The principle of identity as originally formulated by Aristotle states that everything exists in its own way, which is its identity. However, if we take the word "existence" to mean precisely the way by which anything exists, which is the same as its "being," then we have that anything must be the precise way of its being, hence that anything must be identical to itself. And since, as the principle of identity refers to anything, it must refer to falsehoods, what is the necessary identity between a falsehood and itself? What means the statement "a falsehood must be precisely that falsehood"? First of all, we have an ambiguity here, since a falsehood can be taken as meaning:
1. Something false.
2. The circumstance of something being false.
If a falsehood is taken as meaning something false, then that something becomes the same as its own falsity, so:
1. If we take what is false as a being, then being false becomes the same as just being, so everything becomes false, including the falsity of everything.
2. If we take what is false as a nonbeing, then being false becomes nothing, being destroyed altogether.
To escape such unsolvable contradictions, we must choose the meaning according to which a falsehood is the circumstance of something being false. Then, that circumstance must be true since, if it were false, then it would be the circumstance of whatever is false being true rather than false: the circumstance of something being false is the truth of the falsity of that something, as independent of the truth or falsity to which it refers -- it is a truth rather than a falsehood. Hence, the only way for us to formulate the principle of identity regarding falsehoods -- without running into unsolvable contradictions -- is by taking any falsehood as meaning rather a truth -- the truth of the falsity of the truth it falsifies -- by which the principle of identity becomes "every truth must be identical to itself." However, to be faithful to the original meaning of the principle of identity, which is that anything exists in its own way, we should rather say that "every truth must be true," since the way of a truth to exist is to be true.