Albert-Ludwigs-Universitat FreiburgFakultat fur Mathematik und Physik

Mathematisches Institut

The model completion for the theoryof Heyting algebras

Bachelor Thesis

Dennis Muller

Matrikel-Nr. 3104758

August 11, 2013Advisor: Dr. Markus Junker

Contents

0 Introduction and Notation 2

1 Intuitionistic logic 3

1.1 Intuitionistic propositional logic (IpC) . . . . . . . . . . . . . . . . . . . . . 3

1.2 IpC2 and Pitts’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Heyting algebras 14

2.1 Lattices and Heyting algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Heyting algebras as models of IpC . . . . . . . . . . . . . . . . . . . . . . . 18

3 Model companions and completions 23

4 The model completion of TH 27

5 Appendix 31

1

0 Introduction and Notation

The goal of this bachelor thesis is to prove the existence of a model completion for the

theory of Heyting algebras (as outlined in [GZ97]). Heyting algebras can be interpreted

as models of intuitionistic propositional logic (IpC); a fact that will be used extensively.

The proof is based on the main theorem of [Pit92], which states that every second order

propositional formula (with quantification over propositional variables) is intuitionistically

equivalent to a first order formula. We can use this to show that the class of existentially

closed Heyting algebras is an elementary class, proving that the theory has a model com-

panion. The fact that Heyting algebras have the amalgamation property implies, that this

model companion is in deed a model completion.

Some rudimentary knowledge of basic concepts in mathematical logic is expected;

however, the prerequisites are attempted to be held at a reasonable minimum. Proofs

taken from different authors are referenced as such; all others are my own work and

should therefore be escpecially subject to critical consideration.

Intuitionistic logic and Pitts’ Theorem are presented in section 1. In section 2, we will

define lattices and Heyting algebras and explore their relation to IpC. In section 3 we will

define model completions and present related model theoretic concepts and results, before

we will take on the main proof in section 4.

In formulae (both in the propositional and predicate logical calculus), we will use the

symbols ¬,∀,∃,∧,∨,→,>,⊥ in that order of connective strength from strongest to weak-

est (e.g. A ∧ B → C is to be read as (A ∧ B) → C). The symbols ⇒,⇔ will be used

for implication and equivalence of statements outside of formulae. In lattices and Heyting

algebras we will use the symbols u,t, ↪→ for join, meet and the relative pseudocomplement

respectively. Their connective strength is analogous to the logical symbols. The letters

A,B,C, ... will represent structures, whereas A,B,C, ... will denote their underlying uni-

verses. If a formula ϕ holds in a model A or a theory T , we will write A |= ϕ or T |= ϕ.

Substituting a variable x by some term or formula t in some formula ϕ will be written as

ϕ[tx

]. We write ϕ(x, y) to express that the formula ϕ contains the free variables x and y,

where y will denote a tuple of variables. Consequently, for a parameter or set of parameters

a, we will sometimes write ϕ(a) for ϕ(x)[ax

], if it is clear which variables are supposed to

be substituted. In propositional logic, every propositional variable is considered to be free.

To avoid confusion, we denote equality within formulae by.=. We write A ≺ B to express

that A is an elementary substructure of B. A ≡ B will denote elementary equivalence.

For a subset B ⊆ A, the pair (A, B) will denote the structure A in the language extended

by constant symbols from B.

All used notations are listed again in table 3.

2

1 Intuitionistic logic

Intuitionism as a philosophy goes back to the beginning of the twentieth century, particu-

larly to L.E.J. Brouwer.1 For him, all mathematical objects and properties were mental

constructions and consequently, proofs ought to be similarly constructive. In particular,

he considered proofs by contradiction (especially for the existence of objects with certain

properties) to be unsatisfactory, and he rejected the law of the excluded middle, i.e. the

proposition that “each particular mathematical problem can be solved in the sense that

the question under consideration can either be affirmed, or refuted”.2

Intuitionistic logic was first fully formalized by Arend Heyting in 1928, even though

Brouwer himself did not consider doing so useful in any way. Heyting also gave an in-

tuitionistic formalization for arithmetic as well as for a kind of set theory (based on the

notion of a “species”3). Several useful interpretations of intuitionistic logic have since

been discovered, showing connections to the closure operator in topology, lattices and the

lambda calculus4, which has caused some interest among computer scientists.

1.1 Intuitionistic propositional logic (IpC)

The syntax in intuitionstic logic is the same as in classical logic. But unlike in classical

logic, we will need all of the connectives ¬,∧,∨,→ in our axioms, as in IpC none of these

is definable in terms of the others. We can however (as we will show) add the symbols >and ⊥ (for “true” and “false”, respectively) and then replace ¬ϕ with ϕ→ ⊥.

We will use a Hilbert-type calculus, similar to the formalization given by Heyting5, but

clearer with regard to the rules of inference. We write `I ϕ if the formula ϕ is derivable

in IpC.

Definition 1.1 (IpC). 6 Intuitionistic propositional logic has the following axioms (for all

formulae ϕ,ψ, χ):

IpC1 `I ϕ→ (ψ → ϕ)

IpC2 `I (ϕ→ ψ)→ ((ϕ→ (ψ → χ))→ (ϕ→ χ))

IpC3 `I ϕ→ (ψ → (ϕ ∧ ψ))

IpC4 `I (ϕ ∧ ψ)→ ϕ `I (ϕ ∧ ψ)→ ψ

IpC5 `I ϕ→ (ϕ ∨ ψ) `I ψ → (ϕ ∨ ψ)

IpC6 `I (ϕ→ χ)→ ((ψ → χ)→ ((ϕ ∨ ψ)→ χ))

1For more on the history of intuitionism see [VD86] or for intuitionism itself [Hey71].2 Attributed to Hilbert in [VD86, p.227]3[Hey71, p.37]4 via the curry-howard isomorphism, see [SU98]5[Hey71, p.105]6This formalization is from [VD86, p.234]

3

IpC7 `I (ϕ→ ψ)→ ((ϕ→ ¬ψ)→ ¬ϕ)

IpC8 `I ϕ→ (¬ϕ→ ψ)

and the following rule of inference:

Modus Ponens (M.P.): If `I ϕ and `I ϕ→ ψ, then `I ψ.

If we add `I (ϕ∨¬ϕ) we get classical propositional logic. In fact, Godel and Gentzen

showed that a formula ϕ is provable in classical propositional logic, iff `I ¬¬ϕ.7 We call

formulae that are provable directly from axioms tautologies.

Let Γ be a propositional theory (i.e. a set of formulae). We write Γ `I ϕ if

• ϕ ∈ Γ or

• ϕ is derivable from formulas in Γ and axioms using Modus Ponens.

Obviously, if Γ′ `I ϕ and Γ′ ⊆ Γ, then Γ `I ϕ. We write ψ `I ϕ for {ψ} `I ϕ. Also, it

is useful to note that by this definition proofs in IpC are finite, which allows us to restrict

ourselves in many cases to finite subsets of theories. We call two formulae ϕ,ψ logically

equivalent (in Γ), if Γ∪{ϕ} `I ψ and Γ∪{ψ} `I ϕ and denote this by writing Γ `I ϕ↔ ψ.

This notation will be justified by theorem 1.1, since it implies that ϕ and ψ are logically

equivalent iff Γ `I (ϕ → ψ) ∧ (ψ → ϕ). As usual in mathematical logic, we are (almost

always) only interested in formulae up to logical equivalence.

Before we prove the deduction theorem, which will be of enormous utility, we will first

prove the following statements:

Lemma 1.1.

(i) `I (ϕ ∧ ψ) iff `I (ψ ∧ ϕ)

(ii) `I (ϕ ∧ ψ) ∧ χ iff `I ϕ ∧ (ψ ∧ χ)

(iii) `I ϕ→ ϕ

Proof.

(i)

`I ϕ ∧ ψ (1)

`I (ϕ ∧ ψ)→ ϕ (2) (IpC4)

`I (ϕ ∧ ψ)→ ψ (3) (IpC4)

`I ϕ (4) (M.P. on (1) and (2))

`I ψ (5) (M.P. on (1) and (3))

`I ψ → (ϕ→ (ψ ∧ ϕ)) (6) (IpC3)

`I ϕ→ (ψ ∧ ϕ) (7) (M.P. on (5) and (6))

`I ψ ∧ ϕ (8) (M.P. on (4) and (7))

7[VD86, p.229]

4

(ii)

`I (ϕ ∧ ψ) ∧ χ (1)

`I ((ϕ ∧ ψ) ∧ χ)→ (ϕ ∧ ψ) (2) (IpC4)

`I ((ϕ ∧ ψ) ∧ χ)→ χ (3) (IpC4)

`I (ϕ ∧ ψ) (4) (M.P. on (1) and (2))

`I χ (5) (M.P. on (1) and (3))

`I (ϕ ∧ ψ)→ ϕ (6) (IpC4)

`I (ϕ ∧ ψ)→ ψ (7) (IpC4)

`I ϕ (8) (M.P. on (4) and (6))

`I ψ (9) (M.P. on (4) and (7))

`I ψ → (χ→ (ψ ∧ χ)) (10) (IpC3)

`I ψ ∧ χ (11) (M.P. on (9),(5) and (10))

`I ϕ→ ((ψ ∧ χ)→ (ϕ ∧ (ψ ∧ χ))) (12) (IpC3)

`I ϕ ∧ (ψ ∧ χ) (13) (M.P. on (8),(11) and (12))

The other direction can be proven analogously.

