Admissible rules of propositional dependence logic Fan Yang Utrecht - - PowerPoint PPT Presentation

Admissible rules of propositional dependence logic

Fan Yang

Utrecht University

Les Diablerets Jan 30 - Feb 2, 2015 ———————————————– Joint work with Rosalie Iemhoff

1/19

Outline

1

propositional dependence logic

2

flat formulas and projective formulas

3

structural completeness

2/19

Dependence between first-order variables

First Order Quantifiers: ∀x1∃y1∀x2∃y2φ

3/19

Dependence between first-order variables

First Order Quantifiers: ∀x1∃y1∀x2∃y2φ

3/19

Dependence between first-order variables

First Order Quantifiers: ∀x1∃y1∀x2∃y2φ

3/19

Dependence between first-order variables

First Order Quantifiers: ∀x1∃y1∀x2∃y2φ Henkin Quantifiers (Henkin, 1961): ∀x1 ∃y1 ∀x2 ∃y2

• φ

3/19

Dependence between first-order variables

First Order Quantifiers: ∀x1∃y1∀x2∃y2φ Henkin Quantifiers (Henkin, 1961): ∀x1 ∃y1 ∀x2 ∃y2

• φ

Independence Friendly Logic (Hintikka, Sandu, 1989): ∀x1∃y1∀x2∃y2/{x1}φ

3/19

Dependence between first-order variables

First Order Quantifiers: ∀x1∃y1∀x2∃y2φ Henkin Quantifiers (Henkin, 1961): ∀x1 ∃y1 ∀x2 ∃y2

• φ

Independence Friendly Logic (Hintikka, Sandu, 1989): ∀x1∃y1∀x2∃y2/{x1}φ First-order dependence Logic (Väänänen 2007): ∀x1∃y1∀x2∃y2( =(x2, y2) ∧ φ)

3/19

Dependence between first-order variables

First Order Quantifiers: ∀x1∃y1∀x2∃y2φ Henkin Quantifiers (Henkin, 1961): ∀x1 ∃y1 ∀x2 ∃y2

• φ

Independence Friendly Logic (Hintikka, Sandu, 1989): ∀x1∃y1∀x2∃y2/{x1}φ First-order dependence Logic (Väänänen 2007): ∀x1∃y1∀x2∃y2( =(x2, y2) ∧ φ) Theorem (Enderton, Walkoe). Over sentences, all of the above extensions of first-order logic are equivalent to Σ1

1.

3/19

Propositional dependence logic = propositional logic + =( p, q)

I will be absent depending on whether he shows up or not. Whether it rains depends completely on whether it is summer or not. summer

rainy windy v1 1

4/19

Propositional dependence logic = propositional logic + =( p, q)

I will be absent depending on whether he shows up or not. Whether it rains depends completely on whether it is summer or not. summer

rainy windy v1 1 =(p, q)

4/19

Propositional dependence logic = propositional logic + =( p, q)

I will be absent depending on whether he shows up or not. Whether it rains depends completely on whether it is summer or not. summer

rainy windy v1 1 =(p, q)

4/19

Propositional dependence logic = propositional logic + =( p, q)

I will be absent depending on whether he shows up or not. Whether it rains depends completely on whether it is summer or not. summer

rainy windy v1 1 =(p, q)

4/19

Propositional dependence logic = propositional logic + =( p, q)

I will be absent depending on whether he shows up or not. Whether it rains depends completely on whether it is summer or not.

Team semantics (Hodges 1997)

summer

rainy windy v1 1 =(p, q)

4/19

Propositional dependence logic = propositional logic + =( p, q)

I will be absent depending on whether he shows up or not. Whether it rains depends completely on whether it is summer or not.

Team semantics (Hodges 1997)

summer

rainy windy v1 1 v1 | = =(p, q)?

4/19

Propositional dependence logic = propositional logic + =( p, q)

I will be absent depending on whether he shows up or not. Whether it rains depends completely on whether it is summer or not.

Team semantics (Hodges 1997)

summer

rainy windy v1 1 v2 1 1 v3 1 1 v4 1 v1 | = =(p, q)?

4/19

Propositional dependence logic = propositional logic + =( p, q)

I will be absent depending on whether he shows up or not. Whether it rains depends completely on whether it is summer or not.

Team semantics (Hodges 1997)

X{

summer

rainy windy v1 1 v2 1 1 v3 1 1 v4 1 =(p, q)

4/19

Propositional dependence logic = propositional logic + =( p, q)

I will be absent depending on whether he shows up or not. Whether it rains depends completely on whether it is summer or not.

Team semantics (Hodges 1997)

X A team {

summer

rainy windy v1 1 v2 1 1 v3 1 1 v4 1 X | = =(p, q)

4/19

Propositional dependence logic = propositional logic + =( p, q)

I will be absent depending on whether he shows up or not. Whether it rains depends completely on whether it is summer or not.

Team semantics (Hodges 1997)

X A team {

summer

rainy windy v1 1 v2 1 1 v3 1 1 v4 1 X | = =(p, q)

4/19

Propositional dependence logic = propositional logic + =( p, q)

I will be absent depending on whether he shows up or not. Whether it rains depends completely on whether it is summer or not.

Team semantics (Hodges 1997)

X A team {

summer

rainy windy v1 1 v2 1 1 v3 1 1 v4 1 X | = =(p, q)

4/19

Propositional dependence logic = propositional logic + =( p, q)

I will be absent depending on whether he shows up or not. Whether it rains depends completely on whether it is summer or not.

Team semantics (Hodges 1997)

X A team {

summer

rainy windy v1 1 v2 1 1 v3 1 1 v4 1 X | = =(p, q)

4/19

Propositional dependence logic = propositional logic + =( p, q)

I will be absent depending on whether he shows up or not. Whether it rains depends completely on whether it is summer or not.

Team semantics (Hodges 1997)

X A team {

summer

rainy windy v1 1 v2 1 1 v3 1 1 v4 1 X | = =(p, q)

4/19

Propositional dependence logic = propositional logic + =( p, q)

I will be absent depending on whether he shows up or not. Whether it rains depends completely on whether it is summer or not.

Team semantics (Hodges 1997)

X A team {

summer

rainy windy v1 1 v2 1 1 v3 1 1 v4 1 v5 1 1 1

Y

{

X | = =(p, q), Y | = =(p, q)

4/19

Propositional dependence logic and its variants

Well-formed formulas of propositional dependence logic (PD) are given by the following grammar φ ::= pi | ¬pi | =(pi1, . . . , pin−1, pin) | φ ∧ φ | φ ⊗ φ Well-formed formulas of propositional intuitionistic dependence logic (PID) are given by the following grammar: φ ::= pi | ⊥ | =(pi1, . . . , pin−1, pin) | φ ∧ φ | φ ∨ φ | φ → φ (¬φ := φ → ⊥) PD∨ is the logic extended from PD by adding the connective ∨.

p0 p1 p2 . . . v1 1 . . .

5/19

Propositional dependence logic and its variants

Well-formed formulas of propositional dependence logic (PD) are given by the following grammar φ ::= pi | ¬pi | =(pi1, . . . , pin−1, pin) | φ ∧ φ | φ ⊗ φ Well-formed formulas of propositional intuitionistic dependence logic (PID) are given by the following grammar: φ ::= pi | ⊥ | =(pi1, . . . , pin−1, pin) | φ ∧ φ | φ ∨ φ | φ → φ (¬φ := φ → ⊥) PD∨ is the logic extended from PD by adding the connective ∨.

p0 p1 p2 . . . v1 1 . . .

5/19

Propositional dependence logic and its variants

Well-formed formulas of propositional dependence logic (PD) are given by the following grammar φ ::= pi | ¬pi | =(pi1, . . . , pin−1, pin) | φ ∧ φ | φ ⊗ φ Well-formed formulas of propositional intuitionistic dependence logic (PID) are given by the following grammar: φ ::= pi | ⊥ | =(pi1, . . . , pin−1, pin) | φ ∧ φ | φ ∨ φ | φ → φ (¬φ := φ → ⊥) PD∨ is the logic extended from PD by adding the connective ∨.

p0 p1 p2 . . . v1 1 . . .

