Bridges Between Logic and Algebra Part 1: Intuitionistic Logic

Home , Heyting algebra

Bridges between Logic and Algebra Part 1: Intuitionistic Logic

George Metcalfe

Mathematical Institute University of Bern

TACL 2019 Summer School, Île de Porquerolles, June 2019

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 1 / 32 variables of α variables of β

`L γ(y)

A Problem in Logic

Does some logic L admit interpolation?

α(x, y) `L β(y, z)

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 2 / 32 `L γ(y)

A Problem in Logic

Does some logic L admit interpolation?

variables of α variables of β

α(x, y) `L β(y, z)

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 2 / 32 A Problem in Logic

Does some logic L admit interpolation?

variables of α variables of β

α(x, y) `L γ(y) `L β(y, z)

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 2 / 32 A Problem in Algebra

Does some class of algebras K have the amalgamation property?

B C

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 3 / 32 A Problem in Algebra

Does some class of algebras K have the amalgamation property?

B C

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 3 / 32 A Problem in Algebra

Does some class of algebras K have the amalgamation property?

B C

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 3 / 32 A Problem in Algebra

Does some class of algebras K have the amalgamation property?

B C

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 3 / 32 A Bridge Theorem

L admits interpolation ⇐⇒K L has the amalgamation property

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 4 / 32 This Tutorial

How can we build and cross bridges between logic and algebra?

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 5 / 32 Today

How can we do this for intuitionistic logic and Heyting algebras?

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 6 / 32 The Brouwer-Heyting-Kolmogorov interpretation presents the validity of formulas in intuitionistic logic in terms of the construction of proofs, e.g.,

“A proof of α ∨ β is given via a proof of α or a proof of β.”

Intuitionistic logic may be presented syntactically via axiom systems, natural deduction, tableau or sequent calculi, etc. or semantically via Heyting algebras, Kripke models, topological semantics, etc.

Intuitionistic Logic

Intuitionistic logic was introduced by Heyting in the 1930s to formalize certain principles used in Brouwer’s constructive mathematics.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 7 / 32 e.g.,

“A proof of α ∨ β is given via a proof of α or a proof of β.”

Intuitionistic Logic

Intuitionistic logic was introduced by Heyting in the 1930s to formalize certain principles used in Brouwer’s constructive mathematics. The Brouwer-Heyting-Kolmogorov interpretation presents the validity of formulas in intuitionistic logic in terms of the construction of proofs,

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 7 / 32 Intuitionistic logic may be presented syntactically via axiom systems, natural deduction, tableau or sequent calculi, etc. or semantically via Heyting algebras, Kripke models, topological semantics, etc.

Intuitionistic Logic

“A proof of α ∨ β is given via a proof of α or a proof of β.”

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 7 / 32 or semantically via Heyting algebras, Kripke models, topological semantics, etc.

Intuitionistic Logic

“A proof of α ∨ β is given via a proof of α or a proof of β.”

Intuitionistic logic may be presented syntactically via axiom systems, natural deduction, tableau or sequent calculi, etc.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 7 / 32 Intuitionistic Logic

“A proof of α ∨ β is given via a proof of α or a proof of β.”

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 7 / 32 We write T `IL α to denote that a formula α is derivable from a set of formulas T using the axiom schema

α → (β → α)(α → (β → γ)) → ((α → β) → (α → γ)) (α ∧ β) → α (α ∧ β) → β α → (α ∨ β) β → (α ∨ β) α → (β → (α ∧ β)) (α → γ) → ((β → γ) → ((α ∨ β) → γ)) α → > ⊥ → α together with the modus ponens rule: from α and α → β, infer β.

An Axiom System

Formulas α, β, γ . . . are deﬁned inductively for a propositional language with binary connectives ∧, ∨, → and constants ⊥, > over a countably inﬁnite set of variables x, y, z ..., where α ↔ β := (α → β) ∧ (β → α).

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 8 / 32 the axiom schema

An Axiom System

We write T `IL α to denote that a formula α is derivable from a set of formulas T using

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 8 / 32 An Axiom System

We write T `IL α to denote that a formula α is derivable from a set of formulas T using the axiom schema

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 8 / 32 that is, for any set of formulas T ∪ T 0 ∪ {α},

(i) if α ∈ T , then T ÌL α (reflexivity); 0 0 (ii) if T ÌL α and T ⊆ T , then T ÌL α (monotonicity); 0 0 (iii) if T ÌL α and T ÌL β for every β ∈ T , then T ÌL α (transitivity);

(iv) if T ÌL α, then σ[T ] ÌL σ(α) for any substitution σ (structurality); 0 0 (v) if T ÌL α, then T ÌL α for some finite T ⊆ T (finitarity).

Consequence

It is easy to check that `IL is a ﬁnitary structural consequence relation;

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 9 / 32 0 0 (ii) if T ÌL α and T ⊆ T , then T ÌL α (monotonicity); 0 0 (iii) if T ÌL α and T ÌL β for every β ∈ T , then T ÌL α (transitivity);

(iv) if T ÌL α, then σ[T ] ÌL σ(α) for any substitution σ (structurality); 0 0 (v) if T ÌL α, then T ÌL α for some finite T ⊆ T (finitarity).

Consequence

It is easy to check that `IL is a ﬁnitary structural consequence relation; that is, for any set of formulas T ∪ T 0 ∪ {α},

(i) if α ∈ T , then T `IL α (reﬂexivity);

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 9 / 32 0 0 (iii) if T ÌL α and T ÌL β for every β ∈ T , then T ÌL α (transitivity);

(iv) if T ÌL α, then σ[T ] ÌL σ(α) for any substitution σ (structurality); 0 0 (v) if T ÌL α, then T ÌL α for some finite T ⊆ T (finitarity).

Consequence

It is easy to check that `IL is a ﬁnitary structural consequence relation; that is, for any set of formulas T ∪ T 0 ∪ {α},

(i) if α ∈ T , then T ÌL α (reflexivity); 0 0 (ii) if T ÌL α and T ⊆ T , then T ÌL α (monotonicity);

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 9 / 32 (iv) if T ÌL α, then σ[T ] ÌL σ(α) for any substitution σ (structurality); 0 0 (v) if T ÌL α, then T ÌL α for some finite T ⊆ T (finitarity).

Consequence

It is easy to check that `IL is a ﬁnitary structural consequence relation; that is, for any set of formulas T ∪ T 0 ∪ {α},

(i) if α ∈ T , then T ÌL α (reflexivity); 0 0 (ii) if T ÌL α and T ⊆ T , then T ÌL α (monotonicity); 0 0 (iii) if T ÌL α and T ÌL β for every β ∈ T , then T ÌL α (transitivity);

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 9 / 32 0 0 (v) if T ÌL α, then T ÌL α for some finite T ⊆ T (finitarity).

Consequence

It is easy to check that `IL is a ﬁnitary structural consequence relation; that is, for any set of formulas T ∪ T 0 ∪ {α},

(i) if α ∈ T , then T ÌL α (reflexivity); 0 0 (ii) if T ÌL α and T ⊆ T , then T ÌL α (monotonicity); 0 0 (iii) if T ÌL α and T ÌL β for every β ∈ T , then T ÌL α (transitivity);

(iv) if T `IL α, then σ[T ] `IL σ(α) for any substitution σ (structurality);

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 9 / 32 Consequence

It is easy to check that `IL is a ﬁnitary structural consequence relation; that is, for any set of formulas T ∪ T 0 ∪ {α},

(i) if α ∈ T , then T ÌL α (reflexivity); 0 0 (ii) if T ÌL α and T ⊆ T , then T ÌL α (monotonicity); 0 0 (iii) if T ÌL α and T ÌL β for every β ∈ T , then T ÌL α (transitivity);

(iv) if T ÌL α, then σ[T ] ÌL σ(α) for any substitution σ (structurality); 0 0 (v) if T ÌL α, then T ÌL α for some finite T ⊆ T (finitarity).

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 9 / 32 Proof.

(⇒) Suppose that T `IL α → β. By monotonicity, T ∪ {α} `IL α → β and,

by reflexivity, T ∪ {α} ÌL α. So, by modus ponens, T ∪ {α} ÌL β. (⇐) By induction on the length of a derivation of β from T ∪ {α} in IL.

For β = α, note that T `IL α → α. If β is in T or an axiom, then T `IL β

and, since T `IL β → (α → β), also T `IL α → β. For the induction step,

suppose that T ∪ {α} `IL γ and T ∪ {α} `IL γ → β. By the induction

hypothesis, T `IL α → γ and T `IL α → (γ → β). Since also

T `IL (α → (γ → β)) → ((α → γ) → (α → β)), we get T `IL α → β.

A Deduction Theorem

Theorem For any set of formulas T ∪ {α, β},

T `IL α → β ⇐⇒ T ∪ {α} `IL β.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 10 / 32 By monotonicity, T ∪ {α} `IL α → β and,

by reflexivity, T ∪ {α} ÌL α. So, by modus ponens, T ∪ {α} ÌL β. (⇐) By induction on the length of a derivation of β from T ∪ {α} in IL.

For β = α, note that T `IL α → α. If β is in T or an axiom, then T `IL β

and, since T `IL β → (α → β), also T `IL α → β. For the induction step,

suppose that T ∪ {α} `IL γ and T ∪ {α} `IL γ → β. By the induction

hypothesis, T `IL α → γ and T `IL α → (γ → β). Since also

T `IL (α → (γ → β)) → ((α → γ) → (α → β)), we get T `IL α → β.

A Deduction Theorem

Theorem For any set of formulas T ∪ {α, β},

T `IL α → β ⇐⇒ T ∪ {α} `IL β.

Proof.

(⇒) Suppose that T `IL α → β.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 10 / 32 and,

by reflexivity, T ∪ {α} ÌL α. So, by modus ponens, T ∪ {α} ÌL β. (⇐) By induction on the length of a derivation of β from T ∪ {α} in IL.

For β = α, note that T `IL α → α. If β is in T or an axiom, then T `IL β

and, since T `IL β → (α → β), also T `IL α → β. For the induction step,

suppose that T ∪ {α} `IL γ and T ∪ {α} `IL γ → β. By the induction

hypothesis, T `IL α → γ and T `IL α → (γ → β). Since also

T `IL (α → (γ → β)) → ((α → γ) → (α → β)), we get T `IL α → β.

A Deduction Theorem

Theorem For any set of formulas T ∪ {α, β},

T `IL α → β ⇐⇒ T ∪ {α} `IL β.

Proof.

(⇒) Suppose that T `IL α → β. By monotonicity, T ∪ {α} `IL α → β

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 10 / 32 So, by modus ponens, T ∪ {α} `IL β. (⇐) By induction on the length of a derivation of β from T ∪ {α} in IL.

For β = α, note that T `IL α → α. If β is in T or an axiom, then T `IL β

and, since T `IL β → (α → β), also T `IL α → β. For the induction step,

suppose that T ∪ {α} `IL γ and T ∪ {α} `IL γ → β. By the induction

hypothesis, T `IL α → γ and T `IL α → (γ → β). Since also

T `IL (α → (γ → β)) → ((α → γ) → (α → β)), we get T `IL α → β.

A Deduction Theorem

Theorem For any set of formulas T ∪ {α, β},

T `IL α → β ⇐⇒ T ∪ {α} `IL β.

Proof.

(⇒) Suppose that T `IL α → β. By monotonicity, T ∪ {α} `IL α → β and,

by reﬂexivity, T ∪ {α} `IL α.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 10 / 32 (⇐) By induction on the length of a derivation of β from T ∪ {α} in IL.

For β = α, note that T `IL α → α. If β is in T or an axiom, then T `IL β

and, since T `IL β → (α → β), also T `IL α → β. For the induction step,

suppose that T ∪ {α} `IL γ and T ∪ {α} `IL γ → β. By the induction

hypothesis, T `IL α → γ and T `IL α → (γ → β). Since also

T `IL (α → (γ → β)) → ((α → γ) → (α → β)), we get T `IL α → β.

A Deduction Theorem

Theorem For any set of formulas T ∪ {α, β},

T `IL α → β ⇐⇒ T ∪ {α} `IL β.

Proof.

(⇒) Suppose that T `IL α → β. By monotonicity, T ∪ {α} `IL α → β and,

by reflexivity, T ∪ {α} ÌL α. So, by modus ponens, T ∪ {α} ÌL β.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 10 / 32 For β = α, note that T `IL α → α. If β is in T or an axiom, then T `IL β

and, since T `IL β → (α → β), also T `IL α → β. For the induction step,

suppose that T ∪ {α} `IL γ and T ∪ {α} `IL γ → β. By the induction

hypothesis, T `IL α → γ and T `IL α → (γ → β). Since also

T `IL (α → (γ → β)) → ((α → γ) → (α → β)), we get T `IL α → β.

