Moving Beyond Basic Modal Logic directedness Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer : reflexive symmetric transitive models. : reflexive transitive models : reflexive models : all models . . . . . . 12 of 62 symmetry transitivity reflexivity property condition Restrictions on the accessibility relation lead to properties for □ and ♢ . □ A ⊢ A A ⊢ ♢ A . wRw wRv ∧ vRu ⊃ wRu □ A ⊢ □□ A ♢♢ A ⊢ ♢ A . wRv ⊃ vRw A ⊢ □♢ A ♢□ A ⊢ A .
Moving Beyond Basic Modal Logic . Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer : reflexive symmetric transitive models. : reflexive transitive models : reflexive models : all models . . . . . directedness 12 of 62 transitivity condition property reflexivity symmetry Restrictions on the accessibility relation lead to properties for □ and ♢ . □ A ⊢ A A ⊢ ♢ A . wRw wRv ∧ vRu ⊃ wRu □ A ⊢ □□ A ♢♢ A ⊢ ♢ A . wRv ⊃ vRw A ⊢ □♢ A ♢□ A ⊢ A . ( ∃ v ) wRv □ ⊥ ⊢ ⊢ ♢ ⊤
Moving Beyond Basic Modal Logic . Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer : reflexive symmetric transitive models. : reflexive transitive models : reflexive models : all models . . . . . directedness 12 of 62 transitivity condition property reflexivity symmetry Restrictions on the accessibility relation lead to properties for □ and ♢ . □ A ⊢ A A ⊢ ♢ A . wRw wRv ∧ vRu ⊃ wRu □ A ⊢ □□ A ♢♢ A ⊢ ♢ A . wRv ⊃ vRw A ⊢ □♢ A ♢□ A ⊢ A . ( ∃ v ) wRv □ ⊥ ⊢ ⊢ ♢ ⊤
Moving Beyond Basic Modal Logic symmetry Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer . . . . . . directedness 12 of 62 transitivity condition property reflexivity Restrictions on the accessibility relation lead to properties for □ and ♢ . □ A ⊢ A A ⊢ ♢ A . wRw wRv ∧ vRu ⊃ wRu □ A ⊢ □□ A ♢♢ A ⊢ ♢ A . wRv ⊃ vRw A ⊢ □♢ A ♢□ A ⊢ A . ( ∃ v ) wRv □ ⊥ ⊢ ⊢ ♢ ⊤ K : all models T : reflexive models S4 : reflexive transitive models S5 : reflexive symmetric transitive models.
modal sequent systems
[ L ] [ L ] [ R ] These rules characterise the modal logic . Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 14 of 62 What could L/R rules for □ and ♢ look like? ??? ⊢ ??? X ⊢ □ A, Y
[ L ] [ L ] [ R ] These rules characterise the modal logic . Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 14 of 62 What could L/R rules for □ and ♢ look like? X ⊢ A □ X ⊢ □ A
[ L ] [ L ] [ R ] These rules characterise the modal logic . Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 14 of 62 What could L/R rules for □ and ♢ look like? □ X ⊢ A □ X ⊢ □ A
[ L ] [ L ] [ R ] These rules characterise the modal logic . Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 14 of 62 What could L/R rules for □ and ♢ look like? □ X ⊢ A, ♢ Y [ □ R ] □ X ⊢ □ A, ♢ Y
[ L ] [ R ] These rules characterise the modal logic . Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 14 of 62 What could L/R rules for □ and ♢ look like? X, A ⊢ Y [ □ L ] □ X ⊢ A, ♢ Y [ □ R ] X, □ A ⊢ Y □ X ⊢ □ A, ♢ Y
14 of 62 These rules characterise the modal logic Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer . What could L/R rules for □ and ♢ look like? X, A ⊢ Y [ □ L ] □ X ⊢ A, ♢ Y [ □ R ] X, □ A ⊢ Y □ X ⊢ □ A, ♢ Y □ X, A ⊢ ♢ Y X ⊢ A, Y [ ♢ R ] [ ♢ L ] X ⊢ ♢ A, Y □ X, ♢ A ⊢ ♢ Y
Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer 14 of 62 What could L/R rules for □ and ♢ look like? X, A ⊢ Y [ □ L ] □ X ⊢ A, ♢ Y [ □ R ] X, □ A ⊢ Y □ X ⊢ □ A, ♢ Y □ X, A ⊢ ♢ Y X ⊢ A, Y [ ♢ R ] [ ♢ L ] X ⊢ ♢ A, Y □ X, ♢ A ⊢ ♢ Y These rules characterise the modal logic S4 .
