Not Incompatible Logics Olivier Hermant MINES ParisTech, PSL - PowerPoint PPT Presentation

Not Incompatible Logics Olivier Hermant MINES ParisTech, PSL Research University Inria October, 18th October, 18th O. Hermant Not Incompatible 1 / 10

Two Incompatible Logics ◮ Constructivism (BHK) ⋆ first-order approximation: intuitionistic logic ◮ Classicism ⋆ first-order classical logic ◮ in particuliar, arithmetic: ⋆ Peano and Heyting versions ⋆ same axioms, differents inference rules October, 18th O. Hermant Not Incompatible 2 / 10

Two Incompatible Logics ◮ Constructivism (BHK) ⋆ first-order approximation: intuitionistic logic ◮ Classicism ⋆ first-order classical logic ◮ in particuliar, arithmetic: ⋆ Peano and Heyting versions ⋆ same axioms, differents inference rules ◮ very pernicious conflict: ⋆ same syntax, different semantic ⋆ at least, sound ◮ two confronting schools for a long time: ⋆ incompatible properties ⋆ incompatible persons ◮ how can we conciliate them ? October, 18th O. Hermant Not Incompatible 2 / 10

The Root of the Problem Disjunction Property A proof of ⊢ i A ∨ B can be turned into a proof of ⊢ i A or a proof of ⊢ i B . ◮ in classical logic ⊢ c A ∨ ¬ A provable whatever is A ◮ another formulation: ⊢ c ¬¬ A ⇒ A ◮ similarly for the ∃ quantifier: ⋆ witness property ⋆ Drinker’s paradox ◮ lot of solutions ⋆ depends of what we are expecting ⋆ as discussed yesterday ⋆ and may be today October, 18th O. Hermant Not Incompatible 3 / 10

Double Negation Translations Kolmogorov (1925) 1 B Ko = ¬¬ B (atoms) ( B ∧ C ) Ko = ¬¬ ( B Ko ∧ C Ko ) ( B ∨ C ) Ko = ¬¬ ( B Ko ∨ C Ko ) ( B ⇒ C ) Ko = ¬¬ ( B Ko ⇒ C Ko ) ( ∀ xA ) Ko = ¬¬ ( ∀ xA Ko ) ( ∃ xA ) Ko = ¬¬ ( ∃ xA Ko ) Theorem Γ ⊢ ∆ classical provable iff Γ Ko , ¬ ∆ Ko ⊢ intuitionistically provable. October, 18th O. Hermant Not Incompatible 4 / 10

Double Negation Translations Kolmogorov (1925) 1 Gödel and Gentzen (1931) 2 ⋆ ∨ and ∃ , are the conflicting connective/quantifiers ⋆ leave the rest unchanged B gg = ¬¬ B (atoms) ( A ∧ B ) gg = A gg ∧ B gg ( A ∨ B ) gg = ¬ ( ¬ A gg ∧ ¬ B gg ) ( A ⇒ B ) gg = A gg ⇒ B gg ( ∀ xA ) gg = ∀ xA gg ( ∃ xA ) gg = ¬∀ x ¬ A gg Theorem Γ ⊢ ∆ classical provable iff Γ gg , ¬ ∆ gg ⊢ intuitionistically provable. October, 18th O. Hermant Not Incompatible 4 / 10

Double Negation Translations Kolmogorov (1925) 1 Gödel and Gentzen (1931) 2 ⋆ ∨ and ∃ , are the conflicting connective/quantifiers ⋆ leave the rest unchanged Glivenko (1929): head negation enough in the propositional case 3 Kuroda (1951): extension to FO: reset after each ∀ quantifier 4 B Ku = B (atoms) ( A ∧ B ) Ku = A Ku ∧ B Ku ( A ∨ B ) Ku = A Ku ∨ B Ku ( A ⇒ B ) Ku = A Ku ⇒ B Ku ( ∀ xA ) Ku = ∀ x ¬¬ A Ku ( ∃ xA ) Ku = ∀ x A Ku Theorem Γ ⊢ ∆ classical provable iff Γ Ku , ¬ ∆ Ku ⊢ intuitionistically provable. October, 18th O. Hermant Not Incompatible 4 / 10

