"This statement is false." In propositional logic, a proposition is a statement that is either true or false, but not both. In a text I am reading and in many others, "this statement is false" is not considered a proposition. But is this because it is both true and false or because it is neither true or false, i.e., doesn't have a truth value?  
EDIT 1:
The text I am reading says that the truth or falsity of a proposition may be clearly understood or arbitrarily assigned, which I interpret as meaning that what is important is that a proposition must be able to hold a single "stable" truth value. When we attempt to assign a truth value to "this statement is false" what is the problem?  
EDIT 2: I understand that in order for an assertion to be considered a proposition, we must be able to associate a truth value to it. I have seen the following terse reasoning about why the assertion "this sentence is false" is not a proposition: If it is true, then it is false, and if it is false, then it is true. Does this mean that assigning it a truth value leads to it being both true and false, or that assigning it a truth value leads to a contradiction? Please explain.
 A: Statements are very specific type of strings which are constructed recursively:


*

*If $p$ is an atomic proposition, then $p$ is a statement.

*If $p,q$ are statements, then: $p\land q$, $p\lor q$, $p\to q$ and $\lnot p$ are also statements.


If you try to construct $p$ which states "$p$ is true" or "$p$ is false" then you quickly realize this is not a well-formed formula.
And just like in natural language, not every string of characters has any actual meaning. The vile bile danced for a while, as the liver was guile. 
A: It's a paradox, meaning any interpretation of it's truth value results in a contradiction. The statement is neither true nor false. If "this statement is false" were false, then it would be true, contradicting the assumption that it's false. Similarly if it were true. It can be shown using the rules of logic that a contradiction $A \wedge \neg A$ implies $B$, for any proposition $B$. Hence when constructing a formal logical system we want to guarantee that contradictions are excluded, and therefore define propositions in ways that do not allow for the kind of paradox you mentioned.
A: You can use propositional logic to prove this statement is not a proposition:
Suppose the statement, $S$, "This statement is false", is a proposition 
That is  $S\iff\lnot S$
Then $S$ is true or false, but both of those lead to a contradiction, so our supposition is false:
"This statement is false" is not a proposition.