Example Derivations Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 15 of 62 A ⊢ A B ⊢ B A ⊢ A [ □ L ] [ ∧ R ] A, B ⊢ A ∧ B [ □ L ] □ A ⊢ A [ □ R ] □ A, B ⊢ A ∧ B [ □ L ] □ A ⊢ □ A [ □ R ] □ A, □ B ⊢ A ∧ B □ A ⊢ □□ A [ □ R ] □ A, □ B ⊢ □ ( A ∧ B ) [ ∧ L ] □ A ∧ □ B ⊢ □ ( A ∧ B )
16 of 62 [ Cut ] Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer rule?) (How could you apply a has no cut-free proof. The sequent [ L ] [ R ] [ R ] What about S5 ? □ X ⊢ A, □ Y ♢ X, A ⊢ ♢ Y [ □ R ′ ] [ ♢ L ′ ] □ X ⊢ □ A, □ Y ♢ X, ♢ A ⊢ ♢ Y
16 of 62 [ Cut ] Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer rule?) (How could you apply a has no cut-free proof. The sequent What about S5 ? □ X ⊢ A, □ Y ♢ X, A ⊢ ♢ Y [ □ R ′ ] [ ♢ L ′ ] □ X ⊢ □ A, □ Y ♢ X, ♢ A ⊢ ♢ Y □ p ⊢ □ p [ ¬ R ] ⊢ □ p, ¬ □ p p ⊢ p [ □ R ′ ] [ □ L ] ⊢ □ p, □ ¬ □ p □ p ⊢ p ⊢ p, □ ¬ □ p
16 of 62 [ Cut ] Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer What about S5 ? □ X ⊢ A, □ Y ♢ X, A ⊢ ♢ Y [ □ R ′ ] [ ♢ L ′ ] □ X ⊢ □ A, □ Y ♢ X, ♢ A ⊢ ♢ Y □ p ⊢ □ p [ ¬ R ] ⊢ □ p, ¬ □ p p ⊢ p [ □ R ′ ] [ □ L ] ⊢ □ p, □ ¬ □ p □ p ⊢ p ⊢ p, □ ¬ □ p The sequent ⊢ p, □ ¬ □ p has no cut-free proof. (How could you apply a □ rule?)
17 of 62 L and Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer Hard/impossible to generalise. rules. rules and right the left — all the work is done by R are weak Problems with these □ and ♢ rules X, A ⊢ Y [ □ L ] □ X ⊢ A, ♢ Y □ X ⊢ A, □ Y [ □ R ′ ] [ □ R ] X, □ A ⊢ Y □ X ⊢ □ A, □ Y □ X ⊢ □ A, ♢ Y □ X, A ⊢ ♢ Y ♢ X, A ⊢ ♢ Y X ⊢ A, Y [ ♢ R ] [ ♢ L ] [ ♢ L ′ ] X ⊢ ♢ A, Y □ X, ♢ A ⊢ ♢ Y ♢ X, ♢ A ⊢ ♢ Y Entanglement between □ and ♢ .
17 of 62 — all the work is done by Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer Hard/impossible to generalise. Problems with these □ and ♢ rules X, A ⊢ Y [ □ L ] □ X ⊢ A, ♢ Y □ X ⊢ A, □ Y [ □ R ′ ] [ □ R ] X, □ A ⊢ Y □ X ⊢ □ A, □ Y □ X ⊢ □ A, ♢ Y □ X, A ⊢ ♢ Y ♢ X, A ⊢ ♢ Y X ⊢ A, Y [ ♢ R ] [ ♢ L ] [ ♢ L ′ ] X ⊢ ♢ A, Y □ X, ♢ A ⊢ ♢ Y ♢ X, ♢ A ⊢ ♢ Y Entanglement between □ and ♢ . □ L and ♢ R are weak the left ♢ rules and right □ rules.