(iii)

`I ϕ→ (ϕ→ ϕ) (1) (IpC1)

`I (ϕ→ (ϕ→ ϕ))→ ((ϕ→ ((ϕ→ ϕ)→ ϕ))→ (ϕ→ ϕ)) (2) (IpC2)

`I (ϕ→ ((ϕ→ ϕ)→ ϕ))→ (ϕ→ ϕ) (3) (M.P. on (1) and (2))

`I ϕ→ ((ϕ→ ϕ)→ ϕ) (4) (IpC1)

`I ϕ→ ϕ (5) (M.P. on (4) and (3))

Remark 1. Commutativity and associativity now allow us to ignore parentheses and the or-

der in conjunctive (sub-)formulae. Hence, they also justifiy writing formulae ϕ1 ∧ ϕ2 ∧ ... ∧ ϕn

simply as

n∧i=1

ϕi.

Theorem 1.1 (Deduction Theorem). 8 {φ1, ..., φn} `I ϕ iff {φ1, ..., φn−1} `I φn → ϕ.

Proof. The implication from right to left follows directly from Modus Ponens.

Let {φ1, ..., φn} `I ϕ, then there is a finite sequence of formulae S = (S1, ..., Sm) such

that Sm = ϕ and each Si is either one of the φj , an axiom or follows by Modus Ponens

from two previous formulae. We will now define a sequence S∗ inductively:

• If Si is an axiom or one of the φj (j 6= n), we let S∗i = φn → Si, which is provable

from Si and IpC1 using M.P.

8The idea for this proof is well known.

5

• If Si = φn, we let S∗i = φn → φn, which is provable by Lemma 1.1.

• If Si follows by Modus Ponens from Sj and Sk = Sj → Si (j, k < i), then by

induction S∗j = φn → Sj and S∗k = φn → (Sj → Si) are provable from Γ. By IpC2

we have

(φn → Sj)→ ((φn → (Sj → Si))→ (φn → Si))

and by applying Modus Ponens twice we get φn → Si =: S∗i .

It follows, that every formula in S∗ is provable from {φ1, ..., φn−1} and by definition we

have S∗m = φn → ϕ.

Corollary 1.1. Γ `I ϕ iff there is a finite subset Γ′ ⊂ Γ such that `I∧φ∈Γ′

φ→ ϕ.

Proof. First, note that if Γ `I ϕ, then there is a finite sequence of formulae proving ϕ, as

in the previous proof. Hence, there is a finite subset Γ′ ⊂ Γ such, that Γ′ `I ϕ (Γ′ contains

only the formulae occuring in said finite sequence).

By remark 1 and using IpC3 and IpC4 repeatedly, we get

Γ′ `I∧φ∈Γ′

φ and∧φ∈Γ′

φ `i ψ for all ψ ∈ Γ′

and hence

Γ′ `I ϕ iff∧φ∈Γ′

φ `I ϕ iff `I∧φ∈Γ′

φ→ ϕ.

We can now define > := ϕ → ϕ (or any other IpC-tautology, since they are logically

equivalent) for any formula ϕ and ⊥ := ¬>. We call a propositional theory Γ inconsistent,

if Γ `I ⊥. This will allow us to prove some further simple, but useful statements:

Lemma 1.2.

(i) `I ϕ→ ¬ψ iff {ϕ,ψ} is inconsistent,

(ii) `I ¬ϕ iff `I ϕ→ ⊥,

(iii) `I ϕ→ ¬¬ϕ,

(iv) ϕ→ ψ `I ¬ψ → ¬ϕ,

(v) `I ¬ϕ iff `I ¬¬¬ϕ.

Remark 2. For (i), note that the same holds only in one direction if we take ϕ → ψ and

{ϕ,¬ψ} (in general, {ϕ,¬ψ} `I ⊥ does not imply `I ϕ → ψ). (ii) allows us to define

negation in terms of ⊥ and implication, which will be useful when we work with Heyting

algebras later on. Also, note that the converse statements to (iii) and (iv) do not hold,

which corresponds to rejecting (certain) proofs by contradiction.

6

Proof of lemma.

(i) Let `I ϕ → ¬ψ, then by M.P. {ϕ,ψ} `I ¬ψ and trivially {ϕ,ψ} `I ψ. By IpC8 we

have `I ψ → (¬ψ → ⊥) and by applying M.P. twice we get {ϕ,ψ} `I ⊥.

For the converse, let {ϕ,ψ} `I ⊥, then with the deduction theorem ϕ `I ψ → ¬>.

By IpC1 we have `I > → (ψ → >) and by IpC7 we have

`I (ψ → >)→ ((ψ → ¬>)→ ¬ψ).

Applying M.P. twice gives us ϕ `I ¬ψ and thus `I ϕ→ ¬ψ.

(ii) Let `I ¬ϕ, then by IpC8 and M.P. we get ϕ `I ⊥ and thus `I ϕ→ ⊥.

Conversely, let `I ϕ → ⊥, then {ϕ,>} is inconsistent and thus by (i) we have

`I > → ¬ϕ and by M.P. `I ¬ϕ.

(iii) Follows from the fact that {ϕ,¬ϕ} is inconsistent and (i).

(iv) We have by IpC7

`I (ϕ→ ψ)→ ((ϕ→ ¬ψ)→ ¬ϕ)

and by IpC1 ¬ψ `I ϕ→ ¬ψ. Hence, we get {ϕ→ ψ,¬ψ} `I ¬ϕ and thus

ϕ→ ψ `I ¬ψ → ¬ϕ.

(v) Implication from left to right follows from the fact that {¬ϕ,¬¬ϕ} is inconsistent.

For the converse, if `I ¬¬¬ϕ, then {>,¬¬ϕ} is inconsistent. From (iii) we get that

{>, ϕ} must be inconsistent as well and by (i) and M.P. we have `I ¬ϕ.

Also, only one of De Morgan’s Laws holds in IpC; the other one only holds for one

direction:

Lemma 1.3.

(i) ¬(ϕ ∨ ψ) `I ¬ϕ ∧ ¬ψ

(ii) ¬ϕ ∧ ¬ψ `I ¬(ϕ ∨ ψ)

(iii) ¬ϕ ∨ ¬ψ `I ¬(ϕ ∧ ψ)

Proof.

(i) We have `I ϕ→ (ϕ ∨ ψ) and thus `I ¬(ϕ ∨ ψ)→ ¬ϕ. We can do the same with ψ

and the claim follows.

(ii) We have ¬ϕ `I ϕ→ ⊥ and ¬ψ `I ψ → ⊥. By IpC8

`I (ϕ→ ⊥)→ ((ψ → ⊥)→ ((ϕ ∨ ψ)→ ⊥))

and hence ¬ϕ ∧ ¬ψ `I (ϕ ∨ ψ)→ ⊥.

7

(iii) By IpC6, we have

`I (¬ϕ→ ⊥)→ ((¬ψ → ⊥)→ ((¬ϕ ∨ ¬ψ)→ ⊥))

and by IpC8

ϕ ∧ ψ `I ¬ϕ→ ⊥ and ϕ ∧ ψ `I ¬ψ → ⊥

and thus {¬ϕ ∨ ¬ψ,ϕ ∧ ψ} `I ⊥. Therefore, the two formulae are inconsistent,

which implies the claim.

1.2 IpC2 and Pitts’ Theorem

Before we look at Pitts’ Theorem (which lies at the heart of the proof for our main

theorem) we need to introduce second order intuitionistic propositional logic (IpC2), which

is IpC extended by the usual quantifiers ∃ and ∀ ranging over propositional variables. The

grammar of IpC2 is consequently the usual one for propositional logic, extended by the

following rule:

Given a propositional variable P and an IpC2-formula ϕ, then ∀Pϕ is an IpC2-formula.

A sequent calculus for IpC2 can be found in [Pit92], however an actual calculus is not

necessary for our needs and thus omitted. The existential quantifier is then defined by

∃Pϕ := ∀Q(∀P (ϕ→ Q)→ Q),

where Q is any new propositional variable not occuring in any other formula under consi-

deration. This definition might seem unusual at first, however the classical definition of the

existential quantifier (i.e. ∃Pϕ = ¬∀P¬ϕ) would not work in IpC, since the introduction

of negations would lead to problems. We will see in corollary 1.2, that the definition given

here actually captures the intended meaning.

Given a first order propositional formula ϕ, we will denote by Var(ϕ) the set of all

propositional variables occuring in ϕ. Analogously we define Var for propositional theories.

Andrew Pitts showed in [Pit92] that IpC2 is already contained in IpC, in the sense

that for every second order propositional formula there is a logically equivalent first order

propositional formula. This result is trivial in classical propositional logic - since classical

logic is two valued, the formula ∃Pϕ is equivalent to ϕ[>P

]∨ϕ

[⊥P

]and ∀Pϕ is equivalent

to ϕ[>P

]∧ ϕ

[⊥P

].

Formally, what Pitts showed is the following:

Pitts’ Theorem. 9 Given a propositional variable P , for each first order intuitionistic

proposition ϕ there is a first order intuitionistic proposition ∀Pϕ with

Var(∀Pϕ) ⊆ Var(ϕ) \ {P} and satisfying:

(i) If Γ `I ϕ, then Γ `I ∀Pϕ, provided P /∈ Var(Γ) and

9[Pit92, Theorem 1]

8

(ii) If Γ `I ∀Pϕ, then for all ψ, Γ `I ϕ[ψP

].