5/19

Propositional dependence logic and its variants

Well-formed formulas of propositional dependence logic (PD) are given by the following grammar φ ::= pi | ¬pi | =(pi1, . . . , pin−1, pin) | φ ∧ φ | φ ⊗ φ Well-formed formulas of propositional intuitionistic dependence logic (PID) are given by the following grammar: φ ::= pi | ⊥ | =(pi1, . . . , pin−1, pin) | φ ∧ φ | φ ∨ φ | φ → φ (¬φ := φ → ⊥) PD∨ is the logic extended from PD by adding the connective ∨.

p0 p1 p2 . . . v1 1 . . .

5/19

Propositional dependence logic and its variants

Well-formed formulas of propositional dependence logic (PD) are given by the following grammar φ ::= pi | ¬pi | =(pi1, . . . , pin−1, pin) | φ ∧ φ | φ ⊗ φ Well-formed formulas of propositional intuitionistic dependence logic (PID) are given by the following grammar: φ ::= pi | ⊥ | =(pi1, . . . , pin−1, pin) | φ ∧ φ | φ ∨ φ | φ → φ (¬φ := φ → ⊥) PD∨ is the logic extended from PD by adding the connective ∨. Team semantics: A valuation is a function v : N → {0, 1}.

p0 p1 p2 . . . v1 1 . . .

5/19

Propositional dependence logic and its variants

Well-formed formulas of propositional dependence logic (PD) are given by the following grammar φ ::= pi | ¬pi | =(pi1, . . . , pin−1, pin) | φ ∧ φ | φ ⊗ φ Well-formed formulas of propositional intuitionistic dependence logic (PID) are given by the following grammar: φ ::= pi | ⊥ | =(pi1, . . . , pin−1, pin) | φ ∧ φ | φ ∨ φ | φ → φ (¬φ := φ → ⊥) PD∨ is the logic extended from PD by adding the connective ∨. Team semantics: A valuation is a function v : N → {0, 1}. A team is a set of valuations.

p0 p1 p2 . . . v1 1 . . . v2 1 1 . . . v4 1 . . . . . . . . . . . . . . . . . .

5/19

Team Semantics

Let X be a team. X | = pi iff for all v ∈ X, v(i) = 1; X | = ¬pi iff for all v ∈ X, v(i) = 0; X | = ⊥ iff X = ∅; X | = =(pi1, . . . , pin) iff for all v, v′ ∈ X

• v(i1) = v′(i1), . . . , v(in−1) = v′(in−1)
• =

⇒ v(in) = v′(in); X | = φ ∧ ψ iff X | = φ and X | = ψ; X | = φ⊗ψ iff there exist teams Y, Z ⊆ X with X = Y ∪Z such that Y | = φ and Z | = ψ; p0 p1 p2 v1 1 v2 1 1 v3 1 v4 1 1 X | = φ ∨ ψ iff X | = φ or X | = ψ; X | = φ → ψ iff for any team Y ⊆ X,

6/19

Team Semantics

Let X be a team. X | = pi iff for all v ∈ X, v(i) = 1; X | = ¬pi iff for all v ∈ X, v(i) = 0; X | = ⊥ iff X = ∅; X | = =(pi1, . . . , pin) iff for all v, v′ ∈ X

• v(i1) = v′(i1), . . . , v(in−1) = v′(in−1)
• =

⇒ v(in) = v′(in); X | = φ ∧ ψ iff X | = φ and X | = ψ; X | = φ⊗ψ iff there exist teams Y, Z ⊆ X with X = Y ∪Z such that Y | = φ and Z | = ψ; X { X | = p0 p0 p1 p2 v1 1 v2 1 1 v3 1 v4 1 1 X | = φ ∨ ψ iff X | = φ or X | = ψ; X | = φ → ψ iff for any team Y ⊆ X,

6/19

Team Semantics

Let X be a team. X | = pi iff for all v ∈ X, v(i) = 1; X | = ¬pi iff for all v ∈ X, v(i) = 0; X | = ⊥ iff X = ∅; X | = =(pi1, . . . , pin) iff for all v, v′ ∈ X

• v(i1) = v′(i1), . . . , v(in−1) = v′(in−1)
• =

⇒ v(in) = v′(in); X | = φ ∧ ψ iff X | = φ and X | = ψ; X | = φ⊗ψ iff there exist teams Y, Z ⊆ X with X = Y ∪Z such that Y | = φ and Z | = ψ; X { X | = p0 Y | = ¬p0 Y { p0 p1 p2 v1 1 v2 1 1 v3 1 v4 1 1 X | = φ ∨ ψ iff X | = φ or X | = ψ; X | = φ → ψ iff for any team Y ⊆ X,

6/19

Team Semantics

Let X be a team. X | = pi iff for all v ∈ X, v(i) = 1; X | = ¬pi iff for all v ∈ X, v(i) = 0; X | = ⊥ iff X = ∅; X | = =(pi1, . . . , pin) iff for all v, v′ ∈ X

• v(i1) = v′(i1), . . . , v(in−1) = v′(in−1)
• =

⇒ v(in) = v′(in); X | = φ ∧ ψ iff X | = φ and X | = ψ; X | = φ⊗ψ iff there exist teams Y, Z ⊆ X with X = Y ∪Z such that Y | = φ and Z | = ψ; X { X | = p0 Y | = ¬p0 X ∪ Y | = p0 X ∪ Y | = ¬p0 Y { p0 p1 p2 v1 1 v2 1 1 v3 1 v4 1 1 X | = φ ∨ ψ iff X | = φ or X | = ψ; X | = φ → ψ iff for any team Y ⊆ X,

6/19

Team Semantics

Let X be a team. X | = pi iff for all v ∈ X, v(i) = 1; X | = ¬pi iff for all v ∈ X, v(i) = 0; X | = ⊥ iff X = ∅; X | = =(pi1, . . . , pin) iff for all v, v′ ∈ X

• v(i1) = v′(i1), . . . , v(in−1) = v′(in−1)
• =

⇒ v(in) = v′(in); X | = φ ∧ ψ iff X | = φ and X | = ψ; X | = φ⊗ψ iff there exist teams Y, Z ⊆ X with X = Y ∪Z such that Y | = φ and Z | = ψ; p0 p1 p2 v1 1 v2 1 1 v3 1 v4 1 1 X | = φ ∨ ψ iff X | = φ or X | = ψ; X | = φ → ψ iff for any team Y ⊆ X,

6/19

Team Semantics

Let X be a team. X | = pi iff for all v ∈ X, v(i) = 1; X | = ¬pi iff for all v ∈ X, v(i) = 0; X | = ⊥ iff X = ∅; X | = =(pi1, . . . , pin) iff for all v, v′ ∈ X

• v(i1) = v′(i1), . . . , v(in−1) = v′(in−1)
• =

⇒ v(in) = v′(in); X | = φ ∧ ψ iff X | = φ and X | = ψ; X | = φ⊗ψ iff there exist teams Y, Z ⊆ X with X = Y ∪Z such that Y | = φ and Z | = ψ; X | = =(p0, p1) p0 p1 p2 v1 1 v2 1 1 v3 1 v4 1 1 X | = φ ∨ ψ iff X | = φ or X | = ψ; X | = φ → ψ iff for any team Y ⊆ X,

6/19

Team Semantics

Let X be a team. X | = pi iff for all v ∈ X, v(i) = 1; X | = ¬pi iff for all v ∈ X, v(i) = 0; X | = ⊥ iff X = ∅; X | = =(pi1, . . . , pin) iff for all v, v′ ∈ X

• v(i1) = v′(i1), . . . , v(in−1) = v′(in−1)
• =

⇒ v(in) = v′(in); X | = φ ∧ ψ iff X | = φ and X | = ψ; X | = φ⊗ψ iff there exist teams Y, Z ⊆ X with X = Y ∪Z such that Y | = φ and Z | = ψ; X | = =(p0, p1) p0 p1 p2 v1 1 v2 1 1 v3 1 v4 1 1 X | = φ ∨ ψ iff X | = φ or X | = ψ; X | = φ → ψ iff for any team Y ⊆ X,

6/19

Team Semantics

Let X be a team. X | = pi iff for all v ∈ X, v(i) = 1; X | = ¬pi iff for all v ∈ X, v(i) = 0; X | = ⊥ iff X = ∅; X | = =(pi1, . . . , pin) iff for all v, v′ ∈ X

• v(i1) = v′(i1), . . . , v(in−1) = v′(in−1)
• =

⇒ v(in) = v′(in); X | = φ ∧ ψ iff X | = φ and X | = ψ; X | = φ⊗ψ iff there exist teams Y, Z ⊆ X with X = Y ∪Z such that Y | = φ and Z | = ψ; X | = φ ∨ ψ iff X | = φ or X | = ψ; X | = φ → ψ iff for any team Y ⊆ X, Y | = φ = ⇒ Y | = ψ.