A Deduction Theorem

Theorem For any set of formulas T ∪ {α, β},

T `IL α → β ⇐⇒ T ∪ {α} `IL β.

Proof.

(⇒) Suppose that T `IL α → β. By monotonicity, T ∪ {α} `IL α → β and,

by reflexivity, T ∪ {α} ÌL α. So, by modus ponens, T ∪ {α} ÌL β. (⇐) By induction on the length of a derivation of β from T ∪ {α} in IL.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 10 / 32 If β is in T or an axiom, then T `IL β

and, since T `IL β → (α → β), also T `IL α → β. For the induction step,

suppose that T ∪ {α} `IL γ and T ∪ {α} `IL γ → β. By the induction

hypothesis, T `IL α → γ and T `IL α → (γ → β). Since also

T `IL (α → (γ → β)) → ((α → γ) → (α → β)), we get T `IL α → β.

A Deduction Theorem

Theorem For any set of formulas T ∪ {α, β},

T `IL α → β ⇐⇒ T ∪ {α} `IL β.

Proof.

(⇒) Suppose that T `IL α → β. By monotonicity, T ∪ {α} `IL α → β and,

by reflexivity, T ∪ {α} ÌL α. So, by modus ponens, T ∪ {α} ÌL β. (⇐) By induction on the length of a derivation of β from T ∪ {α} in IL.

For β = α, note that T `IL α → α.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 10 / 32 and, since T `IL β → (α → β), also T `IL α → β. For the induction step,

suppose that T ∪ {α} `IL γ and T ∪ {α} `IL γ → β. By the induction

hypothesis, T `IL α → γ and T `IL α → (γ → β). Since also

T `IL (α → (γ → β)) → ((α → γ) → (α → β)), we get T `IL α → β.

A Deduction Theorem

Theorem For any set of formulas T ∪ {α, β},

T `IL α → β ⇐⇒ T ∪ {α} `IL β.

Proof.

(⇒) Suppose that T `IL α → β. By monotonicity, T ∪ {α} `IL α → β and,

by reflexivity, T ∪ {α} ÌL α. So, by modus ponens, T ∪ {α} ÌL β. (⇐) By induction on the length of a derivation of β from T ∪ {α} in IL.

For β = α, note that T `IL α → α. If β is in T or an axiom, then T `IL β

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 10 / 32 For the induction step,

suppose that T ∪ {α} `IL γ and T ∪ {α} `IL γ → β. By the induction

hypothesis, T `IL α → γ and T `IL α → (γ → β). Since also

T `IL (α → (γ → β)) → ((α → γ) → (α → β)), we get T `IL α → β.

A Deduction Theorem

Theorem For any set of formulas T ∪ {α, β},

T `IL α → β ⇐⇒ T ∪ {α} `IL β.

Proof.

(⇒) Suppose that T `IL α → β. By monotonicity, T ∪ {α} `IL α → β and,

by reflexivity, T ∪ {α} ÌL α. So, by modus ponens, T ∪ {α} ÌL β. (⇐) By induction on the length of a derivation of β from T ∪ {α} in IL.

For β = α, note that T `IL α → α. If β is in T or an axiom, then T `IL β

and, since T `IL β → (α → β), also T `IL α → β.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 10 / 32 By the induction

hypothesis, T `IL α → γ and T `IL α → (γ → β). Since also

T `IL (α → (γ → β)) → ((α → γ) → (α → β)), we get T `IL α → β.

A Deduction Theorem

Theorem For any set of formulas T ∪ {α, β},

T `IL α → β ⇐⇒ T ∪ {α} `IL β.

Proof.

(⇒) Suppose that T `IL α → β. By monotonicity, T ∪ {α} `IL α → β and,

by reflexivity, T ∪ {α} ÌL α. So, by modus ponens, T ∪ {α} ÌL β. (⇐) By induction on the length of a derivation of β from T ∪ {α} in IL.

For β = α, note that T `IL α → α. If β is in T or an axiom, then T `IL β

and, since T `IL β → (α → β), also T `IL α → β. For the induction step,

suppose that T ∪ {α} `IL γ and T ∪ {α} `IL γ → β.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 10 / 32 Since also

T `IL (α → (γ → β)) → ((α → γ) → (α → β)), we get T `IL α → β.

A Deduction Theorem

Theorem For any set of formulas T ∪ {α, β},

T `IL α → β ⇐⇒ T ∪ {α} `IL β.

Proof.

(⇒) Suppose that T `IL α → β. By monotonicity, T ∪ {α} `IL α → β and,

by reflexivity, T ∪ {α} ÌL α. So, by modus ponens, T ∪ {α} ÌL β. (⇐) By induction on the length of a derivation of β from T ∪ {α} in IL.

For β = α, note that T `IL α → α. If β is in T or an axiom, then T `IL β

and, since T `IL β → (α → β), also T `IL α → β. For the induction step,

suppose that T ∪ {α} `IL γ and T ∪ {α} `IL γ → β. By the induction

hypothesis, T `IL α → γ and T `IL α → (γ → β).

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 10 / 32 A Deduction Theorem

Theorem For any set of formulas T ∪ {α, β},

T `IL α → β ⇐⇒ T ∪ {α} `IL β.

Proof.

(⇒) Suppose that T `IL α → β. By monotonicity, T ∪ {α} `IL α → β and,

by reflexivity, T ∪ {α} ÌL α. So, by modus ponens, T ∪ {α} ÌL β. (⇐) By induction on the length of a derivation of β from T ∪ {α} in IL.

For β = α, note that T `IL α → α. If β is in T or an axiom, then T `IL β

and, since T `IL β → (α → β), also T `IL α → β. For the induction step,

suppose that T ∪ {α} `IL γ and T ∪ {α} `IL γ → β. By the induction

hypothesis, T `IL α → γ and T `IL α → (γ → β). Since also

T `IL (α → (γ → β)) → ((α → γ) → (α → β)), we get T `IL α → β.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 10 / 32 (i) hA, ∧, ∨, ⊥, >i is a bounded lattice with a ≤ b :⇐⇒ a ∧ b = a; (ii) a ≤ b → c ⇐⇒ a ∧ b ≤ c for all a, b, c ∈ A.

The class HA of Heyting algebras forms a variety.

Examples: 1. any Boolean algebra; 2. letting U be the set of upsets of a poset hX , ≤i, hU, ∩, ∪, →, ∅, X i where Y → Z = {a ∈ X | a ≤ b ∈ Y =⇒ b ∈ Z};

3. letting O be the set of open subsets of R with the usual topology, c hO, ∩, ∪, →, ∅, Ri where Y → Z = int(Y ∪ Z).

Heyting Algebras

A Heyting algebra is an algebraic structure hA, ∧, ∨, →, ⊥, >i such that

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 11 / 32 (ii) a ≤ b → c ⇐⇒ a ∧ b ≤ c for all a, b, c ∈ A.

The class HA of Heyting algebras forms a variety.

Examples: 1. any Boolean algebra; 2. letting U be the set of upsets of a poset hX , ≤i, hU, ∩, ∪, →, ∅, X i where Y → Z = {a ∈ X | a ≤ b ∈ Y =⇒ b ∈ Z};

3. letting O be the set of open subsets of R with the usual topology, c hO, ∩, ∪, →, ∅, Ri where Y → Z = int(Y ∪ Z).

Heyting Algebras

A Heyting algebra is an algebraic structure hA, ∧, ∨, →, ⊥, >i such that

(i) hA, ∧, ∨, ⊥, >i is a bounded lattice with a ≤ b :⇐⇒ a ∧ b = a;

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 11 / 32 The class HA of Heyting algebras forms a variety.

Examples: 1. any Boolean algebra; 2. letting U be the set of upsets of a poset hX , ≤i, hU, ∩, ∪, →, ∅, X i where Y → Z = {a ∈ X | a ≤ b ∈ Y =⇒ b ∈ Z};

3. letting O be the set of open subsets of R with the usual topology, c hO, ∩, ∪, →, ∅, Ri where Y → Z = int(Y ∪ Z).

Heyting Algebras

A Heyting algebra is an algebraic structure hA, ∧, ∨, →, ⊥, >i such that

(i) hA, ∧, ∨, ⊥, >i is a bounded lattice with a ≤ b :⇐⇒ a ∧ b = a; (ii) a ≤ b → c ⇐⇒ a ∧ b ≤ c for all a, b, c ∈ A.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 11 / 32 Examples: 1. any Boolean algebra; 2. letting U be the set of upsets of a poset hX , ≤i, hU, ∩, ∪, →, ∅, X i where Y → Z = {a ∈ X | a ≤ b ∈ Y =⇒ b ∈ Z};

3. letting O be the set of open subsets of R with the usual topology, c hO, ∩, ∪, →, ∅, Ri where Y → Z = int(Y ∪ Z).

Heyting Algebras

A Heyting algebra is an algebraic structure hA, ∧, ∨, →, ⊥, >i such that

(i) hA, ∧, ∨, ⊥, >i is a bounded lattice with a ≤ b :⇐⇒ a ∧ b = a; (ii) a ≤ b → c ⇐⇒ a ∧ b ≤ c for all a, b, c ∈ A.

The class HA of Heyting algebras forms a variety.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 11 / 32 2. letting U be the set of upsets of a poset hX , ≤i, hU, ∩, ∪, →, ∅, X i where Y → Z = {a ∈ X | a ≤ b ∈ Y =⇒ b ∈ Z};

3. letting O be the set of open subsets of R with the usual topology, c hO, ∩, ∪, →, ∅, Ri where Y → Z = int(Y ∪ Z).

Heyting Algebras

A Heyting algebra is an algebraic structure hA, ∧, ∨, →, ⊥, >i such that

(i) hA, ∧, ∨, ⊥, >i is a bounded lattice with a ≤ b :⇐⇒ a ∧ b = a; (ii) a ≤ b → c ⇐⇒ a ∧ b ≤ c for all a, b, c ∈ A.

The class HA of Heyting algebras forms a variety.

Examples: 1. any Boolean algebra;

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 11 / 32 3. letting O be the set of open subsets of R with the usual topology, c hO, ∩, ∪, →, ∅, Ri where Y → Z = int(Y ∪ Z).

Heyting Algebras

A Heyting algebra is an algebraic structure hA, ∧, ∨, →, ⊥, >i such that

(i) hA, ∧, ∨, ⊥, >i is a bounded lattice with a ≤ b :⇐⇒ a ∧ b = a; (ii) a ≤ b → c ⇐⇒ a ∧ b ≤ c for all a, b, c ∈ A.

The class HA of Heyting algebras forms a variety.

Examples: 1. any Boolean algebra; 2. letting U be the set of upsets of a poset hX , ≤i, hU, ∩, ∪, →, ∅, X i where Y → Z = {a ∈ X | a ≤ b ∈ Y =⇒ b ∈ Z};

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 11 / 32 Heyting Algebras

A Heyting algebra is an algebraic structure hA, ∧, ∨, →, ⊥, >i such that

(i) hA, ∧, ∨, ⊥, >i is a bounded lattice with a ≤ b :⇐⇒ a ∧ b = a; (ii) a ≤ b → c ⇐⇒ a ∧ b ≤ c for all a, b, c ∈ A.

The class HA of Heyting algebras forms a variety.

Examples: 1. any Boolean algebra; 2. letting U be the set of upsets of a poset hX , ≤i, hU, ∩, ∪, →, ∅, X i where Y → Z = {a ∈ X | a ≤ b ∈ Y =⇒ b ∈ Z};

3. letting O be the set of open subsets of R with the usual topology, c hO, ∩, ∪, →, ∅, Ri where Y → Z = int(Y ∪ Z).

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 11 / 32 Then ΘT is an equivalence relation satisfying for ? ∈ {∧, ∨, →},

α1 ΘT β1 and α2 ΘT β2 =⇒ α1 ? α2 ΘT β1 ? β2,

and we obtain a Heyting algebra

AT = hAT , ∧T , ∨T , →T , [⊥]T , [>]T i

where AT is the set of ΘT -equivalence classes [α]T and for ? ∈ {∧, ∨, →},

[α]T ?T [β]T = [α ? β]T .

In particular, `IL α if and only if A∅ |= α ≈ >.

