European Review of Phi-losophy, (Ed. A.C. Varzi).4, 1999, 45–78.

The Intuitionistic Conception of Logic

Dirk van dalen

Introduction

If anything, the last decennia have brought the scientific world a proliferationof logics. Almost any calculus seems to be a logic. In order to remove possibleconfusion let us agree that by logic we mean the science or art of drawingcorrect conclusions. This definition leaves us enough room to incorporate mostlogics. The burden of this convention is carried by the adjective ‘correct’. Wemight consider “correct according to the two-valued truth tables”, or “correctaccording to the action of some machine”. The latter is not as strange as itseems, the motivation of Girard’s linear logic has certain features of the cigarettevending machine [Girard 1990].

In the present exposition we will look at ‘correct’ from the position of aconstructivist, that is to say, we will require arguments to preserve ‘constructivetruth’ or ‘constructibility’. In the course of this our discussion these expressionswill be clarified.

The problem of constructive argument has been with us for over one-hundredyears. It was Kronecker who insisted on effective constructions instead of ab-stract promises. One of his followers, Jules Molk, expressed the demands of theKronecker school as follows: “definitions should be algebraic and not just logical.It is not sufficient to say: something is or is not”. The central issue in the con-structive tradition has always been ‘existence’, this notion was under pressureever since Hilbert’s miraculous proof of the existence of a finite basis for a classof invariants. It was the prototype of the so-called ‘pure existence proofs’. Suchproofs showed that something existed, without giving any indication how to findit. This technique had the advantage of simplicity, it provided short surveyableproofs; and it became the new standard in mathematics. Although the pureexistence proofs caused some surprise, they were quickly incorporated in thearsenal of mathematical techniques. The event that caused the mathematicalworld to think twice about ‘existence’ was Zermelo’s proof of the well-orderingtheorem. Zermelo had shown that any set can be well-ordered; in the proofhe relied on the axiom of choice. However, in the testcase of the continuum(the real line), nobody succeeded in defining the claimed well-ordering. Earlyconstructivists, such as Borel, demanded that objects could only be granted inexistence if they could be defined by finitely many words. None of them went,

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however, so far as to scrutinise the role of logic.

The first persons to extend the criticism of abstract non-constructive practicein mathematics was L.E.J. Brouwer. In a revolutionary article, “The unrelia-bility of the logical principles” (1908), he questioned the validity of the logicalarguments.Already in his dissertation (1907) he had rejected the traditional place of logicin the context of mathematics. Whereas it was usually assumed that one neededlogic in order to practice mathematics, Brouwer boldly reversed the order:‘Mathematics is independent of logic’, but ‘logic depends on mathematics’.

In constructive mathematics, one constructs the objects of mathematics andoperates on large conglomerates of mathematical objects by means of construc-tive operations. A simple equation like 5 + 7 = 12 is, for example, proved byfirst constructing 5, 7 and 12. Next the sum of 5 and 7 is constructed and theoutcome is compared (by means of the comparison construction) with 12. Thelast-mentioned construction tells us that the left hand and right hand sides areequal.

For the intuitionist mathematical objects, such as numbers, triangles, func-tions, . . . , are mental constructs. That is to say, we construct our own math-ematical universe and hence we establish the properties of our mathematicalobjects also by means of constructions. Since the possibility of the construc-tions is on Brouwer’s view anchored in the intuition of time, or mathematicalur-intuition (cf. [Brouwer 1907], p. 8), he had coined the name ‘intuitionism’ forhis particular philosophy and mathematics.1

Now we observe that the mathematical objects and its properties are de-scribed by language. In the use of the language of mathematics we observecertain regularities, which connect mathematical states of affairs. These reg-ularities constitute logic, and the expression of logic have a well-determinedmeaning in mathematics. It is a fact of experience that the language of mathe-matical logic is extremely modest, one needs a few connectives and quantifiersto describe almost all situations and procedures of mathematics. In fact it suf-fices to use ‘and’ (∧), ‘or (∨),‘if . . . then’ (→), ‘not’(¬), ‘for all’ (∀), and ‘thereexists’ (∃).

Shaping intuitionistic mathematical logic

Since logic leads in the hands of intuitionists to phenomena that markedlydiffer from that of classical two-valued logic, intuitionistic logic has to possessits own constructive interpretation. This interpretation, which is implicit inBrouwer’s writings, was made explicit by Kolmogorov and Heyting. Heyting’sinterpretations is closest to Brouwer’s (not surprisingly), Kolmogorov’s inter-pretation is almost isomorphic to that of Heyting. To start with the problem-

1The term ‘neo-intuitionism’ has also been used for Brouwerian intuitionism.

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interpretation of Kolmogorov, we see that Kolmogorov considers mathematicalstatements as problems to be solved. A statement is true if it has a solutionwhen considered as a problem. In terms of problem and solutions, one can givea meaning to the connectives: a solution of A ∧B is a pair (s1, s2) of solutionsto A and to B; a solution of ∃xA(x) is a pair (a, S) such that S is a solution ofA(a). Etc.

We will now consider Heyting’s proof-interpretation. The basic idea is thata proof is a construction, this is in agreement with Brouwer’s claim that theobjects of mathematics are our own mental constructions. We have already metthe construction that proves an atomic statement like ‘5+7 = 12’, it remains toexplain the constructions that establish composite statements. The formulationsare essentially parallel to those of Kolmogorov’s problem-interpretation. E.g. aproof p of A → B is a construction that converts any proof q of A into a proofp(q) of B; a proof of ∀xA(x) is a construction that converts any element a of thedomain under consideration into a proof of A(a). The advantage of Heyting’sformulation is that one needs only one kind of object, whereas Kolmogorov’sinterpretation requires two classes: solutions and operations.

We will introduce a suitable abbreviation in order to give a concise formu-lation of the proof-interpretation, also called the Brouwer-Heyting-Kolmogorov(BHK) interpretation.Let ‘a : A’ stand for ‘a is a proof for A’, then the following table captures theinterpretation of composite statements in terms of the interpretations of theirparts. For convenience we use a few standard notations: (a1, a2) is the orderedpair of the objects a1 and a2, the particular codification is not important here.In the disjunction clause the first component of (a1, a2) plays the role of casedistinguisher – “if (a1 = 0 then . . . , if (a1 = 1 then . . . ”.

a : A conditionsa : A ∧B a = (a1, a2), where a1 : A and a2 : Ba : A ∨B a = (a1, a2), where a1 = 0 and a2 : A or a1 = 1 and a2 : Ba : A → B for all p with p : A a(p) : Ba : ∃xA(x) a = (a1, a2) and a2 : A(a1)a : ∀xA(x) for all d ∈ D a(d) : A(d) where D is a given domaina : ¬A for all p : A a(p) :⊥

For an intuitionist “A is true” means “A has a proof”, i.e. ‘there is an asuch that a : A. ‘Constructive truth’ thus means ‘provability’, albeit no formalprovability. In order to interpret the negation of a statement A, we have toconsider “the false statement”. This rather elusive statement is not as exoticas one might think, it simply is a patently false statement, such as ‘0 = 1’. Letus use ⊥ as a symbol for ‘falsity’. Now intuitionists consider negation as animpossibility - as a ‘negative’ item. Thus ‘not-A’ is taken to stand for “from Aa contradiction (i.e. ⊥) can be derived.” In symbols: ¬A := A →⊥.

