Depending on your perspective, On Denoting is either an example of the best or the worst that philosophy has to offer.
I’m not sure if I understood Russell exactly, so in this post I’m going to try and explain Russell to myself and if I’m right, I’ve explained it to you, too. Feel free to correct me if you find errors. I will mostly use the examples Russell provides.
Russell is providing a new theory of denotation that is meant to be simpler than Frege’s. Frege distinguishes two elements in a denoting phrase: denotation (the thing an expression refers to) and meaning (the way a phrase asserts its reference). Russell writes: “Thus `the center of mass of the solar system at the beginning of the twentieth century’ is highly complex in meaning, but its denotation is a certain point, which is simple.” In the phrase ‘the center of mass of the solar system at the beginning of the twentieth century,’ the denotation has no constituents, and the constituents of meaning are ‘the solar system,’ ‘twentieth century,’ etc. The advantage of this theory is an identity of denotation with a difference in meaning. If I say Jane Austen is the author of Emma, I haven’t made the statement a is a, because although ‘Jane Austen’ denotes the same person x as does ‘the author of Emma,‘ the meaning of the two are different. They denote the same person but the sense in which they denote that person is different.
Russell’s first problem with Frege’s theory are phrases where the denotation appears to be absent, such as in the denoting phrase: ‘the present King of France is bald.’ This phrase has meaning, but ‘the present King of France’ does not denote anything, and therefore the phrase should be nonsense. It is not, however, nonsense (says Russell)1.
Regarding denoting phrases with terms that do not have denotation, e.g. the present King of France is bald, Frege will say that they are neither true nor false. Russell wants to say that they are false; his argument for this is extremely complicated. Note that Russell, in the case of denoting phrases where the denotation does not obviously exist, finds explanations that assume there really is a denotation and those that assume there is not a denotation wanting.
First, I will lay out Russell’s theory, mostly by his words and exposition; I will then list the three puzzles Russell presents that he claims ought to be solved by a theory of denoting. I will attempt to explain how each Frege and Russell might think they solve these puzzles with their respective theories.
Russell’s theory, by his words:
“I take the notion of the variable as fundamental; I use `C(x)‘ to mean a proposition in which x is a constituent, where x, the variable, is essentially and wholly undetermined. Then we can consider the two notions `C(x) is always true’ and `C(x) is sometimes true’. Then everything and nothing and something (which are the most primitive of denoting phrases) are to be interpreted as follows:
C(everything) means `C(x) is always true’;
C(nothing) means ` “C(x) is false” is always true’;
C(something) means `It is false that “C(x) is false” is always true.‘
… This is the principle of the theory of denoting I wish to advocate: that denoting phrases never have any meaning in themselves, but that every proposition in whose verbal expression they occur has a meaning.2” (emphasis mine)
The Russellian translation of the proposition ‘I met a man’ goes as follows (the real meaning of the denoting phrase):
” ‘ “I met x, and x is human” is not always false.’
Generally, defining the class of men as the class of objects having the predicate human, we say that:
‘C(a man)’ means ‘”C(x) and x is human” is not always false.’
This leaves ‘a man’ by itself, wholly destitute of meaning, but gives a meaning to every proposition in whose verbal expression ‘a man’ occurs.”
Thus the proposition ‘all men are mortal,’ in fact a hypothetical proposition, ‘if x is a man, x is mortal,’ can be translated as: ` “If x is human, x is mortal” is always true.’ : `C(all men)’ means ` “If x is human, then C(x) is true” is always true.’
Russell moves on to the most difficult type of denoting phrase, denoting phrases with definite descriptions, ie using the word the. Russell’s proposition is: ‘the father of Charles II was executed.’ Thus ‘the father of Charles II was executed’ becomes ‘it is not always false of x that x begat Charles II and that x was exectured and that “if y begat Charles II, y is identical with x” is always true of y.’
To interpret ‘C(the father of Charles II)’ where C stands for any statement about him, we have only to substitute C(x) for ‘x was executed’ in the above. Observe that, according to the above interpretation, whatever statement C may be, ‘C(the father of Charles II)’ implies:
‘It is not always false of x that “if y begat Charles II, y is identical with x” is always true of y,’
Which is what is expressed in common language by ‘Charles II had one father and no more.’ Consequently if this condition fails, every proposition of the form ‘C(the father of Charles II)’ is false.” (Note that this will be crucial for the famous ‘the present King of France puzzle.’)
