What is the the truth value, if they have one, of propositions whose subject do not exist? "The current king of France is bald" is the famous example. Is that true or false, or neither? I have a hard time understanding how the current king of France can be neither bald nor not bald, even though I have no trouble understanding that there is no current king of France.

Philosophers have given various answers to questions like yours. See, for example, this SEP entry.

Here's one approach: "The current king of France is bald" is false because it implies the existence of a current king of France when in fact there isn't one. "The current king of France is not bald" is likewise false if it's construed as implying the existence of a current king of France (and asserting of him that he's not bald). On a possible but perhaps less likely interpretation, the second quoted sentence is simply the wide-scope negation of the first quoted sentence: i.e., "It's false that the current king of France is bald." On that interpretation, the second quoted sentence comes out true since it simply asserts that the first quoted sentence is false. On neither interpretation is anyone neither bald nor not bald, so that particular claim of classical logic -- everything is either bald or not bald -- is preserved.

There are (at least) 3 ways to handle the assignment of a truth value to sentences with non-referring subjects, like "The current king of France is bald":

1. Bertrand Russell's solution (as Stephen Maitzen's response points out) was to argue that the subject-predicate (or noun-phrase/verb-phrase) "surface" structure of the sentence was not its real, "deeper", logical structure, and that its truth value could only be determined by examining that logical structure, which would be a conjunction of three propositions:

(a) There is at least one current king of France,

and (b) there is at most one current king of France,

and (c ) he is bald.

Because (a) is false, the entire conjunction (and hence the original sentence) is false.

It's apparent negation, "The current king of France is not bald", can then be seen to be ambiguous between:

(i) It is not the case that the current king of France is bald,

i.e.: It is not the case that: (a) & (b) & (c )

and

(ii) The current king of France is not bald,

i.e.: (a) & (b) & it is not the case that (c )

Then (i) is true (as the negation of a false sentence should be), but (ii) is also false(!), for the same reason that the original sentence is false on this analysis.

2. Peter Strawson objected to Russell's analysis, and argued that such sentences have no truth value.

3. Followers of Alexius Meinong would argue that some such sentences can be true, such as "Pegasus the flying horse is a horse and can fly" or "The golden mountain is golden and is a mountain").

To read more about this, take a look at:

Bertrand Russell, 'On Denoting', Mind 14 (1905): 479-493

Peter F. Strawson, 'On Referring', Mind 59 (1950): 320-344

On Meinong, see http://plato.stanford.edu/entries/meinong/

esp. sect. 4.4 ("Beingless Objects---Russell versus Meinong")

Read another response by Stephen Maitzen, William Rapaport
Read another response about Truth