Remark 3. It is important to keep in mind that ∀Pϕ is a first order formula not to be

confused with the second order formula ∀Pϕ. It might be helpful to think of ∀P as a unary

function on the set of first order formulae.

The proof for Pitts’ Theorem is purely proof-theoretical, but quite long and technical.

Consequently, I leave out the details:

Proof (sketch). The proof is based on a specific form of a cut-free Gentzen-style sequent

calculus labelled LJ∗ given in table 1. A sequent has the form ∆ � ϕ for a propositional

formula ϕ and a finite multiset (i.e. a set, where each element is assigned a certain mul-

tiplicity) of formulae ∆. We then have Γ `I ϕ iff there is a finite multiset ∆ built up

from formulae in Γ such that the sequent ∆ � ϕ is provable in this calculus, denoted by

` ∆ � ϕ.

We define an order <wt on formulae via the following weight-function:

• wt(⊥) = wt(P ) = 1 for any propositional variable P ,

• wt(ϕ ∨ ψ) = wt(ϕ→ ψ) = wt(ϕ) + wt(ψ) + 1,

• wt(ϕ ∧ ψ) = wt(ϕ) + wt(ψ) + 2.

We can then extend <wt to a relation between finite multisets, such that:

Γ <wt ∆ iff there are multisets ∆1,∆2,Γ′, ∆2 6= ∅ such that

• ∆ = ∆1 ∪∆2 and Γ = ∆1 ∪ Γ′,

• For all ϕ ∈ Γ′ there exists ψ ∈ ∆2 with ϕ <wt ψ.

Finally, we can extend <wt to sequents by letting

(∆1 � ϕ) <wt (∆2 � ψ) if (∆1 ∪ {ϕ}) <wt (∆2 ∪ {ψ}).10

LJ∗ is constructed such that the premise in each rule is always smaller with respect to

<wt than the conclusion (which is also the reason why we work with multisets instead of

“normal” sets).

Now, given a multiset ∆, formula ϕ and propositional variable P , we define finite sets

of formulae EP (∆) and AP (∆, ϕ) and corresponding formulae EP (∆) :=∧φ∈EP (∆) φ and

AP (∆, ϕ) :=∨φ∈AP (∆,ϕ) φ simultaniously via mutual recursion as in table 2.

We now have to show the following:

(i) Var(EP (∆)) ⊆ Var(∆) \ {P} and Var(AP (∆, ϕ)) ⊆ Var(∆ ∪ {ϕ}) \ {P}

(ii) ` ∆ � EP (∆) and ` ∆ ∪ {AP (∆, ϕ)} � ϕ10To clarify: Γ ∪ {ψ} means, the multiplicity of ψ in Γ is to be increased by 1. If ψ /∈ Γ, ψ is considered

to have multiplicity 0. Analogously for the union of two arbitrary multisets – the mutliplicities are simplyadded.

9

Γ ∪ {P} � P(Atom)

Γ ∪ {⊥} � ϕ(⊥ �)

Γ � ϕ Γ � ψΓ � ϕ ∧ ψ

(� ∧)Γ ∪ {ϕ,ψ} � χ

Γ ∪ {ϕ ∧ ψ} � χ(∧ �)

Γ � ϕΓ � ϕ ∨ ψ

(� ∨1)Γ � ψ

Γ � ϕ ∨ ψ(� ∨2)

Γ ∪ {ϕ} � χ Γ ∪ {ψ} � χΓ ∪ {ϕ ∨ ψ} � χ

(∨ �)Γ ∪ {ϕ} � ψΓ � ϕ→ ψ

(�→)

Γ ∪ {P,ϕ} � ψΓ ∪ {P, P → ϕ} � ψ

(→�1)Γ ∪ {ϕ→ (φ→ ψ)} � χΓ ∪ {(ϕ ∧ φ)→ ψ} � χ

(→�2)

Γ ∪ {ϕ→ φ, ψ → φ} � χΓ ∪ {(ϕ ∨ ψ)→ φ} � χ

(→�3)Γ ∪ {φ→ ψ} � ϕ→ φ Γ ∪ {ψ} � χ

Γ ∪ {(ϕ→ φ)→ ψ} � χ(→�4)

Table 1: The rules of LJ∗, for any formulae ϕ,ψ, χ, φ and propositional variable P .

∆ = EP (∆) contains

(E0) ∆′ ∪ {⊥} ⊥(E1) ∆′ ∪ {Q} EP (∆′) ∧Q(E2) ∆′ ∪ {φ ∧ ψ} EP (∆′ ∪ {φ, ψ})(E3) ∆′ ∪ {φ ∨ ψ} EP (∆′ ∪ {φ}) ∨ EP (∆′ ∪ {ψ})(E4) ∆′ ∪ {Q→ φ} Q→ EP (∆′ ∪ {φ})(E5) ∆′ ∪ {P, P → φ} EP (∆′ ∪ {P, φ})(E6) ∆′ ∪ {((φ ∧ ψ)→ χ)} EP (∆′ ∪ {(φ→ (ψ → χ))})(E7) ∆′ ∪ {((φ ∨ ψ)→ χ)} EP (∆′ ∪ {φ→ χ, ψ → χ})(E8) ∆′ ∪ {((φ→ ψ)→ χ)} (EP (∆′ ∪ {ψ → χ})→ AP (∆′ ∪ {ψ → χ} , φ→ ψ))

→ EP (∆′ ∪ {χ})

(∆, ϕ) = AP (∆, ϕ) contains

(A1) (∆′ ∪ {Q} , ϕ) AP (∆′, ϕ)(A2) (∆′ ∪ {φ ∧ ψ} , ϕ) AP (∆′ ∪ {φ, ψ} , ϕ)(A3) (∆′ ∪ {φ ∨ ψ} , ϕ) (EP (∆′ ∪ {φ})→ AP (∆′ ∪ {φ} , ϕ))

∧ (EP (∆′ ∪ {ψ})→ AP (∆′ ∪ {ψ} , ϕ))(A4) (∆′ ∪ {Q→ φ} , ϕ) Q ∧AP (∆′ ∪ {φ} , ϕ)(A5) (∆′ ∪ {P, P → φ} , ϕ) AP (∆′ ∪ {P, φ} , ϕ)(A6) (∆′ ∪ {((φ ∧ ψ)→ χ)} , ϕ) AP (∆′ ∪ {(φ→ (ψ → χ))} , ϕ)(A7) (∆′ ∪ {((φ ∨ ψ)→ χ)} , ϕ) AP (∆′ ∪ {φ→ χ, ψ → χ} , ϕ)(A8) (∆′ ∪ {((φ→ ψ)→ χ)} , ϕ) (EP (∆′ ∪ {ψ → χ})→ AP (∆′ ∪ {ψ → χ} , φ→ ψ))

∧AP (∆′ ∪ {χ} , ϕ)(A9) (∆, Q) Q(A10) (∆′ ∪ {P} , P ) >(A11) (∆, φ ∧ ψ) AP (∆, φ) ∧AP (∆, ψ)(A12) (∆, φ ∨ ψ) AP (∆, φ) ∨AP (∆, ψ)(A13) (∆, φ→ ψ) EP (∆ ∪ {φ})→ AP (∆ ∪ {φ} , ψ)

Table 2: The definitions of EP (∆) and AP (∆, ϕ), for formulae φ, ψ, χ 6= ϕ and proposi-tional variables Q 6= P

10

(iii) If ` ∆1 ∪∆2 � ϕ and P /∈ Var(∆1), then

(a) If P /∈ Var(ϕ), then ` ∆1 ∪ {EP (∆2)} � ϕ

(b) ` ∆1 ∪ {EP (∆2)} � AP (∆2, ϕ).

Having done so, we can define ∀Pϕ := AP (∅, ϕ) and Pitts’ Theorem follows.

(i) can be easily seen from table 2 and proven via induction on <wt. (ii) is proven via

simultaneous <wt-induction on ∆∪ϕ, by showing at each step, that for all ψ ∈ EP (∆) and

χ ∈ AP (∆, ϕ) we have ` ∆ � ψ and ` ∆∪{χ} � ϕ for each case in table 2. Finally, (iii) is

quite extensive and proven via induction on the rules of LJ∗. We look at three examplary

cases:

(Atom): ϕ is a propositional variable and in ∆1 ∪∆2. We have two cases:

ϕ = P : (a) holds trivially. For (b), since P /∈ ∆1, we have ∆2 = ∆′ ∪ {P}. Then from

table 2 (A10) we have > ∈ Ap(∆2, P ) and thus Ap(∆2, P ) = >.

ϕ 6= P : If ϕ ∈ ∆1, (a) holds trivially via (Atom) and with ∆1 = ∆′∪{ϕ} and (A9) and

since (a) holds, (b) follows.

If ϕ ∈ ∆2 =: ∆′ ∪ {ϕ}, then (a) follows by (A1) and as in the previous case,

(b) follows with (A9) and (a).

(⊥ �): If ⊥ ∈ ∆1, both (a) and (b) follow immediately by (⊥ �).

If ⊥ ∈ ∆2 both (a) and (b) follow immediately by (A0).