The Lindenbaum-Tarski Construction

Given any set of formulas T , deﬁne a binary relation on formulas by

α ΘT β :⇐⇒ T `IL α ↔ β.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 12 / 32 satisfying for ? ∈ {∧, ∨, →},

α1 ΘT β1 and α2 ΘT β2 =⇒ α1 ? α2 ΘT β1 ? β2,

and we obtain a Heyting algebra

AT = hAT , ∧T , ∨T , →T , [⊥]T , [>]T i

where AT is the set of ΘT -equivalence classes [α]T and for ? ∈ {∧, ∨, →},

[α]T ?T [β]T = [α ? β]T .

In particular, `IL α if and only if A∅ |= α ≈ >.

The Lindenbaum-Tarski Construction

Given any set of formulas T , deﬁne a binary relation on formulas by

α ΘT β :⇐⇒ T `IL α ↔ β.

Then ΘT is an equivalence relation

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 12 / 32 and we obtain a Heyting algebra

AT = hAT , ∧T , ∨T , →T , [⊥]T , [>]T i

where AT is the set of ΘT -equivalence classes [α]T and for ? ∈ {∧, ∨, →},

[α]T ?T [β]T = [α ? β]T .

In particular, `IL α if and only if A∅ |= α ≈ >.

The Lindenbaum-Tarski Construction

Given any set of formulas T , deﬁne a binary relation on formulas by

α ΘT β :⇐⇒ T `IL α ↔ β.

Then ΘT is an equivalence relation satisfying for ? ∈ {∧, ∨, →},

α1 ΘT β1 and α2 ΘT β2 =⇒ α1 ? α2 ΘT β1 ? β2,

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 12 / 32 and for ? ∈ {∧, ∨, →},

[α]T ?T [β]T = [α ? β]T .

In particular, `IL α if and only if A∅ |= α ≈ >.

The Lindenbaum-Tarski Construction

Given any set of formulas T , deﬁne a binary relation on formulas by

α ΘT β :⇐⇒ T `IL α ↔ β.

Then ΘT is an equivalence relation satisfying for ? ∈ {∧, ∨, →},

α1 ΘT β1 and α2 ΘT β2 =⇒ α1 ? α2 ΘT β1 ? β2,

and we obtain a Heyting algebra

AT = hAT , ∧T , ∨T , →T , [⊥]T , [>]T i

where AT is the set of ΘT -equivalence classes [α]T

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 12 / 32 In particular, `IL α if and only if A∅ |= α ≈ >.

The Lindenbaum-Tarski Construction

Given any set of formulas T , deﬁne a binary relation on formulas by

α ΘT β :⇐⇒ T `IL α ↔ β.

Then ΘT is an equivalence relation satisfying for ? ∈ {∧, ∨, →},

α1 ΘT β1 and α2 ΘT β2 =⇒ α1 ? α2 ΘT β1 ? β2,

and we obtain a Heyting algebra

AT = hAT , ∧T , ∨T , →T , [⊥]T , [>]T i

where AT is the set of ΘT -equivalence classes [α]T and for ? ∈ {∧, ∨, →},

[α]T ?T [β]T = [α ? β]T .

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 12 / 32 The Lindenbaum-Tarski Construction

Given any set of formulas T , deﬁne a binary relation on formulas by

α ΘT β :⇐⇒ T `IL α ↔ β.

Then ΘT is an equivalence relation satisfying for ? ∈ {∧, ∨, →},

α1 ΘT β1 and α2 ΘT β2 =⇒ α1 ? α2 ΘT β1 ? β2,

and we obtain a Heyting algebra

AT = hAT , ∧T , ∨T , →T , [⊥]T , [>]T i

where AT is the set of ΘT -equivalence classes [α]T and for ? ∈ {∧, ∨, →},

[α]T ?T [β]T = [α ? β]T .

In particular, `IL α if and only if A∅ |= α ≈ >.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 12 / 32 if for any homomorphism e from the formula algebra to a Heyting algebra,

e(γ) = e(δ) for all γ ≈ δ ∈ Σ =⇒ e(α) = e(β).

Note. |=HA is a ﬁnitary structural equational consequence relation.

Equational Consequence

For any set of equations Σ ∪ {α ≈ β}, we write

Σ |=HA α ≈ β

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 13 / 32 Note. |=HA is a ﬁnitary structural equational consequence relation.

Equational Consequence

For any set of equations Σ ∪ {α ≈ β}, we write

Σ |=HA α ≈ β

if for any homomorphism e from the formula algebra to a Heyting algebra,

e(γ) = e(δ) for all γ ≈ δ ∈ Σ =⇒ e(α) = e(β).

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 13 / 32 Equational Consequence

For any set of equations Σ ∪ {α ≈ β}, we write

Σ |=HA α ≈ β

if for any homomorphism e from the formula algebra to a Heyting algebra,

e(γ) = e(δ) for all γ ≈ δ ∈ Σ =⇒ e(α) = e(β).

Note. |=HA is a ﬁnitary structural equational consequence relation.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 13 / 32 with transformers τ(α) = α ≈ > and ρ(α ≈ β) = α ↔ β.

(i) For any set of formulas T ∪ {α},

T `IL α ⇐⇒ τ[T ] |=HA τ(α). (ii) For any set of equations Σ ∪ {α ≈ β},

Σ |=HA α ≈ β ⇐⇒ ρ[T ] `IL ρ(α ≈ β). (iii) For any formulas α, β,

α a`IL ρ(τ(α)) and α ≈ β =||=HA τ(ρ(α ≈ β)).

A First Bridge Theorem

Theorem HA is an equivalent algebraic semantics for IL

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 14 / 32 (i) For any set of formulas T ∪ {α},

T `IL α ⇐⇒ τ[T ] |=HA τ(α). (ii) For any set of equations Σ ∪ {α ≈ β},

Σ |=HA α ≈ β ⇐⇒ ρ[T ] `IL ρ(α ≈ β). (iii) For any formulas α, β,

α a`IL ρ(τ(α)) and α ≈ β =||=HA τ(ρ(α ≈ β)).

A First Bridge Theorem

Theorem HA is an equivalent algebraic semantics for IL with transformers τ(α) = α ≈ > and ρ(α ≈ β) = α ↔ β.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 14 / 32 (ii) For any set of equations Σ ∪ {α ≈ β},

Σ |=HA α ≈ β ⇐⇒ ρ[T ] `IL ρ(α ≈ β). (iii) For any formulas α, β,

α a`IL ρ(τ(α)) and α ≈ β =||=HA τ(ρ(α ≈ β)).

A First Bridge Theorem

Theorem HA is an equivalent algebraic semantics for IL with transformers τ(α) = α ≈ > and ρ(α ≈ β) = α ↔ β.

(i) For any set of formulas T ∪ {α},

T `IL α ⇐⇒ τ[T ] |=HA τ(α).

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 14 / 32 (iii) For any formulas α, β,

α a`IL ρ(τ(α)) and α ≈ β =||=HA τ(ρ(α ≈ β)).

A First Bridge Theorem

Theorem HA is an equivalent algebraic semantics for IL with transformers τ(α) = α ≈ > and ρ(α ≈ β) = α ↔ β.

(i) For any set of formulas T ∪ {α},

T `IL α ⇐⇒ τ[T ] |=HA τ(α). (ii) For any set of equations Σ ∪ {α ≈ β},

Σ |=HA α ≈ β ⇐⇒ ρ[T ] `IL ρ(α ≈ β).

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 14 / 32 A First Bridge Theorem

Theorem HA is an equivalent algebraic semantics for IL with transformers τ(α) = α ≈ > and ρ(α ≈ β) = α ↔ β.

(i) For any set of formulas T ∪ {α},

T `IL α ⇐⇒ τ[T ] |=HA τ(α). (ii) For any set of equations Σ ∪ {α ≈ β},

Σ |=HA α ≈ β ⇐⇒ ρ[T ] `IL ρ(α ≈ β). (iii) For any formulas α, β,

α a`IL ρ(τ(α)) and α ≈ β =||=HA τ(ρ(α ≈ β)).

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 14 / 32 (⇒) A straightforward induction on the length of a derivation of α from T in IL using properties of Heyting algebras.

(⇐) Suppose that T 6`IL α and consider the homomorphism e from the formula algebra to the Heyting algebra AT given by e(γ) = [γ]T . Since

T `IL γ ⇐⇒ [γ]T = [>]T ,

we have [γ]T = [>]T for all γ ∈ T and [α]T 6= [>]T , so

{γ ≈ > | γ ∈ T } 6|=HA α ≈ >.

(iii) is easy to check, and (ii) follows directly from (i) and (iii).

Proof Sketch

For (i), we need to prove

T `IL α ⇐⇒ {γ ≈ > | γ ∈ T } |=HA α ≈ >.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 15 / 32 (⇐) Suppose that T 6`IL α and consider the homomorphism e from the formula algebra to the Heyting algebra AT given by e(γ) = [γ]T . Since

T `IL γ ⇐⇒ [γ]T = [>]T ,

we have [γ]T = [>]T for all γ ∈ T and [α]T 6= [>]T , so

{γ ≈ > | γ ∈ T } 6|=HA α ≈ >.

(iii) is easy to check, and (ii) follows directly from (i) and (iii).

Proof Sketch

For (i), we need to prove

T `IL α ⇐⇒ {γ ≈ > | γ ∈ T } |=HA α ≈ >.

(⇒) A straightforward induction on the length of a derivation of α from T in IL using properties of Heyting algebras.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 15 / 32 and consider the homomorphism e from the formula algebra to the Heyting algebra AT given by e(γ) = [γ]T . Since

T `IL γ ⇐⇒ [γ]T = [>]T ,

we have [γ]T = [>]T for all γ ∈ T and [α]T 6= [>]T , so

{γ ≈ > | γ ∈ T } 6|=HA α ≈ >.

(iii) is easy to check, and (ii) follows directly from (i) and (iii).

Proof Sketch

For (i), we need to prove

T `IL α ⇐⇒ {γ ≈ > | γ ∈ T } |=HA α ≈ >.

(⇒) A straightforward induction on the length of a derivation of α from T in IL using properties of Heyting algebras.

(⇐) Suppose that T 6`IL α

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 15 / 32 Since

T `IL γ ⇐⇒ [γ]T = [>]T ,

we have [γ]T = [>]T for all γ ∈ T and [α]T 6= [>]T , so

{γ ≈ > | γ ∈ T } 6|=HA α ≈ >.

(iii) is easy to check, and (ii) follows directly from (i) and (iii).

Proof Sketch

For (i), we need to prove

T `IL α ⇐⇒ {γ ≈ > | γ ∈ T } |=HA α ≈ >.

(⇒) A straightforward induction on the length of a derivation of α from T in IL using properties of Heyting algebras.

(⇐) Suppose that T 6`IL α and consider the homomorphism e from the formula algebra to the Heyting algebra AT given by e(γ) = [γ]T .

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 15 / 32 we have [γ]T = [>]T for all γ ∈ T and [α]T 6= [>]T , so

{γ ≈ > | γ ∈ T } 6|=HA α ≈ >.

(iii) is easy to check, and (ii) follows directly from (i) and (iii).

Proof Sketch

For (i), we need to prove

T `IL α ⇐⇒ {γ ≈ > | γ ∈ T } |=HA α ≈ >.

(⇒) A straightforward induction on the length of a derivation of α from T in IL using properties of Heyting algebras.

(⇐) Suppose that T 6`IL α and consider the homomorphism e from the formula algebra to the Heyting algebra AT given by e(γ) = [γ]T . Since

T `IL γ ⇐⇒ [γ]T = [>]T ,

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 15 / 32 so

{γ ≈ > | γ ∈ T } 6|=HA α ≈ >.

(iii) is easy to check, and (ii) follows directly from (i) and (iii).

Proof Sketch

For (i), we need to prove

T `IL α ⇐⇒ {γ ≈ > | γ ∈ T } |=HA α ≈ >.

(⇒) A straightforward induction on the length of a derivation of α from T in IL using properties of Heyting algebras.

(⇐) Suppose that T 6`IL α and consider the homomorphism e from the formula algebra to the Heyting algebra AT given by e(γ) = [γ]T . Since

T `IL γ ⇐⇒ [γ]T = [>]T ,

we have [γ]T = [>]T for all γ ∈ T and [α]T 6= [>]T ,

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 15 / 32 (iii) is easy to check, and (ii) follows directly from (i) and (iii).

Proof Sketch

For (i), we need to prove

T `IL α ⇐⇒ {γ ≈ > | γ ∈ T } |=HA α ≈ >.

(⇒) A straightforward induction on the length of a derivation of α from T in IL using properties of Heyting algebras.

