Russell - A Very Short Indroduction (7 page)

Russell’s more complicated version of type theory is called the ‘ramified theory of types’. (The correct way to understand this theory is a matter of controversy – see, for example, Hylton 1990, chapter 7 – but the following sketch can serve as a first approximation.) Russell’s reason for introducing ‘ramification’ – which means the internal subdivision of types into ‘orders’ – was that he thought a solution to the paradox problem specifically needed it. He took it that the paradox problem results from attempting to define properties by means of expressions which contain reference to ‘all properties’, so talk of ‘all properties’ must be strictly controlled. Properties of, say, type 1 are therefore to be subdivided into orders: a first order of properties in whose definition the expression ‘all properties’ does not occur; a second order of properties in whose definition the expression ‘all properties of the first order’ occurs; a third order in whose definition ‘all properties of the second order’ occurs; and so on. Since there is never reference to ‘all properties’ which does not anchor it to a definite order, no property is ever defined in such a way that reference is made to the totality it belongs to. And this avoids paradox.

But it does so at a major cost. It introduces difficulties into the theory of real numbers by blocking its most important definitions and theorems. It was to overcome this problem that Russell introduced the axiom of reducibility, which tries to engineer a way of reducing orders within a type to the lowest order. This manoeuvre, likened by one commentator to using ‘brute force’ to salvage real number theory, was abandoned by Russell in the second edition of
PM
(1927). But because he could not accept that there is any alternative to a ramified theory of types he was left in quandary. It was in response to this that Ramsey put forward the ‘simple’ theory of types sketched above. (It is as well to note that Ramsey’s theory invites debate on its own account. It makes the controversial claim that the circularity in definitions which ascribe properties to themselves is harmless; and it demands an equally controversial realism about the existence of totalities before they are defined.)

Russell’s logicist ambitions ran into difficulties partly on their own account and partly because, as later developments in mathematics – especially the work of Kurt Gödel – suggest, logicism itself is unfeasible. Gödel showed that in any formal system adequate for number theory there is an undecidable formula, that is, a formula such that neither it nor its negation can be proved. A corollary of this is that the consistency of such a system cannot be established within the system, so one cannot assume that mathematics (or anyway large parts of it) can be provided with a set of axioms sufficient for generating all its truths. His work shows that the axiomatic method has profound inherent limitations, and that the only way to prove the consistency of many kinds of deductive systems is to use a system of reasoning so complicated that its own consistency is equally doubtful.

What Russell required to see his logicist project through was a formal systematization which excludes the possibility of contradiction. Gödel’s work says this is impossible. It has to be concluded that the achievement of
PoM
and particularly
PM
is not to be found in the degree to which they realize their stated aims, but in what might be called their many significant ‘spin-offs’ for logic and philosophy.

The Theory of Descriptions

One of the most influential spin-offs was Russell’s ‘Theory of Descriptions’. In working out this important theory Russell achieved a number of different goals. One lesson he had learned from arguing against idealism is that the surface grammar of language can mislead us about the meaning of what we say. As noted above, he thought that the reason philosophers had been led into adopting a metaphysics of substance and attribute – a view that, as debate in the history of philosophy shows, runs into deep difficulties – was that they had taken all propositions to be fundamentally subject-predicate in form. ‘The table is made of wood’ and ‘the table is to the left of the door’ were both treated as having the expression ‘the table’ as subject, and as predicate the expressions which in each case follow the copula ‘is’. But whereas the first sentence might express a proposition of that form, the second expresses something quite different, namely, a relational proposition; it in fact has two subjects (‘the table’ and ‘the door’) and it asserts that they stand in a particular relation to one another. So the logical form of the second sentence is quite different from the logical form of the first. In Russell’s view, the need, therefore, is for a method of revealing the true underlying form of what we say to help us avoid philosophical mistakes.

The next important step Russell took was to apply the new logic to this task. Just as it serves to define the concepts and operations of mathematics, so we can use it to analyse what we say about the world, thus getting a correct picture of reality.