(� ∧): We have ϕ = ϕ1 ∧ ϕ2, ` ∆1 ∪∆2 � ϕi and by induction (a) and (b) hold for both

` ∆1 ∪∆2 � ϕi.

(a) If P /∈ Var(ϕi) then, since (a) holds for ϕi we have ` ∆1 ∪ {Ep(∆2)} � ϕi and

the claim follows by (� ∧).

(b) Since (b) holds for the ϕi, we have

` ∆1 ∪ {EP (∆2)} � AP (∆2, ϕi)

and by (� ∧) we have

` ∆1 ∪ {EP (∆2)} � AP (∆2, ϕ1) ∧AP (∆2, ϕ2)

By (A11) we have

` {AP (∆2, ϕ1) ∧AP (∆2, ϕ2)} � AP (∆2, ϕ)

and thus (b) holds for ϕ.

The cases (∧ �), (� ∨1) and (� ∨2) work similarly, the implicative cases however involve

more subcases and are thus a lot more extensive.

11

As mentioned above, we define

∃Pϕ := ∀Q(∀P (ϕ→ Q)→ Q).

Remark 4. As the previous proof is constructive, the functions ∃P and ∀P are in fact

computable and referred to as Pitts quantifiers.

The existence of the existential Pitts quantifier already follows from – and is used

extensively in – the proof for Pitts’ Theorem. However, since the existence of the universal

quantifier is sufficient, we will only take this one as “given” and proceed to show that

defining the existential quantifier as above actually works as intended.

Corollary 1.2. For each propositional variable P , intuitionistic propositional formula ϕ

and propositional theory Γ with P /∈ Var(Γ), we have

(i) `I ϕ[ψP

]→ ∃Pϕ for any formula ψ,

(ii) `I ∀P(ϕ[ψQ

])iff `I (∀Pϕ)

[ψQ

], provided P 6= Q and P,Q /∈ Var(ψ),11

(iii) Γ `I ∃Pϕ→ ψ iff Γ `I ϕ→ ψ, provided P /∈ Var(ψ) and

(iv) Γ `I ψ → ∀Pϕ iff Γ `I ψ → ϕ.

Proof.

(i) We have {ϕ(ψ), ∀P (ϕ→ Q)} `I ϕ → Q[ψP

]and thus {ϕ(ψ),∀P (ϕ→ Q)} `I Q.

Therefore

ϕ(ψ) `I ∀P (ϕ→ Q)→ Q

and since Q does not occur in ϕ (or ψ - remember that Q is supposed to be a generic

new variable not occuring anywhere)

ϕ(ψ) `I ∀Q(∀P (ϕ→ Q)→ Q)

which means `I ϕ[ψP

]→ ∃Pϕ.

(ii) We have ∀Pϕ `I ϕ and therefore (∀Pϕ)[ψQ

]`I ϕ

[ψQ

]. Since P /∈ Var

((∀Pϕ)

[ψQ

])we get (∀Pϕ)

[ψQ

]`I ∀P

(ϕ[ψQ

]).

For the converse, note that for any formulae φ, φ′ and χ, we have

φ↔ φ′ `I χ[φ

Q

]↔ χ

[φ′

Q

].

By definition ∀P(ϕ[ψQ

])`I ϕ

[ψQ

]and thus{

ψ ↔ Q,∀P(ϕ

[ψ

Q

])}`I ϕ

[ψ

Q

]↔ ϕ

[Q

Q

]11The proof for this is taken directly from [Pit92]

12

and hence {ψ ↔ Q,∀P

(ϕ

[ψ

Q

])}`I ϕ.

Since P does not occur on the left side, we get{ψ ↔ Q,∀P

(ϕ

[ψ

Q

])}`I ∀Pϕ.

Now we can substitue Q by ψ throughout and get ∀P(ϕ[ψQ

])`I (∀Pϕ)

[ψQ

].

(iii) Let Γ `I ϕ→ ψ. We have

Γ ∪ {∃Pϕ} `I ∀Q(∀P (ϕ→ Q)→ Q),

therefore

Γ ∪ {∃Pϕ} `I ∀P (ϕ→ Q)→ Q

[ψ

Q

]and by (ii), since P /∈ Var(ψ),

Γ ∪ {∃Pϕ} `I (ϕ→ ψ)→ ψ

[P

P

]and thus Γ ∪ {∃Pϕ} `I ψ, which means Γ `I ∃Pϕ→ ψ.

For the converse, let Γ `I ∃Pϕ → ψ, then by (i) we have `I ϕ → ∃Pϕ and thus

Γ `I ϕ→ ψ.

(iv) The equivalency can be proven directly: We have

Γ `I ψ → ∀Pϕ if and only if

Γ ∪ {ψ} `I ∀Pϕ if and only if (by Pitts’ Theorem)

Γ ∪ {ψ} `I ϕ[PP

]if and only if

Γ `I ψ → ϕ.

Interestingly, Pitts’ Theorem allows us to easily adapt a well known proof for Craig’s

interpolation theorem for propositional logic to IpC. We will need interpolation later on

to prove the amalgamation property for Heyting algebras.

Theorem 1.2 (Craig’s interpolation theorem for IpC). If `I ϕ → ψ, then there is an

interpolant, meaning a formula χ such that

`I ϕ→ χ and `I χ→ ψ

with Var(χ) ⊆ Var(ϕ) ∩ Var(ψ).

Proof. Let `I ϕ→ ψ. We will proceed by induction on |Var(ϕ) \ Var(ψ)| =: n.

For n = 0, ϕ itself is suitable.

Let n = m + 1 and let the hypothesis hold for all k ≤ m. Pick one P ∈ Var(ϕ) \ Var(ψ).

By corollary 1.2 we have `I ϕ → ∃Pϕ and `I ∃Pϕ → ψ. Note that ∃Pϕ doesn’t contain

P anymore, which means |Var(∃Pϕ) \ Var(ψ)| ≤ m. Hence, by the induction hypothesis

there is an interpolant χ for ∃Pϕ → ψ and since `I ϕ → ∃Pϕ we can use χ as a suitable

interpolant for ϕ→ ψ.

13

2 Heyting algebras

Heyting algebras are to IpC as boolean algebras are to classical logic, a relationship that

we will explore more detailed later on. In particular, Heyting algebras (and thus boolean

algebras as well) are special kinds of lattices, so it makes sense to start with these.

2.1 Lattices and Heyting algebras

We can define lattices in two ways: As algebraic structures and via partial orderings.

A partial order is a reflexive, antisymmetric and transitive relation.12 All proofs in this

section are taken from either [BS] or [BD74].

Definition 2.1. 13 A lattice is a structure in the language LL = (u,t) that satisfies the

(universal closure of the)14 following theory:

L1 (x u y) u z .= x u (y u z) (x t y) t z .

= x t (y t z) Associativity

L2 x u y .= y u x x t y .

= y t x Commutativity

L3 x u x .= x x t x .

= x Idempotence

L4 x u (x t y).= x x t (x u y)

.= x Absorption

Remark 5. The axioms are dual in the sense that for each axiom, if we consistently replace

u by t and vice versa, the resulting equation is again an axiom. This implies that any

proof for a certain proposition yields a proof for the dual proposition by consistently taking

for each statement in the proof the dual statement.

The second (equivalent) definition of lattices is the following:

Definition 2.2. 15 A lattice is a partially ordered set, in which for all elements a, b both

sup {a, b} and inf {a, b} exist.

Theorem 2.1. Definition 2.1 and definition 2.2 coincide by defining the partial order

a ≤ b iff a = a u b or, respectively, the operations a t b = sup {a, b} and a u b = inf {a, b}.

Proof.

• Let L be a lattice by definition 2.1 and ≤ be defined as above. By idempotence

a = a u a, so ≤ is reflexive.

If a ≤ b and b ≤ a, we have a = a u b = b u a = b, thus ≤ is antisymmetric.

If a ≤ b and b ≤ c we have a = a u b = a u (b u c) = (a u b) u c = a u c and thus

a ≤ c. So ≤ is also transitiv and thus is a partial order.

12[BS, p.6]13[BS, p.5]14i.e. every variable is to be thought of as universally quantified15[BS, p.8]

14

We have by absorption a = a u (a t b) and b = b u (a t b), so a, b ≤ (a t b) and

thus a t b is an upper bound. Assume a, b ≤ u, then a t u = (a u u) t u = u and

analogously b t u = u. We have (a t u) t (b t u) = u t u and by associativity and

idempotence (at b)tu = u. Thus, (at b)uu = (at b)u ((at b)tu). By absorption,

this is equal to a t b and therefore a t b ≤ u. Thus, a t b = sup {a, b}. Analogously

we can show, that a u b = inf {a, b}.

• Let L be a lattice by definition 2.2. Commutativity, associativity and idempotence

follow directly from the definitions of sup and inf. Absorption follows easily by

observing, that a = sup {a, inf {a, b}} = inf {a, sup {a, b}}.

Definition 2.3. 16 A distributive lattice is a lattice that satisfies the axiom

D1 ∀x∀y∀z x u (y t z) .= (x u y) t (x u z) or its dual

D2 ∀x∀y∀z x t (y u z) .= (x t y) u (x t z).

One of both axioms is sufficient, since one implies the other:

Lemma 2.1. D1 and D2 are equivalent.

