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Admissible Rules and Beyond

George Metcalfe

Mathematics Institute University of Bern

WARU II, Les Diablerets, February 2015

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 1 / 28

A First Question

What is an admissible rule?

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 2 / 28

Two Informal Answers

(A) “A rule is admissible in a system if the set of theorems does not change when the rule is added to the system.” (B) “A rule is admissible in a system if any substitution sending its premises to theorems, sends its conclusion to a theorem.”

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 3 / 28

Two Informal Answers

(A) “A rule is admissible in a system if the set of theorems does not change when the rule is added to the system.” (B) “A rule is admissible in a system if any substitution sending its premises to theorems, sends its conclusion to a theorem.”

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 3 / 28

Admissibility in Intuitionistic Logic

The “independence of premises” rule {¬p → (q ∨ r)} ⇒ (¬p → q) ∨ (¬p → r) is not derivable in intuitionistic logic, but it is admissible because. . . (A) adding it to an axiomatization gives no new theorems (B) if ¬ϕ → (ψ ∨ χ) is a theorem, so is (¬ϕ → ψ) ∨ (¬ϕ → χ).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 4 / 28

Admissibility in Intuitionistic Logic

The “independence of premises” rule {¬p → (q ∨ r)} ⇒ (¬p → q) ∨ (¬p → r) is not derivable in intuitionistic logic, but it is admissible because. . . (A) adding it to an axiomatization gives no new theorems (B) if ¬ϕ → (ψ ∨ χ) is a theorem, so is (¬ϕ → ψ) ∨ (¬ϕ → χ).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 4 / 28

Admissibility in Intuitionistic Logic

The “independence of premises” rule {¬p → (q ∨ r)} ⇒ (¬p → q) ∨ (¬p → r) is not derivable in intuitionistic logic, but it is admissible because. . . (A) adding it to an axiomatization gives no new theorems (B) if ¬ϕ → (ψ ∨ χ) is a theorem, so is (¬ϕ → ψ) ∨ (¬ϕ → χ).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 4 / 28

Multiple-Conclusion Rules

The “disjunction property” {p ∨ q} ⇒ {p, q} is admissible in intuitionistic logic because. . . (A) adding it to an axiomatization gives no new theorems (B) if ϕ ∨ ψ is a theorem, either ϕ or ψ is a theorem.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 5 / 28

Multiple-Conclusion Rules

The “disjunction property” {p ∨ q} ⇒ {p, q} is admissible in intuitionistic logic because. . . (A) adding it to an axiomatization gives no new theorems (B) if ϕ ∨ ψ is a theorem, either ϕ or ψ is a theorem.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 5 / 28

Multiple-Conclusion Rules

The “disjunction property” {p ∨ q} ⇒ {p, q} is admissible in intuitionistic logic because. . . (A) adding it to an axiomatization gives no new theorems (B) if ϕ ∨ ψ is a theorem, either ϕ or ψ is a theorem.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 5 / 28

A Splitting of the Notions

The “linearity property” ⇒ {p → q, q → p} is admissible in Gödel logic according to. . . (A) because adding it to an axiomatization gives no new theorems but not according to. . . (B) because it may be that neither ϕ → ψ nor ψ → ϕ is a theorem.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 6 / 28

A Splitting of the Notions

The “linearity property” ⇒ {p → q, q → p} is admissible in Gödel logic according to. . . (A) because adding it to an axiomatization gives no new theorems but not according to. . . (B) because it may be that neither ϕ → ψ nor ψ → ϕ is a theorem.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 6 / 28

A Splitting of the Notions

The “linearity property” ⇒ {p → q, q → p} is admissible in Gödel logic according to. . . (A) because adding it to an axiomatization gives no new theorems but not according to. . . (B) because it may be that neither ϕ → ψ nor ψ → ϕ is a theorem.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 6 / 28

A More Exotic Example

The “Takeuti-Titani density rule” {((ϕ → p) ∨ (p → ψ)) ∨ χ} ⇒ (ϕ → ψ) ∨ χ where p does not occur in ϕ, ψ, or χ is admissible in Gödel logic because. . . (A) adding it to an axiomatization gives no new theorems (B) if ((ϕ → p) ∨ (p → ψ)) ∨ χ is a theorem, (ϕ → ψ) ∨ χ is a theorem.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 7 / 28

A More Exotic Example

The “Takeuti-Titani density rule” {((ϕ → p) ∨ (p → ψ)) ∨ χ} ⇒ (ϕ → ψ) ∨ χ where p does not occur in ϕ, ψ, or χ is admissible in Gödel logic because. . . (A) adding it to an axiomatization gives no new theorems (B) if ((ϕ → p) ∨ (p → ψ)) ∨ χ is a theorem, (ϕ → ψ) ∨ χ is a theorem.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 7 / 28

A More Exotic Example

The “Takeuti-Titani density rule” {((ϕ → p) ∨ (p → ψ)) ∨ χ} ⇒ (ϕ → ψ) ∨ χ where p does not occur in ϕ, ψ, or χ is admissible in Gödel logic because. . . (A) adding it to an axiomatization gives no new theorems (B) if ((ϕ → p) ∨ (p → ψ)) ∨ χ is a theorem, (ϕ → ψ) ∨ χ is a theorem.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 7 / 28

More Generally. . .