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Hence a : ¬A if a is a construction that turns every proof p of A into a proofa(p) of ⊥. Now, ⊥ has no proof – e.g. there is no construction that shows 0 = 1- so a : ¬A stands for “if p : A then a(p) :⊥”, hence A has no proof.

The reader should note that ‘a : A′ can also be read as “a is a solution toA”; the clauses remain intact for the problem-interpretation.The clause for the disjunction indeed has the constructive character that oneshould expect: A∨B means more than “one of the two, but I don’t know whichone” – which is the usual interpretation. It requires that we can determinewhich of the two disjuncts is true (has a proof); one has only to check whetherA or B is true.This is the main reason why the Principle of the Excluded Third (PEM) failsconstructively. Suppose a : A∨¬A, then a = (a1, a2) and if a1 = 0 then a2 : A,if a1 = 1 then a2 : ¬A, i.e. A has no proof. Since the actual possession of aproof, or at least an algorithm to get it, is the main thing, it is clear that ingeneral we have no evidence for A ∨ ¬A.

It has always been an act of faith that a statement is true or its negation istrue, but if one asks for positive evidence, there is little to offer. As Brouwerput it, the belief in the validity of PEM is an unwarranted extrapolation of itsvalidity in finite domains – it is a historical artefact. He illustrated the dubiouscharacter of PEM as follows: consider some problem which has not yet beensettled, e.g. “there is a sequence of 30 digits 9 in the decimal expansion of π”.2

If we take the above statement for A in PEM, then a proof of A∨¬A should(1) decide between A and ¬A, and (2) for the chosen case provide a proof. I.e.,we should either have a proof of “there exists a sequence . . . ”, which means weshould now indicate an index n, such that the decimals an through an+30 are9, or we should exhibit a proof that no such sequence can occur.At the present stage of mathematical development neither case is known to betrue. So we cannot produce the requested proof. Hence there is no evidence forPEM.

The difference between a constructive and a non-constructive proofs can bestbe observed in the case of existence statements. The following example nicelydemonstrates the issue.Claim: there are irrational numbers a and b such that ab is rational.Classical proof: suppose the contrary, then ab is irrational for all irrational a

and b. In particular√

2√

2is irrational, and therefore also (

√2√

2)√

2 is irra-

tional. But (√

2√

2)√

2 = 2 Contradiction. Hence ∃a, b ∈ R − Q.(ab ∈ Q). Thisis a proof by contradiction, it gives no clue as to the values of a and b, hence itcannot be accepted by an intuitionist.

2Properties of the decimal expansion of π were for the first time considered by Brouwer in1908. In the literature various “sequences in π” have been considered. The popular sequence0123 . . . 9 can no longer be used, see [Brouwer 1908], [Borwein 1998]

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The claim happens to be intuitionistically correct, but it requires a good dealof number theory.

The actual development of intuitionistic logic took some time. AlthoughBrouwer established the first theorem in intuitionistic propositional logic:¬¬¬A → ¬A (1923),3 he was not interested in the development of logic. Asystematic development had to wait for Heyting’s prize winning essay in 1928.The formalization was published in 1930.4.Heyting’s formalisation was not the first one, Kolmogorov had already in 1925formalized a fragment of intuitionistic predicate logic with only →,¬,∀ and ∃.In 1929 Glivenko had given an adequate formalization of propositional logic,moreover he established a striking connection between classical propositionallogic and intuitionistic logic: `c A ↔`i ¬¬A, where `c,`i stand for “derivablein classical, resp. intuitionistic propositional logic”.5

Heyting’s formalization opened the way for metamathematical research ofintuitionistic logic. The two major achievements of Heyting were his meticulousformalization and the formulation of the proof interpretation.6

Progress, after Heyting, was made by Gentzen and Godel. Both found struc-tural properties of intuitionistic logic that made clear the rich possibilities ofthis particular logic.By adding to Heyting’s axioms a suitable non-intuitionistic principle, such asPEM or the Double Negation Principle (DNP), one obtains an axiom systemfor classical logic. Hence IQC (Intuitionistic Quantifier (Predicate) Calculus) isa sub-system of CQC (Classical Quantifier Calculus), similarly for the propo-sitional versions IPC and CPC. Although Brouwer’s informal arguments hadshown that one could not expect PEM to be true intuitionistically, more wasneeded to show IPC 6` A ∨ ¬A in general. Heyting accomplished this in his[Heyting 1930A], by means of a suitable three-valued truth table.

Heyting’s goal at the time was a modest one: to give a systematic accountof the logical formulas which are true under the intended proof interpretation.No completeness in the modern sense with respect to a particular semantics wasintended, on the contrary, Heyting rejected the idea.

Intuitionistic mathematics, being a constructional activity, was not like ax-iomatic (or formal) mathematics in need of a consistency proof. Since the intu-itionist deals directly with constructed or generated objects he never gets into

3In his terminology “Absurdity of absurdity of absurdity is equivalent to absurdity”.4[Heyting 1930A, Heyting 1930B].5[Glivenko 1928], [Glivenko 1929] At the time of writing the latter paper Glivenko had seen

Heyting’s formalization, cf. [Troelstra 1978].6[Heyting 1930A], [Heyting 1934]. Heyting told me that he obtained his formalization by

checking the axioms of the Principia Mathematica, and eliminating those that did not fit theintended (proof) interpretation.

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the situation where 0 = 1 (or a similar disaster) is the case, whereas the math-ematician who plays a game with symbols, may by some inherent fault stumbleinto 0 = 1. Hence intuitionistic mathematics was considered as the more de-pendable way of doing mathematics. The same held for intuitionistic logic, itwas consistent by its nature. This extended to intuitionistic formalized arith-metic, as the axioms of arithmetic were justified by Heyting’s interpretation.So the consistency of intuitionistic arithmetic was, rightly or wrongly, not indoubt, whereas the full force of Hilbert’s program was directed at a consistencyproof for classical arithmetic. This somewhat mysterious situation was clarifiedby Godel and Gentzen, who provided translations of classical logic into intu-itionistic logic. As Godel’s translation has become best known, we will spell outhis translation. To each formula A of predicate logic we assign another formulaAo, its Godel translation. The inductive definition runs as follows:

⊥o:=⊥Ao := ¬¬A for atomic A distinct from ⊥(A ∧B)o := Ao ∧Bo

(A ∨B)o := ¬(¬Ao ∧ ¬Bo)(A → B)o := Ao → Bo

∀xA(x) := ∀xAo(x)∃xA(x) := ¬∀x¬Ao(x)

In fact, if we erase the superscript the lefthand and righthand formulas are clas-sically equivalent, but for an intuitionist the right hand formulas are weakerthan the left hand formulas, e.g. for the atoms ¬¬A says that is impossible thatA is not the case, or it is impossible that we will never find a proof, but that isnot the same thing as having a proof. Similarly ¬∀x¬A(x) says that it is impos-sible that A(a) is false for all a, but that does not give us the a we were lookingfor. Hermann Weyl expressed something like this, which we roughly paraphraseas follows: a classical existence statement announces the presence of a treasure,without giving the precise location, thus it is of limited – mostly heuristic –value. An intuitionistic existence statement, on the other hand, really producesthe treasure.

