So let's start with Exists and Type-Raising by Brian Weatherson over at Thoughts Arguments and Rants:
Back in the day there was this old problem about how we could make sense of propositions like (1).With all due respect to Weatherson, Frege had already solved this problem by distinguishing between the sense and the denotation of a noun phrase. Russell's solution was a nominalist re-working of Frege's solution. All of the problems discussed in Wetherson's article and the comments (well worth reading, by the way) are more properly viewed as problems with nominalism (or at least with Russell's nominalist solution to the original problem).
(1) The King of France does not exist.
Very roughly, the worry (or at least a worry) was that if this is to really express a proposition then the denoting term in it must denote something...Just as roughly, Russell solved this problem by noting that the (putatively) denoting term is a quantifier, not a proper name, and quantifiers can make contributions to propositions without actually denoting anything.
The denotation of a noun phrase is the thing that you refer to when you say the phrase. For example "the president of the United States" is a noun phrase that denotes George Bush (for the moment). Consider this sentence:
(2) The president of the United States likes peanuts.
Sentence (2) can be analyzed like this: "The president of the United States" denotes an individual x. "likes peanuts" denotes a proposition P. Sentence (2) means that P is true of x.
This sort of analysis doesn't work for sentence (1). The phrase, "the King of France" doesn't denote anyone so there isn't any x to use in the analysis.
Frege didn't deal with this problem directly (AFAIK). Instead he dealt with the problem of why "is" is useful. It isn't very useful to say
(3) George Bush is George Bush.
Sentence (3) doesn't provide any information. But this
(4) George Bush is president of the United States.
does tell us something useful. But under the analysis I described above using denotations, this is impossible to account for. Sentence (4) would be analyzed like this: "George Bush" denotes some object x. "president of the United States" denotes some object x. Sentence (4) asserts that x=x.
You might have expected that I would say "president of the United States" denotes some object y and then (4) asserts that x=y, but that would just push the problem down to why "x=y" is meaningful. The point is that the denotation is extensional and x is just the name of this extensional object.
Frege's solution is that noun phrases have both a denotation and a sense. The denotation is the object that the phrase refers to and the sense is the concept that the phrase expresses. "George Bush" is a noun phrase that expresses the concept of some guy named George Bush (the context helps to refine this concept to a single George Bush). "president of the United States" expresses the concept of being the top political executive of the United States.
Sentence (4) is now analyzed like this: George Bush expresses a concept C1, "president of the United States" expresses a concept C2, and (4) asserts that the concept C1 has the same extension as the concept C2.
Now let's go back and analyze (1) with Frege's semantic machinery: "The King of France" expresses a concept C1 and sentence (1) asserts that C1 has no extension.
Now, take a look at Weatherson's problematic example:
(5) Leopold Bloom does not exist.
Weatherson doesn't say who Leopold Bloom is. I googled the name and found out that Leopold Bloom is the fictional character in a James Joyce novel. Presumably that's who he meant. In any case, the analysis remains the same. "Leopold Bloom" expresses a concept C1, sentence (5) asserts that C1 has no extension. I don't see any problem here. The analysis works just as well with the example of a fraudulent character.
Oddly enough, Weatherson claims to have a very "expansionary" ontology, so I don't know why he prefers Russell's stilted nominalist solution to a full-blown realist one that works much better.
Actually, Weatherson seems to be realizing quantifiers in a way that makes them somewhat homomorphic to Frege's concepts (though I must confess, I'm not sure I understand what he's doing). In this homomorphism, Weatherson's type raising would correspond to what in Frege's account is the process whereby the overall meaning of the sentence determines whether a sentences expresses a concept or denotes an object.
In the example
(7) a. Red is my favourite colour.
b. ??Red exists.
c. The colour red exists.
I would argue that concepts are often (but not necessarily) expressed with prefixes to insulate them and keep them from being reduced to an object:
"being George Bush" rather than "George Bush"
"that he would go" rather than "he would go"
"the relation of equality" rather than "equality"
"the color red" rather than "red"