So I get what a proposition is (I think) and I understand why “this statement is false” is not a proposition, however this has led me to believe that maybe “this statement is a proposition” is not a proposition. My reasoning (which I recognize may be flawed) is that if “this statement is a proposition” is a false, then it cannot be assigned the value true or false, thus it cannot be assigned the value false, thus it is paradoxical. If it were assigned the value true, it is not paradoxical, but then it can only be assigned one value, thus it cannot be assigned true or false, it can only be assigned true.
In my mind, the way this differs from something like “x is an even number” is that “x i