Godel’s translation takes a classically true statement and weakens it, so thatit becomes intuitionistically correct. In symbols: `c A ⇔ `i Ao In particularPA `c⊥⇔ HA `⊥, but this is exactly the consistency property.7 Hence PA isconsistent if and only if HA is consistent. It shows that as far as consistency isconcerned intuitionistic formal systems are no better than their classical coun-terparts.Godel furthermore cleared up a point that in the thirties caused some confu-sion, namely, if one drops the principle of the excluded third, does one thereforeaccept an “excluded fourth”. To put it in another way if one gives up the 0− 1truth tables, does one just move on to 3-valued (or possibly n-valued, for somen) logic? Godel showed by a simple but clever argument that this is not the

7PA is obtained from HA by adding PEM .

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case: there are no n-valued truth tables that produce just the intuitionisticallyprovable propositions.8

Gentzen systems and proof terms

So far the axiom-systems for logic had been modelled on the example ofEuclid and Hilbert, that is to say, lots of axiom and few derivation rules. Letus recall that in logic one has to specify not only the axioms – the principlesthat one accepts – but also the means of getting new theorems from previouslyacquired ones. These means are laid down in so-called derivation rules. Theprinciple example is Modus Ponens: from A and A → B derive B. In symbols:A → B,A ` B or A → B,A/B

In the so-called Hilbert-type systems one has few derivation rules (usuallyModus Ponens and Generalization). Somehow these systems are not quite con-venient for making actual derivations, some fairly simple propositions have longderivations. It was an ingenious move of Gerhardt Gentzen to introduce a rad-ically different system, in which there were few axioms and lots of derivationrules. He formulated two systems: the System of Natural Deduction and theSequent Calculus. Each system has its virtues, for our purpose the Natural De-duction system is best suited. It is a system with no axioms at all and for eachconnective two derivation rules, an introduction rule and an elimination rule.These rules are simple, they involve only the connective under consideration,and they specify (i) what one needs to know to infer the composite formula, (ii)what one may immediately infer from the composite formula. An example mayserve better than an abstract description:

∧-introduction A BA ∧B

∧-elimination A ∧B A ∧BA B

One has to know A and B in order to infer A ∧ B (alternatively, A and Bhave to be given to justify A ∧ B, or one has to derive A and B in order toderive A∧B). The elimination rule tells us that if we know A∧B, we also knowthe conjuncts. Observe that there is a precise balance between the introductionand elimination rules. Nothing is lost, nothing is gained. The rules determinein an operational sense the meaning of the connectives, cf. [Prawitz 1977].

One might wonder how the system succeeds in deriving theorems if there areno axioms. The redeeming feature of Natural Deduction is its capacity to cancelhypotheses. In order to facilitate the notation we will write A1, . . . , An ` B for“there is a derivation of B from the hypotheses A1, . . . , An”. The ∧ introduc-tion rule tells us in this notation A,B ` A∧B and the elimination rule tells usA ∧B ` A; A ∧B ` B.

8[Godel 1932]

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Let us now consider the most crucial propositional connective, →. The elimi-nation rule is none but our old Modus Ponens A → B,A ` B.The introduction rule, however, presents us with a new feature: If ΓA ` B,then Γ ` A → B, here Γ is a finite set of formulas. In words: if from thehypotheses Γ and A the formula B is derivable, then from Γ alone A → B isderivable. Strange as it may seem, this is exactly our daily practice, Consider,for example, the well-known theorem of Euclid: if in a triangle two sides areequal, then the opposite angles are equal. In symbols AC = BC → ∠A = ∠B.How do we prove this implication? By assuming that in the triangle ABC thesides AC and BC are equal, and then proceeding via a specific argument toshow ∠A = ∠B. I.e. we show Γ, AC = BC ` ∠A = ∠B and we abbreviate theresult as Γ ` AC = BC → ∠A = ∠B (Γ is the set of Euclid’s axioms).

Thus Gentzen’s systems cleverly exploits (or lays down, if you like) the mean-ing of the implication. We say that in the → introduction rule the hypothesisA is cancelled.

A gratifying aspect of Gentzen’s Natural Deduction system is that it ex-actly fits the proof interpretation. Consider the ∧-rule. Let a : A, b : B, then(a, b) : A ∧B, where (a, b) is some suitable pairing of the proofs a and b. So wecan incorporate the proofs into the Gentzen notation:

a : A, b : B ` (a, b) : A ∧B and(a, b) : A ∧B ` a : A(a, b) : A ∧B ` b : B

In fact, a bit of extra notation may further simplify the exposition. We in-troduce in addition to the pairing operation (a, b), two projection operations π1

and π2, such that π1(a, b) = a, π2(a, b) = b and a = (π1a, π2a).Then the elimination rules look like:p : A ∧B ` π1(p) : A; p : A ∧B ` π2(p) : B.

Similarly we introduce an extra operation in order to handle the implication.Consider for example a term x+y, as used in ordinary algebra. We may considerthis as a function in the variable x, this function is denoted by λx.(x + y), sayf = λx.(x + y). We now evaluate f at (say) 5 by plugging in 5: f(5) = 5 + y.We see that f is a function that accepts numbers as inputs and yields (for eachchoice of y) numbers as out puts. We can make another step by applying λy(this is called λ-abstraction – it makes a function out of a term): λy ·(λx.(x+y).This new function takes numbers as inputs and yields functions as outputs. Ingeneral all kinds of input-output combinations are possible. There is a specificrule for the behaviour of λ-abstraction terms:

(λx.t(x)(a) = t(a)

This is called the rule of λ-conversion.

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We will now show the connection between λ-conversion and implication.If Γ, x : A ` t(x) : B then Γ ` λx.t(x) : A → B.This is the perfect analogue of the proof interpretation of the implication: theleft hand says if x is an arbitrary proof of A (that is why we use a variable inx : A), then t(x) is a proof of B. Hence λx.t(x) is an operation on proofs of Awhich yields a proof of B. This is exactly what we wanted for a proof of A → B.

One can carry out a systematic correspondence between the rules of NaturalDeduction and the clauses of the proof interpretation, cf. [Troelstra–van Dalen 1988],[Girard et al. 1989]. This correspondence was first observed by Curry, and latersystematically extended by Howard. It is called the “propositions–as–types”, orCurry–Howard, interpretation.9

The operations that are introduced above have, in fact, types, this preventsthe danger of self-application – and all disastrous consequences. The typing ofthe operations can be read off for the simple case of implicational fragment ofpropositional logic, from the table below on the right-hand side.

The name may seem a bit strange, but there is a natural explanation. Wehave to return to the beginning of this century, at the time when mathematicswas threatened by the paradoxes. Bertrand Russell looked for a solution tothe Russell paradox by introducing a mathematical universe consisting of types.That is to say he considered an immediately given collection of objects (e.g. thenatural numbers), and next introduced the collection of all subsets. This stepwas iterated arbitrarily often, each time introducing objects of a higher type.One may do the same thing for functions: start with (say) the collection N ofnatural numbers, and continue by considering the class of all functions from Nto N. Now carry out this function-space formation for arbitrary function spacesalready obtained. These function spaces are called types (or, one could say,‘have types’). A convenient, and suggestive notation for these types is borrowedfrom logic: the function space, or type, of functions from class A to a class B isdenoted by A → B. Alternative notations are BA and (B)A. So N → N is thetype of function from N to N, N → (N → N) is the type of functions from N to(N → N) , etc.

Now there happens to be a perfect parallel between the pairs (proof, propo-sition) and (element, type) at least for implication. For full logic one has to addsome more technical devices. Here is the parallel:

9Cf. [Howard 1980], [Bruijn 1995], [Gallier 1995], [de Groote (ed.) 1995].

