Collection Frames for Substructural Logics Greg Restall lancog - PowerPoint PPT Presentation

Left to Right Greg Restall Collection Frames, for Substructural Logics 20 of 47

Right to Left Greg Restall Collection Frames, for Substructural Logics 21 of 47

Compositional Multiset Relations R ⊆ M ( P ) × P is compositional if f for each X, Y ∈ M ( P ) and y ∈ P • [ y ] R y • ( ∃ x )( X R x ∧ [ x ] ∪ Y R y ) ⇐ ⇒ X ∪ Y R y Greg Restall Collection Frames, for Substructural Logics 22 of 47

Examples on M ( ω ) × ω X R y if f ... sum y = Σ X (where Σ [ ] = 0 ) Greg Restall Collection Frames, for Substructural Logics 23 of 47

Examples on M ( ω ) × ω X R y if f ... sum y = Σ X (where Σ [ ] = 0 ) product y = Π X (where Π [ ] = 1 ) Greg Restall Collection Frames, for Substructural Logics 23 of 47

Examples on M ( ω ) × ω X R y if f ... sum y = Σ X (where Σ [ ] = 0 ) product y = Π X (where Π [ ] = 1 ) some sum for some X ′ ≤ X , y = Σ X ′ Greg Restall Collection Frames, for Substructural Logics 23 of 47

Examples on M ( ω ) × ω X R y if f ... sum y = Σ X (where Σ [ ] = 0 ) product y = Π X (where Π [ ] = 1 ) some sum for some X ′ ≤ X , y = Σ X ′ some prod . for some X ′ ≤ X , y = Π X ′ Greg Restall Collection Frames, for Substructural Logics 23 of 47

Examples on M ( ω ) × ω X R y if f ... sum y = Σ X (where Σ [ ] = 0 ) product y = Π X (where Π [ ] = 1 ) some sum for some X ′ ≤ X , y = Σ X ′ some prod . for some X ′ ≤ X , y = Π X ′ maximum y = max ( X ) (where max [ ] = 0 ) Greg Restall Collection Frames, for Substructural Logics 23 of 47

Sum X R y if f y = Σ X Greg Restall Collection Frames, for Substructural Logics 24 of 47

Sum X R y if f y = Σ X refl . n = Σ [ n ] Greg Restall Collection Frames, for Substructural Logics 24 of 47

Sum X R y if f y = Σ X refl . n = Σ [ n ] trans . y = Σ ( X ∪ Y ) = Σ X + Σ Y = Σ ([ Σ X ] ∪ Y ) . Greg Restall Collection Frames, for Substructural Logics 24 of 47

Some Product X R y if f for some X ′ ≤ X , y = Π X ′ Greg Restall Collection Frames, for Substructural Logics 25 of 47

Some Product X R y if f for some X ′ ≤ X , y = Π X ′ refl . n = Π [ n ] Greg Restall Collection Frames, for Substructural Logics 25 of 47

Some Product X R y if f for some X ′ ≤ X , y = Π X ′ refl . n = Π [ n ] trans . Z ≤ X ∪ Y if f for some X ′ ≤ X and Y ′ ≤ Y , Z = X ′ ∪ Y ′ , Greg Restall Collection Frames, for Substructural Logics 25 of 47

Some Product X R y if f for some X ′ ≤ X , y = Π X ′ refl . n = Π [ n ] trans . Z ≤ X ∪ Y if f for some X ′ ≤ X and Y ′ ≤ Y , Z = X ′ ∪ Y ′ , so X ∪ Y R y if f for some X ′ ≤ X and Y ′ ≤ Y , y = Π ( X ′ ∪ Y ′ ) . But Π ( X ′ ∪ Y ′ ) = Π X ′ × Π Y ′ = Π ([ Π X ′ ] ∪ Y ′ ) , and X R Π X ′ . Greg Restall Collection Frames, for Substructural Logics 25 of 47

Membership? X R y if f y ∈ X Greg Restall Collection Frames, for Substructural Logics 26 of 47

Membership? X R y if f y ∈ X refl . n ∈ [ n ] Greg Restall Collection Frames, for Substructural Logics 26 of 47

Membership? X R y if f y ∈ X refl . n ∈ [ n ] trans . Left to right: If x ∈ X and y ∈ ([ x ] ∪ Y ) , then y ∈ X ∪ Y . Greg Restall Collection Frames, for Substructural Logics 26 of 47

Membership? X R y if f y ∈ X refl . n ∈ [ n ] trans . Left to right: If x ∈ X and y ∈ ([ x ] ∪ Y ) , then y ∈ X ∪ Y . Right to left: Suppose y ∈ X ∪ Y . Is there some x ∈ X where y ∈ [ x ] ∪ Y ? Greg Restall Collection Frames, for Substructural Logics 26 of 47

Membership? X R y if f y ∈ X refl . n ∈ [ n ] trans . Left to right: If x ∈ X and y ∈ ([ x ] ∪ Y ) , then y ∈ X ∪ Y . Right to left: Suppose y ∈ X ∪ Y . Is there some x ∈ X where y ∈ [ x ] ∪ Y ? If X is non-empty, sure: pick y if y ∈ X , and an arbitrary member otherwise. Greg Restall Collection Frames, for Substructural Logics 26 of 47

Membership? X R y if f y ∈ X refl . n ∈ [ n ] trans . Left to right: If x ∈ X and y ∈ ([ x ] ∪ Y ) , then y ∈ X ∪ Y . Right to left: Suppose y ∈ X ∪ Y . Is there some x ∈ X where y ∈ [ x ] ∪ Y ? If X is non-empty, sure: pick y if y ∈ X , and an arbitrary member otherwise. But this fails when X = [ ] . Greg Restall Collection Frames, for Substructural Logics 26 of 47

Membership? X R y if f y ∈ X refl . n ∈ [ n ] trans . Left to right: If x ∈ X and y ∈ ([ x ] ∪ Y ) , then y ∈ X ∪ Y . Right to left: Suppose y ∈ X ∪ Y . Is there some x ∈ X where y ∈ [ x ] ∪ Y ? If X is non-empty, sure: pick y if y ∈ X , and an arbitrary member otherwise. But this fails when X = [ ] . Membership is a compositional relation on M ′ ( ω ) × ω , on non-empty multisets. Greg Restall Collection Frames, for Substructural Logics 26 of 47

Between? min ( X ) ≤ y ≤ max ( X ) Greg Restall Collection Frames, for Substructural Logics 27 of 47

Between? min ( X ) ≤ y ≤ max ( X ) This is also compositional on M ′ ( ω ) × ω . Greg Restall Collection Frames, for Substructural Logics 27 of 47

multiset frames

Order Consider the binary relation ⊑ on P given by setting x ⊑ y if f [ x ] R y . This is a preorder on P . Greg Restall Collection Frames, for Substructural Logics 29 of 47

Order Consider the binary relation ⊑ on P given by setting x ⊑ y if f [ x ] R y . This is a preorder on P . [ x ] R x Greg Restall Collection Frames, for Substructural Logics 29 of 47

Order Consider the binary relation ⊑ on P given by setting x ⊑ y if f [ x ] R y . This is a preorder on P . [ x ] R x If [ x ] R y and [ y ] R z , then since [ x ] R y and [ y ] ∪ [ ] R z , we have [ x ] R z , as desired. Greg Restall Collection Frames, for Substructural Logics 29 of 47

R respects order X R y Greg Restall Collection Frames, for Substructural Logics 30 of 47

Propositions If x ⊩ p and [ x ] R y then y ⊩ p Greg Restall Collection Frames, for Substructural Logics 31 of 47

Truth Conditions ◮ x ⊩ A ∧ B if f x ⊩ A and x ⊩ B . Greg Restall Collection Frames, for Substructural Logics 32 of 47

Truth Conditions ◮ x ⊩ A ∧ B if f x ⊩ A and x ⊩ B . ◮ x ⊩ A ∨ B if f x ⊩ A or x ⊩ B . Greg Restall Collection Frames, for Substructural Logics 32 of 47

Truth Conditions ◮ x ⊩ A ∧ B if f x ⊩ A and x ⊩ B . ◮ x ⊩ A ∨ B if f x ⊩ A or x ⊩ B . ◮ x ⊩ A → B if f for each y, z where [ x, y ] Rz , if y ⊩ A then z ⊩ B . Greg Restall Collection Frames, for Substructural Logics 32 of 47

Truth Conditions ◮ x ⊩ A ∧ B if f x ⊩ A and x ⊩ B . ◮ x ⊩ A ∨ B if f x ⊩ A or x ⊩ B . ◮ x ⊩ A → B if f for each y, z where [ x, y ] Rz , if y ⊩ A then z ⊩ B . ◮ x ⊩ A ◦ B if f for some y, z where [ y, z ] Rx , both y ⊩ A and z ⊩ B . Greg Restall Collection Frames, for Substructural Logics 32 of 47

Truth Conditions ◮ x ⊩ A ∧ B if f x ⊩ A and x ⊩ B . ◮ x ⊩ A ∨ B if f x ⊩ A or x ⊩ B . ◮ x ⊩ A → B if f for each y, z where [ x, y ] Rz , if y ⊩ A then z ⊩ B . ◮ x ⊩ A ◦ B if f for some y, z where [ y, z ] Rx , both y ⊩ A and z ⊩ B . ◮ x ⊩ t if f [ ] Rx . Greg Restall Collection Frames, for Substructural Logics 32 of 47

Truth Conditions ◮ x ⊩ A ∧ B if f x ⊩ A and x ⊩ B . ◮ x ⊩ A ∨ B if f x ⊩ A or x ⊩ B . ◮ x ⊩ A → B if f for each y, z where [ x, y ] Rz , if y ⊩ A then z ⊩ B . ◮ x ⊩ A ◦ B if f for some y, z where [ y, z ] Rx , both y ⊩ A and z ⊩ B . ◮ x ⊩ t if f [ ] Rx . This models the logic RW + . Greg Restall Collection Frames, for Substructural Logics 32 of 47

Truth Conditions ◮ x ⊩ A ∧ B if f x ⊩ A and x ⊩ B . ◮ x ⊩ A ∨ B if f x ⊩ A or x ⊩ B . ◮ x ⊩ A → B if f for each y, z where [ x, y ] Rz , if y ⊩ A then z ⊩ B . ◮ x ⊩ A ◦ B if f for some y, z where [ y, z ] Rx , both y ⊩ A and z ⊩ B . ◮ x ⊩ t if f [ ] Rx . This models the logic RW + . Our frames automatically satisfy the RW + conditions: Greg Restall Collection Frames, for Substructural Logics 32 of 47

Truth Conditions ◮ x ⊩ A ∧ B if f x ⊩ A and x ⊩ B . ◮ x ⊩ A ∨ B if f x ⊩ A or x ⊩ B . ◮ x ⊩ A → B if f for each y, z where [ x, y ] Rz , if y ⊩ A then z ⊩ B . ◮ x ⊩ A ◦ B if f for some y, z where [ y, z ] Rx , both y ⊩ A and z ⊩ B . ◮ x ⊩ t if f [ ] Rx . This models the logic RW + . Our frames automatically satisfy the RW + conditions: [ x, y ] Rz ⇔ [ y, x ] Rz Greg Restall Collection Frames, for Substructural Logics 32 of 47

Truth Conditions ◮ x ⊩ A ∧ B if f x ⊩ A and x ⊩ B . ◮ x ⊩ A ∨ B if f x ⊩ A or x ⊩ B . ◮ x ⊩ A → B if f for each y, z where [ x, y ] Rz , if y ⊩ A then z ⊩ B . ◮ x ⊩ A ◦ B if f for some y, z where [ y, z ] Rx , both y ⊩ A and z ⊩ B . ◮ x ⊩ t if f [ ] Rx . This models the logic RW + . Our frames automatically satisfy the RW + conditions: [ x, y ] Rz ⇔ [ y, x ] Rz ( ∃ v )([ x, y ] Rv ∧ [ v, z ] Rw ) ⇔ ( ∃ u )([ y, z ] Ru ∧ [ x, u ] Rw ) Greg Restall Collection Frames, for Substructural Logics 32 of 47

Ternary Relational Frames for RW + 〈 P, N, ⊑ , R 〉 1. N is non-empty. ◮ P : a non-empty set 2. ⊑ is a partial order (or preorder). ◮ N ⊆ P 3. R is downward preserved in the its two positions and upward preserved in the third. 4. y ⊑ y ′ if f ( ∃ x )( Nx ∧ Rxyy ′ ) . ◮ ⊑ ⊆ P × P 5. Rxyz ⇔ Rxyz ◮ R ⊆ P × P × P 6. ( ∃ v )( Rxyv ∧ Rvzw ) ⇔ ( ∃ u )( Ryzu ∧ Rxuw ) Greg Restall Collection Frames, for Substructural Logics 33 of 47

Multiset Frames for RW + 〈 P, R 〉 ◮ P : a non-empty set 1. R is compositional. That is, [ x ] R x and ( ∃ x )( X R x ∧ [ x ] ∪ Y R y ) ⇔ X ∪ Y R y ◮ R ⊆ M ( P ) × P Greg Restall Collection Frames, for Substructural Logics 34 of 47

soundness

Soundness Proof Standard argument, by induction on the length of a proof. It is straightforward in a natural deduction sequent system for RW + . Greg Restall Collection Frames, for Substructural Logics 36 of 47

Soundness Proof Standard argument, by induction on the length of a proof. It is straightforward in a natural deduction sequent system for RW + . Show that if Γ � A is derivable, then for any model, if x ⊩ Γ then x ⊩ A . Greg Restall Collection Frames, for Substructural Logics 36 of 47

Soundness Proof Standard argument, by induction on the length of a proof. It is straightforward in a natural deduction sequent system for RW + . Show that if Γ � A is derivable, then for any model, if x ⊩ Γ then x ⊩ A . Extend ⊩ to structures by setting x ⊩ � if f [ ] R x x ⊩ Γ , Γ ′ if f x ⊩ Γ and x ⊩ Γ ′ x ⊩ Γ ; Γ ′ if f for some y, z where [ y, z ] R x , y ⊩ Γ and y ⊩ Γ ′ Greg Restall Collection Frames, for Substructural Logics 36 of 47

completeness

Completeness Proof The canonical RW + frame is a multiset frame. Greg Restall Collection Frames, for Substructural Logics 38 of 47

beyond multisets

Non-Empty Multisets Membership , Betweenness , . . . Greg Restall Collection Frames, for Substructural Logics 40 of 47