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ProofTheory: Logicaland Philosophical Aspects Class 4: Hypersequents forModal Logics Greg Restall and Shawn Standefer nasslli · july 2016 · rutgers

Our Aim To introduce proof theory , with a focus in its applications in philosophy, linguistics and computer science. Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 2 of 47

Our Aim for Today Explore the behaviour of hypersequent systems for modal logics, including two dimensional modal logic with more than one modal operator. Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 3 of 47

Today's Plan Flat Hypersequents Two Dimensional Modal Logic Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 4 of 47

The Modal Logic s5 The modal logic of equivalence relations. Equivalently, it is the modal logic of universal relations. A model is a pair . iff for every , iff for some , Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 5 of 47

The Modal Logic s5 The modal logic of equivalence relations. Equivalently, it is the modal logic of universal relations. A model is a pair . iff for every , iff for some , Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 5 of 47

The Modal Logic s5 The modal logic of equivalence relations. Equivalently, it is the modal logic of universal relations. Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 5 of 47 A model is a pair ⟨ W, v ⟩ . v w ( □ A ) = 1 iff for every u , v u ( A ) = 1 v w ( ♢ A ) = 1 iff for some u , v u ( A ) = 1

How can we simplify hypersequents for s5 ? Eliminate the arrows ! Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer 6 of 47 H [ X ⊢ Y X ′ , A ⊢ Y ′ ] H [ X ⊢ Y ⊢ A ] [ □ L ] [ □ R ] X ′ ⊢ Y ′ ] H [ X, □ A ⊢ Y H [ X ⊢ □ A, Y ] X ′ ⊢ A, Y ′ ] H [ X ⊢ Y A ⊢ ] H [ X ⊢ Y [ ♢ L ] [ ♢ R ] X ′ ⊢ Y ′ ] H [ ♢ A, X ⊢ Y ] H [ X ⊢ ♢ A, Y

How can we simplify hypersequents for s5 ? Eliminate the arrows ! Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer 6 of 47 H [ X ⊢ Y X ′ , A ⊢ Y ′ ] H [ X ⊢ Y ⊢ A ] [ □ L ] [ □ R ] X ′ ⊢ Y ′ ] H [ X, □ A ⊢ Y H [ X ⊢ □ A, Y ] X ′ ⊢ A, Y ′ ] H [ X ⊢ Y A ⊢ ] H [ X ⊢ Y [ ♢ L ] [ ♢ R ] X ′ ⊢ Y ′ ] H [ ♢ A, X ⊢ Y ] H [ X ⊢ ♢ A, Y

flat hypersequents A flat hypersequent is a non-empty multiset of sequents. Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 7 of 47 X 1 ⊢ Y 1 | X 2 ⊢ Y 2 | · · · | X n ⊢ Y n

flat hypersequents

Modal Rules There is subtlety here—concerning reflexivity. Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer can be the same . and the In 9 of 47 X ′ , A ⊢ Y ′ ] H [ X ⊢ Y H [ X ⊢ Y ⊢ A ] [ □ L ] [ □ R ] X ′ ⊢ Y ′ ] H [ X ⊢ □ A, Y ] H [ X, □ A ⊢ Y X ′ ⊢ A, Y ′ ] H [ X ⊢ Y A ⊢ ] H [ X ⊢ Y [ ♢ L ] [ ♢ R ] X ′ ⊢ Y ′ ] H [ ♢ A, X ⊢ Y ] H [ X ⊢ ♢ A, Y

Modal Rules There is subtlety here—concerning reflexivity. Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer can be the same . and the In 9 of 47 H [ X ⊢ Y | X ′ , A ⊢ Y ′ ] H [ X ⊢ Y | ⊢ A ] [ □ L ] [ □ R ] H [ X, □ A ⊢ Y | X ′ ⊢ Y ′ ] H [ X ⊢ □ A, Y ] H [ X ⊢ Y | X ′ ⊢ A, Y ′ ] H [ X ⊢ Y | A ⊢ ] [ ♢ L ] [ ♢ R ] H [ X ⊢ ♢ A, Y | X ′ ⊢ Y ′ ] H [ ♢ A, X ⊢ Y ]