17 of 62 — all the work is done by Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer Hard/impossible to generalise. Problems with these □ and ♢ rules X, A ⊢ Y [ □ L ] □ X ⊢ A, ♢ Y □ X ⊢ A, □ Y [ □ R ′ ] [ □ R ] X, □ A ⊢ Y □ X ⊢ □ A, □ Y □ X ⊢ □ A, ♢ Y □ X, A ⊢ ♢ Y ♢ X, A ⊢ ♢ Y X ⊢ A, Y [ ♢ R ] [ ♢ L ] [ ♢ L ′ ] X ⊢ ♢ A, Y □ X, ♢ A ⊢ ♢ Y ♢ X, ♢ A ⊢ ♢ Y Entanglement between □ and ♢ . □ L and ♢ R are weak the left ♢ rules and right □ rules.
From Modal to Temporal Logic Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 18 of 62 ▶ v w ( □ A ) = 1 if and only if v u ( A ) = 1 for each u where wRu . ▶ v w ( ♢ A ) = 1 if and only if v u ( A ) = 1 for some u where wRu . ▶ v w ( ■ A ) = 1 if and only if v u ( A ) = 1 for each u where uRw . ▶ v w ( ♦ A ) = 1 if and only if v u ( A ) = 1 for some u where uRw .
From Modal to Temporal Logic Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 18 of 62 ▶ v w ( □ A ) = 1 if and only if v u ( A ) = 1 for each u where wRu . ▶ v w ( ♢ A ) = 1 if and only if v u ( A ) = 1 for some u where wRu . ▶ v w ( ■ A ) = 1 if and only if v u ( A ) = 1 for each u where uRw . ▶ v w ( ♦ A ) = 1 if and only if v u ( A ) = 1 for some u where uRw . A ⊢ □ B ♢ A ⊢ B ♦ A ⊢ B A ⊢ ■ B
Going Forward and Back in a Derivation Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 19 of 62 □ A, □ B ⊢ □ A □ A, □ B ⊢ □ B [ ∧ L ] [ ∧ L ] □ A ∧ □ B ⊢ □ A □ A ∧ □ B ⊢ □ B [ □♦ ] [ □♦ ] ♦ ( □ A ∧ □ B ) ⊢ B ♦ ( □ A ∧ □ B ) ⊢ B [ ∧ R ] ♦ ( □ A ∧ □ B ) ⊢ A ∧ B [ ♦□ ] □ A ∧ □ B ⊢ □ ( A ∧ B )
Generalised Sequents It should have something to do with some but the is evaluated in a different state . We need to record state shifts in sequents. display logic labelled sequents tree hypersequents Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 20 of 62 How do we establish X ⊢ □ A, Y ?
Generalised Sequents We need to record state shifts in sequents. display logic labelled sequents tree hypersequents Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 20 of 62 How do we establish X ⊢ □ A, Y ? It should have something to do with some X ′ ⊢ A, Y ′ but the A is evaluated in a different state .
Generalised Sequents We need to record state shifts in sequents. display logic labelled sequents tree hypersequents Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 20 of 62 How do we establish X ⊢ □ A, Y ? It should have something to do with some X ′ ⊢ A, Y ′ but the A is evaluated in a different state .