6/19

Team Semantics

Let X be a team. X | = pi iff for all v ∈ X, v(i) = 1; X | = ¬pi iff for all v ∈ X, v(i) = 0; X | = ⊥ iff X = ∅; X | = =(pi1, . . . , pin) iff for all v, v′ ∈ X

• v(i1) = v′(i1), . . . , v(in−1) = v′(in−1)
• =

⇒ v(in) = v′(in); X | = φ ∧ ψ iff X | = φ and X | = ψ; X | = φ⊗ψ iff there exist teams Y, Z ⊆ X with X = Y ∪Z such that Y | = φ and Z | = ψ; X | = φ ∨ ψ iff X | = φ or X | = ψ; X | = φ → ψ iff for any team Y ⊆ X, Y | = φ = ⇒ Y | = ψ.

6/19

Team Semantics

Let X be a team. X | = pi iff for all v ∈ X, v(i) = 1; X | = ¬pi iff for all v ∈ X, v(i) = 0; X | = ⊥ iff X = ∅; X | = =(pi1, . . . , pin) iff for all v, v′ ∈ X

• v(i1) = v′(i1), . . . , v(in−1) = v′(in−1)
• =

⇒ v(in) = v′(in); X | = φ ∧ ψ iff X | = φ and X | = ψ; X | = φ⊗ψ iff there exist teams Y, Z ⊆ X with X = Y ∪Z such that Y | = φ and Z | = ψ; X | = φ ∨ ψ iff X | = φ or X | = ψ; X | = φ → ψ iff for any team Y ⊆ X, Y | = φ = ⇒ Y | = ψ.

6/19

Team Semantics

Let X be a team. X | = pi iff for all v ∈ X, v(i) = 1; X | = ¬pi iff for all v ∈ X, v(i) = 0; X | = ⊥ iff X = ∅; X | = =(pi1, . . . , pin) iff for all v, v′ ∈ X

• v(i1) = v′(i1), . . . , v(in−1) = v′(in−1)
• =

⇒ v(in) = v′(in); X | = φ ∧ ψ iff X | = φ and X | = ψ; X | = φ⊗ψ iff there exist teams Y, Z ⊆ X with X = Y ∪Z such that Y | = φ and Z | = ψ; X | = φ ∨ ψ iff X | = φ or X | = ψ; X | = φ → ψ iff for any team Y ⊆ X, Y | = φ = ⇒ Y | = ψ. The logics PD, PD∨, PID are downwards closed, that is, X | = φ and Y ⊆ X = ⇒ Y | = φ; and have the empty team property, that is, ∅ | = φ, for all φ.

6/19

Team Semantics

Let X be a team. X | = pi iff for all v ∈ X, v(i) = 1; X | = ¬pi iff for all v ∈ X, v(i) = 0; X | = ⊥ iff X = ∅; X | = =(pi1, . . . , pin) iff for all v, v′ ∈ X

• v(i1) = v′(i1), . . . , v(in−1) = v′(in−1)
• =

⇒ v(in) = v′(in); X | = φ ∧ ψ iff X | = φ and X | = ψ; X | = φ⊗ψ iff there exist teams Y, Z ⊆ X with X = Y ∪Z such that Y | = φ and Z | = ψ; X | = φ ∨ ψ iff X | = φ or X | = ψ; X | = φ → ψ iff for any team Y ⊆ X, Y | = φ = ⇒ Y | = ψ. The logics PD, PD∨, PID are downwards closed, that is, X | = φ and Y ⊆ X = ⇒ Y | = φ; and have the empty team property, that is, ∅ | = φ, for all φ.

6/19

Team Semantics

Let X be a team. X | = pi iff for all v ∈ X, v(i) = 1; X | = ¬pi iff for all v ∈ X, v(i) = 0; X | = ⊥ iff X = ∅; X | = =(pi1, . . . , pin) iff for all v, v′ ∈ X

• v(i1) = v′(i1), . . . , v(in−1) = v′(in−1)
• =

⇒ v(in) = v′(in); X | = φ ∧ ψ iff X | = φ and X | = ψ; X | = φ⊗ψ iff there exist teams Y, Z ⊆ X with X = Y ∪Z such that Y | = φ and Z | = ψ; X | = φ ∨ ψ iff X | = φ or X | = ψ; X | = φ → ψ iff for any team Y ⊆ X, Y | = φ = ⇒ Y | = ψ. Intuitionistic disjunction ∨ has the disjunction property: | = φ ∨ ψ = ⇒ | = φ or | = ψ.

7/19

Dependence atoms are definable in the fragment of PID and PD∨ without dependence atoms: =(p0, p1) ≡

• p0 ∧
• p1 ∨ ¬p1
• ⊗
• ¬p0 ∧
• p1 ∨ ¬p1
• ≡ (p0 ∨ ¬p0) → (p1 ∨ ¬p1)

p0 p1 p2 v1 1 v2 1 1 v3 1 v4 1 1

8/19

Dependence atoms are definable in the fragment of PID and PD∨ without dependence atoms: =(p0, p1) ≡

• p0 ∧
• p1 ∨ ¬p1
• ⊗
• ¬p0 ∧
• p1 ∨ ¬p1
• ≡ (p0 ∨ ¬p0) → (p1 ∨ ¬p1)

p0 p1 p2 v1 1 v2 1 1 v3 1 v4 1 1 Observation (de Jongh, Litak) PID− is equivalent to inquisitive logic (Ciardelli, Roelofsen 2011).

8/19

Fix N = {i1, . . . , in} ⊆ N. An n-valuation on N is a function s : N → {0, 1}. An n-team on N is a set of n-valuations on N.

01 00 11 10

There are in total 2n distinct n-valuations, and 22n n-teams, among which there exists a biggest team (denoted by 2n) consisting of all n-valuations on N.

9/19

Fix N = {i1, . . . , in} ⊆ N. An n-valuation on N is a function s : N → {0, 1}. An n-team on N is a set of n-valuations on N.

01 00 11 10

There are in total 2n distinct n-valuations, and 22n n-teams, among which there exists a biggest team (denoted by 2n) consisting of all n-valuations on N.

9/19

Fix N = {i1, . . . , in} ⊆ N. An n-valuation on N is a function s : N → {0, 1}. An n-team on N is a set of n-valuations on N.

01 00 11 10

There are in total 2n distinct n-valuations, and 22n n-teams, among which there exists a biggest team (denoted by 2n) consisting of all n-valuations on N.

9/19

Fix N = {i1, . . . , in} ⊆ N. An n-valuation on N is a function s : N → {0, 1}. An n-team on N is a set of n-valuations on N.

01 00 11 10

There are in total 2n distinct n-valuations, and 22n n-teams, among which there exists a biggest team (denoted by 2n) consisting of all n-valuations on N.

9/19

Fix N = {i1, . . . , in} ⊆ N. An n-valuation on N is a function s : N → {0, 1}. An n-team on N is a set of n-valuations on N.

01 00 11 10

There are in total 2n distinct n-valuations, and 22n n-teams, among which there exists a biggest team (denoted by 2n) consisting of all n-valuations on N.

9/19

Fix N = {i1, . . . , in} ⊆ N. An n-valuation on N is a function s : N → {0, 1}. An n-team on N is a set of n-valuations on N.

01 00 11 10

There are in total 2n distinct n-valuations, and 22n n-teams, among which there exists a biggest team (denoted by 2n) consisting of all n-valuations on N.

9/19

Fix N = {i1, . . . , in} ⊆ N. An n-valuation on N is a function s : N → {0, 1}. An n-team on N is a set of n-valuations on N.

01 00 11 10

There are in total 2n distinct n-valuations, and 22n n-teams, among which there exists a biggest team (denoted by 2n) consisting of all n-valuations on N.

9/19

Fix N = {i1, . . . , in} ⊆ N. An n-valuation on N is a function s : N → {0, 1}. An n-team on N is a set of n-valuations on N.

01 00 11 10

There are in total 2n distinct n-valuations, and 22n n-teams, among which there exists a biggest team (denoted by 2n) consisting of all n-valuations on N.

9/19

Fix N = {i1, . . . , in} ⊆ N. An n-valuation on N is a function s : N → {0, 1}. An n-team on N is a set of n-valuations on N.

