The discussion at
http://able2know.org/topic/152707-1 shows the statement "every truth must be true" to be an instance of "A is A."
So every truth must be true. And what about falsehoods? We have a problem here: if we say that every falsehood must be false, then we are saying that every falsehood must be the truth it falsifies. The solution is to say that every true falsehood must be a truth, by which any falsehood, if true, is also a truth. But its truth must be different from that of a truth, since it is a falsehood. And what could such a difference be? The only possible difference between a truth and a falsehood is that a falsehood must have no being: its being consists in referring to nothing, which makes it also nothing, despite being something that refers to nothing. So the statement "every true falsehood must be a truth" possibly means "nothing must be nothing," which in turn possibly means:
1. Anything -- possibly nothing -- can be something.
2. Nothing -- as possibly something -- must be what it is, which is nothing.
Both meanings refer to a falsehood only as the falsity of something, rather than as a truth -- one of a falsehood. Indeed, as a truth, no falsehood can be nothing. So despite making a falsehood nothing we possibly get something, first explicitly -- the first meaning -- then implicitly -- the second meaning. The reason is that, since every true falsehood must also be a truth, even as nothing it still can be something: it always has a possible being. Which becomes clear when we formulate "A must be A" as "nothing can be different from itself." In this form, the so-called "principle of identity" becomes either:
1. Everything must be identical to itself.
2. The being of nothing can be different from itself.
And whenever "nothing" means a falsehood "nothing can be different from itself" cannot mean "everything must be identical to itself" -- since it means "a falsehood can be different from itself." Hence, by being a falsehood, nothing is -- or at least can be -- no longer nothing, so "A must be A" can apply to falsehoods only by applying to their truth -- by which a correct reformulation of the so-called "principle of identity" would be the statement "every truth must be true."