One way of showing how the theory of descriptions carries out this task is to describe how it solves an important problem about meaning and reference. The background to Russell’s treatment of this problem is to be found in the work of Alexius Meinong, an Austrian philosopher whose writings Russell had carefully studied and who had therefore been an early influence on him. Meinong held that denoting expressions – names like ‘Russell’ and descriptions like ‘the author of
The Principles of Mathematics
’
–
can occur significantly in propositions (strictly: in sentences expressing propositions) only if what they denote exists. Suppose, Meinong argued, you say, ‘the golden mountain does not exist’. Obviously, you are talking about something – the golden mountain – when you assert that it does not exist; and since what you say is meaningful, there must therefore in some sense
be
a golden mountain. His theory is that everything that can be talked about – named, referred to – must therefore either exist or have some kind of ‘being’ even if such being does not amount to existence, for otherwise what we say would be meaningless.

Russell accepted this view at first, and indeed held it in
PoM
; which is why, as noted earlier, he there expressed belief in the existence or at least being of ‘numbers, Homeric gods and chimeras’. But the implausibility of this view soon came to offend his ‘vivid sense of reality’, as he put it, for it crowded the universe not just with abstract and mythological entities but also with
impossible
objects like ‘the round square’ – and this Russell could not accept.

Russell used the techniques of logic to devise a beautiful solution. He did not wish to give up the view that a name is meaningful only if there is something it names, but he argued that the only ‘logically proper’ names are those which denote
particular
entities with which one can be
acquainted
. By ‘acquaintance’ Russell meant an immediate and direct relation between a mind and an object; examples include awareness of sense-data in perception (see below) and knowledge of such abstract entities as propositions. Only logically proper names can properly occupy subject-position in sentences. The best examples are the demonstrative pronouns
this
and
that
, for the reason that they are guaranteed a reference every time they are used. All other apparent naming expressions are in fact not naming expressions at all; they are – or when they are analysed they turn out to be – ‘definite descriptions’, that is, expressions of the form ‘the so-and-so’. The importance of this is that when sentences containing descriptions are analysed, the descriptive phrases vanish, and therefore the meaningfulness of what one says does not depend upon the supposed existence or being of some entity which the descriptions appear – according to surface grammar – to denote.

This can be seen by considering an example. Take the sentence ‘The present king of France is bald’, said at a time when France has no king. On the supposition that sentences are always either true or false, what is one to say if asked: is this sentence true or false? It seems obvious to say ‘false’ – not because the present king of France has a fine head of hair, but because he does not exist. This point gave Russell his clue. He argued that sentences with definite descriptions in grammatical subject-place turn out upon analysis to be shorthand for a set of sentences asserting the existence, the uniqueness, and the baldness of something having the property of being the present king of France. Thus ‘the present king of France is bald’ is equivalent to:

(1) there is a king of France,
(2) there is not more than one king of France, (3) whatever is king of France is bald.

Sentence (1) is an existence claim; (2) is a uniqueness claim, that is, it captures the implication of ‘the’ in the description that there is only one thing being talked about; and (3) is the predication. The original sentence ‘the present king of France is bald’ is true when all three are true; it is false if any one of them is false. In the present case it is false because (1) is false.

In none of (1)–(3) does the description ‘the present king of France’ appear. Because the descriptive phrase has vanished – has been analysed away – there is no need to invoke a subsistent king of France to make the sentence meaningful.

Owing to the imperfections of ordinary language, and the fact that the surface forms of sentences can diverge from their underlying logical form, the analysis thus given is not yet, says Russell, good enough. It needs to be expressed in the ‘perfect language’ of symbolic logic. This alone can display with
complete
clarity what is being asserted by ‘the present king of France is bald’. In notation which is now standard, the logical analysis of this sentence is:

(∃x)[Fx & (y)(Fy → y = x) & Gx]

The ‘&’ in this string of symbols stands for ‘and’, dividing the string of symbols into three conjoined formulae, so the three sentences (1)–(3) above are respectively:

(1) (∃x)Fx This is pronounced ‘there is an x such that x is F’. Let ‘F’ be ‘has the property of being king of France’; the formula symbolizes ‘there is something which is king of France’. (The existential quantifier (∃x) binds every occurrence of ‘x’ in the whole string, of course, as the square brackets show.)