Generalised Sequents We need to record state shifts in sequents. Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 20 of 62 How do we establish X ⊢ □ A, Y ? It should have something to do with some X ′ ⊢ A, Y ′ but the A is evaluated in a different state . display logic • labelled sequents • tree hypersequents
display logic
Nuel Belnap
Sequents Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 23 of 62 Sequents are of the form X ⊢ Y , where X and Y are structures Structures are built up out of formulas and the structural connetives ∗ , • (both unary), and ◦ (binary) For example, ∗ ( p ◦ q ) ⊢ • ( r ◦ ∗ s )
Display equivalences (These rules ensure that Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer and its converse possibility the left.) acts like a necessity on the right and is conjunctive on the left and disjunctive on the right, acts like negation , 24 of 62 Certain sequents are stipulated to be equivalent via display equivalences X ⊢ Y ◦ Z ⇐ ⇒ X ◦ ∗ Y ⊢ Z ⇐ ⇒ X ⊢ Z ◦ Y X ⊢ Y ⇐ ⇒ ∗ Y ⊢ ∗ X ⇐ ⇒ X ⊢ ∗ ∗ Y • X ⊢ Y ⇐ ⇒ X ⊢ • Y
Display equivalences Certain sequents are stipulated to be equivalent via display equivalences Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer and its converse possibility the left.) 24 of 62 X ⊢ Y ◦ Z ⇐ ⇒ X ◦ ∗ Y ⊢ Z ⇐ ⇒ X ⊢ Z ◦ Y X ⊢ Y ⇐ ⇒ ∗ Y ⊢ ∗ X ⇐ ⇒ X ⊢ ∗ ∗ Y • X ⊢ Y ⇐ ⇒ X ⊢ • Y (These rules ensure that ∗ acts like negation , ◦ is conjunctive on the left and disjunctive on the right, and • acts like a necessity on the right
Displaying By means of the display equivalences, one can display a formula or structure on one side of the turnstile in isolation This permits the left and right rules to deal with only the displayed formulas and structures Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 25 of 62 A ◦ B ⊢ X X ⊢ A Y ⊢ B [ ∧ L ] [ ∧ R ] A ∧ B ⊢ X X ◦ Y ⊢ A ∧ B
Generality The connectives rules are formulated so that each connective is paired with a structural connective Different logical behaviour is obtained by imposing different rules on the structural connectives A single form of conjunction rule can be used for, say, classical conjunction and relevant fusion, the difference coming out in the structural rules in force Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 26 of 62
Cut Because formulas can always be displayed, a simple form of Cut can be used for a range of logics Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 27 of 62 X ⊢ A A ⊢ Y [ Cut ] X ⊢ Y
Eliminating Cut The Elimination Theorem is proved via a general argument that depends on eight conditions on the rules. If these conditions are satisfied, then it follows that Cut is admissible This argument is due to Haskell Curry and Nuel Belnap . Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 28 of 62
The Structure of the Curry–Belnap Cut Elimination Proof formula in the premises of the cut inference upward to where they first appear. Replace the cut at those instances (either with cuts on subformulas, or by weakening, or the cuts evaprate into identities) and then replay the substitution downward. Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 29 of 62 ▶ It’s a Cut elimination argument (it doesn’t appeal to a Mix rule). ▶ It’s an induction on grade (complexity of the Cut formula), as usual. ▶ To eliminate a Cut on a formula A , trace the parametric occurrences of a
The Crucial Step . Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer . . . . . 30 of 62 · · · A, A · · · · · · A, A · · · · · · A · · · · · · A · · · · · · A · · · · · · A · · · · · · A · · · X ⊢ A A ⊢ Y [ Cut ] X ⊢ Y
The Crucial Step . Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer . . . . . 31 of 62 · · · A, A · · · · · · A, A · · · · · · A · · · · · · A · · · · · · A · · · · · · A · · · · · · A · · · A ⊢ Y
The Crucial Step . Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer . . . . . 31 of 62 · · · X, X · · · · · · X, X · · · · · · X · · · · · · A · · · · · · A · · · · · · A · · · · · · A · · · A ⊢ Y
The Crucial Step . Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer . . . . . 31 of 62 · · · X, X · · · · · · X, X · · · · · · X · · · · · · X · · · · · · X · · · · · · X · · · · · · X · · · X ⊢ Y
The Eight Conditions Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 32 of 62 ▶ c1: Preservation of formulas . ▶ c2: Shape-alikeness of parameters . ▶ c3: Non-proliferation of parameters . ▶ c4: Position-alikeness of parameters . ▶ c5: Display of principal constituents . ▶ c6: Closure under substitution for consequent parameters . ▶ c7: Closure under substitution for antecedent parameters . ▶ c8: Eliminability of matching principal constituents .
