{{Short description|Family of modal logics that extend provability logic}} '''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, Π<sub>1</sub>-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 by adding the unary modal operator <math>\Box</math> and the binary modal operator <math>\triangleright</math> (as usual, <math>\Diamond p</math> is defined as <math>\neg \Box\neg p</math>). The arithmetical interpretation of <math>\Box p</math> is “<math>p</math> is provable in Peano arithmetic (PA)”, and <math>p \triangleright q</math> is understood as “<math>PA+q</math> is interpretable in <math>PA+p</math>”.
# All classical tautologies # <math>\Box(p \rightarrow q) \rightarrow (\Box p \rightarrow \Box q)</math> # <math>\Box(\Box p \rightarrow p) \rightarrow \Box p</math> # <math> \Box (p \rightarrow q) \rightarrow (p \triangleright q)</math> # <math> (p \triangleright q)\rightarrow (\Diamond p \rightarrow \Diamond q) </math> # <math> (p \triangleright q)\wedge (q \triangleright r)\rightarrow (p\triangleright r)</math> # <math> (p \triangleright r)\wedge (q \triangleright r)\rightarrow ((p\vee q)\triangleright r)</math> # <math> \Diamond p \triangleright p </math> # <math> (p \triangleright q)\rightarrow((p\wedge\Box r)\triangleright (q\wedge\Box r)) </math>
# “From <math>p</math> and <math>p\rightarrow q</math> conclude <math>q</math>” # “From <math>p</math> conclude <math>\Box p</math>”.
The completeness of ILM with respect to its arithmetical interpretation was independently proven by Alessandro Berarducci and Vladimir Shavrukov.
=== Logic TOL ===
The language of TOL extends that of classical propositional logic by adding the modal operator <math>\Diamond</math> which is allowed to take any nonempty sequence of arguments. The arithmetical interpretation of <math>\Diamond( p_1,\ldots,p_n)</math> is “<math>(PA+p_1,\ldots,PA+p_n)</math> is a tolerant sequence of theories”. Axioms (with <math>p,q</math> standing for any formulas, <math>\vec{r},\vec{s}</math> for any sequences of formulas, and <math>\Diamond()</math> identified with ⊤):
# All classical tautologies # <math>\Diamond (\vec{r},p,\vec{s})\rightarrow \Diamond (\vec{r}, p\wedge\neg q,\vec{s})\vee \Diamond (\vec{r}, q,\vec{s}) </math> # <math>\Diamond (p)\rightarrow \Diamond (p\wedge \neg\Diamond (p)) </math> # <math>\Diamond (\vec{r},p,\vec{s})\rightarrow \Diamond (\vec{r},\vec{s})</math> # <math>\Diamond (\vec{r},p,\vec{s})\rightarrow \Diamond (\vec{r},p,p,\vec{s})</math> # <math>\Diamond (p,\Diamond(\vec{r}))\rightarrow \Diamond (p\wedge\Diamond(\vec{r}))</math> # <math>\Diamond (\vec{r},\Diamond(\vec{s}))\rightarrow \Diamond (\vec{r},\vec{s})</math>
Rules of inference:
# “From <math>p</math> and <math>p\rightarrow q</math> conclude <math>q</math>” # “From <math>\neg p</math> conclude <math>\neg \Diamond( p)</math>”.
The completeness of TOL with respect to its arithmetical interpretation was proven by Giorgi Japaridze.
==References== * [https://web.archive.org/web/20190419120954/http://www.csc.villanova.edu/~japaridz/ Giorgi Japaridze] and Dick de Jongh, ''The Logic of Provability''. In '''Handbook of Proof Theory''', S. Buss, ed., Elsevier, 1998, pp. 475-546.
Category:Provability logic Category:Interpretation (philosophy)