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Justification logic for constructive modal logic Sonia Marin With Roman Kuznets and Lutz Straburger Inria, LIX, Ecole Polytechnique IMLA17 July 17, 2017 The big picture The big picture Justification logic: G odel: What is the

  1. Justification logic for constructive modal logic Sonia Marin With Roman Kuznets and Lutz Straßburger Inria, LIX, ´ Ecole Polytechnique IMLA’17 July 17, 2017

  2. The big picture

  3. The big picture Justification logic: G¨ odel: What is the classical provability semantics of intuitionistic logic? Artemov: Logic of Proofs gives an operational view of this S4 type of provability. ✷ A ❀ t : A ❀ t is a proof of A

  4. The big picture Justification logic: G¨ odel: What is the classical provability semantics of intuitionistic logic? Artemov: Logic of Proofs gives an operational view of this S4 type of provability. ✷ A ❀ t : A ❀ t is a proof of A Semantics: Peano arithmetics or epistemic possible worlds models Extensions: realisation of logics below and above S4

  5. The big picture Justification logic: G¨ odel: What is the classical provability semantics of intuitionistic logic? Artemov: Logic of Proofs gives an operational view of this S4 type of provability. ✷ A ❀ t : A ❀ t is a proof of A Semantics: Peano arithmetics or epistemic possible worlds models Extensions: realisation of logics below and above S4 Intuitionistic variants: Some investigations toward ◮ realisation theorems (Artemov/Steren and Bonelli), ◮ epistemic semantics (Marti and Studer), ◮ and arithmetical completeness (Artemov and Iemhoff), but where the modal language is restricted to the ✷ modality.

  6. The big picture Justification logic: G¨ odel: What is the classical provability semantics of intuitionistic logic? Artemov: Logic of Proofs gives an operational view of this S4 type of provability. ✷ A ❀ t : A ❀ t is a proof of A Semantics: Peano arithmetics or epistemic possible worlds models Extensions: realisation of logics below and above S4 Intuitionistic variants: Some investigations toward ◮ realisation theorems (Artemov/Steren and Bonelli), ◮ epistemic semantics (Marti and Studer), ◮ and arithmetical completeness (Artemov and Iemhoff), but where the modal language is restricted to the ✷ modality. However, intuitionistically ✸ cannot simply be viewed as the dual of ✷ .

  7. What are we doing here? Justifying ✸ : We start with Artemov’s treatment of the ✷ -fragment of intuitonistic modal logic.

  8. What are we doing here? Justifying ✸ : We start with Artemov’s treatment of the ✷ -fragment of intuitonistic modal logic. ✷ being read as provability, we propose to read ✸ as consistency. µ : A ✸ A ❀ ❀ µ is an witness of A

  9. What are we doing here? Justifying ✸ : We start with Artemov’s treatment of the ✷ -fragment of intuitonistic modal logic. ✷ being read as provability, we propose to read ✸ as consistency. µ : A ✸ A ❀ ❀ µ is an witness of A Intuitionistic modal logic?

  10. What are we doing here? Justifying ✸ : We start with Artemov’s treatment of the ✷ -fragment of intuitonistic modal logic. ✷ being read as provability, we propose to read ✸ as consistency. µ : A ✸ A ❀ ❀ µ is an witness of A Intuitionistic modal logic? The program: represent the operational side of the intuitionistic ✸ .

  11. What are we doing here? Justifying ✸ : We start with Artemov’s treatment of the ✷ -fragment of intuitonistic modal logic. ✷ being read as provability, we propose to read ✸ as consistency. µ : A ✸ A ❀ ❀ µ is an witness of A Intuitionistic modal logic? The program: represent the operational side of the intuitionistic ✸ . The focus: on constructive versions of modal logic.

  12. Constructive modal logic Formulas: A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A Logic CK : Intuitionistic Propositional Logic

  13. Constructive modal logic Formulas: A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A | ✷ A | ✸ A Logic CK : Intuitionistic Propositional Logic A k 1 : ✷ ( A ⊃ B ) ⊃ ( ✷ A ⊃ ✷ B ) + + necessitation: − − − k 2 : ✷ ( A ⊃ B ) ⊃ ( ✸ A ⊃ ✸ B ) ✷ A (Wijesekera/Bierman and de Paiva/Mendler and Scheele)

  14. Justification logic Justification logic adds proof terms directly inside its language. ✷ A t : A t is a proof of A ❀ ❀

  15. Justification logic Justification logic adds proof terms directly inside its language. ✷ A t : A t is a proof of A ❀ ❀ In the constructive version, we also add witness terms into the language. ✸ A ❀ µ : A ❀ µ is a witness of A

  16. Justification logic Modal formulas: A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A | ✷ A A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A | t : A Justification formulas: Grammar of terms: t ::= | | ( t · t ) | ( t + t ) | ! t c x c : proof constants x : proof variables · : application + : sum ! : proof checker

