Road to cut elimination in G3c ◮ From invertibility of logical rules ◮ to admissibility of contraction ◮ to cut elimination 13 / 53
Labelled systems ◮ Variant of Gentzen sequents for modal logics 14 / 53
Labelled systems ◮ Variant of Gentzen sequents for modal logics ◮ (in fact, for any logic with a Kripke semantics) 14 / 53
Labelled systems ◮ Variant of Gentzen sequents for modal logics ◮ (in fact, for any logic with a Kripke semantics) ◮ combination of modal and (a fragment of) first-order language 14 / 53
Labelled systems ◮ Variant of Gentzen sequents for modal logics ◮ (in fact, for any logic with a Kripke semantics) ◮ combination of modal and (a fragment of) first-order language ◮ multisets of labelled formulas x : A or relational atoms xRy 14 / 53
Labelled systems ◮ Variant of Gentzen sequents for modal logics ◮ (in fact, for any logic with a Kripke semantics) ◮ combination of modal and (a fragment of) first-order language ◮ multisets of labelled formulas x : A or relational atoms xRy ◮ logical rules for labelled formulas 14 / 53
Labelled systems ◮ Variant of Gentzen sequents for modal logics ◮ (in fact, for any logic with a Kripke semantics) ◮ combination of modal and (a fragment of) first-order language ◮ multisets of labelled formulas x : A or relational atoms xRy ◮ logical rules for labelled formulas ◮ non-logical rules for relational atoms 14 / 53
Labelled systems ◮ Similarity between � and ∀ Γ ⇒ ∆ , A ( y / x ) xRy , Γ ⇒ ∆ , y : A R � ∗ R ∀∗ Γ ⇒ ∆ , x : � A Γ ⇒ ∆ , ∀ xA A ( t / x ) , ∀ xA , Γ ⇒ ∆ y : A , x : � , xRy , Γ ⇒ ∆ L � L ∀ x : � A , xRy , Γ ⇒ ∆ ∀ xA , Γ ⇒ ∆ 15 / 53
Labelled systems ◮ Similarity between ♦ and ∃ Γ ⇒ ∆ , ∃ xA , A ( y / x ) xRy , Γ ⇒ ∆ , x : ♦ A , y : A R ♦ R ∃ xRy , Γ ⇒ ∆ , x : ♦ A Γ ⇒ ∆ , ∃ xA A ( y / x ) , Γ ⇒ ∆ xRy , y : A , Γ ⇒ ∆ L ♦ ∗ L ∃∗ x : ♦ A , Γ ⇒ ∆ ∃ xA , Γ ⇒ ∆ 16 / 53
G3K x : P , Γ ⇒ ∆ , x : P x : ⊥ , Γ ⇒ ∆ x : A , x : B , Γ ⇒ ∆ Γ ⇒ ∆ , x : A Γ ⇒ ∆ , x : B L ∧ R ∧ x : A ∧ B , Γ ⇒ ∆ Γ ⇒ ∆ , x : A ∧ B x : A , Γ ⇒ ∆ x : B , Γ ⇒ ∆ Γ ⇒ ∆ , x : A , x : B L ∨ R ∨ x : A ∨ B , Γ ⇒ ∆ Γ ⇒ ∆ , x : A ∨ B Γ ⇒ ∆ , x : A x : B , Γ ⇒ ∆ x : A , Γ ⇒ ∆ , x : B L ⊃ R ⊃ x : A ⊃ B , Γ ⇒ ∆ Γ ⇒ ∆ , x : A ⊃ B y : A , x : � A , xRy , Γ ⇒ ∆ xRy , Γ ⇒ ∆ , y : A L � R � ∗ x : � A , xRy , Γ ⇒ ∆ Γ ⇒ ∆ , x : � A xRy , y : A , Γ ⇒ ∆ xRy , Γ ⇒ ∆ , x : ♦ A , y : A L ♦ ∗ R ♦ x : ♦ A , Γ ⇒ ∆ xRy , Γ ⇒ ∆ , x : ♦ A 17 / 53
❷ ❶ Extensions of G3K ◮ Idea: reach stronger logics by adding rules for R 18 / 53
❷ ❶ Extensions of G3K ◮ Idea: reach stronger logics by adding rules for R ◮ Problem: cut-free Gentzen system with new rules, i.e. 18 / 53
Extensions of G3K ◮ Idea: reach stronger logics by adding rules for R ◮ Problem: cut-free Gentzen system with new rules, i.e. ◮ criteria for a new rule to be ❷ good ❶ w.r.t cut elimination. 18 / 53
Extensions of G3K ◮ Idea: reach stronger logics by adding rules for R ◮ Problem: cut-free Gentzen system with new rules, i.e. ◮ criteria for a new rule to be ❷ good ❶ w.r.t cut elimination. ◮ Example: R is an equivalence relation 18 / 53
Extensions of G3K ◮ Idea: reach stronger logics by adding rules for R ◮ Problem: cut-free Gentzen system with new rules, i.e. ◮ criteria for a new rule to be ❷ good ❶ w.r.t cut elimination. ◮ Example: R is an equivalence relation ◮ Reflexivity and Euclideaness of R as axioms ⇒ xRx xRy , xRz ⇒ yRz 18 / 53
