Definite Description - Russell's Analysis

Russell's Analysis

France is presently a republic, and has no king. Bertrand Russell pointed out that this raises a puzzle about the truth value of the sentence "The present King of France is bald."

The sentence does not seem to be true: if we consider all the bald things, the present King of France isn't among them, since there is no present King of France. But if it is false, then one would expect that the negation of this statement, that is, "It is not the case that the present King of France is bald," or its logical equivalent, "The present King of France is not bald," is true. But this sentence doesn't seem to be true either: the present King of France is no more among the things that fail to be bald than among the things that are bald. We therefore seem to have a violation of the Law of Excluded Middle.

Is it meaningless, then? One might suppose so (and some philosophers have; see below) since "the present King of France" certainly does fail to refer. But on the other hand, the sentence "The present King of France is bald" (as well as its negation) seem perfectly intelligible, suggesting that "the Present King of France" can't be meaningless.

Russell proposed to resolve this puzzle via his theory of descriptions. A definite description like "the present King of France", he suggested, isn't a referring expression, as we might naively suppose, but rather an "incomplete symbol" that introduces quantificational structure into sentences in which it occurs. The sentence "the present King of France is bald", for example, is analyzed as a conjunction of the following three quantified statements:

1. there is an x such that x is presently King of France: ∃x (using 'PKoF' for 'presently King of France')
2. for any x and y, if x is presently King of France and y is presently King of France, then x=y (i.e. there is at most one thing which is presently King of France): ∀x∀y → y=x]
3. for every x that is presently King of France, x is bald: ∀x

More briefly put, the claim is that "The present King of France is bald" says that some x is such that x is presently King of France, and that any y is presently King of France only if y = x, and that x is bald:

∃x & B(x)]

This is false, since it is not the case that some x is presently King of France.

The negation of this sentence, i.e. "The present King of France is not bald", is ambiguous. It could mean one of two things, depending on where we place the negation 'not'. On one reading, it could mean that there is no one who is presently King of France and bald:

~∃x & B(x)]

On this disambiguation, the sentence is true (since there is indeed no x that is presently King of France).

On a second, reading, the negation could be construed as attaching directly to 'bald', so that the sentence means that there is presently a King of France, but that this King fails to be bald:

∃x & ~B(x)]

On this disambiguation, the sentence is false (since there is no x that is presently King of France).

Thus, whether "the present King of France is not bald" is true or false depends on how it is interpreted at the level of logical form: if the negation is construed as taking wide scope (as in ~∃x & B(x)]), it is true, whereas if the negation is construed as taking narrow scope (with the existential quantifier taking wide scope, as in ∃x & ~B(x)]), it is false. In neither case does it lack a truth value.

So we do not have a failure of the Law of Excluded Middle: "the present King of France is bald" (i.e. ∃x & B(x)]) is false, because there is no present King of France. The negation of this statement is the one in which 'not' takes wide scope: ~∃x & B(x)]. This statement is true because there does not exist anything which is presently King of France.