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

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

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Collection Frames for Substructural Logics Greg Restall lancog workshop on substructural logic lisbon 26 september 2019 Joint work with Shawn Standefer Our Aims To better understand , to simplify and to generalise the ternary relational

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Collection Frames for Substructural Logics

Greg Restall

lancog workshop on substructural logic lisbon ⋄ 26 september 2019 Joint work with Shawn Standefer

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Our Aims

To better understand, to simplify and to generalise the ternary relational semantics for substructural logics.

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Our Plan

Ternary Relational Frames Multiset Relations Multiset Frames Soundness Completeness Beyond Multisets

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ternary relational frames

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Ternary Relational Frames for Positive Substructural Logics

〈P, N, ⊑, R〉

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Ternary Relational Frames for Positive Substructural Logics

〈P, N, ⊑, R〉

◮ P: a non-empty set ◮ N ⊆ P ◮ ⊑ ⊆ P × P ◮ R ⊆ P × P × P

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Ternary Relational Frames for Positive Substructural Logics

〈P, N, ⊑, R〉

◮ P: a non-empty set ◮ N ⊆ P ◮ ⊑ ⊆ P × P ◮ R ⊆ P × P × P

  • 1. N is non-empty.

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Ternary Relational Frames for Positive Substructural Logics

〈P, N, ⊑, R〉

◮ P: a non-empty set ◮ N ⊆ P ◮ ⊑ ⊆ P × P ◮ R ⊆ P × P × P

  • 1. N is non-empty.
  • 2. ⊑ is a partial order (or preorder).

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Ternary Relational Frames for Positive Substructural Logics

〈P, N, ⊑, R〉

◮ P: a non-empty set ◮ N ⊆ P ◮ ⊑ ⊆ P × P ◮ R ⊆ P × P × P

  • 1. N is non-empty.
  • 2. ⊑ is a partial order (or preorder).
  • 3. R is downward preserved in the its two

positions and upward preserved in the third, i.e. if Rx′y′z and x ⊑ x′, y ⊑ y′, z ⊑ z′, then Rxyz′.

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Ternary Relational Frames for Positive Substructural Logics

〈P, N, ⊑, R〉

◮ P: a non-empty set ◮ N ⊆ P ◮ ⊑ ⊆ P × P ◮ R ⊆ P × P × P

  • 1. N is non-empty.
  • 2. ⊑ is a partial order (or preorder).
  • 3. R is downward preserved in the its two

positions and upward preserved in the third, i.e. if Rx′y′z and x ⊑ x′, y ⊑ y′, z ⊑ z′, then Rxyz′.

  • 4. y ⊑ y′ iff (∃x)(Nx ∧ Rxyy′).

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Modal Frames

〈P, R〉

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Modal Frames

〈P, R〉

◮ P: a non-empty set ◮ R ⊆ P × P

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Modal Frames

〈P, R〉

◮ P: a non-empty set ◮ R ⊆ P × P

No conditions!

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Modal Frames

〈P, R〉

◮ P: a non-empty set ◮ R ⊆ P × P

No conditions! Binary relations are everywhere.

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Intuitionist Frames

〈P, ⊑〉

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Intuitionist Frames

〈P, ⊑〉

◮ P: a non-empty set ◮ ⊑ ⊆ P × P

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Intuitionist Frames

〈P, ⊑〉

◮ P: a non-empty set ◮ ⊑ ⊆ P × P

  • 1. ⊑ is a partial order

(or preorder).

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Intuitionist Frames

〈P, ⊑〉

◮ P: a non-empty set ◮ ⊑ ⊆ P × P

  • 1. ⊑ is a partial order

(or preorder).

Partial orders are everywhere.

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Ternary Relational Frames for Positive Substructural Logics

〈P, N, ⊑, R〉

◮ P: a non-empty set ◮ N ⊆ P ◮ ⊑ ⊆ P × P ◮ R ⊆ P × P × P

  • 1. N is non-empty.
  • 2. ⊑ is a partial order (or preorder).
  • 3. R is downward preserved in the its two

positions and upward preserved in the third, i.e. if Rx′y′z and x ⊑ x′, y ⊑ y′, z ⊑ z′, then Rxyz′.

  • 4. y ⊑ y′ iff (∃x)(Nx ∧ Rxyy′).

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Ternary Relational Frames for Positive Substructural Logics

Where can you find a structure like that?

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One, Two, Three,...

〈P, N, ⊑, R〉

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One, Two, Three,...