Extensions of G3K ◮ Idea: reach stronger logics by adding rules for R ◮ Problem: cut-free Gentzen system with new rules, i.e. ◮ criteria for a new rule to be ❷ good ❶ w.r.t cut elimination. ◮ Example: R is an equivalence relation ◮ Reflexivity and Euclideaness of R as axioms ⇒ xRx xRy , xRz ⇒ yRz ◮ No cut-free derivation of the symmetry of R ⇒ xRx xRy , xRx ⇒ yRx xRy ⇒ yRx 18 / 53
Extensions of G3K ◮ Reflexivity and Euclideaness of R as rules of inference xRx , Γ ⇒ ∆ yRz , xRy , xRz , Γ ⇒ ∆ Ref R Eucl R Γ ⇒ ∆ xRy , xRz , Γ ⇒ ∆ 19 / 53
Extensions of G3K ◮ Reflexivity and Euclideaness of R as rules of inference xRx , Γ ⇒ ∆ yRz , xRy , xRz , Γ ⇒ ∆ Ref R Eucl R Γ ⇒ ∆ xRy , xRz , Γ ⇒ ∆ ◮ Cut-free derivation of the symmetry of R yRx ⇒ yRx Eucl R xRy , xRx ⇒ yRx Ref R xRy ⇒ yRx 19 / 53
Labelled systems ◮ What conditions should R satisfy so as to yield a cut-free extension of G3K? 20 / 53
Labelled systems ◮ What conditions should R satisfy so as to yield a cut-free extension of G3K? ◮ R can be any first-order condition 20 / 53
Labelled systems ◮ What conditions should R satisfy so as to yield a cut-free extension of G3K? ◮ R can be any first-order condition ◮ what modal logics can be captured? 20 / 53
Labelled systems ◮ What conditions should R satisfy so as to yield a cut-free extension of G3K? ◮ R can be any first-order condition ◮ what modal logics can be captured? ◮ any modal logic characterized by a first-order frame condition 20 / 53
Labelled systems ◮ What conditions should R satisfy so as to yield a cut-free extension of G3K? ◮ R can be any first-order condition ◮ what modal logics can be captured? ◮ any modal logic characterized by a first-order frame condition Dyckhoff and Negri. Geometrisation of first-order logic. The Bulletin of Symbolic Logic , 21(2):123-163, 2015. 20 / 53
Extensions of G3K ◮ G3T = G3K + Ref xRx , Γ ⇒ ∆ Ref Γ ⇒ ∆ ◮ G3S4 = G3T + Trans xRz , xRy , yRz , Γ ⇒ ∆ Trans xRy , yRz , Γ ⇒ ∆ ◮ G3S5 = G3S4 + Sym yRx , xRy , Γ ⇒ ∆ Sym xRy , Γ ⇒ ∆ 21 / 53
Cut elimination in (extensions of) G3K Theorem (Negri) In G3KT , G3KS4 and G3KS5 ◮ Logical rules are hp-invertible ◮ Contraction is hp-admissible ◮ Cut is admissible Negri. Proof analysis in modal logic. Journal of Philosophical Logic , 34(5-6):507-544, 2005. 22 / 53
Cut elimination in (extensions of) G3K ◮ Moreover, hyperexponential growth of cut-free derivations Schwichtenberg. Proof theory: some applications of cut elimination. In J. Barwise, editor, Handbook of Mathematical Logic , volume 90, pages 867-895. North-Holland, 1977. 23 / 53
Completeness of (extensions of) G3K ◮ No obvious formula-interpretation of a labelled sequent 24 / 53
Completeness of (extensions of) G3K ◮ No obvious formula-interpretation of a labelled sequent ◮ In ordinary sequent calculi � � Γ ⇒ ∆ iff Γ ⊃ ∆ 24 / 53
Completeness of (extensions of) G3K ◮ No obvious formula-interpretation of a labelled sequent ◮ In ordinary sequent calculi � � Γ ⇒ ∆ iff Γ ⊃ ∆ ◮ Two forms of completeness for a labelled system S ◮ S is weakly complete if it derives all valid labelled formulas ◮ S is strongly complete if it derives all labelled sequents 24 / 53