Modal Rules To give rules for modal operators, you use the modal structure . Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 33 of 62 A ⊢ Y X ⊢ • B [ □ L ] [ □ R ] □ A ⊢ • Y X ⊢ □ B
Example Display Logic Derivation [ W ] Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer [ display ] 34 of 62 A ⊢ A B ⊢ B [ □ L ] [ □ L ] □ A ⊢ • A [ K ] □ B ⊢ • B [ K ] □ A ◦ □ B ⊢ • A [ display ] □ A ◦ □ B ⊢ • B [ display ] • ( □ A ◦ □ B ) ⊢ A • ( □ A ◦ □ B ) ⊢ B [ ∧ R ] • ( □ A ◦ □ B ) ◦ • ( □ A ◦ □ B ) ⊢ A ∧ B • ( □ A ◦ □ B ) ⊢ A ∧ B □ A ◦ □ B ⊢ • ( A ∧ B ) [ ∧ L ] □ A ∧ □ B ⊢ • ( A ∧ B ) [ □ R ] □ A ∧ □ B ⊢ □ ( A ∧ B )
Structural Rules [ trans ] [ sym ] Many more structural rules are possible . Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 35 of 62 X ⊢ • Y [ refl ] A ⊢ A X ⊢ Y [ □ L ] □ A ⊢ • A [ refl ] □ A ⊢ A
Structural Rules Many more structural rules Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer [ display ] are possible . [ trans ] [ sym ] [ trans ] 35 of 62 X ⊢ • Y [ refl ] X ⊢ Y A ⊢ A X ⊢ • Y [ □ L ] □ A ⊢ • A X ⊢ •• Y □ A ⊢ •• A [ display ] • □ A ⊢ • A [ □ R ] • □ A ⊢ □ A □ A ⊢ • □ A [ □ R ] □ A ⊢ □□ A
Structural Rules are possible . Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer [ display ] [ display ] [ display ] 35 of 62 Many more structural rules [ trans ] X ⊢ • Y [ refl ] X ⊢ Y A ⊢ A ∗ A ⊢ ∗ A [ ¬ L ] X ⊢ • Y ¬ A ⊢ ∗ A X ⊢ •• Y [ □ L ] □ ¬ A ⊢ •∗ A [ sym ] □ ¬ A ⊢ ∗• A X ⊢ •∗ Y [ sym ] • A ⊢ ∗ □ ¬ A [ ¬ R ] X ⊢ ∗• Y • A ⊢ ¬ □ ¬ A A ⊢ • ¬ □ ¬ A [ □ R ] A ⊢ □ ¬ □ ¬ A
Structural Rules [ trans ] Many more structural rules are possible . Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 35 of 62 X ⊢ • Y [ refl ] X ⊢ Y X ⊢ • Y X ⊢ •• Y X ⊢ •∗ Y [ sym ] X ⊢ ∗• Y
[ Cut ] [ display ] [ display ] Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 36 of 62 Cut Elimination: The □ Case A cut on a principal □ A may be simplified into a cut on A . X ⊢ • A A ⊢ Y [ □ R ] [ □ L ] X ⊢ □ A □ A ⊢ • Y [ Cut ] X ⊢ • Y
36 of 62 [ display ] Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer Cut Elimination: The □ Case A cut on a principal □ A may be simplified into a cut on A . X ⊢ • A [ display ] X ⊢ • A A ⊢ Y • X ⊢ A A ⊢ Y [ Cut ] [ □ R ] [ □ L ] X ⊢ □ A □ A ⊢ • Y [ Cut ] • X ⊢ Y X ⊢ • Y X ⊢ • Y
