Aren't there definitions of Truth that are not the negation of Falsehood? Can't there be a function True(x) that is not equal to !False(x)? Can't there be a third function Paradox(x) such that these counterexamples can be considered paradoxes and therefore outside of the truth? I'm admittedly not a logician and don't formally study paradoxes, but I never quite understood the whole category of "this sentence is false"…
If terms we use are not well-defined, then sentences using such terms can not have meaning.
But for the sake of argument let's explore, what could the "this" in (so called) "self-referential" sentences refer to?
Do they refer to a specific encoding of the sentence you are reading, as some bits in computer memory perhaps?
That would require that those bits somehow have a unique "identity" and the "this" in a self-referential sentence would have to refer to those bits in specific addresses of a specific memory-chip.
But of course the "this" does not specify which memory chip, which specific (concrete) encoding of its (purported) meaning it is referring to. And if it did, then it would be talking about that specific set of bits in that specific memory-chip, not of "itself".
The fallacy is that what we perceive as a "sentence we read" is somehow "speaking" of something. But no, the sentence is not a subject, a sentence can not speak, and THEREFORE it specifically can not speak of itself.
A sentence can not speak, only actors, only subjects, like humans and AI, can "speak". And their speech must be encoded in some physical medium. A written sentence like "This sentence is ..." gives us the false impressions that somehow the SENTENCE IS SPEAKING of itself!
But speech can not speak, speech is the product of speaking.
Hence, in my view, "self-referential sentences" do not have any meaning and whatever paradoxes they might seem to create are results of confusion between ontological levels of "Subject" vs. "Speech".