What does it mean for a first-order sentence such as (∃x)(∀y)(x ≤ y)

• r

(∀x)(∃y)¬(x ≤ y) to be admissible in a logic or class of algebras?

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 8 / 28

The Main Question

How can these notions of admissibility be characterized?

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 9 / 28

References

We take a “first-order” approach as described in

• G. Metcalfe. Admissible Rules: From Characterizations to Applications.

Proceedings of WoLLIC 2012, LNCS 7456, Springer (2012), 56–69.

A “consequence relations” approach is described in

• R. Iemhoff. A Note on Consequence. Submitted.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 10 / 28

References

We take a “first-order” approach as described in

• G. Metcalfe. Admissible Rules: From Characterizations to Applications.

Proceedings of WoLLIC 2012, LNCS 7456, Springer (2012), 56–69.

A “consequence relations” approach is described in

• R. Iemhoff. A Note on Consequence. Submitted.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 10 / 28

First-Order Logic

Assume the usual terminology of first-order logic with equality, using the symbols ∀, ∃, ⊓, ⊔, ⇒, ∼, 0, 1, and ≈. Fix an algebraic language L with terms Tm(L) and sentences Sen(L). For sets of L-equations Γ and ∆, denote by Γ ⇒ ∆ the L-clause (∀¯ x)(⊓Γ ⇒ ⊔∆) called an L-quasiequation if |∆| = 1 and a positive L-clause if Γ = ∅.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 11 / 28

First-Order Logic

Assume the usual terminology of first-order logic with equality, using the symbols ∀, ∃, ⊓, ⊔, ⇒, ∼, 0, 1, and ≈. Fix an algebraic language L with terms Tm(L) and sentences Sen(L). For sets of L-equations Γ and ∆, denote by Γ ⇒ ∆ the L-clause (∀¯ x)(⊓Γ ⇒ ⊔∆) called an L-quasiequation if |∆| = 1 and a positive L-clause if Γ = ∅.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 11 / 28

First-Order Logic

Assume the usual terminology of first-order logic with equality, using the symbols ∀, ∃, ⊓, ⊔, ⇒, ∼, 0, 1, and ≈. Fix an algebraic language L with terms Tm(L) and sentences Sen(L). For sets of L-equations Γ and ∆, denote by Γ ⇒ ∆ the L-clause (∀¯ x)(⊓Γ ⇒ ⊔∆) called an L-quasiequation if |∆| = 1 and a positive L-clause if Γ = ∅.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 11 / 28

Admissibility Algebraically

Let Tm(L) denote the term algebra of L, and consider a class of L-algebras K and a set of L-equations Γ. A K-unifier of Γ is a homomorphism σ: Tm(L) → Tm(L) such that K | = σ(s) ≈ σ(t) for all s ≈ t ∈ Γ. We say that an L-clause Γ ⇒ ∆ is K-admissible if σ is a K-unifier of Γ = ⇒ σ is a K-unifier of some s ≈ t ∈ ∆.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 12 / 28

Admissibility Algebraically

Let Tm(L) denote the term algebra of L, and consider a class of L-algebras K and a set of L-equations Γ. A K-unifier of Γ is a homomorphism σ: Tm(L) → Tm(L) such that K | = σ(s) ≈ σ(t) for all s ≈ t ∈ Γ. We say that an L-clause Γ ⇒ ∆ is K-admissible if σ is a K-unifier of Γ = ⇒ σ is a K-unifier of some s ≈ t ∈ ∆.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 12 / 28

Admissibility Algebraically

Let Tm(L) denote the term algebra of L, and consider a class of L-algebras K and a set of L-equations Γ. A K-unifier of Γ is a homomorphism σ: Tm(L) → Tm(L) such that K | = σ(s) ≈ σ(t) for all s ≈ t ∈ Γ. We say that an L-clause Γ ⇒ ∆ is K-admissible if σ is a K-unifier of Γ = ⇒ σ is a K-unifier of some s ≈ t ∈ ∆.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 12 / 28

An Algebraic Characterization

For any class of L-algebras K and an L-clause Γ ⇒ ∆, Γ ⇒ ∆ is K-admissible ⇔ FK | = Γ ⇒ ∆ where FK is the free algebra of K on countably many generators.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 13 / 28

Another Notion of Admissibility

But what about notion (A) “A rule is admissible in a system if the set of theorems does not change when the rule is added to the system.” ? Reformulating, consider. . . the “system” as a class of L-algebras K the “rule” as a first-order L-sentence ϕ the “theorems” as a set of L-sentences Σ, when does ϕ preserve Σ in K?

