1, x=y defined: E!x & E!y & (All F)(Fx <-> Fy).
and
2. E!x defined: (Some F)(Fx).

3. (All x)(x=x <-> E!x), is a theorem.
If either or both do not exist then x=y is provably false.

4. (All x)(x=x) is not valid.

example
(The present King if France)=(The present King if France), is false, even though
(All F)(F(The present King if France) <-> F(The present King if France)), is tautologous.

is it wrong to argue that a is not a because one a is on the left side of the copula and the other a is on the right side, and having different properties they are clearly not identical. — jlrinc

IS the sentence about the marks on either side of the "="?

I wanted to get some opinions from people who are more knowledgeable than I am in logic. Regarding the Law of identity "a is a" is it wrong to argue that a is not a because one a is on the left side of the copula and the other a is on the right side, and having different properties they are clearly not identical. I was actually going to use this in an argument but it sounds too cute so I thought I'd ask people who knew the subject better if this is a valid point, Is there some technical reason why it doesnt work and in general what your thoughts were. Has Aristotelian logic been subjected to the same critiques as Euclid's geometry. In other words is there a non Aristotelian logic to be derived by a critical examination of it's axioms? — jlrinc

More generally is it incorrect to point out that some unknown property may exist between "a" and "a" that makes them different? Would claiming no property exists because it cannot be proven otherwise not be committing an argument from ignorance fallacy?

When you write "a" and "a" as two distinct things, and ask about the difference between these two things, you have given us the premise that they are two distinct things. The need here would be to support, justify that premise, that they are distinct. We can see that they are distinct things because they occupy different places. So despite the fact that they look the same, the claim that they are distinct things is justified by that fact, that they occupy different places.

is it wrong to argue that a is not a because one a is on the left side of the copula and the other a is on the right side, and having different properties they are clearly not identical. — jlrinc

That would be confusing use with mention.

If you're not familiar with the use/mention distinction, here are a couple easy examples:

"Dogs" has four letters. Dogs have no letters.

The "mention" is marked off by quotation marks above. The "use" isn't. "Mention" concerns the expression as an expression. "Use" is what the expression is about. It's what the expression "points to," the referent of it.

Another example, courtesy of Wikipedia's page on the distinction (https://en.wikipedia.org/wiki/Use%E2%80%93mention_distinction):

Use: Cheese is derived from milk.
Mention: 'Cheese' is derived from the Old English word ċēse.

So when you talk about one A being on the left and the other on the right, you're talking about the mention.

But the principle of identity isn't about anything in the mention sense. It's about the use sense. In the use sense, there aren't two different As. We're simply required to write or say it that way a la a mention.

Not trying to do that. I can word it a different way like how do you prove an unknown property doesn't exist between this letter ==> "a" <== and this letter without commiting an argument from ignorance fallacy?

What are you saying, that a thing might be different from itself? So I don't get your point. You point to "A", and ask if there is a property of that thing which is also not a property of it?

Yeah at best we can say that it's so obvious that A is A that there's no conceivable reason why it wouldn't be. But I don't see why there couldn't exist some inconceivable reason why it's not.

It's a principle, "A is the same as itself". If there is no conceivable reason why this wouldn't be true then it's a solid principle. If you allow that there might be a "reason" why it is not true, or might not be true in some cases, then by the use of that word, "reason", you allow that it is conceivable. Then we might doubt that principle and seek the reason. But to say "inconceivable reason" is contradictory and doesn't give us any reason to doubt the principle.

The principle serves to help us understand things. And our understanding is only as solid as the principle. if things start going wrong with our understanding of things, evidence comes forward that our understanding might really be a misunderstanding, then we might start to question our principles, to determine where the problem is, why is there an appearance of mistake. If we start at the bottom, and the law of identity is pretty much the bottom, we can consider whether there is any reason to doubt this principle. But it doesn't make sense to look for a reason which is inconceivable. What kind of reason would that be, and how could we ever look for it?

I'm also applying the principle that something is not true because it cannot be proven false. It's an argument from ignorant fallacy to assume that. When I construct theories I allow for the possibility of being wrong for some reason I don't understand currently. Maybe reason is an illusion.

Consider that the principle which we call the law of identity, is not necessarily true, it's just a useful principle. So long as it serves us well, we'll use it. But if we start finding misunderstandings, and mistakes, like I described in the last post, then we might question this principle to make sure that it isn't leading us astray. So we don't really assume that it's true, just because it hasn't been proven false, there's two factors. We assume it true because it has served us well and it hasn't misled us. The latter, "it hasn't misled us" is similar to "it hasn't been proven false", and the former "it has served us well" is similar to being proven true.