Completeness of (extensions of) G3K Theorem (Negri) G3KT , G3KS4 and G3KS5 are strongly complete. Negri. Kripke completeness revisited. in G. Primiero and S. Rahman (eds.), Acts of Knowledge - History, Philosophy and Logic , College Publications, 2009 25 / 53
Completeness of (extensions of) G3K ◮ Weak completeness is proved via derivability of axiom system 26 / 53
Completeness of (extensions of) G3K ◮ Weak completeness is proved via derivability of axiom system ◮ (and completeness of the axiom system itself) 26 / 53
Completeness of (extensions of) G3K ◮ Weak completeness is proved via derivability of axiom system ◮ (and completeness of the axiom system itself) ◮ G3K derives all valid labelled formulas ◮ x : � A , x : � ( A ⊃ B ) ⇒ x : � B (axiom K) ◮ from ⇒ x : A infer ⇒ x : � A (necessitation rule) 26 / 53
Completeness of G3K ◮ In the derivation of K xRy , y : A ⇒ y : B , y : A y : B , y : A ⇒ y : B xRy , y : A , y : A ⊃ B ⇒ y : B L � xRy , y : A , x : � ( A ⊃ B ) ⇒ y : B L � xRy , x : � A , x : � ( A ⊃ B ) ⇒ y : B x : � A , x : � ( A ⊃ B ) ⇒ x : � B ◮ the two applications of L � are on distinct labelled formulas ◮ no labelled formula get analyzed twice by the same rule 27 / 53
Doing away with repetitions ◮ Consider L � without the principal labelled formula repeated y : A , xRy , Γ ⇒ ∆ x : � A , xRy , Γ ⇒ ∆ L � ♯ 28 / 53
Doing away with repetitions ◮ Consider L � without the principal labelled formula repeated y : A , xRy , Γ ⇒ ∆ x : � A , xRy , Γ ⇒ ∆ L � ♯ ◮ and K is still derivable 28 / 53
Doing away with repetitions ◮ Consider L � without the principal labelled formula repeated y : A , xRy , Γ ⇒ ∆ x : � A , xRy , Γ ⇒ ∆ L � ♯ ◮ and K is still derivable ◮ does this count as a proof that G3K ♯ is weakly complete? 28 / 53
Doing away with repetitions ◮ No, modus ponens still needs to be proved admissible 29 / 53
Doing away with repetitions ◮ No, modus ponens still needs to be proved admissible ◮ this require cut elimination of G3K ♯ 29 / 53
Doing away with repetitions ◮ No, modus ponens still needs to be proved admissible ◮ this require cut elimination of G3K ♯ ◮ but cut elimination requires contraction 29 / 53
Doing away with repetitions ◮ No, modus ponens still needs to be proved admissible ◮ this require cut elimination of G3K ♯ ◮ but cut elimination requires contraction ◮ and contraction requires invertibility 29 / 53
Doing away with repetitions ◮ No, modus ponens still needs to be proved admissible ◮ this require cut elimination of G3K ♯ ◮ but cut elimination requires contraction ◮ and contraction requires invertibility ◮ and L � ♯ is not invertible 29 / 53
Doing away with repetitions ◮ Is it possible to get rid of the repetition in L � 30 / 53
Doing away with repetitions ◮ Is it possible to get rid of the repetition in L � ◮ without losing weak completeness? 30 / 53
Doing away with repetitions ◮ Is it possible to get rid of the repetition in L � ◮ without losing weak completeness? ◮ are there modal validities that can be derived using L � 30 / 53
Doing away with repetitions ◮ Is it possible to get rid of the repetition in L � ◮ without losing weak completeness? ◮ are there modal validities that can be derived using L � ◮ and that are not derivable using L � ♯ ? 30 / 53