Proof. Assume D1 holds, then

x t (y u z)

=(x t (x u z)) t (y u z) (Absorption)

=x t ((z u x) t (z u y)) (Associativity and commutativity)

=x t (z u (x t y)) (D1)

=(x u (x t y)) t (z u (x t y)) (Absorption)

=(x t y) u (x t z). (D1)

By duality, (D2) also implies (D1).

Next, we introduce a smallest and a largest element:

Definition 2.4. 17 A bounded lattice is a lattice with two distinguished elements 0 and

1 such that

B1 ∀x x u 1.= x and

B2 ∀x x t 0.= x

hold.

Remark 6. In a bounded lattice, the duality principle again holds, if (in addition to u and

t) we exchange 0 and 1 (or to put it another way: the dual of a bounded lattice is again

a bounded lattice). If existent, 0 and 1 are unique (since if L is a bounded lattice, both

(L,u, 1) and (L,t, 0) are monoids).

16[BS, p.12]17[BD74, p.49]

15

Definition 2.5. 18 Let L be a lattice and x, y ∈ L. If there is a largest element z, such

that xu z ≤ y, we call z the relative pseudocomplement of x with respect to y, denoted by

x ↪→ y.

This finally brings us to:

Definition 2.6. 19 A Heyting algebra is a bounded lattice, where for each two elements

x, y the relative pseudocomplement x ↪→ y exists.

Example 2.1. 20 Examples of Heyting algebras are:

• Every boolean algebra is a Heyting algebra, by defining x ↪→ y = xC t y.

• Every chain with a least and largest element (0 and 1) is a Heyting algebra by

defining x ↪→ y =

{1, if x ≤ y,y otherwise.

• Let T be a topological space over some set X. With U ↪→ V := int(UC ∪ V ) (where

int denotes the interior) and u and t as intersection and union respectively, T is a

Heyting algebra.

• As we will show, the Tarski-Lindenbaum algebra of IpC is a Heyting algebra. More-

over, Heyting algebras are exactly the algebraic models of IpC-theories (in the sense

of theorem 2.4).

Recall that u and t bind stronger than ↪→ (e.g. xuy ↪→ z is to be read as (xuy) ↪→ z).

Lemma 2.2. 21 In a Heyting algebra, the following statements hold for all x, y, z:

(i) x u (x ↪→ y) ≤ y

(ii) x u y ≤ z ⇔ y ≤ x ↪→ z

(iii) x ≤ y ⇔ x ↪→ y = 1

(iv) y ≤ x ↪→ y

(v) x ≤ y ⇒ z ↪→ x ≤ z ↪→ y and y ↪→ z ≤ x ↪→ z

(vi) x ↪→ (y ↪→ z) = x u y ↪→ z

(vii) x u (y ↪→ z) = x u (x u y ↪→ x u z)

(viii) x u (x ↪→ y) = x u y

(ix) (x t y)→ z = (x ↪→ z) u (y ↪→ z)

18[BD74, p.173]19[BD74, p.174]20[BD74, p.177]21[BD74, p.174]

16

(x) x ↪→ y u z = (x ↪→ y) u (x ↪→ z)

(xi) (x ↪→ y) ↪→ 0 = ((x ↪→ 0) ↪→ 0) ∧ (y ↪→ 0)

Proof.

(i) Holds by definition.

(ii) ⇒: holds by definition (x ↪→ z is the largest element with this property).

⇐: If y ≤ x ↪→ z, then x u y ≤ x u x ↪→ z ≤ z.

(iii) We have x ↪→ y = 1⇔ 1 ≤ x ↪→ y ⇔ x u 1 ≤ x u (x ↪→ y) ≤ y.

(iv) x u y ≤ y, therefore by (ii) y ≤ x ↪→ y.

(v) If x ≤ y, then z u (z ↪→ x) ≤ x ≤ y, so by (ii) z ↪→ x ≤ z ↪→ y. Also, we have

x u (y ↪→ z) ≤ y u (y ↪→ z) ≤ z, so again by (ii) we get y ↪→ z ≤ x ↪→ z.

(vi) We have

x u y u (x ↪→ (y ↪→ z)) = y u (x u (x ↪→ (y ↪→ z))) ≤ y u (y ↪→ z) ≤ z,

so by (ii) x ↪→ (y ↪→ z) ≤ x u y ↪→ z.

Conversely, yuxu (xu y ↪→ z) ≤ z, so by (ii) xu (xu y → z) ≤ y ↪→ z and therefore

again by (ii) x u y ↪→ z ≤ x ↪→ (y ↪→ z).

(vii) We have (xuy)uxu (y ↪→ z) ≤ xuz, so with (ii) we get xu (y ↪→ z) ≤ xuy ↪→ xuzand thus

x u x u (y ↪→ z) ≤ x u (x u y ↪→ y u z).

Conversely, x u (x u y ↪→ x u z) ≤ x and y u x u (x u y ↪→ x u z) ≤ x u z ≤ z, so by

(ii) xu (xu y ↪→ xu z) ≤ y ↪→ z and therefore xuxu (xu y ↪→ xu z) ≤ xu (y ↪→ z).

(viii) By definition x u (x ↪→ y) ≤ x, y and x u y ≤ x, x ↪→ y (the latter by (ii)).

(ix) We have x, y ≤ x t y and thus by (v) x t y ↪→ z ≤ x ↪→ z, y ↪→ z.

Conversely,

(x t y) u (x ↪→ z) u (y ↪→ z)

≤(x u (x ↪→ z) u (y ↪→ z)) t (y u (x ↪→ z) u (y ↪→ z))

≤(x u (x ↪→ z)) t (y u (y ↪→ z))

≤z t z = z,

so by (ii) (x ↪→ z) u (y ↪→ z) ≤ (x t y) ↪→ z.

(x) We have y u z ≤ y, z and thus by (v) x ↪→ y ∧ z ≤ x ↪→ y, x ↪→ z. Conversely, we

have

x u (x ↪→ y) u (x ↪→ z) ≤ x u y u (x ↪→ z) ≤ y u z,

so by (ii) we get (x ↪→ y) u (x ↪→ z) ≤ x ↪→ (y u z).

17

(xi) By (iv) y ≤ x ↪→ y and by (v) (x ↪→ y) ↪→ 0 ≤ y ↪→ 0. Since 0 ≤ y we get by

applying (v) twice x ↪→ 0 ≤ x ↪→ y and (x ↪→ y) ↪→ 0 ≤ (x ↪→ 0) ↪→ 0. Combining

both, we get (x ↪→ y) ↪→ 0 ≤ ((x ↪→ 0) ↪→ 0) u (y ↪→ 0).

Conversely,

((x ↪→ 0) ↪→ 0) u (y ↪→ 0) u (x→ y)

≤((x ↪→ 0) ↪→ 0) u (y ↪→ 0) u ((y ↪→ 0) u x→ (y ↪→ 0) u y) (by (vii))

which is equal to

((x ↪→ 0) ↪→ 0) u (y ↪→ 0) u ((y ↪→ 0) u x→ 0)

=((x ↪→ 0) ↪→ 0) u (y ↪→ 0) u ((y ↪→ 0) u x→ (y ↪→ 0) u 0)

=((x ↪→ 0) ↪→ 0) u (y ↪→ 0) u (x ↪→ 0) (again by (vii))

which is equal to 0. So by (ii) we have ((x ↪→ 0) ↪→ 0)u(y ↪→ 0) ≤ (x ↪→ y) ↪→ 0.

With these, we can now give an equational axiomatization for Heyting algebras:

Theorem 2.2. Heyting algebras are exactly the models of the (universal closure of the)

following equational theory TH in the language LH = (1, 0,u,t, ↪→):

The axioms for a bounded distributive lattice (i.e. L1–L4, D1 (or D2), B1 and B2) and

H1 x u (x ↪→ y).= x u y

H2 x u (y ↪→ z).= x u (x u y ↪→ x u z)

H3 x u y ↪→ x.= 1

Proof. We have to show the following:

• Every Heyting algebra satisfies H1–H3:

H1 is lemma 2.2.(viii), H2 is lemma 2.2.(vii). and H3 follows from lemma 2.2.(iii).

• In every bounded distributive lattice satisfying H1–H3, x ↪→ y is the relative pseu-

docomplement:

We have by H1 x u (x ↪→ y) = x u y ≤ y. Suppose x u z ≤ y for some z (and thus

x u y u z = x u z), then by H2 and H3

z u (x ↪→ y) = z u (z u x ↪→ z u y) = z u (x u y u z ↪→ y u z) = z

and therefore z ≤ (x ↪→ y).

2.2 Heyting algebras as models of IpC

We start by looking at the freely generated Heyting algebras. Given a set of elements

(“generators”) A, the set of LH(A)-terms T (A) obviously coincides with the set of propo-

sitional formulae F(A) (with the elements of A as propositional variables) via the following

recursively defined bijection:

18

[·] : F(A)→ T (A)

[P ] = P for P ∈ A

[ϕ ∧ ψ] = [ϕ] u [ψ]

[ϕ ∨ ψ] = [ϕ] t [ψ]

[ϕ→ ψ] = [ϕ] ↪→ [ψ]

[>] = 1 [⊥] = 0

We now define an equivalence relation on T (A), by letting for any terms t1 and t2:

t1 ∼ t2 iff TH |= t1.= t2.

Since TH is equational, the quotient algebra T (A)/ ∼ is a Heyting algebra22, the freely

generated Heyting algebra over A, denoted by HA.