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 14 / 28

Another Notion of Admissibility

But what about notion (A) “A rule is admissible in a system if the set of theorems does not change when the rule is added to the system.” ? Reformulating, consider. . . the “system” as a class of L-algebras K the “rule” as a first-order L-sentence ϕ the “theorems” as a set of L-sentences Σ, when does ϕ preserve Σ in K?

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 14 / 28

Another Notion of Admissibility

But what about notion (A) “A rule is admissible in a system if the set of theorems does not change when the rule is added to the system.” ? Reformulating, consider. . . the “system” as a class of L-algebras K the “rule” as a first-order L-sentence ϕ the “theorems” as a set of L-sentences Σ, when does ϕ preserve Σ in K?

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 14 / 28

Another Notion of Admissibility

But what about notion (A) “A rule is admissible in a system if the set of theorems does not change when the rule is added to the system.” ? Reformulating, consider. . . the “system” as a class of L-algebras K the “rule” as a first-order L-sentence ϕ the “theorems” as a set of L-sentences Σ, when does ϕ preserve Σ in K?

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 14 / 28

Another Notion of Admissibility

But what about notion (A) “A rule is admissible in a system if the set of theorems does not change when the rule is added to the system.” ? Reformulating, consider. . . the “system” as a class of L-algebras K the “rule” as a first-order L-sentence ϕ the “theorems” as a set of L-sentences Σ, when does ϕ preserve Σ in K?

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 14 / 28

Another Notion of Admissibility

But what about notion (A) “A rule is admissible in a system if the set of theorems does not change when the rule is added to the system.” ? Reformulating, consider. . . the “system” as a class of L-algebras K the “rule” as a first-order L-sentence ϕ the “theorems” as a set of L-sentences Σ, when does ϕ preserve Σ in K?

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 14 / 28

Preserving Sentences

Definition

For a class of L-algebras K and Σ ⊆ Sen(L), we set ThΣ(K) = {ψ ∈ Σ : K | = ψ} and say that ϕ ∈ Sen(L) preserves Σ in K if ThΣ(K) = ThΣ({A ∈ K : A | = ϕ}). If Θ ⊆ Sen(L) axiomatizes K, then ϕ preserves Σ in K if for all ψ ∈ Σ: Θ | = ψ ⇔ Θ ∪ {ϕ} | = ψ.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 15 / 28

Preserving Sentences

Definition

For a class of L-algebras K and Σ ⊆ Sen(L), we set ThΣ(K) = {ψ ∈ Σ : K | = ψ} and say that ϕ ∈ Sen(L) preserves Σ in K if ThΣ(K) = ThΣ({A ∈ K : A | = ϕ}). If Θ ⊆ Sen(L) axiomatizes K, then ϕ preserves Σ in K if for all ψ ∈ Σ: Θ | = ψ ⇔ Θ ∪ {ϕ} | = ψ.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 15 / 28

Preserving Sentences

Definition

For a class of L-algebras K and Σ ⊆ Sen(L), we set ThΣ(K) = {ψ ∈ Σ : K | = ψ} and say that ϕ ∈ Sen(L) preserves Σ in K if ThΣ(K) = ThΣ({A ∈ K : A | = ϕ}). If Θ ⊆ Sen(L) axiomatizes K, then ϕ preserves Σ in K if for all ψ ∈ Σ: Θ | = ψ ⇔ Θ ∪ {ϕ} | = ψ.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 15 / 28

Preserving Sentences

Definition

For a class of L-algebras K and Σ ⊆ Sen(L), we set ThΣ(K) = {ψ ∈ Σ : K | = ψ} and say that ϕ ∈ Sen(L) preserves Σ in K if ThΣ(K) = ThΣ({A ∈ K : A | = ϕ}). If Θ ⊆ Sen(L) axiomatizes K, then ϕ preserves Σ in K if for all ψ ∈ Σ: Θ | = ψ ⇔ Θ ∪ {ϕ} | = ψ.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 15 / 28

Preserving Sentences

Definition

For a class of L-algebras K and Σ ⊆ Sen(L), we set ThΣ(K) = {ψ ∈ Σ : K | = ψ} and say that ϕ ∈ Sen(L) preserves Σ in K if ThΣ(K) = ThΣ({A ∈ K : A | = ϕ}). If Θ ⊆ Sen(L) axiomatizes K, then ϕ preserves Σ in K if for all ψ ∈ Σ: Θ | = ψ ⇔ Θ ∪ {ϕ} | = ψ.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 15 / 28