〈P, N, ⊑, R〉

N ⊆ P ⊑ ⊆ P × P R ⊆ P × P × P

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... and more

R2(xy)zw =df (∃v)(Rxyv ∧ Rvzw) R′2x(yz)w =df (∃v)(Ryzv ∧ Rxvw)

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... and more

R2(xy)zw =df (∃v)(Rxyv ∧ Rvzw) R′2x(yz)w =df (∃v)(Ryzv ∧ Rxvw) R2, R′2 ⊆ P × P × P × P

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In RW+

Rxyz ⇐ ⇒ Ryxz R2(xy)zw ⇐ ⇒ R′2x(yz)w

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In RW+ and in R+

Rxyz ⇐ ⇒ Ryxz R2(xy)zw ⇐ ⇒ R′2x(yz)w Rxxx

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The Behaviour of N, ⊑ and R

N z x ⊑ z R xyz

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The Behaviour of N, ⊑ and R

N z x ⊑ z R xyz

◮ The position of an underlined variable is closed downwards along ⊑.

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The Behaviour of N, ⊑ and R

N z x ⊑ z R xyz

◮ The position of an underlined variable is closed downwards along ⊑. ◮ The position of an overlined variable is closed upwards along ⊑.

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The Behaviour of N, ⊑ and R

N z x ⊑ z xy R z

◮ The position of an underlined variable is closed downwards along ⊑. ◮ The position of an overlined variable is closed upwards along ⊑.

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The Behaviour of N, ⊑ and R

R z x R z xy R z

◮ The position of an underlined variable is closed downwards along ⊑. ◮ The position of an overlined variable is closed upwards along ⊑.

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Collection Relations

R z x R z xy R z

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Collection Relations

X R z

X is a finite collection of elements of P; z is in P.

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What kind of fjnite collection?

Leaf-Labelled Trees Lists Multisets Sets more ...

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What kind of fjnite collection?

Leaf-Labelled Trees Lists Multisets Sets more ...

Rxyz ⇐ ⇒ Ryxz R2(xy)zw ⇐ ⇒ R′2x(yz)w

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What kind of fjnite collection?

Leaf-Labelled Trees Lists Multisets Sets more ...

Rxyz ⇐ ⇒ Ryxz R2(xy)zw ⇐ ⇒ R′2x(yz)w

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multiset relations

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(Finite) Multisets

[1, 2] [1, 1, 2] [1, 2, 1] [1] [ ]

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Finding our Target

R ⊆ M(P) × P

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Finding our Target

R ⊆ M(P) × P

R generalises ⊑.

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Finding our Target

R ⊆ M(P) × P

R generalises ⊑. So, it should satisfy analogues of reflexivity and transitivity.

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Refmxivity

[x] R x

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Generalised Transitivity

X R x

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Generalised Transitivity

X R x [x] ∪ Y R y

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Generalised Transitivity

X R x [x] ∪ Y R y X ∪ Y R y

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Generalised Transitivity

(X R x ∧ [x] ∪ Y R y) ⇒ X ∪ Y R y

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Generalised Transitivity

(X R x ∧ [x] ∪ Y R y) ⇒ X ∪ Y R y X ∪ Y R y

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Generalised Transitivity

(X R x ∧ [x] ∪ Y R y) ⇒ X ∪ Y R y X ∪ Y R y X R x

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Generalised Transitivity

(X R x ∧ [x] ∪ Y R y) ⇒ X ∪ Y R y X ∪ Y R y X R x [x] ∪ Y R y

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Generalised Transitivity

(X R x ∧ [x] ∪ Y R y) ⇒ X ∪ Y R y X ∪ Y R y ⇒ (∃x)(X R x ∧ [x] ∪ Y R y)

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Generalised Transitivity

(∃x)(X R x ∧ [x] ∪ Y R y) ⇔ X ∪ Y R y

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Left to Right

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Right to Left

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Compositional Multiset Relations

R ⊆ M(P) × P is compositional iff for each X, Y ∈ M(P) and y ∈ P

  • [y] R y
  • (∃x)(X R x ∧ [x] ∪ Y R y) ⇐

⇒ X ∪ Y R y

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Examples on M(ω) × ω

X R y iff ... sum y = ΣX (where Σ[ ] = 0)

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Examples on M(ω) × ω

X R y iff ... sum y = ΣX (where Σ[ ] = 0) product y = ΠX (where Π[ ] = 1)

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Examples on M(ω) × ω

X R y iff ... sum y = ΣX (where Σ[ ] = 0) product y = ΠX (where Π[ ] = 1) some sum for some X′ ≤ X, y = ΣX′