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a : A a ∈ Aa is a proof of A a is an element of Aa : A a ∈ Aa : A, p : A → B then p(a) : B a ∈ A, p ∈ A → B then p(a) ∈ Bif x : A ⇒ t(x) : B then if x ∈ A ⇒ t(x) ∈ B then λx.t(x) ∈ A → Bλx.t(x) : A → B

By adding a pairing operation one can handle the other propositions in asimilar, parallel way. Nowadays, the analogy between intuitionistic logic andtype theory is satisfactorily worked out by a number of logicians, e.g. Martin-Lof, Girard, Coquand, Barendregt.10.

There is a general feeling that logic influences mathematics, but that mathe-matics influences logic hardly, if not at all. This happens to be an artefact fromclassical mathematics. Classical predicate logic being complete for the standardtwo-valued truth tables leaves little opportunity for mathematics to make itselffelt.In intuitionistic mathematics there are lots of examples of the influence of math-ematics.Here are two:

1. The full axiom of choice implies PEM . (Diaconescu). Since there is avery simple proof, let us have a look at it. Consider a proposition P .Define two sets A and B:

A = {n ∈ N|n = 0 ∨ (n = 1 ∧ P )}

B = {n ∈ N|N = 1 ∨ (n = 0 ∧ P )}

Clearly ∀X ∈ {A,B}∃x ∈ N(x ∈ X).If {A,B} has a choice function f , then f(A) ∈ A, f(B) ∈ B and f(A) =f(B) ∨ f(A) 6= f(B), because f takes natural numbers as values and forthose equality is decidable.f(A) = f(B) implies P and f(A) 6= f(B) implies A 6= B (extensionality),and hence ¬P . This shows P ∨ ¬P .

2. Apartness and inequality on the reals are equivalent iff Markov’s Principleholds. Here Markov’s Principles is formulated for decidable predicates:∀x(A(x)∨¬A(x)∧¬¬∃xA(x) → ∃xA(x). Cf. [Troelstra–Dalen88], p. 205.

10[Barendregt 1992], [Martin-Lof 1984], [Coquand, Th. and G. Huet 1988],[Girard et al. 1989], [Howard 1980], [Troelstra–van Dalen 1988]

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In intuitionistic logic there is a specific phenomenon that has been thecause of some surprise, the closure under derivation rules. We have seen that` ∃xA(x) ⇒` A(t) for a suitable term (cf. [Dalen, D. van 1997] p. §6.3. How-ever, ∃xA(x) ` A(t) may fail for all t.We express the above by saying that IQC is closed under the rule ∃xA(x) ` A(t).Closure under rules is particularly interesting for realistic theories, such as in-tuitionistic arithmetic, HA. Here are some examples.

1. Closure under Church’s rule:HA ` ∀x∃yA(x, y) ⇒ HA ` ∀xA(x, {e}x) for a suitable e.I.e. if HA proves a ∀∃ statement, then there is a recursive function thatpicks the right y values.

2. Closure under Markov’s rule:HA ` ∀x(A(x) ∨ ¬A(x) ∧ ¬¬∃xA(x) ⇒ HA ` ∃xA(x).Markov’s principle is a general formulation of the idea that if it is im-possible that a Turing machine on a certain input never halts, then ithalts.

3. Closure under the rule of Independence of Premiss Principle:HA ` ∀x(A(x) ∨ ¬A(x)) ∧ ∀xA(x) → ∃yB(y) ⇒ HA ` ∃y(∀xA(x) →B(y))(cf. [Smorynski 1973], p.366,369, [Troelstra–van Dalen 1988], p.507.

Choice Aspects

So far we have dealt with the effective aspects of the intuitionistic universe.There is, however, one particular feature that makes intuitionism different frommost versions of constructivism. It was introduced by Brouwer in his 1918paper,[Brouwer 1918]. Since intuitionistic mathematics is a more or less freelydeveloping mental activity of the individual, it makes sense to allow for infinitesequences that are chosen (more or less) freely. Indeed, the notion is an almostinescapable consequence of Brouwer’s philosophical construction of the universe.In his early writings, and again in the Vienna Lectures he points out that thesubject experiences sequences of sensations;11 there is a certain identificationof ‘similar’ sequences, and the more stable ones (i.e. the ones that are not orlittle depending on the free will) play the role of objects. The sequences modulothe above mentioned identification go by the name of causal sequences. Sincethere is no reason whatsoever that these causal sequences are ruled by laws -on the contrary they are partly the product of actions of the free will - thenotion of infinite sequence is from the beginning open-ended. From the causalsequences to mathematical choice sequences, is but one step. It is of the greatestimportance to note that here there is a clear-cut break from tradition. So farall foundational considerations had led to the conclusion that the infinity of

11cf. [Brouwer 1907], p. 81 ff.; [Brouwer 1929], p. 153, [Brouwer 1949], p. 1235.

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a sequence a0, a1, a2, a3, . . . , an, . . . would only be guaranteed by a law (Borel,Holder, Weyl); Brouwer placed the burden of the responsibility for infinity atthe individual’s doorstep. It is the individual who guarantees or promises thathe will go on producing new members of the sequence.The basic question is how to exploit choice sequences mathematically. Some usesare unproblematic, e.g. consider two sequences (an) and (bn) of rational numberswhich determine two real numbers a and b (so-called Cauchy sequences). Thesum of a and b is determined by the sequence (an +bn). Such uses are, however,too specific to bring out the characteristic feature of choice sequences.

Brouwer discovered in 1916 how to make use of the choice-feature of infinitesequences (say of natural numbers). Suppose one has a function F operatingon choice sequences and yielding natural numbers as outputs. Then for eachchoice sequence α the value F (α) is a natural number n that can be computedin finitely many steps. Now, since in general a limited amount of informationabout α is added at each next choice step (not the total infinite behaviour, unlessyou are very lucky), the output is determined when only an initial segment ofα has been produced, and the value F (α) is determined by this initial segment.Hence two choice sequences α and β who share this particular initial segmentyield the same output. In order to formulate this idea in mathematical logic, letus introduce some notation: α(k) denotes the sequence (α(0), α(1), . . . α(k − 1)(the initial segment of length k). The above considerations tell us that F isdetermined by another function, say f , which operates on initial segments asfollows:

∀α∃k(F (α) = f(α(k))

and hence∀α∃k∀β(α(k) = β(k) → F (α) = F (β))

This is Brouwer’s continuity principle for functions.A similar argument yields the general continuity principle:

∀α∃xA(α, x) → ∀α∃x∃k∀β(α(k) = β(k) → A(β, x))

That is to say, if for all choice sequences α there is a number x such thatA(α, x) holds, then there is an x that only depends on an initial segment of α.

The continuity principle for functions is introduced in [Brouwer 1918], p. 13,in just four lines without any comment, even without a name! Brouwer usedthe principle to show that the set of choice sequences is not denumerable. Theargument is simple: let F be a function from the set if choice sequences (NN)to the natural numbers (N), then for each α ∈ NN there is a k such that allβ’s which have the same initial segment α(k) yield the same value F (α). HenceF cannot be a bijection (is not one-one). This proof replaced the traditionaldiagonal argument of Cantor.