  17. Justification logic for constructive modal logic Modal formulas: A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A | ✷ A | ✸ A A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A | t : A | µ : A Justification formulas: Grammar of terms: t ::= | | ( t · t ) | ( t + t ) | ! t c x c : proof constants x : proof variables · : application + : sum ! : proof checker

  18. Justification logic for constructive modal logic Modal formulas: A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A | ✷ A | ✸ A A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A | t : A | µ : A Justification formulas: Grammar of terms: t ::= | | ( t · t ) | ( t + t ) | ! t c x µ ::= α | t ⋆ µ | ( µ ⊔ µ ) c : proof constants x : proof variables · : application + : sum ! : proof checker

  19. Justification logic for constructive modal logic Modal formulas: A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A | ✷ A | ✸ A A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A | t : A | µ : A Justification formulas: Grammar of terms: t ::= | | ( t · t ) | ( t + t ) | ! t c x µ ::= α | t ⋆ µ | ( µ ⊔ µ ) c : proof constants x : proof variables α : witness variables · : application ⋆ : execution + : sum ⊔ : disjoint witness union ! : proof checker

  20. Justification logic for constructive modal logic Axiomatisation JCK : taut: Complete finite set of axioms for intuitionistic propositional logic jk ✷ : t : ( A ⊃ B ) ⊃ ( s : A ⊃ t · s : B ) sum: s : A ⊃ ( s + t ) : A and t : A ⊃ ( s + t ) : A A ⊃ B A A is an axiom instance mp − ian − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − B c 1 : . . . c n : A

  21. Justification logic for constructive modal logic Axiomatisation JCK : taut: Complete finite set of axioms for intuitionistic propositional logic jk ✷ : t : ( A ⊃ B ) ⊃ ( s : A ⊃ t · s : B ) jk ✸ : t : ( A ⊃ B ) ⊃ ( µ : A ⊃ t ⋆ µ : B ) sum: s : A ⊃ ( s + t ) : A and t : A ⊃ ( s + t ) : A union: µ : A ⊃ ( µ ⊔ ν ) : A and ν : A ⊃ ( µ ⊔ ν ) : A A ⊃ B A A is an axiom instance mp − ian − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − B c 1 : . . . c n : A

  22. Justification logic for constructive modal logic Axiomatisation JCK : taut: Complete finite set of axioms for intuitionistic propositional logic jk ✷ : t : ( A ⊃ B ) ⊃ ( s : A ⊃ t · s : B ) jk ✸ : t : ( A ⊃ B ) ⊃ ( µ : A ⊃ t ⋆ µ : B ) sum: s : A ⊃ ( s + t ) : A and t : A ⊃ ( s + t ) : A union: µ : A ⊃ ( µ ⊔ ν ) : A and ν : A ⊃ ( µ ⊔ ν ) : A A ⊃ B A A is an axiom instance mp − ian − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − B c 1 : . . . c n : A

  23. The machinery Application: jk ✷ : t : ( A ⊃ B ) ⊃ ( s : A ⊃ t · s : B ) If t is a proof of A ⊃ B and s is a proof of A , then t · s is a proof of B .

  24. The machinery Application: jk ✷ : t : ( A ⊃ B ) ⊃ ( s : A ⊃ t · s : B ) If t is a proof of A ⊃ B and s is a proof of A , then t · s is a proof of B . jk ✸ : t : ( A ⊃ B ) ⊃ ( µ : A ⊃ t ⋆ µ : B ) Witness execution: If t is a proof of A ⊃ B and µ is a witness for A , then the same model denoted t ⋆ µ is also a witness for B .

  25. The machinery Application: jk ✷ : t : ( A ⊃ B ) ⊃ ( s : A ⊃ t · s : B ) If t is a proof of A ⊃ B and s is a proof of A , then t · s is a proof of B . jk ✸ : t : ( A ⊃ B ) ⊃ ( µ : A ⊃ t ⋆ µ : B ) Witness execution: If t is a proof of A ⊃ B and µ is a witness for A , then the same model denoted t ⋆ µ is also a witness for B . s : A ⊃ ( s + t ) : A , µ : A ⊃ ( µ ⊔ ν ) : B , . . . Sum and union: We adopt Artemov’s + to incorporate monotonicity of reasoning, and also transpose it on the witness side with ⊔ .

  26. The machinery Application: jk ✷ : t : ( A ⊃ B ) ⊃ ( s : A ⊃ t · s : B ) If t is a proof of A ⊃ B and s is a proof of A , then t · s is a proof of B . jk ✸ : t : ( A ⊃ B ) ⊃ ( µ : A ⊃ t ⋆ µ : B ) Witness execution: If t is a proof of A ⊃ B and µ is a witness for A , then the same model denoted t ⋆ µ is also a witness for B . s : A ⊃ ( s + t ) : A , µ : A ⊃ ( µ ⊔ ν ) : B , . . . Sum and union: We adopt Artemov’s + to incorporate monotonicity of reasoning, and also transpose it on the witness side with ⊔ . Iterated axiom necessitation and modus ponens:

  27. The machinery Justification logic can internalise its own reasoning.

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