Epsilon Calculi are extended forms of the predicate calculus that incorporate epsilon terms. Epsilon terms are individual terms of the form ‘xFx’, being defined for all predicates in the language. The epsilon term ‘xFx’ denotes a chosen F, if there are any F’s, and has an arbitrary reference otherwise. Epsilon calculi were originally developed to study certain forms of Arithmetic, and Set Theory; also to prove some important meta-theorems about the predicate calculus. Later formal developments have included a variety of intensional epsilon calculi, of use in the study of necessity, and more general intensional notions, like belief. An epsilon term such as ‘xFx’ was originally read ‘the first F’, and in arithmetical contexts ‘the least F’. More generally it can be read as the demonstrative description ‘that F’, when arising either deictically, i.e. in a pragmatic context where some F is being pointed at, or in linguistic cross-reference situations, as with, for example, ‘There is a red haired man in the room. That red haired man is Caucasian’. The application of epsilon terms to natural language shares some features with the use of iota terms within the theory of descriptions given by Bertrand Russell, but differs in formalising aspects of a slightly different theory of reference, first given by Keith Donnellan. More recently epsilon terms have been used by a number of writers to formalise cross-sentential anaphora, which would arise if ‘that red haired man’ in the linguistic case above was replaced with a pronoun such as ‘he’. There is then also the similar application in intensional cases, like ‘There is a red haired man in the room. Celia believed he was a woman.’

## 1. Introduction

Epsilon terms were introduced by the german mathematician David Hilbert, in Hilbert 1923, 1925, to provide explicit definitions of the existential and universal quantifiers, and resolve some problems in infinitistic mathematics. But it is not just the related formal results, and structures which are of interest. In Hilbert’s major book *Grundlagen der Mathematik*, which he wrote with his collaborator Paul Bernays, epsilon terms were presented as formalising certain natural language constructions, like definite descriptions. And they in fact have a considerably larger range of such applications, for instance in the symbolisation of certain cross-sentential anaphora. Hilbert and Bernays also used their epsilon calculus to prove two important meta-theorems about the predicate calculus. One theorem subsequently led, for instance, to the development of semantic tableaux: it is called the First Epsilon Theorem, and its content and proof will be given later, in section 6 below. A second theorem that Hilbert and Bernays proved, which we shall also look at then, establishes that epsilon calculi are conservative extensions of the predicate calculus, that is, that no more theorems expressible just in the quantificational language of the predicate calculus can be proved in epsilon calculi than can be proved in the predicate calculus itself. But while epsilon calculi do have these further important formal functions, we will not only be concerned to explore them, for we shall also first discuss the natural language structures upon which epsilon calculi have a considerable bearing.

The growing awareness of the larger meaning and significance of epsilon calculi has only come in stages. Hilbert and Bernays introduced epsilon terms for several meta-mathematical purposes, as above, but the extended presentation of an epsilon calculus, as a formal logic of interest in its own right, in fact only first appeared in Bourbaki’s *Éléments de Mathématique* (although see also Ackermann 1937-8). Bourbaki’s epsilon calculus with identity (Bourbaki, 1954, Book 1) is axiomatic, with Modus Ponens as the only primitive inference or derivation rule. Thus, in effect, we get:

(X ∨ X) → X,

X → (X ∨ Y),

(X ∨ Y) → (Y ∨ X),

(X ∨ Y) → ((Z ∨ X) → (Z ∨ Y)),

Fy → FεxFx,

x = y → (Fx ↔ Fy),

(x)(Fx ↔ Gx) → εxFx = εxGx.

This adds to a basis for the propositional calculus an epsilon axiom schema, then Leibniz’ Law, and a second epsilon axiom schema, which is a further law of identity. Bourbaki, though, used the Greek letter tau rather than epsilon to form what are now called ‘epsilon terms’; nevertheless, he defined the quantifiers in terms of his tau symbol in the manner of Hilbert and Bernays, namely:

(∃x)Fx ↔ FεxFx,

(x)Fx ↔ Fεx¬Fx;

and note that, in his system the other usual law of identity, ‘x = x’, is derivable.

The principle purpose Bourbaki found for his system of logic was in his theory of sets, although through that, in the modern manner, it thereby came to be the foundation for the rest of mathematics. Bourbaki’s theory of sets discriminates amongst predicates those which determine sets: thus some, but only some, predicates determine sets, i.e. are ‘collectivisantes’. All the main axioms of classical Set Theory are incorporated in his theory, but he does not have an Axiom of Choice as a separate axiom, since its functions are taken over by his tau symbol. The same point holds in Bernays’ epsilon version of his set theory (Bernays 1958, Ch VIII).