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In 1924 Brouwer further analyzed the behaviour of functions on choice se-quences, and, by implication, real functions. On the basis of a conceptual ar-gument he obtained a form of transfinite induction, called bar-induction. Onlyafter the formalization of intuitionistic logic by Kleene and Kreisel, the principlegot its perspicuous form. We will just consider one particular variant here, thePrinciple of Monotone Bar Induction (BIM):

∀α∃xA(α(x)) (bar)∀n∀m(A(n) → A(n ∗m)) (mon)∀n(∀xA(n ∗ (x) → A(n)) (prog)

⇒ ∀nA(n).

In the above formulation m and n stand for (coded) finite sequences, m ∗ nstands for the concatenation of two finite sequences m and n, (x) stands for thesequence consisting of the single element x.

In order to get a feeling for the principle, it is convenient to use a geometricmetaphor: finite sequences of natural numbers are partially ordered by the re-lation ‘initial segment of’, thus the set of all finite sequences of natural numbersforms a partially ordered set, more specifically a tree. That is there is a topnode (element), the empty sequence, and from each node one can get in a finitenumber of (immediate predecessor-) steps to the top. This particular tree iscalled the universal tree. We will think of the tree as going downwards.A choice sequence gives us an infinite path through the tree.12 A bar in thetree is a set of nodes that ‘bar’ all the paths, that is, each infinite path passesthrough a node of the bar, think of a set that somehow runs from left to right,intersecting all paths. Imagine that A(n) is the predicate ‘red’. Then the barcondition tells us that every choice sequence with eventually pass through a rednode of the tree. The monotonicity condition says that all nodes below a rednode are red. The progressive-condition inverts this: if all nodes immediatelybelow n are red, then n is red. The three conditions together imply that thewhole tree is red. Kleene, Kreisel and Troelstra considered a number of variantsof the principle of bar induction. Kleene showed how delicate the formulationsare, one wrong move and one gets a contradiction ([Kleene–Vesley 1965], p. 112;[Troelstra–van Dalen 1988], p. 220.).

There is an intimate and obvious connection between bar induction andtransfinite induction (cf.[Howard–Kreisel 1966]), indeed the condition bar saysthat the non-red nodes form a well-founded tree within the big tree. The connec-tion between well-founded trees and ordinals is familiar (cf. [Shoenfield 1967]),it was already observed by Brouwer in his early papers ([Brouwer 1918]).The principle of bar induction yielded a simple proof of the fan theorem:

12The set of infinite paths through the universal tree in fact is a topological space with basicopen sets “all paths through a given node”. This topological space goes by the name of Bairespace. It certainly is no coincidence that Brouwer’s basic principles for choice sequences aretopological in nature.

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Consider a finitely branching tree, (a finitary tree, or fan, then:

∀α∃x(A(α(x)) → ∃z∀α∃y ≤ zA(α(y))

In words: if in a fan each infinite path α hits a red node then there is a finitedepth such that there is a red bar above it. Or, alternatively, if A(α(x)) is readas “the sequence α is cut at moment x”, then the fan theorem says that if allpaths α are eventually cut, at a certain depth all paths have been cut.

In topological terms, the fan theorem simply says that the natural topologyon a fan is compact. Observe that the contraposition of the fan theorem isKonig’s infinity lemma: in an infinite fan there is an infinite path. Konig’slemma is intuitionistically not correct, neither is its counterpart in analysis, theBolzano-Weierstrass theorem.

Brouwer showed using the fan theorem, that the continuum is locally com-pact, and that every real valued function on a compact subset of R is uni-formly continuous. This is the famous locally uniform continuity theorem,[Brouwer 1924]. For the details the reader is referred to [Troelstra–van Dalen 1988],vol. 1.

The moral of the above is that formalized intuitionistic mathematics, wasat the first order level a subsystem of the classical system (in particular inarithmetic), but that at the second-order level major departures from classicalmathematics occurred. The notion of choice sequence forced certain principleson the theory, that contradicted classical mathematics. Here is a small exam-ple: in the classical system one has: ∀α(∃x(α(x) = 0) ∨ ∀x(α(x) 6= 0). One canreformulate this as: ∀α∃!y[(y = 0 ∧ ∃x(α(x) = 0)) ∨ (y = 1 ∧ ∀x(α(x) 6= 0))]So there is a function F with:

F (α) ={

0 if α produces a 01 else

According to Brouwer’s continuity theorem this F must be continuous, but thatis impossible: consider F (λx.1) = 1. Continuity of F tells us that there is aninitial segment (1, 1, . . . 1) such that all continuations β of the segment yieldF (β) = 1, i.e. no extension of (1, 1, . . . , 1) can have a 0. Quod non.

So here the classical and intuitionistic systems clearly diverge.

The continuity principle in combination with bar induction gave intuitionis-tic second-order arithmetic (this system with choice sequences is usually calledanalysis) considerable strength, which confused Brouwer’s contemporaries whothought intuitionistic mathematics to be essentially a fragment of classical math-ematics.

There is significant distinction between second-order logic (arithmetic) withfunction variables, and with set variables. The reason is that sets have littleconstructive content, they are basically as good or as bad as propositional logic.

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One may consider S = {x ∈ 0|A}. If A holds S = {0}, if A is false S = ∅, butin general one can say just as little about S as about A. The subsets of {0} (orany singleton set) correspond to the propositions modulo logical equivalence. SoP({0}) can be considered as the set of truth values of intuitionistic propositionallogic. For this reason sets, say of natural numbers, are considerably ‘wilder’than functions. Hence it is no surprise that the familiar correspondence betweenfunctions and sets breaks down: also decidable sets have characteristic functions.For if ∀x[(x ∈ S ↔ f(x) = 1 ∧ x 6∈ S ↔ f(x) = 0)], then the existence ofsuch a characteristic function f implies, in view of the fact that equality onN is decidable (i.e. ∀xy(x = y ∨ x 6= y), ∀x(f(x) = 0 ∨ f(x) 6= 0) and hence∀x(x ∈ S ∨ x 6∈ S). Indeed, the elusive character of the subsets of a given set isthe motivation for Troelstra’s Uniformity Principle:

∀X∃xA(A, x) → ∃x∀XA(X, x)

The traditional example, x = 0 if X = ∅ and x = 1 if X 6= ∅, does not workhere, because it is undecidable whether X is empty, hence the existence of theclaimed x cannot be shown!

Beyond second-order arithmetic, there are the various theories of higher-types and of intuitionistic set theory. Arithmetic of all finite types is a di-rect descendant of Godel’s theory T, which, so to speak, is the logic-free partof higher-type arithmetic. Godel introduced in [Godel 1958] functionals of fi-nite types as a tool for a constructive interpretation of arithmetic. He used alogical translation of HA into type theory, and by a final elimination of thequantifiers, he obtained equations in the higher type calculus T. Given theintuitive constructive character of these functionals, a consistency proof was ob-tained. The particular interpretation is known as the Dialectica Interpretation.In [Troelstra 1973] the proof theory of the functional interpretations is exten-sively elaborated.