(2) (y)(Fy → y = x)

This is pronounced ‘for everything y, if y is F then y and x are identical’. This expresses the uniqueness implied by ‘the’, that is, that only one thing has the property F.

(3) Gx
This is pronounced ‘x is G’. Let ‘G’ be ‘is bald’; the formula symbolizes ‘x is bald’.

Objections to Russell’s theory mainly take the form of resisting his claim that definite descriptions are never referring expressions, and questioning his analysis of sentences containing them in grammatical subject-place. In the latter connection, what some dispute is the claim that definite descriptions embody both uniqueness and existence claims.

The problem about uniqueness is exemplified by someone’s saying, ‘the baby is crying’. Russell’s analysis seems to imply that this can only be true if there is just one baby in the world. The way out is to require that there is an implicit understanding that the context of the remark shows how much of the world is included in its range of application. Suppose the parents of a baby inhabit a block of flats where there are dozens of babies, all crying, and their own begins to follow suit. If one said, ‘the baby is crying’, there would obviously be no misunderstanding because the context restricts reference to the one baby in which they have a special interest. So much seems intuitive, and suggests ways of disposing of the objection by appealing to implicit or explicit delimitations of the ‘domain of discourse’.

The problem about existence is a little more complex. In a much-cited discussion of Russell’s theory, P. F. Strawson argues that in saying ‘the present king of France is bald’ one is not
stating
that a present king of France exists, but presupposing or assuming that it does (‘On Referring’,
Mind
, 1950). This is shown by the fact that if someone uttered this sentence, his interlocutors are not likely to say, ‘that’s false’, but instead, ‘there’s no king of France at present’, thereby making the point that he had not in fact made a statement, that is, he had not succeeded in saying anything true or false. This amounts to saying that descriptions must be referring expressions because an important part of their contribution to the truth-values of sentences containing them is that, unless they refer, the sentences in question do not have a truthvalue at all.

Strawson’s use of a notion of ‘presupposition’ to explain how, on his opposed view, descriptions function in sentences, has prompted much critical debate, and so has his preparedness to allow ‘truth-value gaps’, that is, absence of truth-value in a meaningful sentence – thus breaching the ‘principle of bivalence’ which says that every (declarative) sentence must have one or other of the two truth-values ‘true’ and ‘false’. But the main response to his criticism of Russell is undoubtedly to say that the fact upon which his case turns, namely, that we would not say ‘that’s false’ when someone says ‘the present king of France is bald’, does not mean that the description cannot be treated as making an existential claim. It might be true that we would respond by denying that there is a king of France; after all, merely to say ‘that’s false’ might be misleading, because it could imply something quite different, namely, that there is a hairy king of France. But if we reply ‘there is no king of France at present’ we have in effect acknowledged that use of the description makes an existential claim – for that is exactly what the denial addresses.
Another objection is that Russell did not see that there are two different uses that can be made of descriptions. Consider the following two cases. First, you see a painting you like, and you say, ‘the artist who painted this is a genius’. You do not know who the artist is, but you attribute genius to him. Secondly, the painting is ‘Madonna of the Rocks’, and you know that Leonardo painted it. In admiration you murmur the same sentence. In the first case the description is used ‘attributively’, in the second ‘referentially’. According to Keith Donnellan, who advanced this criticism, Russell’s account concerns only attributive uses. This matters because there are cases where a description can be used successfully to refer to someone even if it does not apply to him – ‘the man drinking champagne over there is bald’ can be used to say something true even if the bald man’s glass contains only fizzy water.