Interpretability Logic
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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 Tolerance or toleration is the state of tolerating, or putting up with, conditionally. Economics, business, and politics * Toleration Party, a historic political party active in Connecticut * Tolerant Systems, the former name of Veritas Software ...
, 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 \Box and the binary modal operator \triangleright (as always, \Diamond p is defined as \neg \Box\neg p). The arithmetical interpretation of \Box p is “p is provable in
Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...
(PA)”, and p \triangleright q is understood as “PA+q is interpretable in PA+p”. Axiom schemata: 1. All classical tautologies 2. \Box(p \rightarrow q) \rightarrow (\Box p \rightarrow \Box q) 3. \Box(\Box p \rightarrow p) \rightarrow \Box p 4. \Box (p \rightarrow q) \rightarrow (p \triangleright q) 5. (p \triangleright q)\wedge (q \triangleright r)\rightarrow (p\triangleright r) 6. (p \triangleright r)\wedge (q \triangleright r)\rightarrow ((p\vee q)\triangleright r) 7. (p \triangleright q)\rightarrow (\Diamond p \rightarrow \Diamond q) 8. \Diamond p \triangleright p 9. (p \triangleright q)\rightarrow((p\wedge\Box r)\triangleright (q\wedge\Box r)) Rules of inference: 1. “From p and p\rightarrow q conclude q” 2. “From p conclude \Box p”. 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 \Diamond which is allowed to take any nonempty sequence of arguments. The arithmetical interpretation of \Diamond( p_1,\ldots,p_n) is “(PA+p_1,\ldots,PA+p_n) is a tolerant sequence of theories”. Axioms (with p,q standing for any formulas, \vec,\vec for any sequences of formulas, and \Diamond() identified with ⊤): 1. All classical tautologies 2. \Diamond (\vec,p,\vec)\rightarrow \Diamond (\vec, p\wedge\neg q,\vec)\vee \Diamond (\vec, q,\vec) 3. \Diamond (p)\rightarrow \Diamond (p\wedge \neg\Diamond (p)) 4. \Diamond (\vec,p,\vec)\rightarrow \Diamond (\vec,\vec) 5. \Diamond (\vec,p,\vec)\rightarrow \Diamond (\vec,p,p,\vec) 6. \Diamond (p,\Diamond(\vec))\rightarrow \Diamond (p\wedge\Diamond(\vec)) 7. \Diamond (\vec,\Diamond(\vec))\rightarrow \Diamond (\vec,\vec{s}) Rules of inference: 1. “From p and p\rightarrow q conclude q” 2. “From \neg p conclude \neg \Diamond( p)”. 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. Modal logic Provability logic