Virtues and Vices of Display Logic display Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer Gentzen-plus Nonredundant Subformula 37 of 62 Separation Systematic Explicit Cut-free + + + + + − −
Virtues and Vices of Display Logic display Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer Gentzen-plus Nonredundant Subformula 37 of 62 Separation Systematic Explicit Cut-free + + + + + − −
labelled sequents
Recall this derivation… [ W ] Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer [ display ] 39 of 62 A ⊢ A B ⊢ B [ □ L ] [ □ L ] □ A ⊢ • A [ K ] □ B ⊢ • B [ K ] □ A ◦ □ B ⊢ • A [ display ] □ A ◦ □ B ⊢ • B [ display ] • ( □ A ◦ □ B ) ⊢ A • ( □ A ◦ □ B ) ⊢ B [ ∧ R ] • ( □ A ◦ □ B ) ◦ • ( □ A ◦ □ B ) ⊢ A ∧ B • ( □ A ◦ □ B ) ⊢ A ∧ B □ A ◦ □ B ⊢ • ( A ∧ B ) [ ∧ L ] □ A ∧ □ B ⊢ • ( A ∧ B ) [ □ R ] □ A ∧ □ B ⊢ □ ( A ∧ B )
Here is another way to represent it Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 40 of 62 v : A ⊢ v : A [ □ L ] v : B ⊢ v : B [ □ L ] wRv, w : □ A ⊢ v : A [ K ] wRv, w : □ B ⊢ v : B [ K ] wRv, w : □ A, w : □ B ⊢ v : A wRv, w : □ A, w : □ B ⊢ v : B [ ∧ R ] wRv, w : □ A, w : □ B, wRv, w : □ A, w : □ B ⊢ v : A ∧ B [ W ] wRv, w : □ A, w : □ B ⊢ A ∧ B [ ∧ L ] wRv, w : □ A ∧ □ B ⊢ v : A ∧ B [ □ R ] w : □ A ∧ □ B ⊢ w : □ ( A ∧ B )
Labelled Sequent Rules: Boolean Connectives Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 41 of 62 (Plus weakening and contraction.) x : A ⊢ x : A x : A, x : B, X ⊢ Y X ⊢ x : A, Y X ⊢ x : B, Y [ ∧ L ] [ ∧ R ] x : A ∧ B, X ⊢ Y X ⊢ x : A ∧ B, Y x : A, X ⊢ Y x : B, X ⊢ Y X ⊢ x : A, x : B, Y [ ∨ L ] [ ∨ R ] x : A ∨ B, X ⊢ Y X ⊢ x : A ∨ B, Y X ⊢ x : A, Y [ ¬ L ] x : A, X ⊢ Y [ ¬ R ] x : ¬ A, X ⊢ Y X ⊢ x : ¬ A, Y
Labelled Sequent Rules: Modal Operators Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 42 of 62 x : A, X ⊢ Y [ □ L ] xRy, X ⊢ y : A, Y [ □ R ] yRx, y : □ A, X ⊢ Y X ⊢ x : □ A, Y xRy, y : A, X ⊢ Y X ⊢ x : A, Y [ ♢ R ] [ ♢ L ] yRx, X ⊢ y : ♢ A, Y x : ♢ A, X ⊢ Y In □ R and ♢ L , the label y must not be present in X , Y or be identical to x .
Labelled Sequents introduced only on the left of the sequent. We may without loss of deductive power, restrict our attention to sequents in Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 43 of 62 In these rules (except for weakenings) relational statements ( xRy ) are X ⊢ Y which relational statements appear only in X and not in Y .