Preserving Sentences

Definition

For a class of L-algebras K and Σ ⊆ Sen(L), we set ThΣ(K) = {ψ ∈ Σ : K | = ψ} and say that ϕ ∈ Sen(L) preserves Σ in K if ThΣ(K) = ThΣ({A ∈ K : A | = ϕ}). If Θ ⊆ Sen(L) axiomatizes K, then ϕ preserves Σ in K if for all ψ ∈ Σ: Θ | = ψ ⇔ Θ ∪ {ϕ} | = ψ.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 15 / 28

Preserving Equations

Theorem

The following are equivalent for any L-quasiequation ϕ: (i) ϕ is K-admissible (ii) FK | = ϕ (iii) ϕ preserves L-equations in K (iv) K ⊆ V({A ∈ K : A | = ϕ}), and if K is a quasivariety, (v) each B ∈ K is a homomorphic image of an A ∈ K such that A | = ϕ.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 16 / 28

Preserving Equations

Theorem

The following are equivalent for any L-quasiequation ϕ: (i) ϕ is K-admissible (ii) FK | = ϕ (iii) ϕ preserves L-equations in K (iv) K ⊆ V({A ∈ K : A | = ϕ}), and if K is a quasivariety, (v) each B ∈ K is a homomorphic image of an A ∈ K such that A | = ϕ.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 16 / 28

Preserving Equations

Theorem

The following are equivalent for any L-quasiequation ϕ: (i) ϕ is K-admissible (ii) FK | = ϕ (iii) ϕ preserves L-equations in K (iv) K ⊆ V({A ∈ K : A | = ϕ}), and if K is a quasivariety, (v) each B ∈ K is a homomorphic image of an A ∈ K such that A | = ϕ.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 16 / 28

Preserving Equations

Theorem

The following are equivalent for any L-quasiequation ϕ: (i) ϕ is K-admissible (ii) FK | = ϕ (iii) ϕ preserves L-equations in K (iv) K ⊆ V({A ∈ K : A | = ϕ}), and if K is a quasivariety, (v) each B ∈ K is a homomorphic image of an A ∈ K such that A | = ϕ.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 16 / 28

Preserving Positive Clauses

Theorem

The following are equivalent for any L-clause ϕ: (i) ϕ is K-admissible (ii) FK | = ϕ (iii) ϕ preserves positive L-clauses in K (iv) K ⊆ U+({A ∈ K : A | = ϕ}), and if K is a universal class, (v) each B ∈ K is a homomorphic image of an A ∈ K such that A | = ϕ.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 17 / 28

Preserving Positive Clauses

Theorem

The following are equivalent for any L-clause ϕ: (i) ϕ is K-admissible (ii) FK | = ϕ (iii) ϕ preserves positive L-clauses in K (iv) K ⊆ U+({A ∈ K : A | = ϕ}), and if K is a universal class, (v) each B ∈ K is a homomorphic image of an A ∈ K such that A | = ϕ.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 17 / 28

Preserving Positive Clauses

Theorem

The following are equivalent for any L-clause ϕ: (i) ϕ is K-admissible (ii) FK | = ϕ (iii) ϕ preserves positive L-clauses in K (iv) K ⊆ U+({A ∈ K : A | = ϕ}), and if K is a universal class, (v) each B ∈ K is a homomorphic image of an A ∈ K such that A | = ϕ.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 17 / 28

Preserving Clauses

Theorem

The following are equivalent for any ϕ ∈ Sen(L): (i) ϕ preserves L-clauses in K (ii) K ⊆ U({A ∈ K : A | = ϕ}), and if K is an elementary class, (iii) each B ∈ K embeds into an A ∈ K such that A | = ϕ.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 18 / 28

Preserving Clauses

Theorem

The following are equivalent for any ϕ ∈ Sen(L): (i) ϕ preserves L-clauses in K (ii) K ⊆ U({A ∈ K : A | = ϕ}), and if K is an elementary class, (iii) each B ∈ K embeds into an A ∈ K such that A | = ϕ.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 18 / 28

Example

For the variety BA of Boolean algebras in a language LBool, ϕ = (∀x)((x ≈ ⊥) ⊔ (x ≈ ⊤)) preserves LBool-equations in BA, but FBA | = ϕ. Note that ¬ϕ, equivalent to (∃x)(¬(x ≈ ⊥) ⊓ ¬(x ≈ ⊤)), also preserves LBool-equations in BA.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 19 / 28

Example

For the variety BA of Boolean algebras in a language LBool, ϕ = (∀x)((x ≈ ⊥) ⊔ (x ≈ ⊤)) preserves LBool-equations in BA, but FBA | = ϕ. Note that ¬ϕ, equivalent to (∃x)(¬(x ≈ ⊥) ⊓ ¬(x ≈ ⊤)), also preserves LBool-equations in BA.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 19 / 28