Theorem 2.3. The freely generated Heyting algebra HA is the Tarski-Lindenbaum algebra

of IpC over A. This means for any formulae a, b ∈ F(A) we have [a] ∼ [b] iff `I a↔ b.

Proof. We have to show the following:

1. For every axiom [t1].= [t2] of TH we have `I t1 ↔ t2.

2. Modus Ponens: If [ϕ] = 1 and [ϕ] ↪→ [ψ] = 1, then [ψ] = 1.

3. For every axiom ϕ(ψ1, ψ2, ψ3) of IpC we have HA |= ∀x, y, z[ϕ(x, y, z)].= 1.

Proof of 1. We have already proven, that in IpC conjunction is commutative and associative, the

same holds for disjunction (follows immediately by IpC5, IpC6 and Modus Ponens).

The axioms for 1 and 0 hold by definition of > and ⊥.

Idempotence: By IpC4 and IpC5 we have `I x→ (x ∨ x) and `I (x ∧ x)→ x. IpC8 gives us

`I (x→ x)→ ((x→ x)→ ((x ∨ x)→ x))

and therefore `I (x∨ x)→ x. By IpC3 we have `I x→ (x→ (x∧ x)) and thus

`I x→ (x ∧ x).

Absorption: We have `I x ∧ (x ∨ y)→ x ( IpC4), `I x→ x ∨ y ( IpC5), `I x→ x and thus

`I x→ (x ∧ (x ∨ y)). The converse is IpC4.

Distributivity : We only need to show one distributive law: We have by IpC8

`I (y → ((x∧y)∨(x∧z)))→ ((z → ((x∧y)∨(x∧z)))→ ((y∨z)→ ((x∧y)∨(x∧z))))

and x ∧ (y ∨ z) `I x and thus x ∧ (y ∨ z) `I y → (x ∧ y). Therefore

x ∧ (y ∨ z) `I y → ((x ∧ y) ∨ (x ∧ z))22For a more detailled exploration of equational theories and (fully invariant) congruence relations, see

[BS, p.99ff]

19

(analogously for z instead of y). Thus x ∧ (y ∨ z) `I ((x ∧ y) ∨ (x ∧ z)).Conversely, we have

`I ((x ∧ y)→ x)→ (((x ∧ z)→ x)→ (((x ∧ y) ∨ (x ∧ z))→ x))

and hence (x∧y)∨(x∧z) `I x. Analogously we can show (x∧y)∨(x∧z) `I y∨zand the claim follows.

H1: We have (x ∧ (x → y)) `I x and x ∧ (x → y) `I x → y and by Modus Ponens

and IpC3 x ∧ (x→ y) `I x ∧ y. The converse follows directly by IpC1.

H2: By Modus Ponens {x ∧ (y → z), x ∧ y} `I z and thus

x ∧ (y → z) `I x ∧ y → x ∧ z.

For the converse,

{x ∧ (x ∧ y → x ∧ z), y} `I x ∧ y

and by Modus Ponens and IpC3

{x ∧ (x ∧ y → x ∧ z), y} `I z.

Hence, x ∧ (x ∧ y → x ∧ z) `I y → z.

H3: Follows from the fact, that x ∧ y → x is an axiom (IpC4).

Proof of 2. Let [ϕ] = [ϕ] ↪→ [ψ] = 1.

We then have [ϕ] u ([ϕ] ↪→ [ψ]) ≤ [ψ], ergo 1 u 1 ≤ [ψ] and thus HA |= 1.= [ψ].

Proof of 3.IpC1: ϕ→ (ψ → ϕ)

We have by lemma 2.2.(iv) x ≤ y ↪→ x and thus by lemma 2.2.(iii) x ↪→ (y ↪→ x) = 1.

IpC2: (ϕ→ ψ)→ ((ϕ→ (ψ → χ))→ (ϕ→ χ))

We have by lemma 2.2.(vi) x ↪→ (y ↪→ z) = (x u y) ↪→ z and

(x ↪→ y) ↪→ (((xuy) ↪→ z) ↪→ (x ↪→ z)) = ((x ↪→ y)u((xuy) ↪→ z)) ↪→ (x ↪→ z)

and thus ((x ↪→ y) u ((x u y) ↪→ z)) ≤ (x ↪→ z) which holds iff

x ↪→ y ≤ ((x u y) ↪→ z) ↪→ (x ↪→ z)

and therefore (x ↪→ y) ↪→ (((x u y) ↪→ z) ↪→ (x ↪→ z)) = 1.

IpC3: ϕ→ (ψ → (ϕ ∧ ψ))

We have (x ↪→ (y ↪→ (x u y)) = x u y ↪→ x u y = 1, since x u y ≤ x u y.

IpC4: (ϕ ∧ ψ)→ ϕ

Holds, since x u y ≤ x.

IpC5: ϕ→ (ϕ ∨ ψ)

Holds, since x ≤ x t y.

20

IpC6: (ϕ→ χ)→ ((ψ → χ)→ ((ϕ ∨ ψ)→ χ))

We have by lemma 2.2.(vi)

(x ↪→ z) ↪→ ((y ↪→ z) ↪→ ((xty) ↪→ z)) = ((x ↪→ z)u(y ↪→ z)) ↪→ ((xty) ↪→ z))

and by lemma 2.2.(ix) (x t y) ↪→ z = (x ↪→ z) u (y ↪→ z).

IpC7: (ϕ→ ψ)→ ((ϕ→ ¬ψ)→ ¬ϕ)

We need to show

1 =(x ↪→ y) ↪→ ((x ↪→ (y ↪→ 0)) ↪→ (x ↪→ 0))

=((x ↪→ y) u (x ↪→ (y ↪→ 0))) ↪→ (x ↪→ 0)

(lemma 2.2.(vi)) which holds iff

(x ↪→ y) u (x ↪→ (y ↪→ 0)) u (x ↪→ 0) = (x ↪→ y) u (x ↪→ (y ↪→ 0))

⇔ x ↪→ (y u (y ↪→ 0) u 0) = x ↪→ (y u (y ↪→ 0)) (lemma 2.2.(x))

⇔ x ↪→ (y u 0) = x ↪→ (y u 0)

IpC8: ϕ→ (¬ϕ→ ψ)

We have

x ↪→ ((x ↪→ 0) ↪→ y) = (x u (x ↪→ 0)) ↪→ y = 0 ↪→ y

(lemma 2.2.(vi)) which holds since 0 ≤ y.

All other (i.e. not freely generated) Heyting algebras are covered by the following

theorem:

Theorem 2.4. Given any IpC-theory Γ over a set of propositional variables A, the set of

all propositional formulae F(A)/Γ over A divided by the equivalence relation

ϕ ∼Γ ψ iff Γ `I ϕ↔ ψ

is a Heyting algebra. Conversely, we can interpret any Heyting algebra H as F(H)/H=1,

where H=1 := {[t] | t ∈ F(H) and H |= [t].= 1}.

Proof. That F(A)/Γ is a Heyting algebra can be easily seen by the fact that `I ϕ⇒ Γ `I ϕ,

and thus ∼⊆∼Γ (where ∼ is the equivalence relation as defined above) - since we only

divide by additional equations, the equations from TH still hold if we divide by ∼Γ.

For a more rigorous proof, we can take the freely generated Heyting algebra HA and

use the well known fact, that the filters in a Heyting algebra H are isomorphic to the

congruence relations on H.23 We can w.l.o.g. assume Γ to be deductively closed (i.e. if

Γ `I ϕ, then ϕ ∈ Γ). We then have:

1. If ϕ ∈ Γ and ϕ→ ψ ∈ Γ, then ψ ∈ Γ and

23See [BD74, p.178f]

21

2. If ϕ ∈ Γ and ψ ∈ Γ, then ϕ ∧ ψ ∈ Γ.

This shows, that the image of Γ in HA is a filter on HA and thus, that the equivalence

relation generated by this filter is a congruence relation.

It remains to show, that H |= [t].= 1 iff H=1 `I t. The implication from left to

right holds by definition, the other direction holds, as (as we have shown) TH entails all

IpC-axioms and Modus Ponens.

Given this translation between IpC-theories and Heyting algebras, we can define:

Definition 2.7. The polynomial Heyting algebra H(X) over a Heyting algebra H is the

Heyting algebra F(H ∪ {X})/H=1 for some X /∈ H.

22

3 Model companions and completions

A model completion for a certain theory (of first order predicative formulae) is a special

kind of model companion, which is a related model complete theory.

Definition 3.1. 24 A theory T is model complete if for every model A |= T , every sub-

structure B which is a model of T is an elementary substructure (i.e. A and B satisfy the

same L(B)-sentences).

Any theory with quantifier elimination is model complete.25 The classic example for

a model complete theory (and the following concepts) is the theory of algebraically closed

fields ACF. Since ACF does not determine the characteristic of a model, it is not complete.

However, given a certain (algebraically closed) field, any algebraically closed extension or

substructure has the same characteristic; all other LField-sentences are already decided by

the theory. As such, ACF is the model completion of the theory of fields:

Definition 3.2.

• 26 A theory T ∗ is a model companion of a theory T if the following conditions are

satisfied:

(a) Each model of T can be extended to a model of T ∗ and vice versa,

(b) T ∗ is model complete.