(⇐) Suppose that T 6`IL α and consider the homomorphism e from the formula algebra to the Heyting algebra AT given by e(γ) = [γ]T . Since

T `IL γ ⇐⇒ [γ]T = [>]T ,

we have [γ]T = [>]T for all γ ∈ T and [α]T 6= [>]T , so

{γ ≈ > | γ ∈ T } 6|=HA α ≈ >.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 15 / 32 Proof Sketch

For (i), we need to prove

T `IL α ⇐⇒ {γ ≈ > | γ ∈ T } |=HA α ≈ >.

(⇒) A straightforward induction on the length of a derivation of α from T in IL using properties of Heyting algebras.

(⇐) Suppose that T 6`IL α and consider the homomorphism e from the formula algebra to the Heyting algebra AT given by e(γ) = [γ]T . Since

T `IL γ ⇐⇒ [γ]T = [>]T ,

we have [γ]T = [>]T for all γ ∈ T and [α]T 6= [>]T , so

{γ ≈ > | γ ∈ T } 6|=HA α ≈ >.

(iii) is easy to check, and (ii) follows directly from (i) and (iii).

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 15 / 32 Proof-search-oriented variants of Gentzen’s sequent calculus for intuitionistic logic were later developed by Ketonen, Kleene, Ono, Vorob’ev, Dragalin, Troelstra, Dyckhoﬀ, Hudelmeier. . .

Sequent calculi (and many variants thereof) have been introduced for many other non-classical logics and classes of algebraic structures.

Sequent Calculi

The ﬁrst sequent calculi for (ﬁrst-order) classical and intuitionistic logic were introduced by Gentzen in the 1930s.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 16 / 32 Sequent calculi (and many variants thereof) have been introduced for many other non-classical logics and classes of algebraic structures.

Sequent Calculi

The ﬁrst sequent calculi for (ﬁrst-order) classical and intuitionistic logic were introduced by Gentzen in the 1930s.

Proof-search-oriented variants of Gentzen’s sequent calculus for intuitionistic logic were later developed by Ketonen, Kleene, Ono, Vorob’ev, Dragalin, Troelstra, Dyckhoﬀ, Hudelmeier. . .

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 16 / 32 Sequent Calculi

The ﬁrst sequent calculi for (ﬁrst-order) classical and intuitionistic logic were introduced by Gentzen in the 1930s.

Proof-search-oriented variants of Gentzen’s sequent calculus for intuitionistic logic were later developed by Ketonen, Kleene, Ono, Vorob’ev, Dragalin, Troelstra, Dyckhoﬀ, Hudelmeier. . .

Sequent calculi (and many variants thereof) have been introduced for many other non-classical logics and classes of algebraic structures.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 16 / 32 We typically write Γ, Π for the multiset sum of Γ and Π, and omit brackets.

A sequent calculus GL consists of a set of rules with instances

S1 ... Sn where S0, S1,..., Sn are sequents. S0 A GL-derivation of a sequent S is a ﬁnite tree of sequents with root S built using the rules of GL. If there exists a GL-derivation of a sequent S of height at most n, we write `n S or just ` S. GL GL

Sequents

A sequent is an ordered pair consisting of a ﬁnite multiset of formulas Γ and a formula α, written Γ ⇒ α.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 17 / 32 A sequent calculus GL consists of a set of rules with instances

Sequents

A sequent is an ordered pair consisting of a ﬁnite multiset of formulas Γ and a formula α, written Γ ⇒ α.

We typically write Γ, Π for the multiset sum of Γ and Π, and omit brackets.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 17 / 32 A GL-derivation of a sequent S is a ﬁnite tree of sequents with root S built using the rules of GL. If there exists a GL-derivation of a sequent S of height at most n, we write `n S or just ` S. GL GL

Sequents

A sequent is an ordered pair consisting of a ﬁnite multiset of formulas Γ and a formula α, written Γ ⇒ α.

We typically write Γ, Π for the multiset sum of Γ and Π, and omit brackets.

A sequent calculus GL consists of a set of rules with instances

S1 ... Sn where S0, S1,..., Sn are sequents. S0

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 17 / 32 If there exists a GL-derivation of a sequent S of height at most n, we write `n S or just ` S. GL GL

Sequents

A sequent is an ordered pair consisting of a ﬁnite multiset of formulas Γ and a formula α, written Γ ⇒ α.

We typically write Γ, Π for the multiset sum of Γ and Π, and omit brackets.

A sequent calculus GL consists of a set of rules with instances

S1 ... Sn where S0, S1,..., Sn are sequents. S0 A GL-derivation of a sequent S is a ﬁnite tree of sequents with root S built using the rules of GL.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 17 / 32 Sequents

A sequent is an ordered pair consisting of a ﬁnite multiset of formulas Γ and a formula α, written Γ ⇒ α.

We typically write Γ, Π for the multiset sum of Γ and Π, and omit brackets.

A sequent calculus GL consists of a set of rules with instances

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 17 / 32 Cut Rule Γ ⇒ α Π,α ⇒ δ (cut) Γ, Π ⇒ δ

Left Operation Rules Right Operation Rules

(⊥⇒) (⇒>) Γ, ⊥ ⇒ δ Γ ⇒ >

Γ, α, β ⇒ δ Γ ⇒ α Γ ⇒ β (∧⇒) (⇒∧) Γ, α ∧ β ⇒ δ Γ ⇒ α ∧ β

Γ, α ⇒ δ Γ, β ⇒ δ Γ ⇒ α Γ ⇒ β (∨⇒) (⇒∨) (⇒∨) Γ, α ∨ β ⇒ δ Γ ⇒ α ∨ β l Γ ⇒ α ∨ β r

Γ, α → β ⇒ α Γ, β ⇒ δ Γ, α ⇒ β (→⇒) (⇒→) Γ, α → β ⇒ δ Γ ⇒ α → β

A Sequent Calculus GIL for Intuitionistic Logic

Identity Axioms

(id) Γ, x ⇒ x

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 18 / 32 Cut Rule Γ ⇒ α Π,α ⇒ δ (cut) Γ, Π ⇒ δ

Γ, α, β ⇒ δ Γ ⇒ α Γ ⇒ β (∧⇒) (⇒∧) Γ, α ∧ β ⇒ δ Γ ⇒ α ∧ β

Γ, α ⇒ δ Γ, β ⇒ δ Γ ⇒ α Γ ⇒ β (∨⇒) (⇒∨) (⇒∨) Γ, α ∨ β ⇒ δ Γ ⇒ α ∨ β l Γ ⇒ α ∨ β r

Γ, α → β ⇒ α Γ, β ⇒ δ Γ, α ⇒ β (→⇒) (⇒→) Γ, α → β ⇒ δ Γ ⇒ α → β

A Sequent Calculus GIL for Intuitionistic Logic

Identity Axioms

(id) Γ, x ⇒ x

Left Operation Rules Right Operation Rules

(⊥⇒) (⇒>) Γ, ⊥ ⇒ δ Γ ⇒ >

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 18 / 32 Cut Rule Γ ⇒ α Π,α ⇒ δ (cut) Γ, Π ⇒ δ

Γ, α ⇒ δ Γ, β ⇒ δ Γ ⇒ α Γ ⇒ β (∨⇒) (⇒∨) (⇒∨) Γ, α ∨ β ⇒ δ Γ ⇒ α ∨ β l Γ ⇒ α ∨ β r

Γ, α → β ⇒ α Γ, β ⇒ δ Γ, α ⇒ β (→⇒) (⇒→) Γ, α → β ⇒ δ Γ ⇒ α → β

A Sequent Calculus GIL for Intuitionistic Logic

Identity Axioms

(id) Γ, x ⇒ x

Left Operation Rules Right Operation Rules

(⊥⇒) (⇒>) Γ, ⊥ ⇒ δ Γ ⇒ >

Γ, α, β ⇒ δ Γ ⇒ α Γ ⇒ β (∧⇒) (⇒∧) Γ, α ∧ β ⇒ δ Γ ⇒ α ∧ β

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 18 / 32 Cut Rule Γ ⇒ α Π,α ⇒ δ (cut) Γ, Π ⇒ δ

Γ, α → β ⇒ α Γ, β ⇒ δ Γ, α ⇒ β (→⇒) (⇒→) Γ, α → β ⇒ δ Γ ⇒ α → β

A Sequent Calculus GIL for Intuitionistic Logic

Identity Axioms

(id) Γ, x ⇒ x

Left Operation Rules Right Operation Rules

(⊥⇒) (⇒>) Γ, ⊥ ⇒ δ Γ ⇒ >

Γ, α, β ⇒ δ Γ ⇒ α Γ ⇒ β (∧⇒) (⇒∧) Γ, α ∧ β ⇒ δ Γ ⇒ α ∧ β

Γ, α ⇒ δ Γ, β ⇒ δ Γ ⇒ α Γ ⇒ β (∨⇒) (⇒∨) (⇒∨) Γ, α ∨ β ⇒ δ Γ ⇒ α ∨ β l Γ ⇒ α ∨ β r

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 18 / 32 Cut Rule Γ ⇒ α Π,α ⇒ δ (cut) Γ, Π ⇒ δ

A Sequent Calculus GIL for Intuitionistic Logic

Identity Axioms

(id) Γ, x ⇒ x

Left Operation Rules Right Operation Rules

(⊥⇒) (⇒>) Γ, ⊥ ⇒ δ Γ ⇒ >

Γ, α, β ⇒ δ Γ ⇒ α Γ ⇒ β (∧⇒) (⇒∧) Γ, α ∧ β ⇒ δ Γ ⇒ α ∧ β

Γ, α ⇒ δ Γ, β ⇒ δ Γ ⇒ α Γ ⇒ β (∨⇒) (⇒∨) (⇒∨) Γ, α ∨ β ⇒ δ Γ ⇒ α ∨ β l Γ ⇒ α ∨ β r

Γ, α → β ⇒ α Γ, β ⇒ δ Γ, α ⇒ β (→⇒) (⇒→) Γ, α → β ⇒ δ Γ ⇒ α → β

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 18 / 32 A Sequent Calculus GIL for Intuitionistic Logic

Identity Axioms Cut Rule Γ ⇒ α Π,α ⇒ δ (id) (cut) Γ, x ⇒ x Γ, Π ⇒ δ

Left Operation Rules Right Operation Rules

(⊥⇒) (⇒>) Γ, ⊥ ⇒ δ Γ ⇒ >

Γ, α, β ⇒ δ Γ ⇒ α Γ ⇒ β (∧⇒) (⇒∧) Γ, α ∧ β ⇒ δ Γ ⇒ α ∧ β

Γ, α ⇒ δ Γ, β ⇒ δ Γ ⇒ α Γ ⇒ β (∨⇒) (⇒∨) (⇒∨) Γ, α ∨ β ⇒ δ Γ ⇒ α ∨ β l Γ ⇒ α ∨ β r

Γ, α → β ⇒ α Γ, β ⇒ δ Γ, α ⇒ β (→⇒) (⇒→) Γ, α → β ⇒ δ Γ ⇒ α → β

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 18 / 32 (id) y, x ⇒ y

(id) (⇒∨) (id) x → y, x ⇒ x y, x ⇒ y ∨ z l x → y, z ⇒ z

(→⇒) (⇒∨)r x → y, x ⇒ y ∨ z x → y, z ⇒ y ∨ z

(∨⇒) x → y, x ∨ z ⇒ y ∨ z

(∧⇒) (x → y) ∧ (x ∨ z) ⇒ y ∨ z

An Example Derivation

(⇒→) ⇒ ((x → y) ∧ (x ∨ z)) → (y ∨ z)

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 19 / 32 (id) y, x ⇒ y

(id) (⇒∨) (id) x → y, x ⇒ x y, x ⇒ y ∨ z l x → y, z ⇒ z

(→⇒) (⇒∨)r x → y, x ⇒ y ∨ z x → y, z ⇒ y ∨ z

(∨⇒) x → y, x ∨ z ⇒ y ∨ z

An Example Derivation

(∧⇒) (x → y) ∧ (x ∨ z) ⇒ y ∨ z (⇒→) ⇒ ((x → y) ∧ (x ∨ z)) → (y ∨ z)

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 19 / 32 (id) y, x ⇒ y

(id) (⇒∨) (id) x → y, x ⇒ x y, x ⇒ y ∨ z l x → y, z ⇒ z

(→⇒) (⇒∨)r x → y, x ⇒ y ∨ z x → y, z ⇒ y ∨ z

An Example Derivation

(∨⇒) x → y, x ∨ z ⇒ y ∨ z (∧⇒) (x → y) ∧ (x ∨ z) ⇒ y ∨ z (⇒→) ⇒ ((x → y) ∧ (x ∨ z)) → (y ∨ z)