Beyond Clauses

The Skolem form sk(ϕ) of a prenex ϕ ∈ Sen(L) results by repeating (∀¯ x)(∃y)ϕ(¯ x, y) = ⇒ (∀¯ x)ϕ(¯ x, f(¯ x)) f new. Then for any Θ ∪ {ψ} ⊆ Sen(L): Θ ∪ {ϕ} | = ψ ⇔ Θ ∪ {sk(ϕ)} | = ψ.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 20 / 28

Beyond Clauses

The Skolem form sk(ϕ) of a prenex ϕ ∈ Sen(L) results by repeating (∀¯ x)(∃y)ϕ(¯ x, y) = ⇒ (∀¯ x)ϕ(¯ x, f(¯ x)) f new. Then for any Θ ∪ {ψ} ⊆ Sen(L): Θ ∪ {ϕ} | = ψ ⇔ Θ ∪ {sk(ϕ)} | = ψ.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 20 / 28

Beyond Clauses

The Skolem form sk(ϕ) of a prenex ϕ ∈ Sen(L) results by repeating (∀¯ x)(∃y)ϕ(¯ x, y) = ⇒ (∀¯ x)ϕ(¯ x, f(¯ x)) f new. Then for any Θ ∪ {ψ} ⊆ Sen(L): Θ ∪ {ϕ} | = ψ ⇔ Θ ∪ {sk(ϕ)} | = ψ.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 20 / 28

Preservation under Skolemization

Let K be a class of L-algebras, L′ an extension of L, and K′ the class

• f L′-algebras whose L-reducts are in K.

Theorem

The following are equivalent for any Σ ∪ {ϕ} ⊆ Sen(L): (1) ϕ preserves Σ in K (2) sk(ϕ) ∈ Sen(L′) preserves Σ in K′.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 21 / 28

Preservation under Skolemization

Let K be a class of L-algebras, L′ an extension of L, and K′ the class

• f L′-algebras whose L-reducts are in K.

Theorem

The following are equivalent for any Σ ∪ {ϕ} ⊆ Sen(L): (1) ϕ preserves Σ in K (2) sk(ϕ) ∈ Sen(L′) preserves Σ in K′.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 21 / 28

Applications?

For a class of algebras K, we often seek a “distinguished subclass” K′ ⊆ K such that for all equations (quasiequations, etc.) ϕ, K′ | = ϕ ⇔ K | = ϕ. For example: Boolean algebras and the two-element Boolean algebra modal algebras and perfect modal algebras Gödel algebras and dense Gödel chains lattice-ordered groups and automorphisms of R. Algebraically, we want to establish V(K) = V(K′) (Q(K) = Q(K′), etc.).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 22 / 28

Applications?

For a class of algebras K, we often seek a “distinguished subclass” K′ ⊆ K such that for all equations (quasiequations, etc.) ϕ, K′ | = ϕ ⇔ K | = ϕ. For example: Boolean algebras and the two-element Boolean algebra modal algebras and perfect modal algebras Gödel algebras and dense Gödel chains lattice-ordered groups and automorphisms of R. Algebraically, we want to establish V(K) = V(K′) (Q(K) = Q(K′), etc.).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 22 / 28

Applications?

For a class of algebras K, we often seek a “distinguished subclass” K′ ⊆ K such that for all equations (quasiequations, etc.) ϕ, K′ | = ϕ ⇔ K | = ϕ. For example: Boolean algebras and the two-element Boolean algebra modal algebras and perfect modal algebras Gödel algebras and dense Gödel chains lattice-ordered groups and automorphisms of R. Algebraically, we want to establish V(K) = V(K′) (Q(K) = Q(K′), etc.).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 22 / 28

Applications?

For a class of algebras K, we often seek a “distinguished subclass” K′ ⊆ K such that for all equations (quasiequations, etc.) ϕ, K′ | = ϕ ⇔ K | = ϕ. For example: Boolean algebras and the two-element Boolean algebra modal algebras and perfect modal algebras Gödel algebras and dense Gödel chains lattice-ordered groups and automorphisms of R. Algebraically, we want to establish V(K) = V(K′) (Q(K) = Q(K′), etc.).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 22 / 28

Applications?

For a class of algebras K, we often seek a “distinguished subclass” K′ ⊆ K such that for all equations (quasiequations, etc.) ϕ, K′ | = ϕ ⇔ K | = ϕ. For example: Boolean algebras and the two-element Boolean algebra modal algebras and perfect modal algebras Gödel algebras and dense Gödel chains lattice-ordered groups and automorphisms of R. Algebraically, we want to establish V(K) = V(K′) (Q(K) = Q(K′), etc.).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 22 / 28

Applications?