Modal Rules There is subtlety here—concerning reflexivity. Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer 9 of 47 H [ X ⊢ Y | X ′ , A ⊢ Y ′ ] H [ X ⊢ Y | ⊢ A ] [ □ L ] [ □ R ] H [ X, □ A ⊢ Y | X ′ ⊢ Y ′ ] H [ X ⊢ □ A, Y ] H [ X ⊢ Y | X ′ ⊢ A, Y ′ ] H [ X ⊢ Y | A ⊢ ] [ ♢ L ] [ ♢ R ] H [ X ⊢ ♢ A, Y | X ′ ⊢ Y ′ ] H [ ♢ A, X ⊢ Y ] In H [ X ⊢ Y | X ′ ⊢ Y ′ ] the X ⊢ Y and X ′ ⊢ Y ′ can be the same .

Modal Rules is a hypersequent Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer are components. and in which 10 of 47 H [ X ⊢ Y | X ′ , A ⊢ Y ′ ] H [ X ′ , A ⊢ Y ′ ] [ □ L ] [ □ L ] H [ X, □ A ⊢ Y | X ′ ⊢ Y ′ ] H [ X ′ , □ A ⊢ Y ′ ] H [ X ⊢ Y | X ′ ⊢ A, Y ′ ] H [ X ′ ⊢ A, Y ′ ] [ ♢ R ] [ ♢ R ] H [ X ⊢ ♢ A, Y | X ′ ⊢ Y ′ ] H [ X ′ ⊢ ♢ A, Y ′ ]

Modal Rules Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 10 of 47 H [ X ⊢ Y | X ′ , A ⊢ Y ′ ] H [ X ′ , A ⊢ Y ′ ] [ □ L ] [ □ L ] H [ X, □ A ⊢ Y | X ′ ⊢ Y ′ ] H [ X ′ , □ A ⊢ Y ′ ] H [ X ⊢ Y | X ′ ⊢ A, Y ′ ] H [ X ′ ⊢ A, Y ′ ] [ ♢ R ] [ ♢ R ] H [ X ⊢ ♢ A, Y | X ′ ⊢ Y ′ ] H [ X ′ ⊢ ♢ A, Y ′ ] H [ X ⊢ Y | X ′ ⊢ Y ′ ] is a hypersequent in which X ⊢ Y and X ′ ⊢ Y ′ are components.

Forms of Weakening [ iKL ] [ iKR ] [ eK ] axK Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 11 of 47 H [ X ⊢ Y ] H [ X ⊢ Y ] H [ X, A ⊢ Y ] H [ X ⊢ A, Y ]

Forms of Weakening [ iKL ] [ iKR ] [ eK ] axK Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 11 of 47 H [ X ⊢ Y ] H [ X ⊢ Y ] H [ X, A ⊢ Y ] H [ X ⊢ A, Y ] H [ X ⊢ Y ] H [ X ⊢ Y | X ′ ⊢ Y ′ ]

Forms of Weakening [ iKL ] [ iKR ] [ eK ] Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 11 of 47 H [ X ⊢ Y ] H [ X ⊢ Y ] H [ X, A ⊢ Y ] H [ X ⊢ A, Y ] H [ X ⊢ Y ] H [ X ⊢ Y | X ′ ⊢ Y ′ ] H [ X, A ⊢ A, Y ] [ axK ]

Forms of Contraction [ iWL ] [ iWR ] [ eWo ] Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 12 of 47 H [ X, A, A ⊢ Y ] H [ X ⊢ A, A, Y ] H [ X, A ⊢ Y ] H [ X ⊢ A, Y ]

Forms of Contraction [ iWL ] [ iWR ] [ eWo ] Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 12 of 47 H [ X, A, A ⊢ Y ] H [ X ⊢ A, A, Y ] H [ X, A ⊢ Y ] H [ X ⊢ A, Y ] H [ X ⊢ Y | X ′ ⊢ Y ′ ] H [ X, X ′ ⊢ Y, Y ′ ]