Doing away with repetitions ◮ In G3c the trade-off between repetition of quantifiers and contraction is unavoidable, 31 / 53
Doing away with repetitions ◮ In G3c the trade-off between repetition of quantifiers and contraction is unavoidable, ◮ (on pain of decidability of first-order logic) 31 / 53
Doing away with repetitions ◮ In G3c the trade-off between repetition of quantifiers and contraction is unavoidable, ◮ (on pain of decidability of first-order logic) ◮ but many modal logics are decidable 31 / 53
Doing away with repetitions ◮ In G3c the trade-off between repetition of quantifiers and contraction is unavoidable, ◮ (on pain of decidability of first-order logic) ◮ but many modal logics are decidable ◮ there could be a way to avoid cumulative modal rules . . . 31 / 53
Doing away with repetitions ◮ In G3c the trade-off between repetition of quantifiers and contraction is unavoidable, ◮ (on pain of decidability of first-order logic) ◮ but many modal logics are decidable ◮ there could be a way to avoid cumulative modal rules . . . ◮ . . . without having contraction rules explicit 31 / 53
Doing away with repetitions ◮ How far we can get without repetition? Not very far. . . 32 / 53
Doing away with repetitions ◮ How far we can get without repetition? Not very far. . . ◮ Notable modal validities do need L � to be derived 32 / 53
Doing away with repetitions ◮ How far we can get without repetition? Not very far. . . ◮ Notable modal validities do need L � to be derived ◮ ( ♦ ( A ⊃ � B ) ∧ ♦ ( A ⊃ � ¬ B )) ⊃ ♦ ¬ A 32 / 53
Doing away with repetitions ◮ How far we can get without repetition? Not very far. . . ◮ Notable modal validities do need L � to be derived ◮ ( ♦ ( A ⊃ � B ) ∧ ♦ ( A ⊃ � ¬ B )) ⊃ ♦ ¬ A ◮ Fitting’s example is derivable in G3S5 but not in G3S5 ♯ 32 / 53
Doing away with repetitions ◮ How far we can get without repetition? Not very far. . . ◮ Notable modal validities do need L � to be derived ◮ ( ♦ ( A ⊃ � B ) ∧ ♦ ( A ⊃ � ¬ B )) ⊃ ♦ ¬ A ◮ Fitting’s example is derivable in G3S5 but not in G3S5 ♯ ◮ G3S5 ♯ is weakly incomplete 32 / 53
Doing away with repetitions ◮ How far we can get without repetition? Not very far. . . ◮ Notable modal validities do need L � to be derived ◮ ( ♦ ( A ⊃ � B ) ∧ ♦ ( A ⊃ � ¬ B )) ⊃ ♦ ¬ A ◮ Fitting’s example is derivable in G3S5 but not in G3S5 ♯ ◮ G3S5 ♯ is weakly incomplete ◮ (and hence also strongly incomplete) 32 / 53
Doing away with repetitions ◮ � ( A ∧ ¬ � A ) ⊃ � ⊥ is der. in G3K4 but not in G3K4 ♯ 33 / 53
Doing away with repetitions ◮ � ( A ∧ ¬ � A ) ⊃ � ⊥ is der. in G3K4 but not in G3K4 ♯ ◮ ♦ ( A ⊃ � A ) is der. in G3T but not in G3X ♯ (X ∈ { T , S4 , S5 } ) 33 / 53
Doing away with repetitions ◮ � ( A ∧ ¬ � A ) ⊃ � ⊥ is der. in G3K4 but not in G3K4 ♯ ◮ ♦ ( A ⊃ � A ) is der. in G3T but not in G3X ♯ (X ∈ { T , S4 , S5 } ) ◮ G3K4 ♯ , G3T ♯ , G3S4 ♯ , G3S5 ♯ are all weakly incomplete 33 / 53
Doing away with repetitions ◮ � ( A ∧ ¬ � A ) ⊃ � ⊥ is der. in G3K4 but not in G3K4 ♯ ◮ ♦ ( A ⊃ � A ) is der. in G3T but not in G3X ♯ (X ∈ { T , S4 , S5 } ) ◮ G3K4 ♯ , G3T ♯ , G3S4 ♯ , G3S5 ♯ are all weakly incomplete ◮ (and hence also strongly incomplete) 33 / 53
Is at least G3K ♯ weakly complete? ◮ Minari: G3K ♯ is weakly complete . . . 34 / 53
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