Epsilon calculi, during this period, were developed without any semantics, but a semantic interpretation was produced by Gunter Asser in 1957, and subsequently published in a book by A.C. Leisenring, in 1969. Even then, readings of epsilon terms in ordinary language were still uncommon. A natural language reading of epsilon terms, however, was present in Hilbert and Bernays’ work. In fact the last chapter of book 1 of the *Grundlagen* is a presentation of a theory of definite descriptions, and epsilon terms relate closely to this. In the more well known theory of definite descriptions by Bertrand Russell (Russell 1905) there are three clauses: with

The king of France is bald

we get, on Russell’s theory, first

there is a king of France,

second

there is only one king of France,

and third

anyone who is king of France is bald.

Russell uses the Greek letter iota to formalise the definite description, writing the whole

BιxKx,

but he recognises the iota term is not a proper individual symbol. He calls it an ‘incomplete symbol’, since, because of the three parts, the whole proposition is taken to have the quantificational analysis,

(∃x)(Kx & (y)(Ky → y = x) & (y)(Ky → By)),

which is equivalent to

(∃x)(Kx & (y)(Ky→ y = x) & Bx).

And that means that it does not have the form ‘Bx’. Russell believed that, in addition to his iota terms, there was another class of individual terms, which he called ‘logically proper names’. These would simply fit into the ‘x’ place in ‘Bx’. He believed that ‘this’ and ‘that’ were in this class, but gave no symbolic characterisation of them.

Hilbert and Bernays, by contrast, produced what is called a ‘pre-suppositional theory’ of definite descriptions. The first two clauses of Russell’s definition were not taken to be part of the meaning of ‘The King of France is bald’: they were merely conditions under which they took it to be permitted to introduce a complete individual term for ‘the King of France’, which then satisfies

Kx & (y)(Ky → y = x).

Hilbert and Bernays continued to use the Greek letter iota in their individual term, although it has a quite different grammar from Russell’s iota term, since, when Hilbert and Bernays’ term can be introduced, it is provably equivalent to the corresponding epsilon term (Kneebone 1963, p102). In fact it was later suggested by many that epsilon terms are not only complete symbols, but can be seen as playing the same role as the ‘logically proper names’ Russell discussed.

It is at the start of book 2 of the *Grundlagen* that we find the definition of epsilon terms. There, Hilbert and Bernays first construct a theory of indefinite descriptions in a similar manner to their theory of definite descriptions. They allow, now, an eta term to be introduced as long as just the first of Russell’s conditions is met. That is to say, given

(∃x)Fx,

one can introduce the term ‘ηxFx’, and say

FηxFx.

But the condition for the introduction of the eta term can be established logically, for certain predicates, since

(∃x)((∃y)Fy → Fx),

is a predicate calculus theorem (Copi 1973, p110). It is the eta term this theorem allows us to introduce which is otherwise called an epsilon term, and its logical basis enables entirely formal theories to be constructed, since such individual terms are invariably defined. Thus we may invariably introduce ‘ηx((∃y)Fy → Fx)’, and this is commonly written ‘εxFx’, about which we can therefore say

(∃y)Fy → FεxFx.

Since it is that F which exists if anything is F, Hilbert read the epsilon term in this case ‘the first F’. For instance, in arithmetic, ‘the first’ may be taken to be the least number operator. However, while if there are F’s then the first F is clearly some chosen one of them, if there are no F’s then ‘the first F’ must be a misnomer. And that form of speech only came to be fully understood in the theories of reference which appeared much later, when reference and denotation came to be more clearly separated from description and attribution. Donnellan (Donnellan 1966) used the example ‘the man with martini in his glass’, and pointed out that, in certain uses, this can refer to someone without martini in his glass. In the terminology Donnellan made popular, ‘the first F’, in the second case above works similarly: it cannot be attributive, and so, while it refers to something, it must refer arbitrarily, from a semantic point of view.

With reference in this way separated from attribution it becomes possible to symbolise the anaphoric cross-reference between, for instance, ‘There is one and only one king of France’ and ‘He is bald’. For, independently of whether the former is true, the ‘he’ in the latter is a pronoun for the epsilon term in the former — by a simple extension of the epsilon definition of the existential quantifier. Thus the pair of remarks may be symbolised

(∃x)(Kx & (y)(Ky → y = x)) & Bεx(Kx & (y)(Ky → y = x)).

Furthermore such cross-reference may occur in connection with intensional constructions of a kind Russell also considered, such as

George IV wondered whether the author of Waverley was Scott.

Thus we can say ‘There is an author of Waverley, and George IV wondered whether he was Scott’. But the epsilon analysis of these cases puts intensional epsilon calculi at odds with Russellian views of such constructions, as we shall see later. The Russellian approach, by not having complete symbols for individuals, tends to confuse cases in which assertions are made about individuals and cases in which assertions are made about identifying properties. As we shall see, epsilon terms enable us to make the discrimination between, for instance,

s = εx(y)(Ay ↔ y = x),

(i.e. ‘Scott is the author of Waverley’), and

(y)(Ay ↔ y = s),

(i.e. ‘there is one and only one author of Waverley and he is Scott’), and so it enables us to locate more exactly the object of George IV’s thought.