Where do you get the nerve to swipe them off the table?
Hey, I gotta lotta nerve, in case you haven’t noticed. But then, you should have met my Dad. Now there was a man with a lotta nerve! They just don't have a known binary truth value. You can't say "This sentence is FALSE" has a value, because none is ever returned when you try to evaluate it. You can therefore, also not say whether such statements are elements in A or NOT(A). So you can't process the law of identity on them.
I don’t want to process the law of identity on them because by definition it has none, it contains a contradiction. All sentences that contain a contradiction have no identity, by definition. Whatever "invalid" means, "this sentence is FALSE" does not poop out of existence because you uttered the word--it is member of the set of predicate statements--said which, we are not normally shy about including in valid domains of discourse about which one may meaningfully issue proclaimations about the Law of Identity.
It is meaningless. It isn’t true, false, up, down, right, wrong, true, untrue, blue, red, green, or anything. It is nothing. It is not valid. None of this has anything to do with anything other than that single statement. Doesn’t mean horses aren’t horses because horses aren’t sentences. It doesn’t mean logic is invalid because logic isn’t a self contradictory sentence. It can contain a self contradictory sentence and that sentence is exactly what it is, one utterly without meaning. It has no other effect outside itself. It has no other implication outside itself. None can be shown, none can be demonstrated. It is an isolated blob meaninglessness.
Ever the burnt hand goes wabbling back to the fire. We already had this discussion on another thread.
Consider the case of the barber who shaves himself. I can make a court case out of this. Suppose the barber is a genuine person, genuinely being sued for false advertisement. How will the judge, using logic, determine whether this is false advertising? The barber was genuine in his commitment to his advertised claim, genuinely lives in the village in question, is genuinely a man who needs shaving. If the judge is constrained to logic alone, he must process the syllogism to determine whether the barber is guilty of false advertising.
As I pointed out previously, there are many real world examples of this problem existent in working programs, which we refuse to trash because they are expensive and useful, but which contain elements that unavoidably recurse in the matter of the barber's paradox. The example I gave you was from a language translator--most existing comprehensive language translators contain such entities--cyclic definitions that cannot therefore be resolved. A literal take on the dictionary would yield the same results. If you tried to expand a dictionary's definition by looking up and substituting all the definions in the dictionary for each word in a given definition, you'd get the same thing.
these problems aren't meaningless--they represent areas in an otherwise valid formally correct engine where you can't go with any hope of returning with a logical result. Since you can't vanquish them, and you can't resolve them, you can't say their universe is confined to either A or NOT(A), so the predicate (A AND NOT(A)) is unknown, and the principal of identity is undemonstratable. Furthermore, they are not meaningless, they are a behavior we can observe that tells us something we can act on to improve our circumstances ie. the code is hung up.
As a matter of intuition, I agree that the principle of Identity is a sound one to maintain, for most practical issues in the gross universe. As a formal basis for developing any overarching notions of how the universe works, it is useless, because it is, by or current lights, demonstrably wrong for things that are very small, and things that are very subjective, and things that are very unamenable to being usefully represented as sets, such as our charming selves' inner workings.