01 00 11 10

There are in total 2n distinct n-valuations, and 22n n-teams, among which there exists a biggest team (denoted by 2n) consisting of all n-valuations on N.

9/19

Fix N = {i1, . . . , in} ⊆ N. An n-valuation on N is a function s : N → {0, 1}. An n-team on N is a set of n-valuations on N.

01 00 11 10

There are in total 2n distinct n-valuations, and 22n n-teams, among which there exists a biggest team (denoted by 2n) consisting of all n-valuations on N.

9/19

Fix N = {i1, . . . , in} ⊆ N. An n-valuation on N is a function s : N → {0, 1}. An n-team on N is a set of n-valuations on N.

01 00 11 10

There are in total 2n distinct n-valuations, and 22n n-teams, among which there exists a biggest team (denoted by 2n) consisting of all n-valuations on N.

9/19

Fix N = {i1, . . . , in} ⊆ N. An n-valuation on N is a function s : N → {0, 1}. An n-team on N is a set of n-valuations on N.

01 00 11 10

There are in total 2n distinct n-valuations, and 22n n-teams, among which there exists a biggest team (denoted by 2n) consisting of all n-valuations on N.

9/19

℘(2n)

00 01 10 11 00 10 11 01 10 11 00 01 11 00 01 10 01 10 01 11 10 11 11 00 10 00 01 00 10 11 01 00

10/19

(℘(2n), ⊇)

00 01 10 11 00 10 11 01 10 11 00 01 11 00 01 10 01 10 01 11 10 11 11 00 10 00 01 00 10 11 01 00

10/19

(℘(2n) \ {∅}, ⊇)

00 01 10 11 00 10 11 01 10 11 00 01 11 00 01 10 01 10 01 11 10 11 11 00 10 00 01 00 10 11 01 00

10/19

A Medvedev frame: (℘(2n) \ {∅}, ⊇)

00 01 10 11 00 10 11 01 10 11 00 01 11 00 01 10 01 10 01 11 10 11 11 00 10 00 01 00 10 11 01 00

10/19

A Medvedev frame: (℘(2n) \ {∅}, ⊇)

00 01 10 11 00 10 11 01 10 11 00 01 11 00 01 10 01 10 01 11 10 11 11 00 10 00 01 00 10 11 01 00

p, q

10/19

A Medvedev frame: (℘(2n) \ {∅}, ⊇)

00 01 10 11 00 10 11 01 10 11 00 01 11 00 01 10 01 10 01 11 10 11 11 00 10 00 01 00 10 11 01 00

p → q p, q

10/19

A Medvedev frame: (℘(2n) \ {∅}, ⊇)

00 01 10 11 00 10 11 01 10 11 00 01 11 00 01 10 01 10 01 11 10 11 11 00 10 00 01 00 10 11 01 00

p → q ¬¬p → p p, q

10/19

A Medvedev frame: (℘(2n) \ {∅}, ⊇)

00 01 10 11 00 10 11 01 10 11 00 01 11 00 01 10 01 10 01 11 10 11 11 00 10 00 01 00 10 11 01 00

p → q ¬¬p → p p, q

[Ciardelli, Roelofsen 2011]: PID− = ML¬ = {φ | τ(φ) ∈ ML, where τ(p) = ¬p}

10/19

A Medvedev frame: (℘(2n) \ {∅}, ⊇)

00 01 10 11 00 10 11 01 10 11 00 01 11 00 01 10 01 10 01 11 10 11 11 00 10 00 01 00 10 11 01 00

p → q ¬¬p → p p, q

[Ciardelli, Roelofsen 2011]: PID− = ML¬ = {φ | τ(φ) ∈ ML, where τ(p) = ¬p} = KP¬ = KP ⊕ ¬¬p → p

10/19

Fix N = {i1, . . . , in}. Put φ(pi1, . . . , pin) := {X ⊆ 2n | X | = φ}, ∇N := {K ⊆ 22n | ∅ ∈ K,

• X ∈ K, Y ⊆ X =

⇒ Y ∈ K

• }.

For each K ∈ ∇N, consider

X∈K ΘX. For any n-team Y on N,

Y | =

• ΘX ⇐

⇒ ∃X ∈ K(Y ⊆ X) ⇐ ⇒ Y ∈ K.

11/19

Fix N = {i1, . . . , in}. Put φ(pi1, . . . , pin) := {X ⊆ 2n | X | = φ}, ∇N := {K ⊆ 22n | ∅ ∈ K,

• X ∈ K, Y ⊆ X =

⇒ Y ∈ K

• }.

For each K ∈ ∇N, consider

X∈K ΘX. For any n-team Y on N,

Y | =

• ΘX ⇐

⇒ ∃X ∈ K(Y ⊆ X) ⇐ ⇒ Y ∈ K.

11/19

Fix N = {i1, . . . , in}. Put φ(pi1, . . . , pin) := {X ⊆ 2n | X | = φ}, ∇N := {K ⊆ 22n | ∅ ∈ K,

• X ∈ K, Y ⊆ X =

⇒ Y ∈ K

• }.

Theorem (Ciardelli, Huuskonen, Y.) PID, PD∨, PD are maximal downwards closed logics, i.e., for L ∈ {PID, PD∨, PD}, ∇N = {φ | φ(pi1, . . . , pin) is an n-formula of L}. In particular, PID ≡ PD∨ ≡ PD. For each K ∈ ∇N, consider

X∈K ΘX. For any n-team Y on N,

Y | =

• ΘX ⇐

⇒ ∃X ∈ K(Y ⊆ X) ⇐ ⇒ Y ∈ K.

11/19

Fix N = {i1, . . . , in}. Put φ(pi1, . . . , pin) := {X ⊆ 2n | X | = φ}, ∇N := {K ⊆ 22n | ∅ ∈ K,

• X ∈ K, Y ⊆ X =

⇒ Y ∈ K

• }.

Theorem (Ciardelli, Huuskonen, Y.) PID, PD∨, PD are maximal downwards closed logics, i.e., for L ∈ {PID, PD∨, PD}, ∇N = {φ | φ(pi1, . . . , pin) is an n-formula of L}. In particular, PID ≡ PD∨ ≡ PD.

Theorem (Y.)

Every instance of ∨ and → is definable in PD, but ∨ and → are not uniformly definable in PD.

11/19

Fix N = {i1, . . . , in}. Put φ(pi1, . . . , pin) := {X ⊆ 2n | X | = φ}, ∇N := {K ⊆ 22n | ∅ ∈ K,

• X ∈ K, Y ⊆ X =

⇒ Y ∈ K

• }.

Theorem (Ciardelli, Huuskonen, Y.) PID, PD∨, PD are maximal downwards closed logics, i.e., for L ∈ {PID, PD∨, PD}, ∇N = {φ | φ(pi1, . . . , pin) is an n-formula of L}. In particular, PID ≡ PD∨ ≡ PD.

• Proof. We only treat PID and PD∨.

For each K ∈ ∇N, consider

X∈K ΘX. For any n-team Y on N,

Y | =

• ΘX ⇐

⇒ ∃X ∈ K(Y ⊆ X) ⇐ ⇒ Y ∈ K.

11/19

Fix N = {i1, . . . , in}. Put φ(pi1, . . . , pin) := {X ⊆ 2n | X | = φ}, ∇N := {K ⊆ 22n | ∅ ∈ K,

• X ∈ K, Y ⊆ X =

⇒ Y ∈ K

• }.

Theorem (Ciardelli, Huuskonen, Y.) PID, PD∨, PD are maximal downwards closed logics, i.e., for L ∈ {PID, PD∨, PD}, ∇N = {φ | φ(pi1, . . . , pin) is an n-formula of L}. In particular, PID ≡ PD∨ ≡ PD.

• Proof. We only treat PID and PD∨. First, consider an n-team:

X{ p1 p2 v1 1 1 v2 1 v3 1

For each K ∈ ∇N, consider

X∈K ΘX. For any n-team Y on N,

Y | =

• ΘX ⇐

⇒ ∃X ∈ K(Y ⊆ X) ⇐ ⇒ Y ∈ K.

11/19

Fix N = {i1, . . . , in}. Put φ(pi1, . . . , pin) := {X ⊆ 2n | X | = φ}, ∇N := {K ⊆ 22n | ∅ ∈ K,

• X ∈ K, Y ⊆ X =

⇒ Y ∈ K

• }.