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 19 / 32 (id) y, x ⇒ y (id) x → y, z ⇒ z

(id) (⇒∨) x → y, x ⇒ x y, x ⇒ y ∨ z l (⇒∨)r x → y, z ⇒ y ∨ z

An Example Derivation

(→⇒) x → y, x ⇒ y ∨ z (∨⇒) x → y, x ∨ z ⇒ y ∨ z (∧⇒) (x → y) ∧ (x ∨ z) ⇒ y ∨ z (⇒→) ⇒ ((x → y) ∧ (x ∨ z)) → (y ∨ z)

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 19 / 32 (id) y, x ⇒ y (id) x → y, z ⇒ z

(⇒∨) y, x ⇒ y ∨ z l (⇒∨)r x → y, z ⇒ y ∨ z

An Example Derivation

(id) x → y, x ⇒ x (→⇒) x → y, x ⇒ y ∨ z (∨⇒) x → y, x ∨ z ⇒ y ∨ z (∧⇒) (x → y) ∧ (x ∨ z) ⇒ y ∨ z (⇒→) ⇒ ((x → y) ∧ (x ∨ z)) → (y ∨ z)

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 19 / 32 (id) x → y, z ⇒ z

(id) y, x ⇒ y

(⇒∨)r x → y, z ⇒ y ∨ z

An Example Derivation

(id) (⇒∨) x → y, x ⇒ x y, x ⇒ y ∨ z l (→⇒) x → y, x ⇒ y ∨ z (∨⇒) x → y, x ∨ z ⇒ y ∨ z (∧⇒) (x → y) ∧ (x ∨ z) ⇒ y ∨ z (⇒→) ⇒ ((x → y) ∧ (x ∨ z)) → (y ∨ z)

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 19 / 32 (id) x → y, z ⇒ z

(⇒∨)r x → y, z ⇒ y ∨ z

An Example Derivation

(id) y, x ⇒ y (id) (⇒∨) x → y, x ⇒ x y, x ⇒ y ∨ z l (→⇒) x → y, x ⇒ y ∨ z (∨⇒) x → y, x ∨ z ⇒ y ∨ z (∧⇒) (x → y) ∧ (x ∨ z) ⇒ y ∨ z (⇒→) ⇒ ((x → y) ∧ (x ∨ z)) → (y ∨ z)

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 19 / 32 (id) x → y, z ⇒ z

An Example Derivation

(id) y, x ⇒ y (id) (⇒∨) x → y, x ⇒ x y, x ⇒ y ∨ z l (→⇒) (⇒∨)r x → y, x ⇒ y ∨ z x → y, z ⇒ y ∨ z (∨⇒) x → y, x ∨ z ⇒ y ∨ z (∧⇒) (x → y) ∧ (x ∨ z) ⇒ y ∨ z (⇒→) ⇒ ((x → y) ∧ (x ∨ z)) → (y ∨ z)

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 19 / 32 An Example Derivation

(id) y, x ⇒ y (id) (⇒∨) (id) x → y, x ⇒ x y, x ⇒ y ∨ z l x → y, z ⇒ z (→⇒) (⇒∨)r x → y, x ⇒ y ∨ z x → y, z ⇒ y ∨ z (∨⇒) x → y, x ∨ z ⇒ y ∨ z (∧⇒) (x → y) ∧ (x ∨ z) ⇒ y ∨ z (⇒→) ⇒ ((x → y) ∧ (x ∨ z)) → (y ∨ z)

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 19 / 32 . . . . Γ, α1, α2 ⇒ α1 Γ, α1, α2 ⇒ α2

(⇒∧) Γ, α1, α2 ⇒ α1 ∧ α2

(∧⇒) Γ, α1 ∧ α2 ⇒ α1 ∧ α2

Identity Lemma

Let GIL◦ be the sequent calculus GIL without the cut rule.

Lemma For any ﬁnite multiset of formulas Γ and any formula α,

`GIL◦ Γ, α ⇒ α.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 20 / 32 . . . . Γ, α1, α2 ⇒ α1 Γ, α1, α2 ⇒ α2

(⇒∧) Γ, α1, α2 ⇒ α1 ∧ α2

(∧⇒) Γ, α1 ∧ α2 ⇒ α1 ∧ α2

The base case Γ, x ⇒ x is an instance of (id). For the inductive step, we consider the principal connective of α; e.g., for α = α1 ∧ α2, we obtain

Identity Lemma

Let GIL◦ be the sequent calculus GIL without the cut rule.

Lemma For any ﬁnite multiset of formulas Γ and any formula α,

`GIL◦ Γ, α ⇒ α.

Proof. By induction on the size (number of occurrences of connectives) |α| of α.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 20 / 32 . . . . Γ, α1, α2 ⇒ α1 Γ, α1, α2 ⇒ α2

(⇒∧) Γ, α1, α2 ⇒ α1 ∧ α2

(∧⇒) Γ, α1 ∧ α2 ⇒ α1 ∧ α2

For the inductive step, we consider the principal connective of α; e.g., for α = α1 ∧ α2, we obtain

Identity Lemma

Let GIL◦ be the sequent calculus GIL without the cut rule.

Lemma For any ﬁnite multiset of formulas Γ and any formula α,

`GIL◦ Γ, α ⇒ α.

Proof. By induction on the size (number of occurrences of connectives) |α| of α. The base case Γ, x ⇒ x is an instance of (id).

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 20 / 32 . . . . Γ, α1, α2 ⇒ α1 Γ, α1, α2 ⇒ α2

(⇒∧) Γ, α1, α2 ⇒ α1 ∧ α2

(∧⇒) Γ, α1 ∧ α2 ⇒ α1 ∧ α2

e.g., for α = α1 ∧ α2, we obtain

Identity Lemma

Let GIL◦ be the sequent calculus GIL without the cut rule.

Lemma For any ﬁnite multiset of formulas Γ and any formula α,

`GIL◦ Γ, α ⇒ α.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 20 / 32 . . . . Γ, α1, α2 ⇒ α1 Γ, α1, α2 ⇒ α2

(⇒∧) Γ, α1, α2 ⇒ α1 ∧ α2

(∧⇒) Γ, α1 ∧ α2 ⇒ α1 ∧ α2

Identity Lemma

Let GIL◦ be the sequent calculus GIL without the cut rule.

Lemma For any ﬁnite multiset of formulas Γ and any formula α,

`GIL◦ Γ, α ⇒ α.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 20 / 32 . . . . Γ, α1, α2 ⇒ α1 Γ, α1, α2 ⇒ α2

(⇒∧) Γ, α1, α2 ⇒ α1 ∧ α2

Identity Lemma

Let GIL◦ be the sequent calculus GIL without the cut rule.

Lemma For any ﬁnite multiset of formulas Γ and any formula α,

`GIL◦ Γ, α ⇒ α.

(∧⇒) Γ, α1 ∧ α2 ⇒ α1 ∧ α2

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 20 / 32 . . . . Γ, α1, α2 ⇒ α1 Γ, α1, α2 ⇒ α2

Identity Lemma

Let GIL◦ be the sequent calculus GIL without the cut rule.

Lemma For any ﬁnite multiset of formulas Γ and any formula α,

`GIL◦ Γ, α ⇒ α.

(⇒∧) Γ, α1, α2 ⇒ α1 ∧ α2 (∧⇒) Γ, α1 ∧ α2 ⇒ α1 ∧ α2

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 20 / 32 Identity Lemma

Let GIL◦ be the sequent calculus GIL without the cut rule.

Lemma For any ﬁnite multiset of formulas Γ and any formula α,

`GIL◦ Γ, α ⇒ α.

Proof. By induction on the size (number of occurrences of connectives) |α| of α. The base case Γ, x ⇒ x is an instance of (id). For the inductive step, we consider the principal connective of α; e.g., for α = α1 ∧ α2, we obtain . . . . Γ, α1, α2 ⇒ α1 Γ, α1, α2 ⇒ α2 (⇒∧) Γ, α1, α2 ⇒ α1 ∧ α2 (∧⇒) Γ, α1 ∧ α2 ⇒ α1 ∧ α2

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 20 / 32 Proof.

(⇒) By induction on the height of a derivation of α1, . . . , αn ⇒ β in GIL. It suﬃces to check that the rules of GIL preserve derivability in IL; e.g.,

ÌL (γ ∧ α) → δ and ÌL (γ ∧ β) → δ =⇒ ÌL (γ ∧ (α ∨ β)) → δ.

(⇐) We prove that `IL α implies `GIL ⇒ α by induction on the length of a derivation of α in IL, showing that the axioms are derivable in GIL and that modus ponens preserves derivability, i.e., cutting twice with β, β → γ ⇒ γ,

`GIL ⇒ β and `GIL ⇒ β → γ =⇒ `GIL ⇒ γ.

Hence if `IL (α1 ∧ ... ∧ αn) → β, then `GIL ⇒ (α1 ∧ ... ∧ αn) → β, and the result follows by cutting with α1, . . . , αn, (α1 ∧ ... ∧ αn) → β ⇒ β.

Soundness and Completeness

Theorem

`GIL α1, . . . , αn ⇒ β ⇐⇒ `IL (α1 ∧ ... ∧ αn) → β.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 21 / 32 It suﬃces to check that the rules of GIL preserve derivability in IL; e.g.,

ÌL (γ ∧ α) → δ and ÌL (γ ∧ β) → δ =⇒ ÌL (γ ∧ (α ∨ β)) → δ.

`GIL ⇒ β and `GIL ⇒ β → γ =⇒ `GIL ⇒ γ.

Hence if `IL (α1 ∧ ... ∧ αn) → β, then `GIL ⇒ (α1 ∧ ... ∧ αn) → β, and the result follows by cutting with α1, . . . , αn, (α1 ∧ ... ∧ αn) → β ⇒ β.

Soundness and Completeness

Theorem

`GIL α1, . . . , αn ⇒ β ⇐⇒ `IL (α1 ∧ ... ∧ αn) → β.

Proof.

(⇒) By induction on the height of a derivation of α1, . . . , αn ⇒ β in GIL.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 21 / 32 e.g.,

ÌL (γ ∧ α) → δ and ÌL (γ ∧ β) → δ =⇒ ÌL (γ ∧ (α ∨ β)) → δ.

`GIL ⇒ β and `GIL ⇒ β → γ =⇒ `GIL ⇒ γ.

Hence if `IL (α1 ∧ ... ∧ αn) → β, then `GIL ⇒ (α1 ∧ ... ∧ αn) → β, and the result follows by cutting with α1, . . . , αn, (α1 ∧ ... ∧ αn) → β ⇒ β.

Soundness and Completeness

Theorem

`GIL α1, . . . , αn ⇒ β ⇐⇒ `IL (α1 ∧ ... ∧ αn) → β.

Proof.

(⇒) By induction on the height of a derivation of α1, . . . , αn ⇒ β in GIL. It suﬃces to check that the rules of GIL preserve derivability in IL;

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 21 / 32 (⇐) We prove that `IL α implies `GIL ⇒ α by induction on the length of a derivation of α in IL, showing that the axioms are derivable in GIL and that modus ponens preserves derivability, i.e., cutting twice with β, β → γ ⇒ γ,

`GIL ⇒ β and `GIL ⇒ β → γ =⇒ `GIL ⇒ γ.

Hence if `IL (α1 ∧ ... ∧ αn) → β, then `GIL ⇒ (α1 ∧ ... ∧ αn) → β, and the result follows by cutting with α1, . . . , αn, (α1 ∧ ... ∧ αn) → β ⇒ β.

Soundness and Completeness

Theorem

`GIL α1, . . . , αn ⇒ β ⇐⇒ `IL (α1 ∧ ... ∧ αn) → β.

Proof.

(⇒) By induction on the height of a derivation of α1, . . . , αn ⇒ β in GIL. It suﬃces to check that the rules of GIL preserve derivability in IL; e.g.,

ÌL (γ ∧ α) → δ and ÌL (γ ∧ β) → δ =⇒ ÌL (γ ∧ (α ∨ β)) → δ.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 21 / 32 showing that the axioms are derivable in GIL and that modus ponens preserves derivability, i.e., cutting twice with β, β → γ ⇒ γ,

`GIL ⇒ β and `GIL ⇒ β → γ =⇒ `GIL ⇒ γ.

Hence if `IL (α1 ∧ ... ∧ αn) → β, then `GIL ⇒ (α1 ∧ ... ∧ αn) → β, and the result follows by cutting with α1, . . . , αn, (α1 ∧ ... ∧ αn) → β ⇒ β.

