I am currently going through a book on discrete mathematics (Rosen), and this was one of the questions in the first chapter. Of course, it was an even-numbered question, so I had to tap the internet to gain more information.
I think Mauro's answer is the most complete, and I'll try to add my thoughts as a pile-on to hopefully clarify what I'm thinking as well (and to potentially gain some feedback from the crowd).
Bottom line, the statement "This sentence is false," is NOT a proposition. The reason for this is because there is no "external reference" (for lack of more precise language) to which the statement makes reference. Or put another way, it is because there is no "variable" which can take the place of a 'word' in the sentence and give it meaning.
For example, the statement "This sentence is false: 'this sentence is false'" IS a proposition, because there is a reference that can 'take the place of a variable (with a value of T or F)' that makes the entire statement T or F. In this case, the variable is the portion of the sentence following the “:” in single quotes. Take as another example the statement/proposition "P(x): x is a four-legged dog." In this example, there is some entity ‘X’ which assumes a role in the sentential function that either makes P(x) a true or false statement. P(Fido): Fido is a four-legged dog, is TRUE, whereas P(Telephone): Telephone is a four-legged dog, is FALSE, for obvious reasons. So here, P(x) is an example of a sentence with “space for a variable,” because it refers to something outside itself.
Now, returning to my earlier statement that “This sentence is false” is NOT a proposition. Let’s try to impose the ‘sentential function’ structure on it. For example, take P(x) to be “This sentence is false,” where ‘sentence’ is the variable (I’m getting a little sloppy here, but I think you can follow the point): Now, P(x) will become “This ‘This sentence is false’ is false.”…hmmm, it looks like the ‘sentence’ variable is still there (but wait, didn’t we just replace it?)…let’s do it again. P(x) after one more “iteration” becomes “This ‘This ‘This sentence is false’ is false’ is false.” See how it becomes “recursive” without a base case defined? A quick thought experiment continues this line of reasoning ad infinitum. Hence our problem, and why P(x) in this case is not a proposition; it makes no reference to anything outside itself.
So the actual “why,” I think, has to do with set and class theory – but I’m not as versed as I’d like to be. Briefly, statements that refer to themselves form what are known as proper classes (I think...), which are disallowed in mathematics and logic, because their entry effectively makes any statement true, which is “hardly a very desirable state of affairs,” according to Thomas Hungerford. The idea underlying all this is that we must place restrictions on what we say since it is possible in “ordinary” language to say things which have no meaning.