Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretability in PRA Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , ∗ Institute for Logic Language and Computation University of Amsterdam and † Department of Logic Charles University; Prague 14th July 2007 Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
Why and how study interpretability Interpretations Proof theoretic characteristics of PRA Interpretability logics Modal matters ◮ We all use the notion T ⊲ S : T interprets S Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
Why and how study interpretability Interpretations Proof theoretic characteristics of PRA Interpretability logics Modal matters ◮ We all use the notion T ⊲ S : T interprets S ◮ T ⊲ S means (modulo some technical details) Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
Why and how study interpretability Interpretations Proof theoretic characteristics of PRA Interpretability logics Modal matters ◮ We all use the notion T ⊲ S : T interprets S ◮ T ⊲ S means (modulo some technical details) ◮ ∃ j ∀ ϕ (Axiom S ( ϕ ) → ∃ p Proof T ( p , � ϕ j � )) Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
Why and how study interpretability Interpretations Proof theoretic characteristics of PRA Interpretability logics Modal matters ◮ We are interested in the structural behavior of the notion of interpretability. Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
Why and how study interpretability Interpretations Proof theoretic characteristics of PRA Interpretability logics Modal matters ◮ We are interested in the structural behavior of the notion of interpretability. ◮ Interpretability can easily be formalized/arithmetized. Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
Why and how study interpretability Interpretations Proof theoretic characteristics of PRA Interpretability logics Modal matters ◮ We are interested in the structural behavior of the notion of interpretability. ◮ Interpretability can easily be formalized/arithmetized. ◮ We shall consider sentential extensions of a base theory Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
Why and how study interpretability Interpretations Proof theoretic characteristics of PRA Interpretability logics Modal matters ◮ We are interested in the structural behavior of the notion of interpretability. ◮ Interpretability can easily be formalized/arithmetized. ◮ We shall consider sentential extensions of a base theory ◮ ϕ ⊲ T ψ stands for Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
Why and how study interpretability Interpretations Proof theoretic characteristics of PRA Interpretability logics Modal matters ◮ We are interested in the structural behavior of the notion of interpretability. ◮ Interpretability can easily be formalized/arithmetized. ◮ We shall consider sentential extensions of a base theory ◮ ϕ ⊲ T ψ stands for ◮ T + ϕ ⊲ T + ψ Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
Why and how study interpretability Interpretations Proof theoretic characteristics of PRA Interpretability logics Modal matters ◮ We are interested in the structural behavior of the notion of interpretability. ◮ Interpretability can easily be formalized/arithmetized. ◮ We shall consider sentential extensions of a base theory ◮ ϕ ⊲ T ψ stands for ◮ T + ϕ ⊲ T + ψ ◮ We are interested in the interpretability logic of a theory T : Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
Why and how study interpretability Interpretations Proof theoretic characteristics of PRA Interpretability logics Modal matters ◮ We are interested in the structural behavior of the notion of interpretability. ◮ Interpretability can easily be formalized/arithmetized. ◮ We shall consider sentential extensions of a base theory ◮ ϕ ⊲ T ψ stands for ◮ T + ϕ ⊲ T + ψ ◮ We are interested in the interpretability logic of a theory T : ◮ The set of all model propositional logical formulas in the language � , ⊲ which are true regardless how you interpret the variables as arithmetical sentences Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
Why and how study interpretability Interpretations Proof theoretic characteristics of PRA Interpretability logics Modal matters ◮ We are interested in the structural behavior of the notion of interpretability. ◮ Interpretability can easily be formalized/arithmetized. ◮ We shall consider sentential extensions of a base theory ◮ ϕ ⊲ T ψ stands for ◮ T + ϕ ⊲ T + ψ ◮ We are interested in the interpretability logic of a theory T : ◮ The set of all model propositional logical formulas in the language � , ⊲ which are true regardless how you interpret the variables as arithmetical sentences ◮ Of course, reading ⊲ as ⊲ T , etc. Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
Why and how study interpretability Interpretations Proof theoretic characteristics of PRA Interpretability logics Modal matters ◮ We are interested in the structural behavior of the notion of interpretability. ◮ Interpretability can easily be formalized/arithmetized. ◮ We shall consider sentential extensions of a base theory ◮ ϕ ⊲ T ψ stands for ◮ T + ϕ ⊲ T + ψ ◮ We are interested in the interpretability logic of a theory T : ◮ The set of all model propositional logical formulas in the language � , ⊲ which are true regardless how you interpret the variables as arithmetical sentences ◮ Of course, reading ⊲ as ⊲ T , etc. ◮ Example: ( ϕ ⊲ ψ ) ∧ ( ψ ⊲ χ ) → ( ϕ ⊲ χ ) Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
Why and how study interpretability Interpretations Proof theoretic characteristics of PRA Interpretability logics Modal matters ◮ For all theories T , IL( T ) contains some sort of core logic IL Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
Why and how study interpretability Interpretations Proof theoretic characteristics of PRA Interpretability logics Modal matters ◮ For all theories T , IL( T ) contains some sort of core logic IL ◮ IL( T ) is characterized by some additional axiom schemes on top of that Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
Why and how study interpretability Interpretations Proof theoretic characteristics of PRA Interpretability logics Modal matters ◮ For all theories T , IL( T ) contains some sort of core logic IL ◮ IL( T ) is characterized by some additional axiom schemes on top of that ◮ For example, for theories with full induction, we have that Montagna’s Axiom holds ( A ⊲ B ) → (( A ∧ � C ) ⊲ ( B ∧ � C )) Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
Why and how study interpretability Interpretations Proof theoretic characteristics of PRA Interpretability logics Modal matters ◮ For all theories T , IL( T ) contains some sort of core logic IL ◮ IL( T ) is characterized by some additional axiom schemes on top of that ◮ For example, for theories with full induction, we have that Montagna’s Axiom holds ( A ⊲ B ) → (( A ∧ � C ) ⊲ ( B ∧ � C )) ◮ It turns out that precisely ILM is, e.g. IL(PA) (Shavrukov 1988; Berarducci 1990) Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
Why and how study interpretability Interpretations Proof theoretic characteristics of PRA Interpretability logics Modal matters ◮ For all theories T , IL( T ) contains some sort of core logic IL ◮ IL( T ) is characterized by some additional axiom schemes on top of that ◮ For example, for theories with full induction, we have that Montagna’s Axiom holds ( A ⊲ B ) → (( A ∧ � C ) ⊲ ( B ∧ � C )) ◮ It turns out that precisely ILM is, e.g. IL(PA) (Shavrukov 1988; Berarducci 1990) ◮ Likewise, the interpretability logic for finitely axiomatized theories is known Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
Why and how study interpretability Interpretations Proof theoretic characteristics of PRA Interpretability logics Modal matters ◮ For all theories T , IL( T ) contains some sort of core logic IL ◮ IL( T ) is characterized by some additional axiom schemes on top of that ◮ For example, for theories with full induction, we have that Montagna’s Axiom holds ( A ⊲ B ) → (( A ∧ � C ) ⊲ ( B ∧ � C )) ◮ It turns out that precisely ILM is, e.g. IL(PA) (Shavrukov 1988; Berarducci 1990) ◮ Likewise, the interpretability logic for finitely axiomatized theories is known ◮ And no other! Marta Bilkova † , Dick de Jongh ∗ , and Joost J. Joosten ∗ , Interpretability in PRA
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