"This statement is false"
Which is "this statement"? That question kills it, dead.
But then we can ponder, and ask "what is a statement"?
Really speaking, there is no such entity as a "statement". There are meanings, to be sure, but what the devil is a "statement" - it's just a pile of shapes.
We might say that at least the pile of shapes is identifiable as the physical ground of the statement.
In which case
"this statement is false" or even "this statement is true"
only points to a position on the page.
How can it do that? Is a pile of shapes, the "statement", animated in some way? Is it like a sign-post? it can't point itself out, there is no identity here. And, in any case, isn't a pile of shapes only identified through the emergence of a specific meaning (and meanings are always specific)?
Does it make sense to say that something can point precisely, even to a position on the page? No. The shapes involved in the act of pointing, the signpost itself, only point significantly through an arbitrary set of publicly endorsed rules.
We might object and say that the pile of shapes, the statement, is at least identifiable or we would not be able to focus on the shapes in an attempt to given them a meaning. But the set of rules that make us attend to a particular set of shapes, rules like a capital letter, letter proximity and full-stop, isn't the same thing as giving them a meaning or allowing a meaning to emerge.
I can find no grounds on which "the statement is false" or even "this statement is true" can be significantly entertained.