Theorem (Ciardelli, Huuskonen, Y.) PID, PD∨, PD are maximal downwards closed logics, i.e., for L ∈ {PID, PD∨, PD}, ∇N = {φ | φ(pi1, . . . , pin) is an n-formula of L}. In particular, PID ≡ PD∨ ≡ PD.

• Proof. We only treat PID and PD∨. First, consider an n-team:

X{ p1 p2 v1 1 1 v2 1 v3 1 Let ΘX :=        for PD; for PID. Then Y | = ΘX ⇐ ⇒ Y ⊆ X, for any n-team Y.

For each K ∈ ∇N, consider

X∈K ΘX. For any n-team Y on N,

Y | =

• ΘX ⇐

⇒ ∃X ∈ K(Y ⊆ X) ⇐ ⇒ Y ∈ K.

11/19

Fix N = {i1, . . . , in}. Put φ(pi1, . . . , pin) := {X ⊆ 2n | X | = φ}, ∇N := {K ⊆ 22n | ∅ ∈ K,

• X ∈ K, Y ⊆ X =

⇒ Y ∈ K

• }.

Theorem (Ciardelli, Huuskonen, Y.) PID, PD∨, PD are maximal downwards closed logics, i.e., for L ∈ {PID, PD∨, PD}, ∇N = {φ | φ(pi1, . . . , pin) is an n-formula of L}. In particular, PID ≡ PD∨ ≡ PD.

• Proof. We only treat PID and PD∨. First, consider an n-team:

X{ p1 p2 v1 1 1 v2 1 v3 1 Let ΘX :=       

• v∈X

(pv(i1)

i1

∧ · · · ∧ pv(in)

in

), for PD; for PID. Then Y | = ΘX ⇐ ⇒ Y ⊆ X, for any n-team Y.

For each K ∈ ∇N, consider

X∈K ΘX. For any n-team Y on N,

Y | =

• ΘX ⇐

⇒ ∃X ∈ K(Y ⊆ X) ⇐ ⇒ Y ∈ K.

11/19

Fix N = {i1, . . . , in}. Put φ(pi1, . . . , pin) := {X ⊆ 2n | X | = φ}, ∇N := {K ⊆ 22n | ∅ ∈ K,

• X ∈ K, Y ⊆ X =

⇒ Y ∈ K

• }.

Theorem (Ciardelli, Huuskonen, Y.) PID, PD∨, PD are maximal downwards closed logics, i.e., for L ∈ {PID, PD∨, PD}, ∇N = {φ | φ(pi1, . . . , pin) is an n-formula of L}. In particular, PID ≡ PD∨ ≡ PD.

• Proof. We only treat PID and PD∨. First, consider an n-team:

X{ p1 p2 v1 1 1 v2 1 v3 1 Let ΘX :=       

• v∈X

(pv(i1)

i1

∧ · · · ∧ pv(in)

in

), for PD; ¬¬

• v∈X

(pv(i1)

i1

∧ · · · ∧ pv(in)

in

), for PID. Then Y | = ΘX ⇐ ⇒ Y ⊆ X, for any n-team Y.

For each K ∈ ∇N, consider

X∈K ΘX. For any n-team Y on N,

Y | =

• ΘX ⇐

⇒ ∃X ∈ K(Y ⊆ X) ⇐ ⇒ Y ∈ K.

11/19

Fix N = {i1, . . . , in}. Put φ(pi1, . . . , pin) := {X ⊆ 2n | X | = φ}, ∇N := {K ⊆ 22n | ∅ ∈ K,

• X ∈ K, Y ⊆ X =

⇒ Y ∈ K

• }.

Theorem (Ciardelli, Huuskonen, Y.) PID, PD∨, PD are maximal downwards closed logics, i.e., for L ∈ {PID, PD∨, PD}, ∇N = {φ | φ(pi1, . . . , pin) is an n-formula of L}. In particular, PID ≡ PD∨ ≡ PD.

• Proof. We only treat PID and PD∨. First, consider an n-team:

X{ p1 p2 v1 1 1 v2 1 v3 1 Let ΘX :=       

• v∈X

(pv(i1)

i1

∧ · · · ∧ pv(in)

in

), for PD; ¬¬

• v∈X

(pv(i1)

i1

∧ · · · ∧ pv(in)

in

), for PID. Then Y | = ΘX ⇐ ⇒ Y ⊆ X, for any n-team Y.

For each K ∈ ∇N, consider

X∈K ΘX. For any n-team Y on N,

Y | =

• ΘX ⇐

⇒ ∃X ∈ K(Y ⊆ X) ⇐ ⇒ Y ∈ K.

11/19

Theorem (Ciardelli, Huuskonen, Y.) PID, PD∨, PD are maximal downwards closed logics, i.e., for L ∈ {PID, PD∨, PD}, ∇N = {φ | φ(pi1, . . . , pin) is an n-formula of L}. In particular, PID ≡ PD∨ ≡ PD.

• Proof. We only treat PID and PD∨. First, consider an n-team:

X{ p1 p2 v1 1 1 v2 1 v3 1 Let ΘX :=       

• v∈X

(pv(i1)

i1

∧ · · · ∧ pv(in)

in

), for PD; ¬¬

• v∈X

(pv(i1)

i1

∧ · · · ∧ pv(in)

in

), for PID. Then Y | = ΘX ⇐ ⇒ Y ⊆ X, for any n-team Y.

For each K ∈ ∇N, consider

X∈K ΘX. For any n-team Y on N,

Y | =

• X∈K

ΘX ⇐ ⇒ ∃X ∈ K(Y ⊆ X) ⇐ ⇒ Y ∈ K. Hence

X∈K ΘX = K.

11/19

• Theorem. PID, PD∨, PD are sound and complete w.r.t. their deductive

systems.

• Corrolary. For every formula φ of PID and PD∨, φ ⊣⊢

i∈I ΘXi.

A Hilbert Style deductive system for PID− (Ciardelli, Roelofsen 2011) Axioms: all substitution instances of IPC axioms all substitution instances of (KP) (¬pi → (pj ∨ pk)) → ((¬pi → pj) ∨ (¬pi → pk)). ¬¬pi → pi for all propositional variables pi Rules: Modus Ponens Natural deduction systems for PD and PD∨ (Väänänen, Y.) For L ∈ {PD, PD∨}, if φ does not contain any ∨ or dependence atoms, then ⊢CPC φ ⇐ ⇒ ⊢L φ.

12/19

• Theorem. PID, PD∨, PD are sound and complete w.r.t. their deductive

systems.

• Corrolary. For every formula φ of PID and PD∨, φ ⊣⊢

i∈I ΘXi.

A Hilbert Style deductive system for PID− (Ciardelli, Roelofsen 2011) Axioms: all substitution instances of IPC axioms all substitution instances of (KP) (¬pi → (pj ∨ pk)) → ((¬pi → pj) ∨ (¬pi → pk)). ¬¬pi → pi for all propositional variables pi Rules: Modus Ponens Natural deduction systems for PD and PD∨ (Väänänen, Y.) For L ∈ {PD, PD∨}, if φ does not contain any ∨ or dependence atoms, then ⊢CPC φ ⇐ ⇒ ⊢L φ.

12/19

• Theorem. PID, PD∨, PD are sound and complete w.r.t. their deductive

systems.

• Corrolary. For every formula φ of PID and PD∨, φ ⊣⊢

i∈I ΘXi.

A Hilbert Style deductive system for PID− (Ciardelli, Roelofsen 2011) Axioms: all substitution instances of IPC axioms all substitution instances of (KP) (¬pi → (pj ∨ pk)) → ((¬pi → pj) ∨ (¬pi → pk)). ¬¬pi → pi for all propositional variables pi Rules: Modus Ponens Natural deduction systems for PD and PD∨ (Väänänen, Y.) For L ∈ {PD, PD∨}, if φ does not contain any ∨ or dependence atoms, then ⊢CPC φ ⇐ ⇒ ⊢L φ.

12/19

Definition

A formula φ is said to be flat iff for all teams X, X | = φ ⇐ ⇒ ∀v ∈ X, {v} | = φ.

13/19

Definition

A formula φ is said to be flat iff for all teams X, X | = φ ⇐ ⇒ ∀v ∈ X, {v} | = φ.

00 01 11 01 11 11 00 01 00 11 01 00

13/19

Definition

A formula φ is said to be flat iff for all teams X, X | = φ ⇐ ⇒ ∀v ∈ X, {v} | = φ.