More Refinments ◮ left intuitionstic and classical sequent rules identical: ⋆ no need to translate anything on LHS of Γ ⊢ c ∆ ⋆ applies to cut-free calculus. Most of the time enough LHS RHS B Ko = B B Ko = B ( B ∧ C ) Ko = ( ¬¬ B Ko ∧ ¬¬ C Ko ) ( B ∧ C ) Ko = ( ¬¬ B Ko ∧ ¬¬ C Ko ) ( B ∨ C ) Ko = ( ¬¬ B Ko ∨ ¬¬ C Ko ) ( B ∨ C ) Ko = ( ¬¬ B Ko ∨ ¬¬ C Ko ) ( B ⇒ C ) Ko = ( ¬¬ B Ko ⇒ ¬¬ C Ko ) ( B ⇒ C ) Ko = ( ¬¬ B Ko ⇒ ¬¬ C Ko ) ( ∀ xA ) Ko = ∀ x ¬¬ A Ko ( ∀ xA ) Ko = ∀ x ¬¬ A Ko ( ∃ xA ) Ko = ∃ x ¬¬ A Ko ( ∃ xA ) Ko = ∃ x ¬¬ A Ko October, 18th O. Hermant Not Incompatible 5 / 10

More Refinments ◮ left intuitionstic and classical sequent rules identical: ⋆ no need to translate anything on LHS of Γ ⊢ c ∆ ⋆ applies to cut-free calculus. Most of the time enough LHS RHS B K + = B B K − = B ( B ∧ C ) K + = ( ( B ∧ C ) K − = ( ¬¬ B K − ∧ ¬¬ C K − ) B K + ∧ C K + ) ( B ∨ C ) K + = ( ( B ∨ C ) K − = ( ¬¬ B K − ∨ ¬¬ C K − ) B K + ∨ C K + ) ( B ⇒ C ) K + = ( ¬¬ B K − ⇒ ( B ⇒ C ) K − = ( B K + ⇒ ¬¬ C K − ) C K + ) ( ∀ xA ) K + = ∀ xA K + ( ∀ xA ) K − = ∀ x ¬¬ A K − ( ∃ xA ) K + = ∃ xA K + ( ∃ xA ) K − = ∃ x ¬¬ A K − October, 18th O. Hermant Not Incompatible 5 / 10

More Refinments ◮ left intuitionstic and classical sequent rules identical: ⋆ no need to translate anything on LHS of Γ ⊢ c ∆ ⋆ applies to cut-free calculus. Most of the time enough ◮ Gilbert: left/right + Kuroda + Gödel-Gentzen. ⋆ Minimal. End of Story ? RHS (gg) LHS RHS (Ku) ϕ ( P ) = ¬¬ P χ ( P ) = P ψ ( P ) = P ϕ ( B ∧ C ) = ϕ ( B ) ∧ ϕ ( C ) χ ( B ∧ C ) = χ ( B ) ∧ χ ( C ) ψ ( B ∧ C ) = ψ ( B ) ∧ ψ ( C ) ϕ ( B ∨ C ) = ¬¬ ( ψ ( B ) ∨ ψ ( C )) χ ( B ∨ C ) = χ ( B ) ∨ χ ( C ) ψ ( B ∨ C ) = ψ ( B ) ∨ ψ ( C ) ϕ ( B ⇒ C ) = χ ( B ) ⇒ ϕ ( C ) χ ( B ⇒ C ) = ψ ( B ) ⇒ χ ( C ) ψ ( B ⇒ C ) = χ ( B ) ⇒ ψ ( C ) ϕ ( ¬ B ) = ¬ χ ( B ) χ ( ¬ B ) = ¬ ψ ( B ) ψ ( ¬ B ) = ¬ χ ( B ) ϕ ( ∀ xA ) = ∀ x ϕ ( A ) χ ( ∀ xA ) = ∀ x χ ( A ) ψ ( ∀ xA ) = ∀ x ϕ ( A ) ϕ ( ∃ xA ) = ¬¬∃ x ψ ( A ) χ ( ∃ xA ) = ∃ x χ ( A ) ψ ( ∃ xA ) = ∃ x ψ ( A ) Theorem Γ ⊢ C classically iff χ (Γ) ⊢ ϕ ( C ) intuitionistically. October, 18th O. Hermant Not Incompatible 5 / 10

More Insights ◮ Chaudhuri, Clerc, Ilik, Miller: ⋆ bijections between proofs of focused calculi ⋆ generating a particular translation by choosing a polarity ◮ Friedman: ⋆ generalize: replace “ ¬ ” with “ ⇒ A ” in translations ⋆ theorem: Theorem Γ ⊢ ∆ classical provable iff Γ Ku , ¬ ∆ Ku ⊢ ⊥ provable. October, 18th O. Hermant Not Incompatible 6 / 10

More Insights ◮ Chaudhuri, Clerc, Ilik, Miller: ⋆ bijections between proofs of focused calculi ⋆ generating a particular translation by choosing a polarity ◮ Friedman: ⋆ generalize: replace “ ¬ ” with “ ⇒ A ” in translations ⋆ theorem: Theorem Γ ⊢ ∆ classical provable iff Γ A , ∆ A ⇒ A ⊢ A provable. October, 18th O. Hermant Not Incompatible 6 / 10