Semantics

From a metamathematical viewpoint, intuitionistic logic is a subsystem ofclassical logic, hence it should have more models (interpretations). This pre-sented a serious problem: the familiar Tarski semantics was so geared to classicallogic, that it automatically validated PEM. So an extension of the notions ofmodel became necessary. The earliest, and most obvious, approach to this prob-lem was via truth tables. One such construction was provided by Jaskowski in1936 ([Jaskowski 1936]). Another semantics was introduced by Tarski in 1935([Tarski 1938]). He generalized the Boolean-algebra semantics to the algebra ofopen sets of a topological space. The trick is to mimic the Boolean interpreta-tion, taking care to stay within the collections of open sets by taking interiorsof sets whenever necessary. Let us use Int(U) for the interior of U , and U c forthe complement of U . The interpretation of propositional logic in a topologicalspace X, with the collection O(X) of open sets is given by:

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[[⊥]] = ∅[[A ∧B]] = [[A]] ∩ [[B]][[A ∨B]] = [[A]] ∪ [[B]]

[[A → B]] = Int([[A]]c ∪ [[B]])[[¬A]] = [[A]]c

The interpretation is extended to predicate calculus logic by putting[[∀xA(x)]] = Int(

⋂d∈D[[A(d)]]

[[∃xA(x)]] =⋃

d∈D[[A(d)]]

where D is the given domain.A sentence A is true under the interpretation if [[A]] = X.

The topological interpretation allows one to manufacture simple counterex-amples to non-intuitionist, classical tautologies. Here are some examples.Consider [[A]] = R− {0}, then [[¬A]] = Int({0}) = ∅ and [[¬¬A]] = R.Observe that [[B → C]] = R iff Int([[B]]c ∪ [[C]] = R i.e. [[B]] ⊆ [[C]].So we see that ¬¬A → A is not true in this model. Similarly [[A ∨ ¬A]] =R− {0} 6= R.The extension of Tarski’s topological interpretation was given by Mostowski in1948 ([Mostowski 1948]).

In view of the algebraic character of the topological interpretation, it comesas no surprise that, in analogy to Boolean algebras, algebras for intuitionisticlogic turned up, see [Stone 1937]. Nowadays these algebras are called Heytingalgebras.A thorough treatment of interpretations by means of Heyting algebras and topo-logical spaces was given by Rasiowa and Sikorski.[Rasiowa–Sikorski 1963].

In spite of their elegance, the above mentioned interpretations could notmatch the 0-1 interpretation for classical logic, as a philosophically motivatedtechnique. The first significant improvement in this respect was given by Bethin 1956 ([Beth 1956]). He considered trees (or partially ordered sets) with cer-tain truth value assignments. This so-called Beth-semantics was more or lessput in the shadow by a similar semantics formulated by Kripke in 1963. SinceKripke’s semantics is somewhat simpler, we will consider it instead of Beth’soriginal notion. The heuristics for Kripke’s semantics is provided by Brouwer’snotion of the creating subject (see below): represent the stages of constructiveactivity of the individual by nodes in a partially ordered set, where ‘<’ standsfor ‘later’. The individual is carrying out two activities simultaneously, he con-structs new objects and establishes new facts about his universe. There is asimple assumption: no objects are destroyed and no basic facts are forgotten.

In order to allow an efficient presentation, we need some notation: (K,≤) isa partially ordered set, the elements are denoted by k, l, . . . , k0, k1, . . . , l1, l2, . . ..For each k ∈ K there is a traditional structure, i.e. a domain with relationsand total functions, denoted by Ak. It is understood that the similarity type(signature) is fixed

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Ak = 〈|Ak|, R1k, R2k..., f1k, f2k, . . . c1k...〉. The structures have the same con-stants.“The individual knows A at stage k” is denoted by ‘k |= A’. The above men-tioned monotonicity aspects of domains and facts is rendered by the followingconditions:

i. |Ak| ⊆ |Al|

ii. Rik ⊆ Ril for k ≤ l

iii. fik ⊆ fil for k ≤ l

This collection of structures, indexed by a partially ordered set and satisfy-ing the monotonicity condition, is called a Kripke model.The structures Ak are often called (possible) worlds.If now remains to define “A holds in world k”, in symbols ‘k ` A’, for all sen-tences A:

k ` A conditionk ` A, for atomic A Ak |= A (in the sense of Tarski)k ` A ∧B k ` A and k ` Bk ` A ∨B k ` A or k ` Bk ` A → B for all l ≥ k(l ` A ⇒ l ` B)k ` ∃xA(x) there exists an a ∈ |Ak| such that k ` A(a)k ` ∀xA(x) for all l ≥ k and all a ∈ |Al| l ` A(a)

By definition we put k ` ¬A ⇔ k ` A →⊥, where, of course, for no k k `⊥.The analogy with the proof interpretation is not exact, in the sense that the‘uniformity’ in, for example, the clause for implication is hidden. The subject‘knows’, so to speak, that if in the future evidence for A turns up, evidence forB will automatically follow.Kripke models are very flexible, and convenient for metamathematical purposes.For a quick demonstration, let us give some counterexamples to classical tautolo-gies. Observe that k ` ¬A means that at no later stage A will be established. Sok ` ¬¬A means that at no later stage l, we will have l ` ¬A, therefore there mustbe a stage k ≥ l, at which A holds. That is k ` ¬¬A ⇔ ∀l ≥ k∃m ≥ l(m ` A).Now we can simply refute the double negation law: consider a model with twonodes k, l and k < l, assume that for an atom A k 6 ` A and l ` A (this is justpart of the specification of the particular model). Then k ` ¬¬A, and hencek 6 ` ¬¬A → A. The same model shows that k 6 ` A ∨ ¬A, for k 6 ` A andk 6 ` ¬A.

It is a little bit more complicated to get a counter model for the followinginstance of De Morgan’s law: ¬(A ∧ B) → ¬A ∨ ¬B. Consider a model withthree nodes, k, l,m,wherek < l and k < m, but l and m are incomparable. Now

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put l ` A and m ` B, then for no node ≥ k A ∧ B holds, so k ` ¬(A ∧ B).But if k ` ¬A ∨ ¬B, we should have k ` ¬A or k ` ¬B. Neither is the case,according to the definition. So k 6 ` ¬(A ∧B) → (¬A ∨ ¬B).

For all of the above mentioned semantics there are completeness theorem,in the sense that IPC ` A ⇔ A holds in all models of the specified class,cf. [Rasiowa–Sikorski 1963], [Kripke 1965], [Fitting 1969], [Dalen, D. van 1997],[Troelstra–van Dalen 1988], [Smorynski 1973].The semantic methods lend themselves to a large variety of metamathematicalresults (which often had been obtained previously by other methods).

The above semantics were further exploited to investigate second-order theo-ries, e.g. arithmetic with set variables (HAS) , arithmetic with choice-sequencevariables (analysis), the theory of the reals, topology, etc.

Dana Scott extended in 1968 the topological interpretation to a model forthe intuitionistic continuum, cf. [Scott 1968], [Scott 1970]. Joan Moschovakisadopted Scott’s methods to give a topological interpretation of intuitionisticanalysis in Kleene’s formalization [Moschovakis 1973], van Dalen did the samething for HAS, [Dalen, D. van 1974]; a model for analysis in Beth semanticsfollowed in 1978, [Dalen, D. van 1978]. In a sense, these papers can be seenas predecessors of a more general sheaf semantics introduced by Scott in theseventies, [Fourman–Scott 1979]. The sheaf models introduced a novelty, thatso far had been beyond existing formalizations and semantics; existence as apredicate and partial operations.