Forms of Cut [ aCut ] [ mCut ] Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 13 of 47 X ⊢ A, Y | H X, A ⊢ Y | H X ⊢ Y | H X ′ , A ⊢ Y ′ | H ′ X ⊢ A, Y | H X, X ′ ⊢ Y, Y ′ | H | H ′

Example Derivation [ K ] Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer 14 of 47 [ K ] A ⊢ A B ⊢ B [ □ L ] [ □ L ] □ A ⊢ | ⊢ A □ B ⊢ | ⊢ B □ A, □ B ⊢ | ⊢ A □ A, □ B ⊢ | ⊢ B [ ∧ R ] □ A, □ B ⊢ | ⊢ A ∧ B [ □ R ] □ A, □ B ⊢ □ ( A ∧ B ) [ ∧ R ] □ A ∧ □ B ⊢ □ ( A ∧ B )

More Example Derivations [ sym ] Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer 15 of 47 A ⊢ A A ⊢ A [ ¬ L ] [ □ L ] □ A ⊢ | ⊢ A ¬ A, A ⊢ [ □ L ] [ □ R ] □ ¬ A ⊢ | A ⊢ □ A ⊢ | ⊢ □ A [ ¬ R ] [ □ R ] ⊢ ¬ □ ¬ A | A ⊢ □ A ⊢ □□ A ⊢ ¬ □ ¬ A | A ⊢ [ □ R ] A ⊢ □ ¬ □ ¬ A

Modifying the Hypersequent Rules for s5 ‘ Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer 16 of 47 H [ X, □ A ⊢ Y | X ′ , A ⊢ Y ′ ] H [ X ⊢ □ A, Y | ⊢ A ] [ □ R ] [ □ L ] H [ X, □ A ⊢ Y | X ′ ⊢ Y ′ ] H [ X ⊢ □ A, Y ] H [ X ⊢ ♢ A, Y | X ′ ⊢ A, Y ′ ] H [ X ⊢ Y | A ⊢ ] [ ♢ L ] [ ♢ R ] H [ X ⊢ ♢ A, Y | X ′ ⊢ Y ′ ] H [ X, ♢ A ⊢ Y ]

Height Preserving Admissibility With these modified rules, internal and external weakening , and internal and external contraction , are height-preserving admissible. The von Plato–Negri cut elimination argument works straightforwardly for this system. (See Poggiolesi 2008.) Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 17 of 47

Height Preserving Admissibility With these modified rules, internal and external weakening , and internal and external contraction , are height-preserving admissible. The von Plato–Negri cut elimination argument works straightforwardly for this system. (See Poggiolesi 2008.) Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 17 of 47

18 of 47 [ mCut ] Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer [ eW ] [ mCut ] simplifies to ( m ) Cut Elimination: the □ Case δ l δ l X ′ ⊢ Y ′ | X ′′ , A ⊢ Y ′′ | H ′ X ⊢ Y | ⊢ A | H [ □ R ] [ □ L ] X ′ , □ A ⊢ Y ′ | X ′′ ⊢ Y ′′ | H ′ X ⊢ □ A, Y | H X, X ′ ⊢ Y, Y ′ | X ′′ ⊢ Y ′′ | H | H ′

18 of 47 [ mCut ] Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer [ eW ] [ mCut ] simplifies to ( m ) Cut Elimination: the □ Case δ l δ l X ′ ⊢ Y ′ | X ′′ , A ⊢ Y ′′ | H ′ X ⊢ Y | ⊢ A | H [ □ R ] [ □ L ] X ′ , □ A ⊢ Y ′ | X ′′ ⊢ Y ′′ | H ′ X ⊢ □ A, Y | H X, X ′ ⊢ Y, Y ′ | X ′′ ⊢ Y ′′ | H | H ′ δ l δ r X ′ ⊢ Y ′ | X ′′ , A ⊢ Y ′′ | H ′ X ⊢ Y | ⊢ A | H X ⊢ Y | X ′ ⊢ Y ′ | X ′′ ⊢ Y ′′ | H | H ′ X, X ′ ⊢ Y, Y ′ | X ′′ ⊢ Y ′′ | H | H ′