Nope. Absolutely not. Here's why:
A proposition is either a true or false whether it's from my perspective or yours, it is a must TRUE or FALSE but not both. If something that is beyond this than it can be considered as into somewhat in the Fuzzy Logic zone.
Now our sentence is "This statement is false." What does this mean?
If the statement is true than the given "This statement is false" is true meaning that the given statement is false.
If the given statement is false than the statement "This statement is false" is false meaning that the given statement is true.
So we get True with false and False with True. Nothing but seems sort of a paradox.
But what is the root of this given "predicate" that is making it so much amazing.
What. Why is this a predicate?
-> "This statement is false" is just a statement until and unless we provide it with some values. And once we provide it with the "True" or "False" values it becomes a proposition that is a "statement".
now, what or where these "True" or "False" values being applied.
These values are being applied on the "this" part of the predicate.
"This statement is false".
We can treat "This" as a free variable that can choose values from the given domain of {True, False}.
As seen earlier, the statement outputs True on input False and output False on input True.
More, the given predicate is a self referencing predicate.
Now if we take "This sentence is false" as F(x), x being the truth value we want to take then we can write predicate as not of F(F(x)).
But this is not a proposition as we can't determine a single truth value. We can't determine what F(x) is. true or false?
Therefore from the given statement "This statement is false" nothing can be concluded that is true or false but not both. Hence not a proposition.