The meaning of a denoting phrase containing the - a definite phrase – must involve: a claim of existence (there is one x), a claim of uniqueness (there is one and only one x), and a predicate (x has property p); for the above example, we see this as ‘there is one and only one x such that x begat Charles II, and for all y, if y begat Charles II, y=x and x was executed.’ The example that will be crucial for Russell is the proposition ‘the present King of France is bald.’ If we translate this: ‘there is one and only one x such that x is presently King of France, and for all y, if y is presently King of France, then y=x, and x is bald.’ For Russell, every proposition of the form C(the present King of France) is false. Let me move on to the puzzles in the hope this will become clear.
“(1) If a is identical with b, whatever is true of the one is true of the other, and either may be substituted for the other in any proposition without altering the truth or falsehood of that proposition. Now George IV wished to know whether Scott was the author of Waverley; and in fact Scott was the author of Waverley. Hence we may substitute Scott for the author of `Waverley’, and thereby prove that George IV wished to know whether Scott was Scott. Yet an interest in the law of identity can hardly be attributed to the first gentleman of Europe.”
‘Frege would say that words that occur in reports of propositional attitudes do not have their customary denotation; rather, they have an indirect denotation, which is their customary meaning3. ‘
The Russellian translation of ‘Scott is the author of Waverely‘ is: ‘there is one and only one x such that x wrote Waverely, and Scott is identical with that x.‘ Or, more fully, as Russell writes it, `It is not always false of x that x wrote Waverley, that it is always true of y that if y wrote Waverley, y is identical with x, and that Scott is identical with x.’
Now to translate the whole proposition (according to Russell there are two ways) – so ‘George IV wished to know whether Scott was the author of Waverely’ becomes:
‘George IV wished to know whether one and only one man wrote Waverley and Scott was that man.’ or ‘One and only one man wrote Waverley, and George IV wished to know whether Scott was that man.’ In these translations, these ways of understanding the proposition, using the principle of identity to substitute ‘Scott’ for ‘the author of Waverely’ is not possible, and the absurdity is avoided. Denoting phrases that appear to be denoting one and the same thing, when accurately translated, no longer pose this problem of identity.
“(2) By the law of the excluded middle, either `A is B‘ or `A is not B‘ must be true. Hence either `the present King of France is bald’ or `the present King of France is not bald’ must be true. Yet if we enumerated the things that are bald, and then the things that are not bald, we should not find the present King of France in either list. Hegelians, who love a synthesis, will probably conclude that he wears a wig.”
‘Frege held that sentences with phrases that lack denotation (e.g., “Odysseus was set ashore at Ithaca while sound asleep”) are neither true nor false. So Frege would say that “The present king of France is bald” is neither true nor false, and that “The present king of France is not bald” is neither true nor false. In other words, Frege’s way of dealing with phrases that lack denotation gives up the law of excluded middle4.’
But Russell says the phrase ‘The present King of France is bald’ is false. The Russellian translation of this proposition would be: ‘There is one and only one x which is presently the King of France, and that x is bald.’ As there is no x that is presently King of France, the proposition is false.
But consider the opposite: ‘The present King of France is not bald.’ For Russell, this proposition is ambiguous, since ‘not’ could modify ‘bald’ or the entire sentence. The two ways we could translate this are: ’There is one and only one x which is presently King of France, and that x is bald,’ and ‘ ’It is not the case that there is one and only one x which is presently King of France and is bald.’ Russell claims that the second of the two translations is how we ought to understand the proposition; this means that for his theory, he wants us to accept, in the cases of phrases that lack denotation, that the proposition ‘X is not Y’ (‘the present King of France is not bald), the ‘not’ modifies the entire proposition, and not merely ‘is Y’ (on top of accepting his exposition of terms that lack denotation). With this, Russell believes he has saved the law of excluded middle.
“(3) Consider the proposition `A differs from B‘. If this is true, there is a difference between A and B, which fact may be expressed in the form `the difference between A and B subsists’. But if it is false that A differs from B, then there is no difference between A and B, which fact may be expressed in the form `the difference between A and B does not subsist’. But how can a non-entity be the subject of a proposition? `I think, therefore I am’ is no more evident than `I am the subject of a proposition, therefore I am’; provided `I am’ is taken to assert subsistence or being, not existence. Hence, it would appear, it must always be self-contradictory to deny the being of anything; but we have seen, in connexion with Meinong, that to admit being also sometimes leads to contradictions. Thus if A and B do not differ, to suppose either that there is, or that there is not, such an object as `the difference between A and B‘ seems equally impossible.”
