with ⊤):

1. All classical tautologies

Rules of inference:

1. ”

2. ”.

The completeness of TOL with respect to its arithmetical interpretation was proven by Giorgi Japaridze.

References

• Giorgi Japaridze and Dick de Jongh, The Logic of Provability. In Handbook of Proof Theory, S. Buss, ed., Elsevier, 1998, pp. 475-546.