For a class of algebras K, we often seek a “distinguished subclass” K′ ⊆ K such that for all equations (quasiequations, etc.) ϕ, K′ | = ϕ ⇔ K | = ϕ. For example: Boolean algebras and the two-element Boolean algebra modal algebras and perfect modal algebras Gödel algebras and dense Gödel chains lattice-ordered groups and automorphisms of R. Algebraically, we want to establish V(K) = V(K′) (Q(K) = Q(K′), etc.).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 22 / 28

Applications?

For a class of algebras K, we often seek a “distinguished subclass” K′ ⊆ K such that for all equations (quasiequations, etc.) ϕ, K′ | = ϕ ⇔ K | = ϕ. For example: Boolean algebras and the two-element Boolean algebra modal algebras and perfect modal algebras Gödel algebras and dense Gödel chains lattice-ordered groups and automorphisms of R. Algebraically, we want to establish V(K) = V(K′) (Q(K) = Q(K′), etc.).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 22 / 28

The Idea

• Question. How can we prove that ϕ preserves Σ in K?

An Answer. (a) Give a proof system that checks for a given ψ ∈ Σ whether Th(K) ∪ {ϕ} | = ψ. (b) Show that “applications of ϕ” can be eliminated from proofs.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 23 / 28

The Idea

• Question. How can we prove that ϕ preserves Σ in K?

An Answer. (a) Give a proof system that checks for a given ψ ∈ Σ whether Th(K) ∪ {ϕ} | = ψ. (b) Show that “applications of ϕ” can be eliminated from proofs.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 23 / 28

The Idea

• Question. How can we prove that ϕ preserves Σ in K?

An Answer. (a) Give a proof system that checks for a given ψ ∈ Σ whether Th(K) ∪ {ϕ} | = ψ. (b) Show that “applications of ϕ” can be eliminated from proofs.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 23 / 28

The Idea

• Question. How can we prove that ϕ preserves Σ in K?

An Answer. (a) Give a proof system that checks for a given ψ ∈ Σ whether Th(K) ∪ {ϕ} | = ψ. (b) Show that “applications of ϕ” can be eliminated from proofs.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 23 / 28

A Proof System GLat for Lattices

Axioms Cut rule s ≤ s

(ID)

s ≤ u u ≤ t s ≤ t

(CUT)

Left rules Right rules si ≤ t s1 ∧ s2 ≤ t

(∧⇒)i (i =1,2)

t ≤ si t ≤ s1 ∨ s2

(⇒∨)i (i =1,2)

s1 ≤ t s2 ≤ t s1 ∨ s2 ≤ t

(∨⇒)

t ≤ s1 t ≤ s2 t ≤ s1 ∧ s2

(⇒∧)

Theorem

(a) ⊢GLat s ≤ t ⇔ Lat | = s ≤ t. (b) GLat admits cut-elimination.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 24 / 28

A Proof System GLat for Lattices

Axioms Cut rule s ≤ s

(ID)

s ≤ u u ≤ t s ≤ t

(CUT)

Left rules Right rules si ≤ t s1 ∧ s2 ≤ t

(∧⇒)i (i =1,2)

t ≤ si t ≤ s1 ∨ s2

(⇒∨)i (i =1,2)

s1 ≤ t s2 ≤ t s1 ∨ s2 ≤ t

(∨⇒)

t ≤ s1 t ≤ s2 t ≤ s1 ∧ s2

(⇒∧)

Theorem

(a) ⊢GLat s ≤ t ⇔ Lat | = s ≤ t. (b) GLat admits cut-elimination.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 24 / 28

A Proof System GLat for Lattices

Axioms Cut rule s ≤ s

(ID)

s ≤ u u ≤ t s ≤ t

(CUT)

Left rules Right rules si ≤ t s1 ∧ s2 ≤ t

(∧⇒)i (i =1,2)

t ≤ si t ≤ s1 ∨ s2

(⇒∨)i (i =1,2)

s1 ≤ t s2 ≤ t s1 ∨ s2 ≤ t

(∨⇒)

t ≤ s1 t ≤ s2 t ≤ s1 ∧ s2

(⇒∧)

Theorem

(a) ⊢GLat s ≤ t ⇔ Lat | = s ≤ t. (b) GLat admits cut-elimination.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 24 / 28

A Proof System GLat for Lattices

Axioms Cut rule s ≤ s

(ID)

s ≤ u u ≤ t s ≤ t

(CUT)

Left rules Right rules si ≤ t s1 ∧ s2 ≤ t

(∧⇒)i (i =1,2)

t ≤ si t ≤ s1 ∨ s2

(⇒∨)i (i =1,2)

s1 ≤ t s2 ≤ t s1 ∨ s2 ≤ t

(∨⇒)

t ≤ s1 t ≤ s2 t ≤ s1 ∧ s2

(⇒∧)

Theorem

(a) ⊢GLat s ≤ t ⇔ Lat | = s ≤ t. (b) GLat admits cut-elimination.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 24 / 28

Example: Boundedness in Lattices

The following LLat-sentence expresses boundedness: ϕBD = (∃x)(∃y)(∀z)((x ≤ z) ⊓ (z ≤ y)). Skolemizing, we obtain (∀z)((⊥ ≤ z) ⊓ (z ≤ ⊤)). We consider GLat extended with the rules: ⊥ ≤ t

(⊥⇒)

and s ≤ ⊤

(⇒⊤).