Frame conditions by a conditional; and universally quantifying over all world labels. is valid on a model if and only if Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 44 of 62 The ‘cash value’ of a labelled sequent X ⊢ Y on a Kripke model is found by replacing x : A by v x ( A ) = 1 ; X by its conjunction; Y by its disjunction; the ⊢
Frame conditions by a conditional; and universally quantifying over all world labels. Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 44 of 62 The ‘cash value’ of a labelled sequent X ⊢ Y on a Kripke model is found by replacing x : A by v x ( A ) = 1 ; X by its conjunction; Y by its disjunction; the ⊢ xRy, x : A ⊢ y : B, x : C is valid on a model if and only if ( ∀ x, y )(( xRy ∧ v x ( A ) = 1 ) ⊃ (( v y ( B ) = 1 ) ∨ v x ( C ) = 1 ))
Translation A systematic translation maps modal display derivations into labelled modal derivations. The translation simplifies the proof structure, erasing display equivalences, which are mapped to identical labelled sequents ( modulo relabelling). For details, see Poggiolesi and Restall “Interpreting and Applying Proof Theory for Modal Logic” (2012). Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 45 of 62
Virtues and Vices Separation Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer Gentzen-plus Nonredundant Subformula display 46 of 62 Systematic Explicit labelled Cut-free + + + + + + + + + +− − +− − +−
Virtues and Vices Separation Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer Gentzen-plus Nonredundant Subformula display 46 of 62 Systematic Explicit labelled Cut-free + + + + + + + + + +− − +− − +−
tree hypersequents
Inspecting the translation Display equivalent sequents correspond to nearly identical labelled sequents. All we care about is that one world accesses the other. We have Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 48 of 62 A ⊢ • B • A ⊢ B
Inspecting the translation Display equivalent sequents correspond to nearly identical labelled sequents. All we care about is that one world accesses the other. We have Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 48 of 62 A ⊢ • B vRw, v : A ⊢ w : B ⇒ • A ⊢ B
Inspecting the translation Display equivalent sequents correspond to nearly identical labelled sequents. All we care about is that one world accesses the other. We have Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 48 of 62 A ⊢ • B vRw, v : A ⊢ w : B ⇒ • A ⊢ B wRv, w : A ⊢ v : B ⇒
Inspecting the translation Display equivalent sequents correspond to nearly identical labelled sequents. All we care about is that one world accesses the other. We have Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 48 of 62 A ⊢ • B vRw, v : A ⊢ w : B ⇒ • A ⊢ B wRv, w : A ⊢ v : B ⇒ A ⊢ ⊢ B
The Recipe consequent of the sequent at the node corresponding to . Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer . to , then place an arc from contains If in the in consequent position, put For every instance of antecedent of the sequent at the node corresponding to . in the in antecedent position, put For every instance of Every node is a sequent. There is one node for every label. 49 of 62 Replace the labelled sequent R , X ⊢ Y by a directed graph of sequents:
The Recipe consequent of the sequent at the node corresponding to . Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer . to , then place an arc from contains If in the in consequent position, put For every instance of antecedent of the sequent at the node corresponding to . in the in antecedent position, put For every instance of Every node is a sequent. 49 of 62 Replace the labelled sequent R , X ⊢ Y by a directed graph of sequents: ▶ There is one node for every label.
The Recipe consequent of the sequent at the node corresponding to . Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer . to , then place an arc from contains If in the in consequent position, put For every instance of antecedent of the sequent at the node corresponding to . in the in antecedent position, put For every instance of 49 of 62 Replace the labelled sequent R , X ⊢ Y by a directed graph of sequents: ▶ There is one node for every label. ▶ Every node is a sequent.
The Recipe If Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer . to , then place an arc from contains consequent of the sequent at the node corresponding to . in the in consequent position, put For every instance of 49 of 62 Replace the labelled sequent R , X ⊢ Y by a directed graph of sequents: ▶ There is one node for every label. ▶ Every node is a sequent. ▶ For every instance of x : A in antecedent position, put A in the antecedent of the sequent at the node corresponding to x .
If The Recipe contains , then place an arc from to . Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 49 of 62 Replace the labelled sequent R , X ⊢ Y by a directed graph of sequents: ▶ There is one node for every label. ▶ Every node is a sequent. ▶ For every instance of x : A in antecedent position, put A in the antecedent of the sequent at the node corresponding to x . ▶ For every instance of x : A in consequent position, put A in the consequent of the sequent at the node corresponding to x .
The Recipe Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 49 of 62 Replace the labelled sequent R , X ⊢ Y by a directed graph of sequents: ▶ There is one node for every label. ▶ Every node is a sequent. ▶ For every instance of x : A in antecedent position, put A in the antecedent of the sequent at the node corresponding to x . ▶ For every instance of x : A in consequent position, put A in the consequent of the sequent at the node corresponding to x . ▶ If R contains Rxy , then place an arc from x to y .
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