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Collection Frames for Substructural Logics Greg Restall melbourne logic seminar / 15 march 2019 J oint work with Shawn Standefer Our Aims T o better understand , to simplify and to generalise the ternary relational semantics for substructural

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Collection Frames for Substructural Logics

Greg Restall

melbourne logic seminar / 15 march 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

◮ Te 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

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

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

◮ Te position of an underlined variable is closed downwards along ⊑. ◮ Te 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 finite collection? Trees Lists Multisets Sets more ...

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What kind of finite collection? Trees Lists Multisets Sets more ...

Rxyz ⇐ ⇒ Ryxz R2(xy)zw ⇐ ⇒ R′2x(yz)w

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What kind of finite collection? 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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Reflxivity

[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 otherwise.

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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 otherwise. 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 otherwise. 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) Tis 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. Tis 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. Tis 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. Tis 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. Tis 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. Tis 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. Tis 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. Tis 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. Tat is, [x] R x and

(∃x)(X R x ∧ [x] ∪ Y R y) ⇔ X ∪ Y R y

Greg Restall Collection Frames, for Substructural Logics 34 of 47

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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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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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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 ǫ 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

Te canonical RW+ frame is a multiset frame.

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beyond multisets

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Non-Empty Multisets Membership, Betweenness, . . .

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

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

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Non-Empty Multisets Membership, Betweenness, . . . (∃x)(X R x ∧ Y(x) R y) ⇔ Y(X) R y

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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].

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

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

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Sets

R ⊆ Pfin(P) × P

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Sets

R ⊆ Pfin(P) × P

{x} R x

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Sets

R ⊆ Pfin(P) × P

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

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

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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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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+?

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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).

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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).

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

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

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Finite Structures Tere is a general mathematical theory of finite structures. (Te theory of species.)

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Finite Structures Tere is a general mathematical theory of finite structures. (Te theory of species.) What other finite structures give rise to natural logics like these?

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The Upshot

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

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The Upshot

◮ Te 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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The Upshot

◮ Te 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, trees or something else.

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thank you!