Theorem

(a) ϕBD preserves LLat-equations in Lat. (b) Lat = V({A ∈ Lat : A is bounded}).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 25 / 28

Example: Boundedness in Lattices

The following LLat-sentence expresses boundedness: ϕBD = (∃x)(∃y)(∀z)((x ≤ z) ⊓ (z ≤ y)). Skolemizing, we obtain (∀z)((⊥ ≤ z) ⊓ (z ≤ ⊤)). We consider GLat extended with the rules: ⊥ ≤ t

(⊥⇒)

and s ≤ ⊤

(⇒⊤).

Theorem

(a) ϕBD preserves LLat-equations in Lat. (b) Lat = V({A ∈ Lat : A is bounded}).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 25 / 28

Example: Boundedness in Lattices

The following LLat-sentence expresses boundedness: ϕBD = (∃x)(∃y)(∀z)((x ≤ z) ⊓ (z ≤ y)). Skolemizing, we obtain (∀z)((⊥ ≤ z) ⊓ (z ≤ ⊤)). We consider GLat extended with the rules: ⊥ ≤ t

(⊥⇒)

and s ≤ ⊤

(⇒⊤).

Theorem

(a) ϕBD preserves LLat-equations in Lat. (b) Lat = V({A ∈ Lat : A is bounded}).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 25 / 28

Example: Boundedness in Lattices

The following LLat-sentence expresses boundedness: ϕBD = (∃x)(∃y)(∀z)((x ≤ z) ⊓ (z ≤ y)). Skolemizing, we obtain (∀z)((⊥ ≤ z) ⊓ (z ≤ ⊤)). We consider GLat extended with the rules: ⊥ ≤ t

(⊥⇒)

and s ≤ ⊤

(⇒⊤).

Theorem

(a) ϕBD preserves LLat-equations in Lat. (b) Lat = V({A ∈ Lat : A is bounded}).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 25 / 28

Example: Boundedness in Lattices

The following LLat-sentence expresses boundedness: ϕBD = (∃x)(∃y)(∀z)((x ≤ z) ⊓ (z ≤ y)). Skolemizing, we obtain (∀z)((⊥ ≤ z) ⊓ (z ≤ ⊤)). We consider GLat extended with the rules: ⊥ ≤ t

(⊥⇒)

and s ≤ ⊤

(⇒⊤).

Theorem

(a) ϕBD preserves LLat-equations in Lat. (b) Lat = V({A ∈ Lat : A is bounded}).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 25 / 28

Example: Unboundedness in Lattices

The following LLat-sentence expresses unboundedness: ϕUNBD = (∀x)(∃y)(∃z)(¬(x ≤ y) ⊓ ¬(z ≤ x)). Skolemizing, we obtain (∀x)(¬(x ≤↓x) ⊓ ¬(↑x ≤ x)). We consider GLat extended with the rules: u ≤ ↓u s ≤ t

(≤↓)

and ↑u ≤ u s ≤ t

(↑≤).

Theorem

(a) ϕUNBD preserves LLat-equations in Lat. (b) Lat = V({A ∈ Lat : A is unbounded}).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 26 / 28

Example: Unboundedness in Lattices

The following LLat-sentence expresses unboundedness: ϕUNBD = (∀x)(∃y)(∃z)(¬(x ≤ y) ⊓ ¬(z ≤ x)). Skolemizing, we obtain (∀x)(¬(x ≤↓x) ⊓ ¬(↑x ≤ x)). We consider GLat extended with the rules: u ≤ ↓u s ≤ t

(≤↓)

and ↑u ≤ u s ≤ t

(↑≤).

Theorem

(a) ϕUNBD preserves LLat-equations in Lat. (b) Lat = V({A ∈ Lat : A is unbounded}).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 26 / 28

Example: Unboundedness in Lattices

The following LLat-sentence expresses unboundedness: ϕUNBD = (∀x)(∃y)(∃z)(¬(x ≤ y) ⊓ ¬(z ≤ x)). Skolemizing, we obtain (∀x)(¬(x ≤↓x) ⊓ ¬(↑x ≤ x)). We consider GLat extended with the rules: u ≤ ↓u s ≤ t

(≤↓)

and ↑u ≤ u s ≤ t

(↑≤).