In order to understand the need for such novelties, one has to go back to or-dinary arithmetic on the reals. Addition, subtraction and multiplication presentno problem, but the inverse asks for some caution. One of the methods math-ematicians use to define real numbers from the rational numbers is that of theCauchy sequences; a real number is defined by a ‘converging sequence’ of ratio-nals, that is to say one requires that for the sequence (ai) the following holds:∀k∃n∀m > n(|an − am| < 2−k) In order to invert a real number a, given bya Cauchy sequence (an), one usually simply inverts all the an’s (except thoseidentical to 0). Classically that is good enough, because if a 6= 0, then from acertain n on all an’s are distinct from 0.Intuitionistically, there is a problem; one can very well imagine a real numbera, distinct from 0, without having any guarantee that (a−1

n ) is again a Cauchysequence, that is to say, the convergence of (a−1

n ) is problematic. In fact itcan only be established if from a certain index on all (|an|) are greater than afixed positive rational number. This notion of ‘strongly distinct from0’ is tradi-tionally called “apart from 0”. The conditions for invertibility is formulated as∃x(a · x = 1) ↔ a#0. Hence, the inverse is essentially a partial operator, it isonly defined for reals apart from zero. And a#0 is not decidable. The classical

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trick of making the inverse total does therefore not work here.13

So, an existence predicate makes sense. Let us write E(x) for “x exists”,then Ea−1 ↔ a#0. Of course, the addition of an existence predicate to logicrequires an overhaul of the system. One has to revise the theory of identity inorder to respect the basic tenet that only existing objects can be identical.Hence one would require x = y → Ex∧Ey. There are similar changes for quan-tification, ∃xA(x) means that there has to be an existing object a such that A(a).This suggests a modified existence introduction rule: A(a) ∧ E(a) ` ∃xA(α).Similarly, one universally quantifies over existing objects: ∀xA(x), E(a) ` A(a).For a complete exposition see [Scott 1979], [Troelstra–van Dalen 1988].

The sheaf semantics fits this approach to existence like a glove. Consider forexample, all continuous functions with open domain in R, they can be consideredas entities in their own right, and they can be put into relation with each other,they can be pointwise added, multiplied etc. In short, they behave as thereals themselves. Now, obviously, such a partial function ‘exists only on itsdomain’, therefore it is plausible to put Ea = {t ∈ R|a(t) is defined}. Nowif we put [[a#0]] = {t ∈ R|a(t) 6= 0} (which is an open set!), then a simplecalculation shows us that [[Ea ↔ a#0]] = R. The objects in these models, ingeneral, exist only partially. This has consequences for the theory of identity,the connection between identuty and existence can be formulated “objects areidentical to themselves precisely in as far a s they exist”– |= a = a ↔ Ea, cf.[Scott 1979], [Troelstra–van Dalen 1988], p. 50 ff., p. 709 ff.

The sheaf semantics has brought out a number of characteristic featuresof intuitionistic systems. There is, for example, a quite simple demonstra-tion that the real numbers defined via Cauchy sequences, are different fromthose defined by Dedekind cuts, cf. [Fourman–Hyland 1979], [Grayson 1981];as a consequence the axiom of countable choice fails in sheaf models oversuitable connected topological spaces. Similarly, it quite straightforward toshow the failure of the fundamental theorem of algebra in the model over R2,[Fourman–Hyland 1979]. Thus there are polynomials that do not have zeros inthe complex numbers, this seems in conflict with Brouwer’s earlier proof of thattheorem, but Brouwer made tacitly use of the countable axiom of choice.

Sheaves, however, are not the end point of our tour of the world of seman-tics. It turns out that the collection of sheaves on a topological space carriesa natural higher-order structure, one can define power sheaves, function spacesheaves, etc. In technical terms: sheaves on a space form a topos. That is to say,it is a special kind of category with convenient closure properties. To be precise,a topos is a category with all finite limits and a subobject classifier. Since wewill not go into the categorical side of intuitionistic logic, we refer the reader fora comprehensive treatment to [Lambek–Scott 1986], [MacLane–Moerdijk 1992].

13Classically one may say “a−1 is the unique b such that ab = 1 if a 6= 0, and 0 else”.

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An important feature of topos theory and its categorical interpretation ofintuitionistic logic, is that it offers a unifying approach to all semantics that havebeen formulated before. The topological interpretation, the sheaf semantics, theBeth and Kripke semantics, all represent specific topos constructions. There iseven a particular interpretation that has not been discussed so far, and thatmore or less to the surprise of earlier researchers, fits nicely into the categoricalframework.A topos can be viewed as an example of the (or ‘a’) intuitionistic universe, andthe categorical properties are closely tied to the logical properties.

The flexibility of the general sheaf-semantics was demonstrated by Moerdijkwho constructed a specific model in which the so-called lawless sequence founda natural interpretation, [Hoeven, G..F. and Moerdijk, I. 1984]. The first se-mantic interpretation of lawless sequences was given by Van Dalen in a Bethsemantics (which happens to be eminently suited for analysis interpretations).[Dalen, D. van 1978].

Lawless sequence go back a long way, Brouwer mentioned the idea (withoutthe name) in a letter to Heyting (26.4.1924, [Troelstra 1982]). Kreisel introducedthe notion in the literature, [Kreisel 1962]. A sequence is lawless if at any stageof the generation only the given initial segment is known, there are and will be norestrictions on future choices. Depending on one’s view, the connection betweenvarious lawless sequences may or may not be allowed. The simplest version doesnot allow for an interplay between lawless sequences. So when a certain sequenceα is lawless, the sequence β with β(2n) = α(2n + 1), β(2n + 1) = α(2n), cannotbe considered as lawless, since it is not completely free with respect to thecontext.

Lawless sequences were used to give intuitionistic versions of logical com-pleteness theorem, [Kreisel 1962], [Troelstra–van Dalen 1988] Ch. 13. Further-more, because of their simple nature, they lent themselves perfectly for theinvestigation (refutation) of classical tautologies. The consistency of the theorywas established proof theoretically long before the first semantics was found.Kreisel discovered that in a sufficiently strong theory of analysis one could elim-inate lawless sequences by means of a translation. The technique was laterextended to wider families of choice sequences, [Kreisel–Troelstra 1970].

One should not get the impression that all semantics of intuitionistic theoriestake place under artificial circumstances. Indeed, one can use a Tarski-style se-mantics, giving the connectives their intended constructive interpretation. Someresults show that a worthwhile model theory can be developed. E.g. Van Dalenshowed that R, Qc Qcc (the negative irrationals and the not-not rationals) areelementarily equivalent with respect to, <,#,=, [Dalen, D. van 1992].

E.g. the topological sheaf semantics has many, if not all, characteristics of auniverse with choice sequences. There happens also to be a specific topos thatreflects the algorithmic aspects of constructive mathematics.

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Recursive functions and intuitionistic logic

In 1945 Kleene presented a specific interpretation of intuitionistic arithmetic,called (Recursive) realizability. The underlying idea was that “truth” of a state-ment should be established by algorithms. For example, if one had an algorithmthat ‘established’ A and another one that ‘established’ B, the pair would estab-lish A ∧ B. Since one would need algorithms operating on algorithms, it wasnatural to consider partial recursive functions (or Turing machines), for eachsuch function f has a description which can be coded as a natural number,usually called the index of f , and instead of having an algorithm operate onan algorithm, one can simulate this effect by letting a partial recursive functionoperate on the index of another such function.