00 01 11 01 11 11 00 01 00 11 01 00

φ

13/19

Definition

A formula φ is said to be flat iff for all teams X, X | = φ ⇐ ⇒ ∀v ∈ X, {v} | = φ.

00 01 11 01 11 11 00 01 00 11 01 00

φ φ φ φ

13/19

Definition

A formula φ is said to be flat iff for all teams X, X | = φ ⇐ ⇒ ∀v ∈ X, {v} | = φ.

00 01 11 01 11 11 00 01 00 11 01 00

φ φ φ φ

Fact: Formulas with no occurrences of dependence atoms or intuitionistic disjunction ∨ are flat.

13/19

Definition

A formula φ is said to be flat iff for all teams X, X | = φ ⇐ ⇒ ∀v ∈ X, {v} | = φ.

00 01 11 01 11 11 00 01 00 11 01 00

φ φ φ φ

Fact: Formulas with no occurrences of dependence atoms or intuitionistic disjunction ∨ are flat. E.g. ΘX =       

• v∈X

(pv(i1)

i1

∧ · · · ∧ pv(in)

in

), for PD; ¬¬

• v∈X

(pv(i1)

i1

∧ · · · ∧ pv(in)

in

), for PID. is flat

13/19

Definition

A formula φ is said to be flat iff for all teams X, X | = φ ⇐ ⇒ ∀v ∈ X, {v} | = φ.

00 01 11 01 11 11 00 01 00 11 01 00

φ φ φ φ

Fact: Formulas with no occurrences of dependence atoms or intuitionistic disjunction ∨ are flat.

• Lemma. In PID, a formula φ is flat if ⊢ φ ↔ ¬¬φ.

E.g. ΘX =       

• v∈X

(pv(i1)

i1

∧ · · · ∧ pv(in)

in

), for PD; ¬¬

• v∈X

(pv(i1)

i1

∧ · · · ∧ pv(in)

in

), for PID. is flat

13/19

Definition

A formula φ is said to be flat iff for all teams X, X | = φ ⇐ ⇒ ∀v ∈ X, {v} | = φ.

Lemma

A formula φ is flat iff φ ⊣⊢ ΘX for some X.

14/19

Definition

A formula φ is said to be flat iff for all teams X, X | = φ ⇐ ⇒ ∀v ∈ X, {v} | = φ.

Lemma

A formula φ is flat iff φ ⊣⊢ ΘX for some X.

• Proof. “=

⇒”: Suppose φ is flat. Note that φ ⊣⊢ ΘX1 ∨ · · · ∨ ΘXk,

14/19

Definition

A formula φ is said to be flat iff for all teams X, X | = φ ⇐ ⇒ ∀v ∈ X, {v} | = φ. Fact: Y | = ΘX ⇐ ⇒ Y ⊆ X, for any n-team Y.

Lemma

A formula φ is flat iff φ ⊣⊢ ΘX for some X.

• Proof. “=

⇒”: Suppose φ is flat. Note that φ ⊣⊢ ΘX1 ∨ · · · ∨ ΘXk, where w.l.o.g. we assume that Xi’s are maximal.

X1 X2 X3 Xk · · · · · ·

14/19

Definition

A formula φ is said to be flat iff for all teams X, X | = φ ⇐ ⇒ ∀v ∈ X, {v} | = φ. Fact: Y | = ΘX ⇐ ⇒ Y ⊆ X, for any n-team Y.

Lemma

A formula φ is flat iff φ ⊣⊢ ΘX for some X.

• Proof. “=

⇒”: Suppose φ is flat. Note that φ ⊣⊢ ΘX1 ∨ · · · ∨ ΘXk, where w.l.o.g. we assume that Xi’s are maximal. If k > 1,

X1 X2 X3 Xk · · · · · ·

14/19

Definition

A formula φ is said to be flat iff for all teams X, X | = φ ⇐ ⇒ ∀v ∈ X, {v} | = φ. Fact: Y | = ΘX ⇐ ⇒ Y ⊆ X, for any n-team Y.

Lemma

A formula φ is flat iff φ ⊣⊢ ΘX for some X.

• Proof. “=

⇒”: Suppose φ is flat. Note that φ ⊣⊢ ΘX1 ∨ · · · ∨ ΘXk, where w.l.o.g. we assume that Xi’s are maximal. If k > 1, then pick

X1 X2 X3 Xk · · · · · · v1

14/19

Definition

A formula φ is said to be flat iff for all teams X, X | = φ ⇐ ⇒ ∀v ∈ X, {v} | = φ. Fact: Y | = ΘX ⇐ ⇒ Y ⊆ X, for any n-team Y.

Lemma

A formula φ is flat iff φ ⊣⊢ ΘX for some X.

• Proof. “=

⇒”: Suppose φ is flat. Note that φ ⊣⊢ ΘX1 ∨ · · · ∨ ΘXk, where w.l.o.g. we assume that Xi’s are maximal. If k > 1, then pick

X1 X2 X3 Xk · · · · · · v1 v2

14/19

Definition

A formula φ is said to be flat iff for all teams X, X | = φ ⇐ ⇒ ∀v ∈ X, {v} | = φ. Fact: Y | = ΘX ⇐ ⇒ Y ⊆ X, for any n-team Y.

Lemma

A formula φ is flat iff φ ⊣⊢ ΘX for some X.

• Proof. “=

⇒”: Suppose φ is flat. Note that φ ⊣⊢ ΘX1 ∨ · · · ∨ ΘXk, where w.l.o.g. we assume that Xi’s are maximal. If k > 1, then pick

X1 X2 X3 Xk · · · · · · v1 v2 vk

14/19

Definition

A formula φ is said to be flat iff for all teams X, X | = φ ⇐ ⇒ ∀v ∈ X, {v} | = φ. Fact: Y | = ΘX ⇐ ⇒ Y ⊆ X, for any n-team Y.

Lemma

A formula φ is flat iff φ ⊣⊢ ΘX for some X.

• Proof. “=

⇒”: Suppose φ is flat. Note that φ ⊣⊢ ΘX1 ∨ · · · ∨ ΘXk, where w.l.o.g. we assume that Xi’s are maximal. If k > 1, then pick

X1 X2 X3 Xk · · · · · · v1 v2 vk

{vi} ⊆ Xi and {v1, . . . , vk} Xi for all 1 ≤ i ≤ k = ⇒ {vi} | = ΘXi and {v1, . . . , vk} | = ΘXi for all 1 ≤ i ≤ k = ⇒ {vi} | = φ for all 1 ≤ i ≤ k whereas {v1, . . . , vk} | = φ

14/19

Definition

A formula φ is said to be flat iff for all teams X, X | = φ ⇐ ⇒ ∀v ∈ X, {v} | = φ. Fact: Y | = ΘX ⇐ ⇒ Y ⊆ X, for any n-team Y.

Lemma

A formula φ is flat iff φ ⊣⊢ ΘX for some X.

• Proof. “=

⇒”: Suppose φ is flat. Note that φ ⊣⊢ ΘX1 ∨ · · · ∨ ΘXk, where w.l.o.g. we assume that Xi’s are maximal. If k > 1, then pick

X1 X2 X3 Xk · · · · · · v1 v2 vk

{vi} ⊆ Xi and {v1, . . . , vk} Xi for all 1 ≤ i ≤ k = ⇒ {vi} | = ΘXi and {v1, . . . , vk} | = ΘXi for all 1 ≤ i ≤ k = ⇒ {vi} | = φ for all 1 ≤ i ≤ k whereas {v1, . . . , vk} | = φ = ⇒ k = 1 and φ ⊣⊢ ΘX1.

14/19

We say that a substitution σ of a logic L is well-behaved, if σ is well-defined: σ(φ) is a well-formed formula of L ⊢L is closed under σ: φ ⊢L ψ implies σ(φ) ⊢L σ(ψ)

15/19

We say that a substitution σ of a logic L is well-behaved, if σ is well-defined: σ(φ) is a well-formed formula of L ⊢L is closed under σ: φ ⊢L ψ implies σ(φ) ⊢L σ(ψ)

Recall: Well-formed formulas of PD are built from the following grammar: φ ::= pi | ¬pi | =(pi1, . . . , pik ) | φ ∧ φ | φ ⊗ φ where pi, pi1, . . . , pik are propositional variables.

