This "problem", or rather (I think) the statement "The present king of France is bald" was used by Bertrand Russell; I don't know whether it originated with him but it sounds like his. It is an illustration of his concept of the definite description.

Russell regarded the definite description "the X" as a conflation or shorthand for "there exists (at least one) X" and "there exists only one X", and "the X is Y" as meaning in addition "if anything is an X it is a Y".

So any statement about the present king of France is false, because one of the truth conditions is false ("there exists a present king of France"). This does not however license you to draw the opposite conclusion as true, because that fails for the same reason.

  1. False: The present king of France is bald.
  2. True: It's not true that the present king of France is bald.
  3. Doubtful: The present king of France is not bald.
  4. False: The present king of France has hair.

Considered only as plain English, the falsity of (1) seems to imply the truth of (4). Russell's analysis shows why it's consistent that neither is true. The law of excluded middle is not violated.

Why (3) is dubious is that in English the word "not" is not a logical negation. To make a strict negation you have to pre-pose an artificial clause like "It's not the case that", because English grammar often has special meanings for particular cases. A clear example is with "like": "I don't like" actually means "I dislike": there is no convenient short way of saying "I neither like nor dislike".

This was Russell's analysis in terms of compound statements. Modern semanticists might say that the word 'the' contains the presuppositions of existence and uniqueness, and that a statement using it would fail to refer if this presupposition was not met.

We could also discuss the sorites paradox here, though that wasn't Russell's point. A person with only one hair on their head is admitted to be bald. And for any person, the difference between having n and n + 1 hairs makes no difference to whether they're bald or not: there's no cut-off point. So inductively any number of hairs should still constitute baldness.

(Later) Thanks to thecarp for jogging my memory. I knew there was something else I wanted to say. Some people would say there is a present king of France: fervent royalists who maintain the claim of the Count of Paris. This could lead to another divagation on what exactly we mean by a name, leading into Saul Kripke's theory of naming. :-)

(Later again) Thanks to forget her for telling me about P.J. Strawson's idea that it's people who do the referring; their utterances don't; and for saying quite rightly that it may be easier to regard many of these claims as lacking a truth value or being nonsensical rather than true or false. This is however one case where we can keep analysing it in terms of true and false and get a fair amount of sense out of it.