To state it quite clearly, Russell is posing this problem: The proposition ‘the round square does not exist’ is a true proposition, and therefore meaningful, but the subject (’round square’) does not exist; therefore the sentence is about nothing. So it is not clear how the proposition can be meaningful.
‘Frege would have to say: since “the round square” has no denotation, the entire proposition in which it occurs has no denotation, and so that proposition is neither true nor false. But this seems incorrect—the proposition certainly seems to be true.’ Therefore Frege’s theory does not give us a satisfactory solution to this puzzle.
Here’s where Russell becomes less explicit and certainly less lucid (it was possible!) than he was before. The Russellian translation of the proposition ‘the round square does not exist’ has to be something like ‘it is not the case that there is one and only one x which is round, and that x is square.’ This is true. The translation cannot be ‘there is one and only one x which is round, and that x is square, and that x does not exist’ because it contradicts itself: it makes an existential claim and also denies it5. (that’s all I have on this one for now…)
Russell briefly expounds the implications of his theory (which I will not try to simplify or break down, because I’ve spent too much uninformed time on this already): “One interesting result of the above theory of denoting is this: when there is an anything with which we do not have immediate acquaintance, but only definition by denoting phrases, then the propositions in which this thing is introduced by means of a denoting phrase do not really contain this thing as a constituent, but contain instead the constituents expressed by the several words of the denoting phrase. Thus in every proposition that we can apprehend (i.e. not only in those whose truth or falsehood we can judge of, but in all that we can think about), all the constituents are really entities with which we have immediate acquaintance. Now such things as matter (in the sense in which matter occurs in physics) and the minds of other people are known to us only by denoting phrases, i.e. we are not acquainted with them, but we know them as what has such and such properties. Hence, although we can form propositional functions C(x) which must hold of such and such a material particle, or of So-and-so’s mind, yet we are not acquainted with the propositions which affirm these things that we know must be true, because we cannot apprehend the actual entities concerned. What we know is `So-and-so has a mind which has such and such properties’ but we do not know `A has such and such properties’, where A is the mind in question. In such a case, we know the properties of a thing without having acquaintance with the thing itself, and without, consequently, knowing any single proposition of which the thing itself is a constituent.”
It was somewhat foolish of me to think I could cruise through this piece with essentially no knowledge of analytic philosophy beyond an intro to formal logic course I took almost four years ago. I have approximately one zillion questions which I had to struggle not to spatter all over this post; I will save them for some poor professor in Belgium next year.
1“If we say `the King of England is bald’, that is, it would seem, not a statement about the complex meaning `the King of England’, but about the actual man denoted by the meaning. But now consider `the king of France is bald’. By parity of form, this also ought to be about the denotation of the phrase `the King of France’. But this phrase, though it has a meaning provided `the King of England’ has a meaning, has certainly no denotation, at least in any obvious sense. Hence one would suppose that `the King of France is bald’ ought to be nonsense; but it is not nonsense, since it is plainly false. “
2The full article can be read here.
3 I am not going to go into detail with Frege’s responses to these questions, as I have not read Frege. I have slightly altered the explanations provided here.
4“It also gives up the principle of bivalence, which is subtly different from LEM:
Principle of Bivalence (df.): every declarative sentence is either true or else false.”
5“The whole realm of non-entities, such as `the round square’, `the even prime other than 2′, `Apollo’, `Hamlet’, etc., can now be satisfactorily dealt with. All these are denoting phrases which do not denote anything. A proposition about Apollo means what we get by substituting what the classical dictionary tells us is meant by Apollo, say `the sun-god’. All propositions in which Apollo occurs are to be interpreted by the above rules for denoting phrases. If `Apollo’ has a primary occurrence, the proposition containing the occurrence is false; if the occurrence is secondary, the proposition may be true. So again `the round square is round’ means `there is one and only one entity x which is round and square, and that entity is round’, which is a false proposition, not, as Meinong maintains, a true one. `The most perfect Being has all perfections; existence is a perfection; therefore the most perfect Being exists’ becomes:
‘There is one and only one entity x which is most perfect; that one has all perfections; existence is a perfection; therefore that one exists.’