Following the tradition we will denote “the function with index e applied tothe number n” as {e}(n). Observe that {e}(n) may be undefined (think of aTuring machine getting into a loop), so some care is required – for example bycalling in the help of the existence predicate.Instead of ‘n establishes A’ we will say ‘n realizes A’. Now here is a way tohandle implication: e realizes A → B ⇔ for any a which realizes A {e}a realizesB.

In spite of the similarity of this clause to the proof-interpretation, Kleenetook his cue from Hilbert–Bernays rather than from Heyting, cf. [Kleene 1973],see also [Dalen, D. van 1997]. The notion ‘n realizes A’ is inductively definedbelow.

Note that in arithmetic closed atoms are decidable (just compute the leftand right hand side of the equation), so the initial clause is offered for free.(n)0 and (n)1 are the two projections of the standard pairing function < p, q >(i.e. (< p, q >)0 = p, (< p, q >)1 = q, < (n)0, (n)1 >= n).

nrA condition

nrA, atomic A a is truenrA ∧B (n)0rA and (n)1rBnrA ∨B (n)0 = 0 ⇒ (n)1rA and (n)0 6= 0 ⇒ (n)1rBnrA → B for all m mrA ⇒ {n}mrBnr∀xA(x) (n)1rA((n0))nr∀xA(x) for all m {n}(m)rA(m)

Surprising as it may seem, the idea worked: HA ` A ⇒ nrA for some n.There is no total success in the sense that the interpretation is complete, thererealizable statements which are not provable in HA. One particularly importantsuch statement is Church’s Thesis. In computability theory the thesis of Churchclaims that every numerical algorithm can be simulated by a Turing machine,or a recursive function. It is truly a thesis in the sense that it connects an infor-mal concept - computability - with a precise mathematical notion - recursivity(or Turing computability). One can either reject the thesis by presenting an

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acknowledged algorithm, which cannot be simulated by a Turing machine, orone can give conceptual arguments in favour of the thesis. The latter was doneby Alan Turing in his [Turing 1937]. Nonetheless, the thesis can never get thestatus of a theorem.

The precise extent of the realizable statements of arithmetic was establishedby Troelstra in his axiomatization, [Troelstra 1973].

In constructive mathematics one can a little bit better by reflecting onthe meaning of the “∀x ∈ N”-combination. According to the proof inter-pretation a : ∀x∃yA(x, y) mean that for all n a(n) : ∃yA(n, y), and thusπ1(a(n)) : A(nπ2(a(n))), hence λx.π1(a(n)) : ∀xA(x, π2(a(x))). Therefore thereis a choice function selecting the required y, and more, if the A contains onlylawlike parameters (in particular no set- or choice parameters) the choice func-tion is itself lawlike. So Church’s Thesis would decree that the choice functionis Turing computable. We can formalize this result:

CT∀x∃yA(x, y) → ∃e∀xA(x, {e}(x))

Here it is tacitly assumed that {e} is total.

We see that in HA Church’s Thesis can be formulated as an arithmeticalschema. Hence it is open to meta-mathematical treatment. Here the realizabil-ity technique offers some help: CT turns out to be realizable, so it is consistentwith HA (observe that it certainly is not derivable!). In other words, there isno objection to carry out arithmetic under the assumption of Church’s Thesis,as long as one avoids PEM .

Hyland extended this result by constructing a topos in which CT holds, theeffective topos.[Hyland 1982].CT has far reaching consequences for the daily practice of mathematics. Speckerhad already pointed out in [Specker 1949] that certain classical theorems of realanalysis were no longer true if one required recursive results.

For example: a recursive continuous real function f with f(0) < 0 andf(1) > 0 need not have a recursive zero. In the setting of the effective topos onecan drop the adjective ‘recursive’, and simply say that there are continuous realfunctions with positive and negative values without zeros. In such a computableuniverse, indeed many traditional theorems lose their validity. More examples:[0, 1] is not compact; a bounded continuous function on [0, 1) need not have asupremum (let alone a maximum); there are functions, positive on [0, 1], withouta positive lower bound, etc. Here also the work of Pour El and Richards on thesolutions of differential equations becomes relevant, [Pour El–Richards 1979].These facts are on the negative side, on the positive side there is the result that(assuming Markov’s Principle) every real function is continuous. So, althoughthe recursive universe and the choice universe are on opposite sides of the spec-trum, they share certain features.

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Another, rather unexpected, fact about CT is that it allows no non-standardmodels for arithmetic, [McCarty 1988]. In view of the rich variety of non-standard models for Peano’s arithmetic, this certainly indicates that life withCT is an exciting experience.

The Creating Subject

A sketch of intuitionistic logic would not be complete without the mentioningof Brouwer’s later extension of his theory of choice sequences. In the earlier pa-pers, Brouwer used the technique of Brouwerian counter examples to show thatcertain classical mathematical facts could not be considered to be intuitionisti-cally true. The technique consisted of a reduction of some theorem to a patentlyunsolved problem. The drawback of this method was that it only allowed weakstatements, like “there is no evidence that . . . ”. In the twenties Brouwer him-self progressed beyond these ‘doubtful’ statements by showing that it is not thecase that each real number is rational or irrational, this was an immediate con-sequence of the continuity theorem. He wanted, however, an extension of theBrouwerian counter example method. In his Berlin Lectures, [Brouwer 1992],Brouwer formulated such an extension. It appeared in print only after the Sec-ond World War, [?]. The method is that of the creating subject.14 The basicidea is that with respect to a certain statement A the subject may, by hook orby crook, try to establish its truth. If A is true, then the creating subject willexperience this in due course, so he can create a function that keeps track ofthe situation by writing down zeros, as long as A has not been established, assoon as he sees that A is true, he writes down a one. This was formalized byKripke’s Schema:

KS ∃α(∃xα(x) = 0 ↔ A

Myhill showed that under Kripke’s Schema the extended continuity principle,“∀α∃β-continuity”, fails, [Myhill 1966]. Quite recently Van Dalen used the ideato extend Brouwer’s undecomposibility of the continuum to the irrationals andsimilar sets, [Dalen, D. van 1997A]. This makes it clear that the method is notjust a tool for refutations of certain principles, but it has structural consequencesas well.

Intuitionistic Completeness

A final remark on the intuitionistic status of metamathematics: the usualtechniques for establishing e.g. completeness results heavily rely on non-intuitionisticprocedures, such as proof by contradiction. The question, therefore, is in howfar can one prove, say, completeness for Kripke models using exclusively intu-itionistic means.

14Erroneously called ‘creative subject’ in the literature.

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Early investigations of Godel and Kreisel, [Kreisel 1962], had shown that thecompleteness proof could be constructivised up to a point: Markov’s principlewas required. This principle asserts that for primitive recursive predicates theimpossibility of the non-existence of an element yielded the existence of anelement.

MP : ¬¬∃xA(x) → ∃xA(x)

or in general, replacing the primitive recursiveness by decidability:

∀x(A(x) ∨ ¬A(x)) ∧ ¬¬∃xA(x) → ∃xA(x)

Since Markov’s principle could not be given a convincing intuitionistic jus-tification (it can be paraphrased as “if it is impossible that a Turing machinewith a given input tape never stops, then it stops”), this seemed to block thedevelopment of a truly intuitionistic completeness theorem. Veldman broke thisstalemate in 1974 this statement by presenting an extension of the standardnotion of Kripke model, [Veldman 1976]. In this new semantics one can indeedgive an intuitionistically acceptable completeness proof. For a systematic surveysee [Dummett 1977], [Troelstra–van Dalen 1988].

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