Soundness and Completeness

Theorem

`GIL α1, . . . , αn ⇒ β ⇐⇒ `IL (α1 ∧ ... ∧ αn) → β.

Proof.

(⇒) By induction on the height of a derivation of α1, . . . , αn ⇒ β in GIL. It suﬃces to check that the rules of GIL preserve derivability in IL; e.g.,

ÌL (γ ∧ α) → δ and ÌL (γ ∧ β) → δ =⇒ ÌL (γ ∧ (α ∨ β)) → δ.

(⇐) We prove that `IL α implies `GIL ⇒ α by induction on the length of a derivation of α in IL,

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 21 / 32 i.e., cutting twice with β, β → γ ⇒ γ,

`GIL ⇒ β and `GIL ⇒ β → γ =⇒ `GIL ⇒ γ.

Hence if `IL (α1 ∧ ... ∧ αn) → β, then `GIL ⇒ (α1 ∧ ... ∧ αn) → β, and the result follows by cutting with α1, . . . , αn, (α1 ∧ ... ∧ αn) → β ⇒ β.

Soundness and Completeness

Theorem

`GIL α1, . . . , αn ⇒ β ⇐⇒ `IL (α1 ∧ ... ∧ αn) → β.

Proof.

(⇒) By induction on the height of a derivation of α1, . . . , αn ⇒ β in GIL. It suﬃces to check that the rules of GIL preserve derivability in IL; e.g.,

ÌL (γ ∧ α) → δ and ÌL (γ ∧ β) → δ =⇒ ÌL (γ ∧ (α ∨ β)) → δ.

(⇐) We prove that `IL α implies `GIL ⇒ α by induction on the length of a derivation of α in IL, showing that the axioms are derivable in GIL and that modus ponens preserves derivability,

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 21 / 32 Hence if `IL (α1 ∧ ... ∧ αn) → β, then `GIL ⇒ (α1 ∧ ... ∧ αn) → β, and the result follows by cutting with α1, . . . , αn, (α1 ∧ ... ∧ αn) → β ⇒ β.

Soundness and Completeness

Theorem

`GIL α1, . . . , αn ⇒ β ⇐⇒ `IL (α1 ∧ ... ∧ αn) → β.

Proof.

(⇒) By induction on the height of a derivation of α1, . . . , αn ⇒ β in GIL. It suﬃces to check that the rules of GIL preserve derivability in IL; e.g.,

ÌL (γ ∧ α) → δ and ÌL (γ ∧ β) → δ =⇒ ÌL (γ ∧ (α ∨ β)) → δ.

`GIL ⇒ β and `GIL ⇒ β → γ =⇒ `GIL ⇒ γ.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 21 / 32 and the result follows by cutting with α1, . . . , αn, (α1 ∧ ... ∧ αn) → β ⇒ β.

Soundness and Completeness

Theorem

`GIL α1, . . . , αn ⇒ β ⇐⇒ `IL (α1 ∧ ... ∧ αn) → β.

Proof.

(⇒) By induction on the height of a derivation of α1, . . . , αn ⇒ β in GIL. It suﬃces to check that the rules of GIL preserve derivability in IL; e.g.,

ÌL (γ ∧ α) → δ and ÌL (γ ∧ β) → δ =⇒ ÌL (γ ∧ (α ∨ β)) → δ.

`GIL ⇒ β and `GIL ⇒ β → γ =⇒ `GIL ⇒ γ.

Hence if `IL (α1 ∧ ... ∧ αn) → β, then `GIL ⇒ (α1 ∧ ... ∧ αn) → β,

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 21 / 32 Soundness and Completeness

Theorem

`GIL α1, . . . , αn ⇒ β ⇐⇒ `IL (α1 ∧ ... ∧ αn) → β.

Proof.

(⇒) By induction on the height of a derivation of α1, . . . , αn ⇒ β in GIL. It suﬃces to check that the rules of GIL preserve derivability in IL; e.g.,

ÌL (γ ∧ α) → δ and ÌL (γ ∧ β) → δ =⇒ ÌL (γ ∧ (α ∨ β)) → δ.

`GIL ⇒ β and `GIL ⇒ β → γ =⇒ `GIL ⇒ γ.

Hence if `IL (α1 ∧ ... ∧ αn) → β, then `GIL ⇒ (α1 ∧ ... ∧ αn) → β, and the result follows by cutting with α1, . . . , αn, (α1 ∧ ... ∧ αn) → β ⇒ β.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 21 / 32 (b) `n Γ, α ∧ β ⇒ δ =⇒ `n Γ, α, β ⇒ δ. GIL◦ GIL◦ (c) `n Γ ⇒ α ∧ β =⇒ `n Γ ⇒ α and `n Γ ⇒ β. GIL◦ GIL◦ GIL◦ (d) `n Γ, α ∨ β ⇒ δ =⇒ `n Γ, α ⇒ δ and `n Γ, β ⇒ δ. GIL◦ GIL◦ GIL◦ (e) `n Γ ⇒ α → β =⇒ `n Γ, α ⇒ β. GIL◦ GIL◦ (f) `n Γ, α → β ⇒ δ =⇒ `n Γ, β ⇒ δ. GIL◦ GIL◦ (g) `n Γ, α, α ⇒ δ =⇒ `n Γ, α ⇒ δ. GIL◦ GIL◦

Proof. Each claim can be proved by a simple (if rather tedious) induction on n.

Weakening, Invertibility, Contraction

Lemma (a) `n Γ ⇒ δ =⇒ `n Γ, α ⇒ δ. GIL◦ GIL◦

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 22 / 32 (c) `n Γ ⇒ α ∧ β =⇒ `n Γ ⇒ α and `n Γ ⇒ β. GIL◦ GIL◦ GIL◦ (d) `n Γ, α ∨ β ⇒ δ =⇒ `n Γ, α ⇒ δ and `n Γ, β ⇒ δ. GIL◦ GIL◦ GIL◦ (e) `n Γ ⇒ α → β =⇒ `n Γ, α ⇒ β. GIL◦ GIL◦ (f) `n Γ, α → β ⇒ δ =⇒ `n Γ, β ⇒ δ. GIL◦ GIL◦ (g) `n Γ, α, α ⇒ δ =⇒ `n Γ, α ⇒ δ. GIL◦ GIL◦

Proof. Each claim can be proved by a simple (if rather tedious) induction on n.

Weakening, Invertibility, Contraction

Lemma (a) `n Γ ⇒ δ =⇒ `n Γ, α ⇒ δ. GIL◦ GIL◦ (b) `n Γ, α ∧ β ⇒ δ =⇒ `n Γ, α, β ⇒ δ. GIL◦ GIL◦

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 22 / 32 (d) `n Γ, α ∨ β ⇒ δ =⇒ `n Γ, α ⇒ δ and `n Γ, β ⇒ δ. GIL◦ GIL◦ GIL◦ (e) `n Γ ⇒ α → β =⇒ `n Γ, α ⇒ β. GIL◦ GIL◦ (f) `n Γ, α → β ⇒ δ =⇒ `n Γ, β ⇒ δ. GIL◦ GIL◦ (g) `n Γ, α, α ⇒ δ =⇒ `n Γ, α ⇒ δ. GIL◦ GIL◦

Proof. Each claim can be proved by a simple (if rather tedious) induction on n.

Weakening, Invertibility, Contraction

Lemma (a) `n Γ ⇒ δ =⇒ `n Γ, α ⇒ δ. GIL◦ GIL◦ (b) `n Γ, α ∧ β ⇒ δ =⇒ `n Γ, α, β ⇒ δ. GIL◦ GIL◦ (c) `n Γ ⇒ α ∧ β =⇒ `n Γ ⇒ α and `n Γ ⇒ β. GIL◦ GIL◦ GIL◦

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 22 / 32 (e) `n Γ ⇒ α → β =⇒ `n Γ, α ⇒ β. GIL◦ GIL◦ (f) `n Γ, α → β ⇒ δ =⇒ `n Γ, β ⇒ δ. GIL◦ GIL◦ (g) `n Γ, α, α ⇒ δ =⇒ `n Γ, α ⇒ δ. GIL◦ GIL◦

Proof. Each claim can be proved by a simple (if rather tedious) induction on n.

Weakening, Invertibility, Contraction

Lemma (a) `n Γ ⇒ δ =⇒ `n Γ, α ⇒ δ. GIL◦ GIL◦ (b) `n Γ, α ∧ β ⇒ δ =⇒ `n Γ, α, β ⇒ δ. GIL◦ GIL◦ (c) `n Γ ⇒ α ∧ β =⇒ `n Γ ⇒ α and `n Γ ⇒ β. GIL◦ GIL◦ GIL◦ (d) `n Γ, α ∨ β ⇒ δ =⇒ `n Γ, α ⇒ δ and `n Γ, β ⇒ δ. GIL◦ GIL◦ GIL◦

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 22 / 32 (f) `n Γ, α → β ⇒ δ =⇒ `n Γ, β ⇒ δ. GIL◦ GIL◦ (g) `n Γ, α, α ⇒ δ =⇒ `n Γ, α ⇒ δ. GIL◦ GIL◦

Proof. Each claim can be proved by a simple (if rather tedious) induction on n.

Weakening, Invertibility, Contraction

Lemma (a) `n Γ ⇒ δ =⇒ `n Γ, α ⇒ δ. GIL◦ GIL◦ (b) `n Γ, α ∧ β ⇒ δ =⇒ `n Γ, α, β ⇒ δ. GIL◦ GIL◦ (c) `n Γ ⇒ α ∧ β =⇒ `n Γ ⇒ α and `n Γ ⇒ β. GIL◦ GIL◦ GIL◦ (d) `n Γ, α ∨ β ⇒ δ =⇒ `n Γ, α ⇒ δ and `n Γ, β ⇒ δ. GIL◦ GIL◦ GIL◦ (e) `n Γ ⇒ α → β =⇒ `n Γ, α ⇒ β. GIL◦ GIL◦

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 22 / 32 (g) `n Γ, α, α ⇒ δ =⇒ `n Γ, α ⇒ δ. GIL◦ GIL◦

Proof. Each claim can be proved by a simple (if rather tedious) induction on n.

Weakening, Invertibility, Contraction

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 22 / 32 Proof. Each claim can be proved by a simple (if rather tedious) induction on n.

Weakening, Invertibility, Contraction

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 22 / 32 Weakening, Invertibility, Contraction

Proof. Each claim can be proved by a simple (if rather tedious) induction on n.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 22 / 32 Proof idea. We push uppermost cuts upwards in GIL-derivations until they reach axioms and disappear, e.g...... (id) . Σ ⇒ δ Π,δ ⇒ δ =⇒ (cut) Σ, Π ⇒ δ Σ, Π ⇒ δ ...... Σ ⇒ α Π,α ⇒ δ Π, β ⇒ δ (⇒∨) (∨⇒) =⇒ Σ ⇒ α Π,α ⇒ δ Σ ⇒ α ∨ β l Π,α ∨ β ⇒ δ (cut) (cut) Σ, Π ⇒ δ Σ, Π ⇒ δ

Cut Elimination

Theorem Any GIL-derivable sequent is GIL◦-derivable.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 23 / 32 . . . (id) . Σ ⇒ δ Π,δ ⇒ δ =⇒ (cut) Σ, Π ⇒ δ Σ, Π ⇒ δ ...... Σ ⇒ α Π,α ⇒ δ Π, β ⇒ δ (⇒∨) (∨⇒) =⇒ Σ ⇒ α Π,α ⇒ δ Σ ⇒ α ∨ β l Π,α ∨ β ⇒ δ (cut) (cut) Σ, Π ⇒ δ Σ, Π ⇒ δ

Cut Elimination

Theorem Any GIL-derivable sequent is GIL◦-derivable.

Proof idea. We push uppermost cuts upwards in GIL-derivations until they reach axioms and disappear, e.g.. . .

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 23 / 32 . =⇒ . Σ, Π ⇒ δ

...... Σ ⇒ α Π,α ⇒ δ Π, β ⇒ δ (⇒∨) (∨⇒) =⇒ Σ ⇒ α Π,α ⇒ δ Σ ⇒ α ∨ β l Π,α ∨ β ⇒ δ (cut) (cut) Σ, Π ⇒ δ Σ, Π ⇒ δ

Cut Elimination

Theorem Any GIL-derivable sequent is GIL◦-derivable.