More Insights ◮ Chaudhuri, Clerc, Ilik, Miller: ⋆ bijections between proofs of focused calculi ⋆ generating a particular translation by choosing a polarity ◮ Friedman: ⋆ generalize: replace “ ¬ ” with “ ⇒ A ” in translations ⋆ theorem: Theorem Γ ⊢ ∆ classical provable iff Γ A , ∆ A ⇒ A ⊢ A provable. ⋆ equiprovability of certain statements ( Π 0 2 ) ⋆ require decidability of some class of formulas ⋆ “Friedman’s trick”: take as A the statement itself. October, 18th O. Hermant Not Incompatible 6 / 10

Mixed Logics ◮ “ On the Unity of Logic ”, Girard (1993) ◮ not the logic is classical/intuitionistic/... ◮ ... but the connectives ◮ problem: ⋆ usual translations negate atoms (no connective here) B Ko = ¬¬ B (atoms) ( B ∧ C ) Ko = ¬¬ ( B Ko ∧ C Ko ) ( B ∨ C ) Ko = ¬¬ ( B Ko ∨ C Ko ) ( B ⇒ C ) Ko = ¬¬ ( B Ko ⇒ C Ko ) ( ∀ xA ) Ko = ¬¬ ( ∀ xA Ko ) ( ∃ xA ) Ko = ¬¬ ( ∃ xA Ko ) Theorem Γ ⊢ ∆ classical provable iff Γ Ko , � \ ∆ Ko ⊢ provable. October, 18th O. Hermant Not Incompatible 7 / 10

Mixed Logics ◮ “ On the Unity of Logic ”, Girard (1993) ◮ not the logic is classical/intuitionistic/... ◮ ... but the connectives ◮ problem: ⋆ usual translations negate atoms (no connective here) ⋆ “light” translations negate the whole (no connective there either) B Ko = B (atoms) ( B ∧ C ) Ko = ( ¬¬ B Ko ∧ ¬¬ C Ko ) ( B ∨ C ) Ko = ( ¬¬ B Ko ∨ ¬¬ C Ko ) ( B ⇒ C ) Ko = ( ¬¬ B Ko ⇒ ¬¬ C Ko ) ( ∀ xA ) Ko = ∀ x ¬¬ A Ko ( ∃ xA ) Ko = ∃ x ¬¬ A Ko Theorem Γ ⊢ ∆ classical provable iff Γ Ko , ¬ ∆ Ko ⊢ provable. October, 18th O. Hermant Not Incompatible 7 / 10

Mixing Logics ◮ Dowek’s translation goes double B Do = B (atoms) ( B ∧ C ) Do = ¬¬ ( ¬¬ B Do ∧ ¬¬ C Do ) ( B ∨ C ) Do = ¬¬ ( ¬¬ B Do ∨ ¬¬ C Do ) ( B ⇒ C ) Do = ¬¬ ( ¬¬ B Do ⇒ ¬¬ C Do ) ( ∀ xA ) Do = ¬¬∀ x ¬¬ A Do ( ∃ xA ) Do = ¬¬∃ x ¬¬ A Do October, 18th O. Hermant Not Incompatible 8 / 10

Mixing Logics ◮ Dowek’s translation goes double B Do = B (atoms) ( B ∧ C ) Do = ¬¬ ( ¬¬ B Do ∧ ¬¬ C Do ) ( B ∨ C ) Do = ¬¬ ( ¬¬ B Do ∨ ¬¬ C Do ) ( B ⇒ C ) Do = ¬¬ ( ¬¬ B Do ⇒ ¬¬ C Do ) ( ∀ xA ) Do = ¬¬∀ x ¬¬ A Do ( ∃ xA ) Do = ¬¬∃ x ¬¬ A Do ◮ gain: no negated atoms, no negated formulas ◮ definition of classical connectives and quantifiers ( B ∧ c C ) = ¬¬ ( ¬¬ B ∧ i ¬¬ C ) ( B ∨ c C ) = ¬¬ ( ¬¬ B ∨ i ¬¬ C ) ( B ⇒ c C ) = ¬¬ ( ¬¬ B ⇒ i ¬¬ C ) ( ∀ c xA ) = ¬¬∀ i x ¬¬ A ( ∃ c xA ) = ¬¬∃ i x ¬¬ A ◮ intuitionistic calculus as a basis October, 18th O. Hermant Not Incompatible 8 / 10