15/19

We say that a substitution σ of a logic L is well-behaved, if σ is well-defined: σ(φ) is a well-formed formula of L ⊢L is closed under σ: φ ⊢L ψ implies σ(φ) ⊢L σ(ψ) For PID: ⊢ ¬¬p → p, but ⊢ ¬¬(p ∨ ¬p) → (p ∨ ¬p)

15/19

We say that a substitution σ of a logic L is well-behaved, if σ is well-defined: σ(φ) is a well-formed formula of L ⊢L is closed under σ: φ ⊢L ψ implies σ(φ) ⊢L σ(ψ) For PID: ⊢ ¬¬p → p, but ⊢ ¬¬(p ∨ ¬p) → (p ∨ ¬p)

Definition

A substitution σ is called a flat substitution if σ(p) is flat for all p.

Lemma

Flat substitutions are well-behaved in PID and PD.

• Proof. For PID, it follows from (Ciardelli). For PD, nontrivial.

15/19

Definition (Projective formula)

Let S be a set of well-behaved substitutions of a logic L. An L-formula φ is said to be S-projective if there exists σ ∈ S such that (1) ⊢L σ(φ) (2) φ, σ(ψ) ⊢L ψ and φ, ψ ⊢L σ(ψ) for all L-formulas ψ.

16/19

Definition (Projective formula)

Let S be a set of well-behaved substitutions of a logic L. An L-formula φ is said to be S-projective if there exists σ ∈ S such that (1) ⊢L σ(φ) (2) φ, σ(ψ) ⊢L ψ and φ, ψ ⊢L σ(ψ) for all L-formulas ψ.

Lemma

Let L ∈ {PD, PID} and F the set of all flat substitutions. An n-formula φ

• f L is F-projective iff φ ⊣⊢ ΘX for some n-team X.

16/19

Definition (Projective formula)

Let S be a set of well-behaved substitutions of a logic L. An L-formula φ is said to be S-projective if there exists σ ∈ S such that (1) ⊢L σ(φ) (2) φ, σ(ψ) ⊢L ψ and φ, ψ ⊢L σ(ψ) for all L-formulas ψ.

Lemma

Let L ∈ {PD, PID} and F the set of all flat substitutions. An n-formula φ

• f L is F-projective iff φ ⊣⊢ ΘX for some n-team X.
• Proof. “=

⇒”: Note that φ ≡

i∈I ΘXi.

16/19

Definition (Projective formula)

Let S be a set of well-behaved substitutions of a logic L. An L-formula φ is said to be S-projective if there exists σ ∈ S such that (1) ⊢L σ(φ) (2) φ, σ(ψ) ⊢L ψ and φ, ψ ⊢L σ(ψ) for all L-formulas ψ.

Lemma

Let L ∈ {PD, PID} and F the set of all flat substitutions. An n-formula φ

• f L is F-projective iff φ ⊣⊢ ΘX for some n-team X.
• Proof. “=

⇒”: Note that φ ≡

i∈I ΘXi. Let σ be the F-projective unifier

• f φ. By (1), |

= σ(φ) = ⇒ | = σ(ΘXi) for some i ∈ I (by DP) = ⇒ φ | = ΘXi (by (2)).

16/19

Definition (Projective formula)

Let S be a set of well-behaved substitutions of a logic L. An L-formula φ is said to be S-projective if there exists σ ∈ S such that (1) ⊢L σ(φ) (2) φ, σ(ψ) ⊢L ψ and φ, ψ ⊢L σ(ψ) for all L-formulas ψ.

Lemma

Let L ∈ {PD, PID} and F the set of all flat substitutions. An n-formula φ

• f L is F-projective iff φ ⊣⊢ ΘX for some n-team X.
• Proof. “=

⇒”: Note that φ ≡

i∈I ΘXi. Let σ be the F-projective unifier

• f φ. By (1), |

= σ(φ) = ⇒ | = σ(ΘXi) for some i ∈ I (by DP) = ⇒ φ | = ΘXi (by (2)). “⇐ =”: Recall: ΘX =       

• v∈X

(pv(i1)

i1

∧ · · · ∧ pv(in)

in

), for PD; ¬¬

• v∈X

(pv(i1)

i1

∧ · · · ∧ pv(in)

in

), for PID.

16/19

Definition (Projective formula)

Let S be a set of well-behaved substitutions of a logic L. An L-formula φ is said to be S-projective if there exists σ ∈ S such that (1) ⊢L σ(φ) (2) φ, σ(ψ) ⊢L ψ and φ, ψ ⊢L σ(ψ) for all L-formulas ψ.

Lemma

Let L ∈ {PD, PID} and F the set of all flat substitutions. An n-formula φ

• f L is F-projective iff φ ⊣⊢ ΘX for some n-team X.
• Proof. “=

⇒”: Note that φ ≡

i∈I ΘXi. Let σ be the F-projective unifier

• f φ. By (1), |

= σ(φ) = ⇒ | = σ(ΘXi) for some i ∈ I (by DP) = ⇒ φ | = ΘXi (by (2)). “⇐ =”: Recall: ΘX =       

• v∈X

(pv(i1)

i1

∧ · · · ∧ pv(in)

in

), for PD; ¬¬

• v∈X

(pv(i1)

i1

∧ · · · ∧ pv(in)

in

), for PID. We only give the proof for PD.

16/19

Definition (Projective formula)

Let S be a set of well-behaved substitutions of a logic L. An L-formula φ is said to be S-projective if there exists σ ∈ S such that (1) ⊢L σ(φ) (2) φ, σ(ψ) ⊢L ψ and φ, ψ ⊢L σ(ψ) for all L-formulas ψ.

Lemma

Let L ∈ {PD, PID} and F the set of all flat substitutions. An n-formula φ

• f L is F-projective iff φ ⊣⊢ ΘX for some n-team X.
• Proof. (ctd.) View φ = ΘX =

v∈X(pv(i1) i1

∧ · · · ∧ pv(in)

in

) as a formula of CPC,

17/19

Definition (Projective formula)

Let S be a set of well-behaved substitutions of a logic L. An L-formula φ is said to be S-projective if there exists σ ∈ S such that (1) ⊢L σ(φ) (2) φ, σ(ψ) ⊢L ψ and φ, ψ ⊢L σ(ψ) for all L-formulas ψ.

Lemma

Let L ∈ {PD, PID} and F the set of all flat substitutions. An n-formula φ

• f L is F-projective iff φ ⊣⊢ ΘX for some n-team X.
• Proof. (ctd.) View φ = ΘX =

v∈X(pv(i1) i1

∧ · · · ∧ pv(in)

in

) as a formula of CPC, then v(φi) = 1 for any v ∈ X.

17/19

Definition (Projective formula)

Let S be a set of well-behaved substitutions of a logic L. An L-formula φ is said to be S-projective if there exists σ ∈ S such that (1) ⊢L σ(φ) (2) φ, σ(ψ) ⊢L ψ and φ, ψ ⊢L σ(ψ) for all L-formulas ψ.

Lemma

Let L ∈ {PD, PID} and F the set of all flat substitutions. An n-formula φ

• f L is F-projective iff φ ⊣⊢ ΘX for some n-team X.
• Proof. (ctd.) View φ = ΘX =

v∈X(pv(i1) i1

∧ · · · ∧ pv(in)

in

) as a formula of CPC, then v(φi) = 1 for any v ∈ X. Define σφ

v as follows:

σφ

v (p) =

• φ ∧ p,

if v(p) = 0; φ → p, if v(p) = 1. (Prucnal’s trick)

17/19

Definition (Projective formula)

Let S be a set of well-behaved substitutions of a logic L. An L-formula φ is said to be S-projective if there exists σ ∈ S such that (1) ⊢L σ(φ) (2) φ, σ(ψ) ⊢L ψ and φ, ψ ⊢L σ(ψ) for all L-formulas ψ.

Lemma

Let L ∈ {PD, PID} and F the set of all flat substitutions. An n-formula φ

• f L is F-projective iff φ ⊣⊢ ΘX for some n-team X.
• Proof. (ctd.) View φ = ΘX =

v∈X(pv(i1) i1

∧ · · · ∧ pv(in)

in

) as a formula of CPC, then v(φi) = 1 for any v ∈ X. Define σφ

v as follows:

σφ

v (p) =

• φ ∧ p,

if v(p) = 0; φ → p, if v(p) = 1. (Prucnal’s trick)

17/19

Definition (Projective formula)

Let S be a set of well-behaved substitutions of a logic L. An L-formula φ is said to be S-projective if there exists σ ∈ S such that (1) ⊢L σ(φ) (2) φ, σ(ψ) ⊢L ψ and φ, ψ ⊢L σ(ψ) for all L-formulas ψ.

