propositional logic , don't know the answer Is the assertion "This statement is false" a proposition?
I think that it is a proposition because this("This Statement is false") may have truth values. the statement may be true or maybe false. therefore its proposition. but i don't know if i am correct. please correct me , if i'm wrong.
please explain the answer.
 A: In modern logic, the predicate $True(\quad)$ is of "difficult usage", because in using it in a formal language we will encouter the problems connected with Self-Reference.
You can see in SEP the entry about the Liar Paradox ; it gives a review of most of philosophical and logical debates about this paradox.
Regarding "standard" logical approach to the Liar, see :

4.2.1 Tarski's hierarchy of languages
Traditionally, the main avenue for resolving the paradox within classical logic is Tarski's hierarchy of languages and metalanguages. Tarski concluded from the paradox that no language could contain its own truth predicate (in his terminology, no language can be ‘semantically closed’).
Instead, Tarski proposed that the truth predicate for a language is to be found only in an expanded metalanguage. For instance, one starts with an interpreted language $L_0$ that contains no truth predicate. One then ‘steps up’ to an expanded language $L_1$, which contains a truth predicate, but one that only applies to sentences of $L_0$. With this restriction, it is easy enough to define a truth predicate which completely accurately states the truth values of every sentence in $L_0$ and yields no paradox. Of course, this process does not stop. If we want to describe truth in $L_1$, we need to step up to $L_2$ to get a truth predicate for $L_1$. And so on. The process goes on indefinitely. At each stage, a new classical interpreted language is produced, which expresses truth for languages below it.
Why is there no Liar paradox in this sort of hierarchy of languages? Because the restriction that no truth predicate can apply to sentences of its own language is enforced as a syntactic one. Any sentence $\varphi$ equivalent to $\lnot Tr(\ulcorner \varphi \urcorner)$ is not syntactically well-formed. There is no Liar paradox because there is no Liar sentence. See the entries on Tarski and Tarski's truth definitions for more on Tarski's views of truth.

A: 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.
-Luke
A: It seems that this sentence ought to be a proposition, because it has the grammatical form of other meaningful sentences, but it is self referential, means little if anything, and is paradoxical. There are many other perfectly grammatical sentences which do not qualify as propositions in classical two-valued logic, because for one reason or other, they cannot be unambiguously classified as true or false. 
A: A very informal analysis by a non-expert on such matters, so your instructor may very well disagree...
Consider the alternative, "This statement is true." No contradictions will arise from it, but it is clearly quite meaningless. It can have no truth-value. If a proposition is required to have meaning or truth-value, then this meaningless statement is not a proposition. 
Now, if "This statement is true" is meaningless, then it stands to reason that "This statement is false" must also be meaningless. If it is meaningless with the word "true", it cannot suddenly have meaning when "true" is changed to "false". And if propositions must have meaning or truth-value, then it, too, must not be a proposition.
A: Statements such as "this statement is false" and "Window is half open" are those which cannot have fixed truth values. Hence these are not propositions.
Reference:
http://mathworld.wolfram.com/Proposition.html
P.S. By not fixed truth values I mean its debatable. The glass is half full ;)