Proof idea. We push uppermost cuts upwards in GIL-derivations until they reach axioms and disappear, e.g...... (id) Σ ⇒ δ Π,δ ⇒ δ (cut) Σ, Π ⇒ δ

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 23 / 32 ...... Σ ⇒ α Π,α ⇒ δ Π, β ⇒ δ (⇒∨) (∨⇒) =⇒ Σ ⇒ α Π,α ⇒ δ Σ ⇒ α ∨ β l Π,α ∨ β ⇒ δ (cut) (cut) Σ, Π ⇒ δ Σ, Π ⇒ δ

Cut Elimination

Theorem Any GIL-derivable sequent is GIL◦-derivable.

Proof idea. We push uppermost cuts upwards in GIL-derivations until they reach axioms and disappear, e.g...... (id) . Σ ⇒ δ Π,δ ⇒ δ =⇒ (cut) Σ, Π ⇒ δ Σ, Π ⇒ δ

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 23 / 32 . . . . =⇒ Σ ⇒ α Π,α ⇒ δ (cut) Σ, Π ⇒ δ

Cut Elimination

Theorem Any GIL-derivable sequent is GIL◦-derivable.

Proof idea. We push uppermost cuts upwards in GIL-derivations until they reach axioms and disappear, e.g...... (id) . Σ ⇒ δ Π,δ ⇒ δ =⇒ (cut) Σ, Π ⇒ δ Σ, Π ⇒ δ

...... Σ ⇒ α Π,α ⇒ δ Π, β ⇒ δ (⇒∨) (∨⇒) Σ ⇒ α ∨ β l Π,α ∨ β ⇒ δ (cut) Σ, Π ⇒ δ

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 23 / 32 Cut Elimination

Theorem Any GIL-derivable sequent is GIL◦-derivable.

Proof idea. We push uppermost cuts upwards in GIL-derivations until they reach axioms and disappear, e.g...... (id) . Σ ⇒ δ Π,δ ⇒ δ =⇒ (cut) Σ, Π ⇒ δ Σ, Π ⇒ δ ...... Σ ⇒ α Π,α ⇒ δ Π, β ⇒ δ (⇒∨) (∨⇒) =⇒ Σ ⇒ α Π,α ⇒ δ Σ ⇒ α ∨ β l Π,α ∨ β ⇒ δ (cut) (cut) Σ, Π ⇒ δ Σ, Π ⇒ δ

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 23 / 32 by induction on the lexicographically ordered pair h|α|, m + ni. E.g., suppose that α is β → γ and the derivations of the premises end with ...... Σ,β ⇒ γ and Π,β → γ ⇒ β Π,γ ⇒ δ (→⇒) (→⇒) Σ ⇒ β → γ Π,β → γ ⇒ δ We apply the induction hypothesis three times: 1. `m Σ ⇒ β → γ and `n−1 Π,β → γ ⇒ β yields ` Σ, Π ⇒ β GIL◦ GIL◦ GIL◦

2. `GIL◦ Σ, Π ⇒ β and Σ,β ⇒ γ yields `GIL◦ Σ, Σ, Π ⇒ γ

3. `GIL◦ Σ, Σ, Π ⇒ γ and Π,γ ⇒ δ yields `GIL◦ Σ, Σ, Π, Π ⇒ δ.

Finally, using the previous lemma, `GIL◦ Σ, Π ⇒ δ.

More formally. . .

We prove (constructively) that `m Σ ⇒ α and `n Π,α ⇒ δ =⇒ ` Σ, Π ⇒ δ, GIL◦ GIL◦ GIL◦

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 24 / 32 E.g., suppose that α is β → γ and the derivations of the premises end with ...... Σ,β ⇒ γ and Π,β → γ ⇒ β Π,γ ⇒ δ (→⇒) (→⇒) Σ ⇒ β → γ Π,β → γ ⇒ δ We apply the induction hypothesis three times: 1. `m Σ ⇒ β → γ and `n−1 Π,β → γ ⇒ β yields ` Σ, Π ⇒ β GIL◦ GIL◦ GIL◦

2. `GIL◦ Σ, Π ⇒ β and Σ,β ⇒ γ yields `GIL◦ Σ, Σ, Π ⇒ γ

3. `GIL◦ Σ, Σ, Π ⇒ γ and Π,γ ⇒ δ yields `GIL◦ Σ, Σ, Π, Π ⇒ δ.

Finally, using the previous lemma, `GIL◦ Σ, Π ⇒ δ.

More formally. . .

We prove (constructively) that `m Σ ⇒ α and `n Π,α ⇒ δ =⇒ ` Σ, Π ⇒ δ, GIL◦ GIL◦ GIL◦ by induction on the lexicographically ordered pair h|α|, m + ni.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 24 / 32 We apply the induction hypothesis three times: 1. `m Σ ⇒ β → γ and `n−1 Π,β → γ ⇒ β yields ` Σ, Π ⇒ β GIL◦ GIL◦ GIL◦

2. `GIL◦ Σ, Π ⇒ β and Σ,β ⇒ γ yields `GIL◦ Σ, Σ, Π ⇒ γ

3. `GIL◦ Σ, Σ, Π ⇒ γ and Π,γ ⇒ δ yields `GIL◦ Σ, Σ, Π, Π ⇒ δ.

Finally, using the previous lemma, `GIL◦ Σ, Π ⇒ δ.

More formally. . .

We prove (constructively) that `m Σ ⇒ α and `n Π,α ⇒ δ =⇒ ` Σ, Π ⇒ δ, GIL◦ GIL◦ GIL◦ by induction on the lexicographically ordered pair h|α|, m + ni. E.g., suppose that α is β → γ and the derivations of the premises end with ...... Σ,β ⇒ γ and Π,β → γ ⇒ β Π,γ ⇒ δ (→⇒) (→⇒) Σ ⇒ β → γ Π,β → γ ⇒ δ

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 24 / 32 1. `m Σ ⇒ β → γ and `n−1 Π,β → γ ⇒ β yields ` Σ, Π ⇒ β GIL◦ GIL◦ GIL◦

2. `GIL◦ Σ, Π ⇒ β and Σ,β ⇒ γ yields `GIL◦ Σ, Σ, Π ⇒ γ

3. `GIL◦ Σ, Σ, Π ⇒ γ and Π,γ ⇒ δ yields `GIL◦ Σ, Σ, Π, Π ⇒ δ.

Finally, using the previous lemma, `GIL◦ Σ, Π ⇒ δ.

More formally. . .

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 24 / 32 2. `GIL◦ Σ, Π ⇒ β and Σ,β ⇒ γ yields `GIL◦ Σ, Σ, Π ⇒ γ

3. `GIL◦ Σ, Σ, Π ⇒ γ and Π,γ ⇒ δ yields `GIL◦ Σ, Σ, Π, Π ⇒ δ.

Finally, using the previous lemma, `GIL◦ Σ, Π ⇒ δ.

More formally. . .

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 24 / 32 3. `GIL◦ Σ, Σ, Π ⇒ γ and Π,γ ⇒ δ yields `GIL◦ Σ, Σ, Π, Π ⇒ δ.

Finally, using the previous lemma, `GIL◦ Σ, Π ⇒ δ.

More formally. . .

2. `GIL◦ Σ, Π ⇒ β and Σ,β ⇒ γ yields `GIL◦ Σ, Σ, Π ⇒ γ

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 24 / 32 Finally, using the previous lemma, `GIL◦ Σ, Π ⇒ δ.

More formally. . .

2. `GIL◦ Σ, Π ⇒ β and Σ,β ⇒ γ yields `GIL◦ Σ, Σ, Π ⇒ γ

3. `GIL◦ Σ, Σ, Π ⇒ γ and Π,γ ⇒ δ yields `GIL◦ Σ, Σ, Π, Π ⇒ δ.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 24 / 32 More formally. . .

2. `GIL◦ Σ, Π ⇒ β and Σ,β ⇒ γ yields `GIL◦ Σ, Σ, Π ⇒ γ

3. `GIL◦ Σ, Σ, Π ⇒ γ and Π,γ ⇒ δ yields `GIL◦ Σ, Σ, Π, Π ⇒ δ.

Finally, using the previous lemma, `GIL◦ Σ, Π ⇒ δ.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 24 / 32 2. Use GIL◦ to prove the disjunction property for intuitionistic logic

ÌL α ∨ β =⇒ ÌL α or ÌL β.

3. Give an algorithm to check if formulas α1, . . . , αn are independent in intuitionistic logic, that is, to check if for any formula β(y1,..., yn),

`IL β(α1, . . . , αn) =⇒ `IL β.

A First Quiz

◦ 1. Explain how GIL can be used to decide if Γ `IL α for Γ ﬁnite.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 25 / 32 3. Give an algorithm to check if formulas α1, . . . , αn are independent in intuitionistic logic, that is, to check if for any formula β(y1,..., yn),

`IL β(α1, . . . , αn) =⇒ `IL β.

A First Quiz

◦ 1. Explain how GIL can be used to decide if Γ `IL α for Γ ﬁnite. 2. Use GIL◦ to prove the disjunction property for intuitionistic logic

ÌL α ∨ β =⇒ ÌL α or ÌL β.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 25 / 32 A First Quiz

◦ 1. Explain how GIL can be used to decide if Γ `IL α for Γ ﬁnite. 2. Use GIL◦ to prove the disjunction property for intuitionistic logic

ÌL α ∨ β =⇒ ÌL α or ÌL β.

3. Give an algorithm to check if formulas α1, . . . , αn are independent in intuitionistic logic, that is, to check if for any formula β(y1,..., yn),

`IL β(α1, . . . , αn) =⇒ `IL β.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 25 / 32 Proof.

To decide Γ ÌL α for a finite set of formulas Γ ∪ {α}, we search for a derivation of Γ ⇒ α in GIL◦. If the left hand sides of sequents are viewed as sets, which is possible because of the admissibility of contraction, then there are only finitely many sequents that can occur in such a derivation. Hence, loop-checking and backtracking can be used to check all potential derivations.

Corollary The quasi-equational theory of Heyting algebras is decidable.

Decidability

Theorem (Gentzen 1935) Finitary consequence in intuitionistic logic is decidable.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 26 / 32 If the left hand sides of sequents are viewed as sets, which is possible because of the admissibility of contraction, then there are only ﬁnitely many sequents that can occur in such a derivation. Hence, loop-checking and backtracking can be used to check all potential derivations.

Corollary The quasi-equational theory of Heyting algebras is decidable.

Decidability

Theorem (Gentzen 1935) Finitary consequence in intuitionistic logic is decidable.

Proof.

To decide Γ `IL α for a ﬁnite set of formulas Γ ∪ {α}, we search for a derivation of Γ ⇒ α in GIL◦.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 26 / 32 which is possible because of the admissibility of contraction, then there are only ﬁnitely many sequents that can occur in such a derivation. Hence, loop-checking and backtracking can be used to check all potential derivations.

Corollary The quasi-equational theory of Heyting algebras is decidable.

Decidability

Theorem (Gentzen 1935) Finitary consequence in intuitionistic logic is decidable.

Proof.

To decide Γ `IL α for a ﬁnite set of formulas Γ ∪ {α}, we search for a derivation of Γ ⇒ α in GIL◦. If the left hand sides of sequents are viewed as sets,

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 26 / 32 then there are only ﬁnitely many sequents that can occur in such a derivation. Hence, loop-checking and backtracking can be used to check all potential derivations.

Corollary The quasi-equational theory of Heyting algebras is decidable.

Decidability

Theorem (Gentzen 1935) Finitary consequence in intuitionistic logic is decidable.

Proof.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 26 / 32 Hence, loop-checking and backtracking can be used to check all potential derivations.

Corollary The quasi-equational theory of Heyting algebras is decidable.

Decidability

Theorem (Gentzen 1935) Finitary consequence in intuitionistic logic is decidable.

Proof.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 26 / 32 Corollary The quasi-equational theory of Heyting algebras is decidable.

Decidability

Theorem (Gentzen 1935) Finitary consequence in intuitionistic logic is decidable.

Proof.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 26 / 32 Decidability

Theorem (Gentzen 1935) Finitary consequence in intuitionistic logic is decidable.

Proof.

Corollary The quasi-equational theory of Heyting algebras is decidable.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 26 / 32 Proof. Just consider the last step of a derivation of ⇒ α ∨ β in GIL◦.