• 27 A model completion T ∗ of a theory T is a model companion of T with the following

additional property:

For all models A |= T and A1,A2 |= T ∗:

If A ⊆ A1,A2, then (A1, A) ≡ (A2, A).

If a model companion exists, it is unique (up to equivalence, of course). Consequently,

we are only interested in whether one exists or not.

Theorem 3.1. Any theory has at most one model companion.

Proof. 28 Assume a theory T has two model companions T1 and T2 and let A0 |= T1,

then A0 can be embedded in a model B0 |= T2, which can in turn be embedded in a

model A1 |= T1 and so on, resulting in two elementary chains (Ai)i∈ω and (Bi)i∈ω. Since

Ai ≺ Bi and Bi ≺ Ai+1, we have⋃i∈ω Ai =

⋃i∈ωBi = C. Since A0 and B0 are elementary

substructures of C, we have A0 ≡ B0 and thus A0 |= T2. Analogously we can show that

every model of T2 is a model of T1.

24[TZ12, p.34]25[TZ12, p.34]26[TZ12, p.35]27[Pot81, p.106]28[TZ12, p.35]

23

Other examples for model companions or completions are:

Example 3.1. 29

• The theory of differentially closed fields is the model companion of the theory of

differential fields.

• The theory of the random graph is the model completion of the theory of graphs.

• 30 The theory of atomless boolean algebras is the model completion of the theory of

boolean algebras.

If a model companion of T exists, its models are exactly the T -existentially closed

structures, as in the following definition:

Definition 3.3. 31

• A substructure A ⊆ B is called existentially closed in B if for every existential

L(A)-sentence ϕ,

B |= ϕ⇒ A |= ϕ.

• A structure A is called T -existentially-closed (T -e.c.) if A can be embedded in a

model of T and is existentially closed in every extension which is a model of T .

The previous definition results in the following useful criterion for the existence of a

model companion:

Theorem 3.2. T has a model companion iff the class of T -existentially-closed structures

is an elementary (i.e. axiomatizable) class.

Proof. The proof uses quite a lot of model theory. The details can be found in [TZ12,

p.35ff].

Assume T has a model companion T ∗. Since T ∗ is model complete, it is in particular

inductive, and thus axiomatizable by ∀∃-formulae. This implies that T ∗ is contained in

the Kaiser hull of T , which is the biggest inductive theory TKH with T∀ = TKH∀ (where

T∀ is the universal part of T ). The Kaiser hull happens to be exactly the ∀∃-part of the

theory of all T -e.c. structures.

So, let M |= T ∗ and A |= T an extension of M. A can be embedded in a model N |= T ∗

and since M ≺ N, M is existentially closed in A. This shows that all models of TKH are

T -e.c., and since all T -e.c. structures are models of TKH , the Kaiser hull serves as an

axiomatization of the class of T -e.c. structures.

For the converse, let T+ be an axiomatization of the class of T -e.c. structures. Robin-

son’s test then tells us that T+ is model complete (since all of its models are by definition

existentially closed) and thus serves as the model companion.

29[TZ12, p.37ff]30[CK90, 197]31[TZ12, p.35]

24

To show that a model companion is in deed a model completion, we can use the

following property:

Definition 3.4. 32

• A class K of structures has the amalgamation property if for all A,B1,B2 ∈ K and

embeddings fi : A → Bi, (i = 1, 2), there is some D ∈ K and two embeddings

gi : Bi → D such that g1 ◦ f1 = g2 ◦ f2, i.e. the following diagram commutes.

B1

g1

A

f1>>

f2

D

B2

g2

>>

• A theory T has the amalgamation property if the class Mod(T ) of all models of T

has the amalgamation property.

Remark 7. Since the fi in the above definition are embeddings, we can always w.l.o.g.

assume them to be the identity, i.e. we can assume A is a common substructure of B1

and B2.

As can be easily seen, we have:

Theorem 3.3. Let T ∗ be the model companion of T . Then T ∗ is a model completion of

T iff T has the amalgamation property.

Proof. Assume T has the amalgamation property and let A |= T be a common substructure

of A1,A2 |= T ∗. We can w.l.o.g. assume A1,A2 |= T (since they can be extended to models

of T ). So there is a model D |= T with A1,A2 ⊆ D. We can again w.l.o.g. assume D |= T ∗

and since T ∗ is model complete, we have A1,A2 ≺ D and thus (A1, A) ≡ (A2, A).

For the converse, assume T ∗ is a model completion and let A be w.l.o.g. the largest

common substructure of A1,A2 |= T . We can w.l.o.g. assume A1,A2 |= T ∗. Then

(A1, A) ≡ (A2, A), which means Th(A, A) = Th(A1, A1) ∩ Th(A2, A2). The claim then

follows immediately by joint consistency ; a corollary to Craig’s interpolation theorem (in

this case for the predicate calculus), which states that for any complete (and consistent)

theory T with two complete (and consistent) extensions T1 and T2, the union of T1 and

T2 is again consistent:

Assume there were some ϕ such that T1 ∪ T2 |= ϕ ∧ ¬ϕ. Then (by the compactness

theorem) there are finite subset Γ1 ⊂ T1 and Γ2 ⊂ T2 with

|=∧ψ∈Γ1

ψ →

∧ψ∈Γ2

ψ → ϕ ∧ ¬ϕ

.

32[TZ12, p.56]

25

By the interpolation theorem there is an interpolant χ in the language of T , which – since

T1 is an extension of T and the theories are complete – has to hold in T1 and T and thus

also in T2. This implies T2 |= ϕ ∧ ¬ϕ, ergo T2 is inconsistent, contradiction.

26

4 The model completion of TH

In order to show that TH has a model completion, we need to show, that the class of exis-

tentially closed Heyting algebras is an elementary class and that TH has the amalgamation

property. We start with the latter:

Theorem 4.1. TH has the amalgamation property.

Proof. Let A |= TH be a common substructure of two Heyting algebras B1 and B2, and

let fi : A → Bi the corresponding embeddings. We may w.l.o.g. assume A to be the

largest common substructure of B1 and B2. We can interpret B1 and B2 as F(B1)/B=11

and F(B2)/B=12 respectively, and have A ≺ B1,B2 and A=1 ⊆ B=1

1 , B=12 . Now consider

F(B1 ∪ B2)/(B=11 ∪ B=1

2 ) =: D. We have to show, that the resulting canonic maps

gi : Bi → D are injective.

So, assume (w.l.o.g.) g1([b1]) = g1([b2]) for some [b1], [b2] ∈ B1. Consequently,

(B=11 ∪B=1

2 ) `I b1 ↔ b2

and by corollary 1.1 there are finite subsets Γ1 ⊆ B=11 and Γ2 ⊆ B=1

2 such that

`I∧ϕ∈Γ2

ϕ︸ ︷︷ ︸∈F(B2)

→

∧ϕ∈Γ1

ϕ→ (b1 ↔ b2)

︸ ︷︷ ︸

∈F(B1)

.

By theorem 1.2 (interpolation) we get a formula ψ ∈ F(B1 ∩B2) = F(A) such that

`I∧ϕ∈Γ2

ϕ→ ψ and `I ψ →

∧ϕ∈Γ1

ϕ→ (b1 ↔ b2)

Then B2 |=

[∧ϕ∈Γ2

ϕ].= 1 and thus B2 |= f2([ψ])

.= 1, which means (since f2 is an

embedding) A |= [ψ].= 1 and hence B1 |= [b1]

.= [b2].

Therefore, the gi are injective and hence embeddings.

To axiomatize the class of e.c. Heyting algebras, we will rely heavily on Pitts’ theorem,

so first, we will look at its consequences for TH .

Remark 8. In the rest of this section, we deliberately do not differentiate between proposi-

tional formulae and LH -terms, since doing so would lead to obfuscation rather than clarity.

From the context it should always be clear, which of both is the intended meaning.

Recall that the Pitts quantifiers are computable, so extending LH (and accordingly

TH) by the binary function symbols ∀x and ∃x is an extension by definition and hence

conservative and unproblematic.

As mentioned in the introduction, our proof for the existence of the model companion

is outlined in [GZ97].

27

Theorem 4.2. For a propositional formula t(a, x) with propositional variables a from a

Heyting algebra H we have

(i) H |= ∃x(t).= 1 iff H(X)/t(X) is an extension of H, where H(X)/t(X) is the poly-

nomial Heyting algebra H(X) divided by the congruence generated by the equation

t.= 1

[Xx

],

(ii) H |= ∀x(t).= 1 iff the equation t

.= 1

[Xx

]holds in H(X).

Proof. We can interpret H as F(H)/H=1 and w.l.o.g. assume H 6|= 0.= 1.

(i) Let H |= ∃x(t).= 1, π : H → H(X)/t(X) the canonic map and a, b ∈ H with

π(a) = π(b), then

(H=1 ∪ {t(X)}) `I a↔ b

and thus

H=1 `I t(X)→ (a↔ b).

By corollary 1.2 we get

H=1 `I ∃xt→ (a↔ b)

and since ∃x(t).= 1 holds in H we have H |= a

.= b. Thus π is injective and hence

H(X)/t(X) an extension.

For the converse, let H(X)/t(X) be an extension of H and π the corresponding

embedding, then with corollary 1.2

(H=1 ∪ {t(X)}) `I ∃xt

and therefore

H(X)/t(X) |= ∃x(t).= 1.