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Examples on M(ω) × ω

X R y iff ... sum y = ΣX (where Σ[ ] = 0) product y = ΠX (where Π[ ] = 1) some sum for some X′ ≤ X, y = ΣX′ some prod. for some X′ ≤ X, y = ΠX′

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Examples on M(ω) × ω

X R y iff ... sum y = ΣX (where Σ[ ] = 0) product y = ΠX (where Π[ ] = 1) some sum for some X′ ≤ X, y = ΣX′ some prod. for some X′ ≤ X, y = ΠX′ maximum y = max(X) (where max [ ] = 0)

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Sum

X R y iff y = ΣX

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Sum

X R y iff y = ΣX

  • refl. n = Σ[n]

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Sum

X R y iff y = ΣX

  • refl. n = Σ[n]
  • trans. y = Σ(X ∪ Y) = ΣX + ΣY = Σ([ΣX] ∪ Y).

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Some Product

X R y iff for some X′ ≤ X, y = ΠX′

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Some Product

X R y iff for some X′ ≤ X, y = ΠX′

  • refl. n = Π[n]

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Some Product

X R y iff for some X′ ≤ X, y = ΠX′

  • refl. n = Π[n]
  • trans. Z ≤ X ∪ Y iff for some X′ ≤ X and Y ′ ≤ Y, Z = X′ ∪ Y ′,

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Some Product

X R y iff for some X′ ≤ X, y = ΠX′

  • refl. n = Π[n]
  • trans. Z ≤ X ∪ Y iff for some X′ ≤ X and Y ′ ≤ Y, Z = X′ ∪ Y ′,

so X ∪ Y R y iff for some X′ ≤ X and Y ′ ≤ Y, y = Π(X′ ∪ Y ′). But Π(X′ ∪ Y ′) = ΠX′ × ΠY ′ = Π([ΠX′] ∪ Y ′), and X R ΠX′.

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Membership?

X R y iff y ∈ X

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Membership?

X R y iff y ∈ X

  • refl. n ∈ [n]

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Membership?

X R y iff y ∈ X

  • refl. n ∈ [n]
  • trans. Left to right: If x ∈ X and y ∈ ([x] ∪ Y), then y ∈ X ∪ Y.

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Membership?

X R y iff 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?

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Membership?

X R y iff 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

  • therwise.

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Membership?

X R y iff 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

  • therwise.

But this fails when X = [ ].

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Membership?

X R y iff 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

  • therwise.

But this fails when X = [ ]. Membership is a compositional relation on M′(ω) × ω,

  • n non-empty multisets.

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Between?

min (X) ≤ y ≤ max(X)

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Between?

min (X) ≤ y ≤ max(X) This is also compositional on M′(ω) × ω.

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multiset frames

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Order

Consider the binary relation ⊑ on P given by setting x ⊑ y iff [x] R y. This is a preorder on P.

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Order

Consider the binary relation ⊑ on P given by setting x ⊑ y iff [x] R y. This is a preorder on P. [x] R x

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Order

Consider the binary relation ⊑ on P given by setting x ⊑ y iff [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.

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R respects order

X R y

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Propositions

If x ⊩ p and [x] R y then y ⊩ p

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Truth Conditions

◮ x ⊩ A ∧ B iff x ⊩ A and x ⊩ B.

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Truth Conditions

◮ x ⊩ A ∧ B iff x ⊩ A and x ⊩ B. ◮ x ⊩ A ∨ B iff x ⊩ A or x ⊩ B.

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Truth Conditions

◮ x ⊩ A ∧ B iff x ⊩ A and x ⊩ B. ◮ x ⊩ A ∨ B iff x ⊩ A or x ⊩ B. ◮ x ⊩ A → B iff for each y, z where [x, y]Rz, if y ⊩ A then z ⊩ B.

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Truth Conditions

◮ x ⊩ A ∧ B iff x ⊩ A and x ⊩ B. ◮ x ⊩ A ∨ B iff x ⊩ A or x ⊩ B. ◮ x ⊩ A → B iff for each y, z where [x, y]Rz, if y ⊩ A then z ⊩ B. ◮ x ⊩ A ◦ B iff for some y, z where [y, z]Rx, both y ⊩ A and z ⊩ B.

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Truth Conditions

◮ x ⊩ A ∧ B iff x ⊩ A and x ⊩ B. ◮ x ⊩ A ∨ B iff x ⊩ A or x ⊩ B. ◮ x ⊩ A → B iff for each y, z where [x, y]Rz, if y ⊩ A then z ⊩ B. ◮ x ⊩ A ◦ B iff for some y, z where [y, z]Rx, both y ⊩ A and z ⊩ B. ◮ x ⊩ t iff [ ]Rx.

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Truth Conditions

◮ x ⊩ A ∧ B iff x ⊩ A and x ⊩ B. ◮ x ⊩ A ∨ B iff x ⊩ A or x ⊩ B. ◮ x ⊩ A → B iff for each y, z where [x, y]Rz, if y ⊩ A then z ⊩ B. ◮ x ⊩ A ◦ B iff for some y, z where [y, z]Rx, both y ⊩ A and z ⊩ B. ◮ x ⊩ t iff [ ]Rx. This models the logic RW+.

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Truth Conditions

◮ x ⊩ A ∧ B iff x ⊩ A and x ⊩ B. ◮ x ⊩ A ∨ B iff x ⊩ A or x ⊩ B. ◮ x ⊩ A → B iff for each y, z where [x, y]Rz, if y ⊩ A then z ⊩ B. ◮ x ⊩ A ◦ B iff for some y, z where [y, z]Rx, both y ⊩ A and z ⊩ B. ◮ x ⊩ t iff [ ]Rx. This models the logic RW+. Our frames automatically satisfy the RW+ conditions:

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Truth Conditions

◮ x ⊩ A ∧ B iff x ⊩ A and x ⊩ B. ◮ x ⊩ A ∨ B iff x ⊩ A or x ⊩ B. ◮ x ⊩ A → B iff for each y, z where [x, y]Rz, if y ⊩ A then z ⊩ B. ◮ x ⊩ A ◦ B iff for some y, z where [y, z]Rx, both y ⊩ A and z ⊩ B. ◮ x ⊩ t iff [ ]Rx. This models the logic RW+. Our frames automatically satisfy the RW+ conditions: [x, y]Rz ⇔ [y, x]Rz

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Truth Conditions

◮ x ⊩ A ∧ B iff x ⊩ A and x ⊩ B. ◮ x ⊩ A ∨ B iff x ⊩ A or x ⊩ B. ◮ x ⊩ A → B iff for each y, z where [x, y]Rz, if y ⊩ A then z ⊩ B. ◮ x ⊩ A ◦ B iff for some y, z where [y, z]Rx, both y ⊩ A and z ⊩ B. ◮ x ⊩ t iff [ ]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)

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Ternary Relational Frames for RW+

〈P, N, ⊑, R〉

◮ P: a non-empty set ◮ N ⊆ P ◮ ⊑ ⊆ P × P ◮ R ⊆ P × P × P

  • 1. N is non-empty.
  • 2. ⊑ is a partial order (or preorder).
  • 3. R is downward preserved in the its two

positions and upward preserved in the third.

  • 4. y ⊑ y′ iff (∃x)(Nx ∧ Rxyy′).
  • 5. Rxyz ⇔ Rxyz
  • 6. (∃v)(Rxyv∧Rvzw) ⇔ (∃u)(Ryzu∧Rxuw)

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Multiset Frames for RW+

〈P, R〉

◮ P: a non-empty set ◮ R ⊆ M(P) × P

  • 1. R is compositional. That is, [x] R x and

(∃x)(X R x ∧ [x] ∪ Y R y) ⇔ X ∪ Y R y

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soundness

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Soundness Proof

Standard argument, by induction on the length of a proof. It is straightforward in a natural deduction sequent system for RW+.

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SLIDE 95

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

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SLIDE 96

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 ⊩ iff [ ] R x x ⊩ Γ, Γ ′ iff x ⊩ Γ and x ⊩ Γ ′ x ⊩ Γ; Γ ′ iff for some y, z where [y, z] R x, y ⊩ Γ and y ⊩ Γ ′

Greg Restall Collection Frames, for Substructural Logics 36 of 47

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completeness

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Completeness Proof

The canonical RW+ frame is a multiset frame.

Greg Restall Collection Frames, for Substructural Logics 38 of 47

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beyond multisets

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Non-Empty Multisets

Membership, Betweenness, . . .