Lemma

Let L ∈ {PD, PID} and F the set of all flat substitutions. An n-formula φ

• f L is F-projective iff φ ⊣⊢ ΘX for some n-team X.
• Proof. (ctd.) View φ = ΘX =

v∈X(pv(i1) i1

∧ · · · ∧ pv(in)

in

) as a formula of CPC, then v(φi) = 1 for any v ∈ X. Define σφ

v as follows:

σφ

v (p) =

• φ ∧ p,

if v(p) = 0; φ∼ ⊗ p, if v(p) = 1. (Prucnal’s trick)

17/19

Definition (Projective formula)

Let S be a set of well-behaved substitutions of a logic L. An L-formula φ is said to be S-projective if there exists σ ∈ S such that (1) ⊢L σ(φ) (2) φ, σ(ψ) ⊢L ψ and φ, ψ ⊢L σ(ψ) for all L-formulas ψ.

Lemma

Let L ∈ {PD, PID} and F the set of all flat substitutions. An n-formula φ

• f L is F-projective iff φ ⊣⊢ ΘX for some n-team X.
• Proof. (ctd.) View φ = ΘX =

v∈X(pv(i1) i1

∧ · · · ∧ pv(in)

in

) as a formula of CPC, then v(φi) = 1 for any v ∈ X. Define σφ

v as follows:

σφ

v (p) =

• φ ∧ p,

if v(p) = 0; φ∼ ⊗ p, if v(p) = 1. (Prucnal’s trick) Put σ = σφ

v . Then ⊢CPC σ(φ) as v(φ) = 1.

17/19

Definition (Projective formula)

Let S be a set of well-behaved substitutions of a logic L. An L-formula φ is said to be S-projective if there exists σ ∈ S such that (1) ⊢L σ(φ) (2) φ, σ(ψ) ⊢L ψ and φ, ψ ⊢L σ(ψ) for all L-formulas ψ.

Lemma

Let L ∈ {PD, PID} and F the set of all flat substitutions. An n-formula φ

• f L is F-projective iff φ ⊣⊢ ΘX for some n-team X.
• Proof. (ctd.) View φ = ΘX =

v∈X(pv(i1) i1

∧ · · · ∧ pv(in)

in

) as a formula of CPC, then v(φi) = 1 for any v ∈ X. Define σφ

v as follows:

σφ

v (p) =

• φ ∧ p,

if v(p) = 0; φ∼ ⊗ p, if v(p) = 1. (Prucnal’s trick) Put σ = σφ

v . Then ⊢CPC σ(φ) as v(φ) = 1. Clearly, σ ∈ F

17/19

Definition (Projective formula)

Let S be a set of well-behaved substitutions of a logic L. An L-formula φ is said to be S-projective if there exists σ ∈ S such that (1) ⊢L σ(φ) (2) φ, σ(ψ) ⊢L ψ and φ, ψ ⊢L σ(ψ) for all L-formulas ψ.

Lemma

Let L ∈ {PD, PID} and F the set of all flat substitutions. An n-formula φ

• f L is F-projective iff φ ⊣⊢ ΘX for some n-team X.
• Proof. (ctd.) View φ = ΘX =

v∈X(pv(i1) i1

∧ · · · ∧ pv(in)

in

) as a formula of CPC, then v(φi) = 1 for any v ∈ X. Define σφ

v as follows:

σφ

v (p) =

• φ ∧ p,

if v(p) = 0; φ∼ ⊗ p, if v(p) = 1. (Prucnal’s trick) Put σ = σφ

v . Then ⊢CPC σ(φ) as v(φ) = 1. Clearly, σ ∈ F and σ(φ) is

classical, thus ⊢PD σ(φ).

17/19

Definition (Projective formula)

Let S be a set of well-behaved substitutions of a logic L. An L-formula φ is said to be S-projective if there exists σ ∈ S such that (1) ⊢L σ(φ) (2) φ, σ(ψ) ⊢L ψ and φ, ψ ⊢L σ(ψ) for all L-formulas ψ.

Lemma

Let L ∈ {PD, PID} and F the set of all flat substitutions. An n-formula φ

• f L is F-projective iff φ ⊣⊢ ΘX for some n-team X.
• Proof. (ctd.) View φ = ΘX =

v∈X(pv(i1) i1

∧ · · · ∧ pv(in)

in

) as a formula of CPC, then v(φi) = 1 for any v ∈ X. Define σφ

v as follows:

σφ

v (p) =

• φ ∧ p,

if v(p) = 0; φ∼ ⊗ p, if v(p) = 1. (Prucnal’s trick) Put σ = σφ

v . Then ⊢CPC σ(φ) as v(φ) = 1. Clearly, σ ∈ F and σ(φ) is

classical, thus ⊢PD σ(φ). Moreover, (2) is satisfied by the choice of σ.

17/19

Definition (Exact formula)

Let S be a set of well-behaved substitutions of a logic L. An L-formula φ is said to be S-exact if there exists σ ∈ S such that (1) ⊢L σ(φ) (2) ⊢L σ(ψ) = ⇒ φ ⊢L ψ for all L-formulas ψ

Lemma

Let L ∈ {PD, PID} and F the set of all flat substitutions. An n-formula φ

• f L is F-exact iff φ ⊣⊢ ΘX for some n-team X.

Corollary

For formulas φ of the logics PD and PID, TFAE: φ is flat; φ is F-projective; φ is F-exact; φ ⊣⊢ ΘX for some X

18/19

Definition (Exact formula)

Let S be a set of well-behaved substitutions of a logic L. An L-formula φ is said to be S-exact if there exists σ ∈ S such that (1) ⊢L σ(φ) (2) ⊢L σ(ψ) = ⇒ φ ⊢L ψ for all L-formulas ψ

Lemma

Let L ∈ {PD, PID} and F the set of all flat substitutions. An n-formula φ

• f L is F-exact iff φ ⊣⊢ ΘX for some n-team X.

Corollary

For formulas φ of the logics PD and PID, TFAE: φ is flat; φ is F-projective; φ is F-exact; φ ⊣⊢ ΘX for some X

18/19

Definition

Let S be a set of well-behaved substitutions of a logic L. A rule φ/ψ of L is said to be a S-admissible rule, in symbols φ | ∼S

L ψ or simply φ |

∼S ψ, if for all σ ∈ S, ⊢L σ(φ) = ⇒ ⊢L σ(ψ). L is said to be S-structurally complete if every S-admissible rule of L is derivable, i.e., φ | ∼S

L ψ =

⇒ φ ⊢L ψ.

19/19

Definition

Let S be a set of well-behaved substitutions of a logic L. A rule φ/ψ of L is said to be a S-admissible rule, in symbols φ | ∼S

L ψ or simply φ |

∼S ψ, if for all σ ∈ S, ⊢L σ(φ) = ⇒ ⊢L σ(ψ). L is said to be S-structurally complete if every S-admissible rule of L is derivable, i.e., φ | ∼S

L ψ =

⇒ φ ⊢L ψ.

19/19

Definition

Let S be a set of well-behaved substitutions of a logic L. A rule φ/ψ of L is said to be a S-admissible rule, in symbols φ | ∼S

L ψ or simply φ |

∼S ψ, if for all σ ∈ S, ⊢L σ(φ) = ⇒ ⊢L σ(ψ). L is said to be S-structurally complete if every S-admissible rule of L is derivable, i.e., φ | ∼S

L ψ =

⇒ φ ⊢L ψ. Clearly, if L is S-structurally complete, then φ | ∼S

L ψ ⇐

⇒ φ ⊢L ψ.

19/19

Definition

Let S be a set of well-behaved substitutions of a logic L. A rule φ/ψ of L is said to be a S-admissible rule, in symbols φ | ∼S

L ψ or simply φ |

∼S ψ, if for all σ ∈ S, ⊢L σ(φ) = ⇒ ⊢L σ(ψ). L is said to be S-structurally complete if every S-admissible rule of L is derivable, i.e., φ | ∼S

L ψ =

⇒ φ ⊢L ψ. Clearly, if L is S-structurally complete, then φ | ∼S

L ψ ⇐

⇒ φ ⊢L ψ.

Theorem

PID and PD are F-structurally complete, where F is the set of all flat substitutions.

• Proof. Since φ ≡

i∈I ΘXi, where each ΘXi is S-projective, for every

formula φ of the logics.

19/19