Theorem

(a) ϕUNBD preserves LLat-equations in Lat. (b) Lat = V({A ∈ Lat : A is unbounded}).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 26 / 28

Example: Unboundedness in Lattices

The following LLat-sentence expresses unboundedness: ϕUNBD = (∀x)(∃y)(∃z)(¬(x ≤ y) ⊓ ¬(z ≤ x)). Skolemizing, we obtain (∀x)(¬(x ≤↓x) ⊓ ¬(↑x ≤ x)). We consider GLat extended with the rules: u ≤ ↓u s ≤ t

(≤↓)

and ↑u ≤ u s ≤ t

(↑≤).

Theorem

(a) ϕUNBD preserves LLat-equations in Lat. (b) Lat = V({A ∈ Lat : A is unbounded}).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 26 / 28

Example: Unboundedness in Lattices

The following LLat-sentence expresses unboundedness: ϕUNBD = (∀x)(∃y)(∃z)(¬(x ≤ y) ⊓ ¬(z ≤ x)). Skolemizing, we obtain (∀x)(¬(x ≤↓x) ⊓ ¬(↑x ≤ x)). We consider GLat extended with the rules: u ≤ ↓u s ≤ t

(≤↓)

and ↑u ≤ u s ≤ t

(↑≤).

Theorem

(a) ϕUNBD preserves LLat-equations in Lat. (b) Lat = V({A ∈ Lat : A is unbounded}).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 26 / 28

Linearity and Density in Gödel Algebras

The following LLat-sentence expresses linearity and density:

ϕDC = (∀x)(∀y)(∃z)(((x ≤ y) ⊔ (y ≤ x)) ⊓ (((x ≤ z) ⊔ (z ≤ y)) ⇒ (x ≤ y))).

Skolemizing, we obtain:

(∀x)(∀y)(((x ≤ y) ⊔ (y ≤ x)) ⊓ (((x ≤ d(x, y)) ⊔ (d(x, y) ≤ y)) ⇒ (x ≤ y))).

Theorem

(a) ϕDC preserves L-equations in the variety G of Gödel algebras. (b) G = V({A ∈ G : A is linearly and densely ordered}).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 27 / 28

Linearity and Density in Gödel Algebras

The following LLat-sentence expresses linearity and density:

ϕDC = (∀x)(∀y)(∃z)(((x ≤ y) ⊔ (y ≤ x)) ⊓ (((x ≤ z) ⊔ (z ≤ y)) ⇒ (x ≤ y))).

Skolemizing, we obtain:

(∀x)(∀y)(((x ≤ y) ⊔ (y ≤ x)) ⊓ (((x ≤ d(x, y)) ⊔ (d(x, y) ≤ y)) ⇒ (x ≤ y))).

Theorem

(a) ϕDC preserves L-equations in the variety G of Gödel algebras. (b) G = V({A ∈ G : A is linearly and densely ordered}).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 27 / 28

Linearity and Density in Gödel Algebras

The following LLat-sentence expresses linearity and density:

ϕDC = (∀x)(∀y)(∃z)(((x ≤ y) ⊔ (y ≤ x)) ⊓ (((x ≤ z) ⊔ (z ≤ y)) ⇒ (x ≤ y))).

Skolemizing, we obtain:

(∀x)(∀y)(((x ≤ y) ⊔ (y ≤ x)) ⊓ (((x ≤ d(x, y)) ⊔ (d(x, y) ≤ y)) ⇒ (x ≤ y))).

Theorem

(a) ϕDC preserves L-equations in the variety G of Gödel algebras. (b) G = V({A ∈ G : A is linearly and densely ordered}).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 27 / 28

Linearity and Density in Gödel Algebras

The following LLat-sentence expresses linearity and density:

ϕDC = (∀x)(∀y)(∃z)(((x ≤ y) ⊔ (y ≤ x)) ⊓ (((x ≤ z) ⊔ (z ≤ y)) ⇒ (x ≤ y))).

Skolemizing, we obtain:

(∀x)(∀y)(((x ≤ y) ⊔ (y ≤ x)) ⊓ (((x ≤ d(x, y)) ⊔ (d(x, y) ≤ y)) ⇒ (x ≤ y))).

Theorem

(a) ϕDC preserves L-equations in the variety G of Gödel algebras. (b) G = V({A ∈ G : A is linearly and densely ordered}).

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 27 / 28

Concluding Remarks

Differing notions of admissibility can be presented and compared in a first-order framework. Establishing the admissibility of a rule (e.g., by elimination) can be used to determine properties of classes of algebras.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 28 / 28

Concluding Remarks

Differing notions of admissibility can be presented and compared in a first-order framework. Establishing the admissibility of a rule (e.g., by elimination) can be used to determine properties of classes of algebras.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 28 / 28

Concluding Remarks

Differing notions of admissibility can be presented and compared in a first-order framework. Establishing the admissibility of a rule (e.g., by elimination) can be used to determine properties of classes of algebras.

George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 28 / 28