Greg Restall Collection Frames, for Substructural Logics 40 of 47

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Non-Empty Multisets

Membership, Betweenness, . . . (∃x)(X R x ∧ [x] ∪ Y R y) ⇔ X ∪ Y R y

Greg Restall Collection Frames, for Substructural Logics 40 of 47

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SLIDE 102

Non-Empty Multisets

Membership, Betweenness, . . . (∃x)(X R x ∧ [x] ∪ [ ] R y) ⇔ X ∪ [ ] R y

Greg Restall Collection Frames, for Substructural Logics 40 of 47

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SLIDE 103

Non-Empty Multisets

Membership, Betweenness, . . . (∃x)(X R x ∧ Y(x) R y) ⇔ Y(X) R y

Greg Restall Collection Frames, for Substructural Logics 40 of 47

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Non-Empty Multisets

Membership, Betweenness, . . . (∃x)(X R x ∧ Y(x) R y) ⇔ Y(X) R y

If Y(x) is a multiset containing x and X is a multiset, Y(X) is the multiset found by replacing x in Y(x) by X, in the natural way. e.g., if Y(x) is [1, 2, 3, x] then Y([3, 4]) is [1, 2, 3, 3, 4].

Greg Restall Collection Frames, for Substructural Logics 40 of 47

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Frames on non-empty multisets model RW+ without t. There are no normal points.

Greg Restall Collection Frames, for Substructural Logics 41 of 47

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Frames on non-empty multisets model RW+ without t. There are no normal points. They model entailment but not logical truth. (Sequents Γ A with a non-empty right hand side.)

Greg Restall Collection Frames, for Substructural Logics 41 of 47

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SLIDE 107

Sets

R ⊆ Pfin(P) × P

Greg Restall Collection Frames, for Substructural Logics 42 of 47

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Sets

R ⊆ Pfin(P) × P

{x} R x

Greg Restall Collection Frames, for Substructural Logics 42 of 47

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Sets

R ⊆ Pfin(P) × P

{x} R x (∃x)(X R x ∧ Y(x) R y) ⇔ Y(X) R y

Greg Restall Collection Frames, for Substructural Logics 42 of 47

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Contraction

Since {x} R x, we have {x, x} R x.

Greg Restall Collection Frames, for Substructural Logics 43 of 47

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Contraction

Since {x} R x, we have {x, x} R x. Set frames are models of R+.

Greg Restall Collection Frames, for Substructural Logics 43 of 47

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SLIDE 112

Contraction

Since {x} R x, we have {x, x} R x. Set frames are models of R+.

  • pen question: Is the logic of set frames exactly R+?

Greg Restall Collection Frames, for Substructural Logics 43 of 47

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Lists, Trees

We can take collections to be lists (order matters)

  • r leaf-labelled binary trees (associativity matters),

and the generalisation works well.

We can model the Lambek Calculus (lists),

  • r the basic substructural logic B+ (trees).

Greg Restall Collection Frames, for Substructural Logics 44 of 47

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SLIDE 114

Lists, Trees

We can take collections to be lists (order matters)

  • r leaf-labelled binary trees (associativity matters),

and the generalisation works well.

We can model the Lambek Calculus (lists),

  • r the basic substructural logic B+ (trees).

The empty list is straightforward and natural. The empty tree is less straightforward.

(To get the logic B+ take the empty tree to be a left but not a right identity.)

Greg Restall Collection Frames, for Substructural Logics 44 of 47

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Finite Structures

There is a general mathematical theory of finite structures. (The theory of species.)

Greg Restall Collection Frames, for Substructural Logics 45 of 47

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Finite Structures

There is a general mathematical theory of finite structures. (The theory of species.) What other finite structures give rise to natural logics like these?

Greg Restall Collection Frames, for Substructural Logics 45 of 47

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The Upshot

◮ The collection of conditions on N, ⊑, R in ternary frames are not ad hoc, but arise out of a single underlying phenomenon, the compositional relation.

Greg Restall Collection Frames, for Substructural Logics 46 of 47

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SLIDE 118

The Upshot

◮ The collection of conditions on N, ⊑, R in ternary frames are not ad hoc, but arise out of a single underlying phenomenon, the compositional relation. ◮ Identifying compositional relations on structures is a way to look for natural models of substructural logics.

Greg Restall Collection Frames, for Substructural Logics 46 of 47

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SLIDE 119

The Upshot

◮ The collection of conditions on N, ⊑, R in ternary frames are not ad hoc, but arise out of a single underlying phenomenon, the compositional relation. ◮ Identifying compositional relations on structures is a way to look for natural models of substructural logics. ◮ Different logics are found by varying the collections being related, whether sets, multisets, lists, leaf-labelled binary trees or something else.

Greg Restall Collection Frames, for Substructural Logics 46 of 47

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SLIDE 120

thank you!