Note. It follows similarly that the following Visser rules are admissible in intuitionistic logic: Vn i=1(αi → βi ) → (αn+1 ∨ αn+2) Wn+2 Vn n = 0, 1, 2,... j=1 ( i=1(αi → βi ) → αj )

The Disjunction Property

Corollary For any formulas α, β,

ÌL α ∨ β =⇒ ÌL α or ÌL β.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 27 / 32 Note. It follows similarly that the following Visser rules are admissible in intuitionistic logic: Vn i=1(αi → βi ) → (αn+1 ∨ αn+2) Wn+2 Vn n = 0, 1, 2,... j=1 ( i=1(αi → βi ) → αj )

The Disjunction Property

Corollary For any formulas α, β,

ÌL α ∨ β =⇒ ÌL α or ÌL β.

Proof. Just consider the last step of a derivation of ⇒ α ∨ β in GIL◦.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 27 / 32 The Disjunction Property

Corollary For any formulas α, β,

ÌL α ∨ β =⇒ ÌL α or ÌL β.

Proof. Just consider the last step of a derivation of ⇒ α ∨ β in GIL◦.

Note. It follows similarly that the following Visser rules are admissible in intuitionistic logic: Vn i=1(αi → βi ) → (αn+1 ∨ αn+2) Wn+2 Vn n = 0, 1, 2,... j=1 ( i=1(αi → βi ) → αj )

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 27 / 32 Proof sketch. We prove that for any sequent Σ(x, y), Π(y, z) ⇒ δ(y, z),

there exists a formula γ(y) such that `n Σ, Π ⇒ δ =⇒ GIL◦ `GIL◦ Σ ⇒ γ and `GIL◦ Π, γ ⇒ δ, by induction on n. Base case. E.g., if Σ is Σ0, δ, let γ = δ; if Π is Π0, δ, let γ = >.

Interpolation

Theorem (Schütte 1962)

If α(x, y) and β(y, z) are formulas such that α `IL β, then there exists a

formula γ(y) such that α `IL γ and γ `IL β.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 28 / 32 there exists a formula γ(y) such that `n Σ, Π ⇒ δ =⇒ GIL◦ `GIL◦ Σ ⇒ γ and `GIL◦ Π, γ ⇒ δ, by induction on n. Base case. E.g., if Σ is Σ0, δ, let γ = δ; if Π is Π0, δ, let γ = >.

Interpolation

Theorem (Schütte 1962)

If α(x, y) and β(y, z) are formulas such that α `IL β, then there exists a

formula γ(y) such that α `IL γ and γ `IL β.

Proof sketch. We prove that for any sequent Σ(x, y), Π(y, z) ⇒ δ(y, z),

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 28 / 32 by induction on n. Base case. E.g., if Σ is Σ0, δ, let γ = δ; if Π is Π0, δ, let γ = >.

Interpolation

Theorem (Schütte 1962)

If α(x, y) and β(y, z) are formulas such that α `IL β, then there exists a

formula γ(y) such that α `IL γ and γ `IL β.

Proof sketch. We prove that for any sequent Σ(x, y), Π(y, z) ⇒ δ(y, z),

there exists a formula γ(y) such that `n Σ, Π ⇒ δ =⇒ GIL◦ `GIL◦ Σ ⇒ γ and `GIL◦ Π, γ ⇒ δ,

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 28 / 32 Base case. E.g., if Σ is Σ0, δ, let γ = δ; if Π is Π0, δ, let γ = >.

Interpolation

Theorem (Schütte 1962)

If α(x, y) and β(y, z) are formulas such that α `IL β, then there exists a

formula γ(y) such that α `IL γ and γ `IL β.

Proof sketch. We prove that for any sequent Σ(x, y), Π(y, z) ⇒ δ(y, z),

there exists a formula γ(y) such that `n Σ, Π ⇒ δ =⇒ GIL◦ `GIL◦ Σ ⇒ γ and `GIL◦ Π, γ ⇒ δ, by induction on n.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 28 / 32 if Π is Π0, δ, let γ = >.

Interpolation

Theorem (Schütte 1962)

If α(x, y) and β(y, z) are formulas such that α `IL β, then there exists a

formula γ(y) such that α `IL γ and γ `IL β.

Proof sketch. We prove that for any sequent Σ(x, y), Π(y, z) ⇒ δ(y, z),

there exists a formula γ(y) such that `n Σ, Π ⇒ δ =⇒ GIL◦ `GIL◦ Σ ⇒ γ and `GIL◦ Π, γ ⇒ δ, by induction on n. Base case. E.g., if Σ is Σ0, δ, let γ = δ;

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 28 / 32 Interpolation

Theorem (Schütte 1962)

If α(x, y) and β(y, z) are formulas such that α `IL β, then there exists a

formula γ(y) such that α `IL γ and γ `IL β.

Proof sketch. We prove that for any sequent Σ(x, y), Π(y, z) ⇒ δ(y, z),

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 28 / 32 then by the induction hypothesis twice, there exist formulas γ1(y), γ2(y) such that the following sequents are GIL-derivable:

0 0 Π ⇒ γ1;Σ , α → β, γ1 ⇒ α; Σ , β ⇒ γ2; and Π, γ2 ⇒ δ.

We obtain an interpolant γ1 → γ2 with derivations ending with . . . . 0 0 . . Σ , α → β, γ1 ⇒ α Σ , β, γ1 ⇒ γ2 . . (→⇒) . . 0 Σ , α → β, γ1 ⇒ γ2 Π, γ → γ ⇒ γ Π, γ ⇒ δ (⇒→) 1 2 1 2 0 (→⇒) Σ , α → β ⇒ γ1 → γ2 Π, γ1 → γ2 ⇒ δ

Interpolation

Inductive step. E.g., if Σ is Σ0, α → β and the derivation ends with . . . . Σ0, α → β, Π ⇒ α Σ0, β, Π ⇒ δ (→⇒) Σ0, α → β, Π ⇒ δ

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 29 / 32 0 0 Π ⇒ γ1;Σ , α → β, γ1 ⇒ α; Σ , β ⇒ γ2; and Π, γ2 ⇒ δ.

Interpolation

Inductive step. E.g., if Σ is Σ0, α → β and the derivation ends with . . . . Σ0, α → β, Π ⇒ α Σ0, β, Π ⇒ δ (→⇒) Σ0, α → β, Π ⇒ δ

then by the induction hypothesis twice, there exist formulas γ1(y), γ2(y) such that the following sequents are GIL-derivable:

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 29 / 32 0 Σ , β ⇒ γ2; and Π, γ2 ⇒ δ.

Interpolation

Inductive step. E.g., if Σ is Σ0, α → β and the derivation ends with . . . . Σ0, α → β, Π ⇒ α Σ0, β, Π ⇒ δ (→⇒) Σ0, α → β, Π ⇒ δ

then by the induction hypothesis twice, there exist formulas γ1(y), γ2(y) such that the following sequents are GIL-derivable:

0 Π ⇒ γ1;Σ , α → β, γ1 ⇒ α;

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 29 / 32 We obtain an interpolant γ1 → γ2 with derivations ending with . . . . 0 0 . . Σ , α → β, γ1 ⇒ α Σ , β, γ1 ⇒ γ2 . . (→⇒) . . 0 Σ , α → β, γ1 ⇒ γ2 Π, γ → γ ⇒ γ Π, γ ⇒ δ (⇒→) 1 2 1 2 0 (→⇒) Σ , α → β ⇒ γ1 → γ2 Π, γ1 → γ2 ⇒ δ

Interpolation

Inductive step. E.g., if Σ is Σ0, α → β and the derivation ends with . . . . Σ0, α → β, Π ⇒ α Σ0, β, Π ⇒ δ (→⇒) Σ0, α → β, Π ⇒ δ

then by the induction hypothesis twice, there exist formulas γ1(y), γ2(y) such that the following sequents are GIL-derivable:

0 0 Π ⇒ γ1;Σ , α → β, γ1 ⇒ α; Σ , β ⇒ γ2; and Π, γ2 ⇒ δ.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 29 / 32 . . . . 0 0 . . Σ , α → β, γ1 ⇒ α Σ , β, γ1 ⇒ γ2 . . (→⇒) . . 0 Σ , α → β, γ1 ⇒ γ2 Π, γ → γ ⇒ γ Π, γ ⇒ δ (⇒→) 1 2 1 2 0 (→⇒) Σ , α → β ⇒ γ1 → γ2 Π, γ1 → γ2 ⇒ δ

Interpolation

Inductive step. E.g., if Σ is Σ0, α → β and the derivation ends with . . . . Σ0, α → β, Π ⇒ α Σ0, β, Π ⇒ δ (→⇒) Σ0, α → β, Π ⇒ δ

then by the induction hypothesis twice, there exist formulas γ1(y), γ2(y) such that the following sequents are GIL-derivable:

0 0 Π ⇒ γ1;Σ , α → β, γ1 ⇒ α; Σ , β ⇒ γ2; and Π, γ2 ⇒ δ.

We obtain an interpolant γ1 → γ2 with derivations ending with

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 29 / 32 . . . . Π, γ1 → γ2 ⇒ γ1 Π, γ2 ⇒ δ (→⇒) Π, γ1 → γ2 ⇒ δ

Interpolation

Inductive step. E.g., if Σ is Σ0, α → β and the derivation ends with . . . . Σ0, α → β, Π ⇒ α Σ0, β, Π ⇒ δ (→⇒) Σ0, α → β, Π ⇒ δ

then by the induction hypothesis twice, there exist formulas γ1(y), γ2(y) such that the following sequents are GIL-derivable:

0 0 Π ⇒ γ1;Σ , α → β, γ1 ⇒ α; Σ , β ⇒ γ2; and Π, γ2 ⇒ δ.

We obtain an interpolant γ1 → γ2 with derivations ending with . . . . 0 0 Σ , α → β, γ1 ⇒ α Σ , β, γ1 ⇒ γ2 (→⇒) 0 Σ , α → β, γ1 ⇒ γ2 (⇒→) 0 Σ , α → β ⇒ γ1 → γ2

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 29 / 32 Interpolation

Inductive step. E.g., if Σ is Σ0, α → β and the derivation ends with . . . . Σ0, α → β, Π ⇒ α Σ0, β, Π ⇒ δ (→⇒) Σ0, α → β, Π ⇒ δ

then by the induction hypothesis twice, there exist formulas γ1(y), γ2(y) such that the following sequents are GIL-derivable:

0 0 Π ⇒ γ1;Σ , α → β, γ1 ⇒ α; Σ , β ⇒ γ2; and Π, γ2 ⇒ δ.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 29 / 32 Proof. By construction or as a consequence of interpolation (shown later).

An Algebraic Consequence

Theorem (Day 1972) HA admits the amalgamation property; that is, for any A, B, C ∈ HA and embeddings i : A → B and j : A → C, there exist D ∈ HA and embeddings h: B → D and k : C → D satisfying hi = kj. B i h

A D

j k C

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 30 / 32 An Algebraic Consequence

A D

j k C

Proof. By construction or as a consequence of interpolation (shown later).

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 30 / 32 explain some of the nuts and bolts of universal algebra

deﬁne and interpret consequence in classes of algebraic structures

Tomorrow

We will. . . consider Pitts’ uniform interpolation theorem for intuitionistic logic

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 31 / 32 deﬁne and interpret consequence in classes of algebraic structures

Tomorrow

We will. . . consider Pitts’ uniform interpolation theorem for intuitionistic logic

explain some of the nuts and bolts of universal algebra

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 31 / 32 Tomorrow

We will. . . consider Pitts’ uniform interpolation theorem for intuitionistic logic

explain some of the nuts and bolts of universal algebra

deﬁne and interpret consequence in classes of algebraic structures

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 31 / 32 Tomorrow

We will. . . consider Pitts’ uniform interpolation theorem for intuitionistic logic

explain some of the nuts and bolts of universal algebra

deﬁne and interpret consequence in classes of algebraic structures

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 31 / 32 References

A. Day, Varieties of Heyting algebras, II (Amalgamation and injectivity). Unpublished note (1972).

D. de Jongh and L.A. Chagrova. The decidability of dependency in intuitionistic propositional logic. Journal of Symbolic Logic 60 (1995), no. 2, 498–504.

R. Dyckhoﬀ. Intuitionistic decision procedures since Gentzen. Advances in Proof Theory, Birkhäuser (2016), 245–267.

S. Ghilardi and M. Zawadowski. Sheaves, Games and Model Completions, Kluwer (2002).

A.M. Pitts. On an interpretation of second-order quantiﬁcation in ﬁrst-order intuitionistic propositional logic. Journal of Symbolic Logic 57 (1992), 33–52.

K. Schütte. Der Interpolationssatz der intuitionistischen Pradikatenlogik. Mathematische Annalen 148 (1962), 192–200.

George Metcalfe (University of Bern) Bridges between Logic and Algebra June 2019 32 / 32