Editing Definite description (section)

==Russell's analysis==
{{main|Theory of descriptions}}
As [[France]] is [[French Fifth Republic|currently a republic]], it 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."<ref name="ondenoting">{{Cite journal |last=Russell |first=Bertrand |date=1905 |title=On Denoting |journal=Mind |language=en |volume=14 |issue=4 |pages=479–493 |doi=10.1093/mind/XIV.4.479}}</ref>

The sentence does not seem to be true: if we consider all the bald things, the present King of France is not among them, since there is [[List of French monarchs|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 equivalence|logical equivalent]], "The present King of France is not bald", is true. But this sentence does not 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){{who|date=October 2021}} since "the present King of France" certainly does [[Failure to refer|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" cannot be meaningless.

Russell proposed to resolve this puzzle via his [[theory of descriptions]]. A definite description like "the present King of France", he suggested, is not a [[reference|referring]] expression, as we might naively suppose, but rather an "incomplete symbol" that introduces [[Quantifier (logic)|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 [[Quantifier (logic)|quantified]] statements:

# there is an x such that x is currently King of France: <math>\exists xKx</math> (using 'Kx' for 'x is currently King of France')
# for any x and y, if x is currently King of France and y is currently King of France, then x=y (i.e. there is at most one thing which is currently King of France): <math>\forall x \forall y ((Kx \land Ky) \rightarrow x=y)</math>
# for every x that is currently King of France, x is bald: <math>\forall x (Kx \rightarrow Bx)</math> (using 'B' for 'bald')

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

{{block indent|<math>\exists x((Kx \land \forall y(Ky \rightarrow y =x)) \land Bx)</math>}}

This is ''false'', since it is ''not'' the case that some {{var|x}} is currently 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 currently King of France and bald:

{{block indent|<math>\lnot \exists x ((Kx \land \forall y (Ky \rightarrow y = x)) \land Bx)</math>}}

On this disambiguation, the sentence is ''true'' (since there is indeed no x that is currently 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 currently a King of France, but that this King fails to be bald:

{{block indent|<math>\exists x ((Kx \land \forall y (Ky \rightarrow y = x)) \land \lnot Bx)</math>}}

On this disambiguation, the sentence is ''false'' (since there is no x that is currently 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 the first of the above), it is true, whereas if the negation is construed as taking narrow scope (as in the second of the above), 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. <math>\exists x((Kx \land \forall y(Ky \rightarrow y =x)) \land Bx)</math>) is false, because there is no present King of France. 

The negation of this statement is the one in which 'not' takes wide scope: <math>\lnot \exists x ((Kx \land \forall y (Ky \rightarrow y = x)) \land Bx)</math>. This statement is ''true'' because there does not exist anything which is currently King of France.