Hence, we can conclude that π(∃x(t)) = π(1H) and since π is an embedding we have

H |= ∃x(t).= 1.

(ii) The equivalency can be shown directly: We have H |= ∀x(t).= 1 if and only if

H=1 `I ∀xt, which by theorem 1.2 (Pitts’ Theorem) holds if and only if H=1 `I t(X),

which is equivalent to H(X) |= t(X).= 1.

Now we can look at how to determine within a given Heyting algebra, whether a

given existential formula has a solution in some extension. For this we will first need the

following definition:

Definition 4.1. A primitive existential formula has the form ∃xϕ, where ϕ is a quantifier-

free conjunction of atomic formulae or their negations. In languages without relation

symbols, primitive existential formulae are thus exactly the systems of equations (and

inequations) in one variable.

28

Remark 9. Every equation t1 = t2 can be expressed in the form (t1 ↪→ t2)u (t2 ↪→ t1) = 1,

so we can w.l.o.g. restrict ourselves to equations of the form t = 1.

Theorem 4.3. Let A |= TH and

∃xϕ := ∃x(t1(x).= 1 ∧ ... ∧ tn(x)

.= 1 ∧ ¬u1(x)

.= 1 ∧ ... ∧ ¬um(x)

.= 1)

some primitive existential LH(A)-sentence. Then ∃xϕ holds in some extension of A iff

the following quantifier-free formulae hold in A:

∃x

(nl

i=1

ti

).= 1 (1)

¬∀x

(nl

i=1

ti ↪→ uj

).= 1 (2)

for all j < m.

Proof. Assume the above formulae hold in A, then (as (1) holds and by theorem 4.2)

A(X)/nl

i=1

ti(X) =: B

is an extension of A in which the formula t1(X).= 1 ∧ ... ∧ tn(X)

.= 1 holds33. Assume

ϕ(X) does not hold in B, then there is some uj such that B |= uj(X).= 1 and hence

A(X) |=

(nl

i=1

ti ↪→ uj

)[X

x

].= 1

which means

A |= ∀x

(nl

i=1

ti ↪→ uj

).= 1,

a contradiction to A |= (2).

For the converse, let ϕ be satisfied by some element a in some extension B′ of A. Then

B′ |=nl

i=1

ti

[ax

].= 1

and thus

B′ |= ∃x

(nl

i=1

ti

).= 1.

Since B′ is an extension of A, we have

A |= ∃x

(nl

i=1

ti

).= 1. (1)

33 Since t1.= 1 ∧ ... ∧ tn

.= 1 iff t1 u ... u tn

.= 1

29

Assume for some j

A |= ∀x

(nl

i=1

ti ↪→ uj

).= 1.

Then

B′ |= ∀x

(nl

i=1

ti ↪→ uj

).= 1

and by theorem 1.2 (Pitts’ theorem)

B′ |=

(nl

i=1

ti ↪→ uj

).= 1

[ax

]and therefore B′ |= uj(a), a contradiction to B′ |= ϕ(a). Therefore

A |= ¬∀x

(nl

i=1

ti ↪→ uj

).= 1 (2)

Remark 10. Since in an existentially closed structure any existential formula has a solution

iff it has a solution in some extension, the previous theorem yields a method to eliminate

the quantifier in primitive existential formulae in e.c. Heyting algebras. As is well known,

it follows that e.c. Heyting algebras have quantifier elimination:

Since

1. any formula is equivalent to a formula in prenex normal form (i.e. with all quantifiers

at the beginning) with quantifier-free part in disjunctive normal form and

2. existential quantifiers are distributive over disjunctions,

we can inductively eliminate one existential quantifier (and hence universal quantifiers as

well) after another in any formula with multiple quantifiers.

It follows that – if we want to axiomatize the class of e.c. Heyting algebras – we can

restrict ourselves to the primitive existential formulae:

Corollary 4.1. The class of existentially closed Heyting algebras is axiomatizable in the

language LH ∪ {∃x, ∀x} with TH extended by the (universal closure of the) following for-

mulae:

∃x

(nl

i=1

ti.= 1 ∧

m∧i=1

¬ui.= 1

)↔

(∃x

(nl

i=1

ti

).= 1 ∧

m∧i=1

¬∀x

(nl

i=1

ti ↪→ ui

).= 1

)

for every finite set of terms {t1(x, y), ..., tn(x, y), u1(x, y), ..., um(x, y)};

which means the class of TH -e.c. structures is an elementary class, and since we have

already shown that TH has the amalgamation property, we finally get our intended result:

Corollary 4.2. The theory of Heyting algebras has a model completion.

30

5 Appendix

References

[BD74] R. Balbes and P. Dwinger. Distributive Lattices. University of Missouri Press,

1974.

[BS] S. Burris and H.P. Sankappanavar. A Course in Universal Algebra (Millennium

Edition). none.

[CK90] C.C. Chang and H.J. Keisler. Model Theory. Studies in Logic and the Foundations

of Mathematics. Elsevier Science, 1990.

[GZ97] S. Ghilardi and M. Zawadowski. Model completions and r-heyting categories.

Annals of Pure and Applied Logic, 88:27–46, 1997.

[Hey71] A. Heyting. Intuitionism - An Introduction. North-Holland Pub. Co., 1971.

[Pit92] A. Pitts. On an interpretation of second order quantification in first order intu-

itionistic propositional logic. The Journal of Symbolic Logic, 57:33–52, 1992.

[Pot81] K. Potthoff. Einfuhrung in die Modelltheorie und ihre Anwendungen. Die Math-

ematik. Wissenschaftliche Buchgesellschaft, 1981.

[SU98] M.H.B. Sørensen and P. Urzyczyn. Lectures on the curry-howard isomorphism,

1998.

[TZ12] K. Tent and M. Ziegler. A Course in Model Theory. Lecture Notes in Logic.

Cambridge University Press, 2012.

[VD86] D. Van Dalen. Intuitionistic logic. In D. Gabbay and F. Guenthner, editors,

Handbook of Philosophical Logic Volume III. D. Reidel Publishing Company, 1986.

Order of connective strength of logical symbols ¬,∀,∃,∧,∨,→,↔,>,⊥Order of connective strength in Heyting algebras u,t, ↪→, 1, 0Structures A,B,C, ...Their underlying universes A,B,C, ...Propositional theories Γ,∆Formulae ϕ,ψ, χ, φϕ holds in a structure (or theory) A A |= ϕϕ is provable from Γ in IpC Γ `I ϕϕ is an intuitionistic tautology `I ϕ{ψ} `I ϕ ψ `I ϕThe set of propositional variables in ϕ Var(ϕ)Sequents in the sequent calculus for IpC Γ � ϕThe sequent Γ � ϕ is provable in IpC ` Γ � ϕTuple of variables xA formula ϕ has the free variables x, y ϕ(x, y)

Substituting x by some term t in ϕ(x) ϕ

[t

x

]or ϕ(t)

A is an elementary substructure of B A ≺ BA and B are elementarily equivalent A ≡ BA extended by constant symbols from B ⊆ A (A, B)The language of Heyting algebras LH = (1, 0,u,t, ↪→)The theory of Heyting algebras THThe class of models of a theory T Mod(T )The propositional formula ϕ interpreted as LH -term [ϕ]The freely generated Heyting algebra over A HAThe set of LH -terms t with H |= t

.= 1 H=1

The theory of A (i.e. {ϕ | A |= ϕ}) Th(A)Some language L extended by constant symbols from a set A L(A)

Table 3: List of notations used

Zusammenfassung

Das Ziel dieser Bachelorarbeit ist es, die Existenz einer Modellvervollstandigung (model

completion) der Theorie der Heyting-Algebren zu beweisen. Dazu interpretieren wir diese

als algebraische Modelle der intuitionistischen Aussagenlogik (IpC - intuitionistic proposi-

tional calculus). Dies erlaubt uns, mit dem zentralen Satz aus [Pit92] – welcher besagt,

dass jede zweitstufige aussagenlogische Formel in IpC aquivalent zu einer erstufigen ist – zu

zeigen, dass die Klasse der existentiell abgeschlossenen Heyting-Algebren eine elementare

Klasse ist, woraus die Existenz eines Modellbegleiters (model companion) folgt. Dass dieser

eine Modellvervollstandigung darstellt folgt dann aus der Tatsache, dass die Theorie der

Heyting-Algebren die Amalgamationseigenschaft (amalgamation property) hat.

In Abschnitt Eins wird intuitionistische Aussagenlogik und der Satz von Pitts vorgestellt.

Abschnitt Zwei widmet sich Verbanden (lattices) und Heyting-Algebren und das Verhaltnis

dieser zu IpC wird untersucht. Abschnitt Drei stellt die notwendigen modelltheoretischen

Definitionen und Resultate vor, bevor in Abschnitt Vier der zentrale Beweis dieser Arbeit

gegeben wird.

Selbststandigkeitserklarung

Hiermit erklare ich, dass ich diese Arbeit selbstandig verfasst habe, keine anderen als

die angegebenen Quellen/Hilfsmittel verwendet habe und alle Stellen, die wortlich oder

sinngemaß aus veroffentlichten Schriften entnommen wurden, als solche kenntlich gemacht

habe. Daruber hinaus erklare ich, dass diese Abschlussarbeit nicht, auch nicht auszugsweise,

bereits fur eine andere Prufung angefertigt wurde.

Datum: Unterschrift: