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One of the properties of identity is that everything is identical to itself. But, does "everything" mean literally everything, or merely every thing that exists? For example, I don't think 1/0 = 1/0, because 1/0 is undefined, so it can't equal anything, not even itself. But what have philosophers written about this issue?

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    Per Quine's ontological commitment expressed in his 'On what there is' your mentioned law of identify only applies to existent objects, preferably only in the sense of FOL as its basic knowledge. Usually your 1/0 is undefined in most formal theory, except perhaps Robinson's nonstandard real analysis... Commented Oct 20, 2023 at 0:09
  • 1/0 is malformed and not even all well-formed expressions represent objects in a theory. That a = a does not apply to those does not subtract from "literally everything". But the "thing" it applies to does not need to exist, it just has to have to be an object in the theory. It applies to fictional characters, for example.
    – Conifold
    Commented Oct 20, 2023 at 0:27
  • Sometimes the identity law is expressed as a conditional, so we might still say, "If 1 is divided by 0, then 1 is divided by zero," per, "If A, then A," and what is amiss with that? Likewise we might say, "If A is not A, then A is not A," and so on, although we might add, "If A is not A, then A is A and not A is not A, plus not A is A," as well. Commented Oct 20, 2023 at 0:40
  • When we say "there is no such number as 1/0" we mean that there exists no mathematical object which instantiates the idea of that arithmetic operation, not that there exists something called a 1/0 which does something called not-existing.
    – g s
    Commented Oct 20, 2023 at 0:49
  • Reference: see Identity. Your puzzle is not about identity, but about "thing": what are non-existing things? Commented Oct 20, 2023 at 6:57

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Some philosophers treat x=x as coextensive with "x exists". In some formal work, they actually define "x exists" as x=x. On the other hand, this could also be viewed as pushing the question of existence down into the interpretation, because x=x only has meaning if the interpretation assigns a value to x. If so, one could argue that x=x is only being used because it is a convenient predicate available in any logic with equality, and one that is true of everything. Any other predicate that is true of everything would work just as well in a theory that had the predicate. Examples include "Is a member of the universal set" in a theory that has a universal set or "has parts" in an atomless theory of mereology.

As to your question,

does "everything" mean literally everything, or merely every thing that exists?

What would it even mean to say that there is something that doesn't exist? Your example of 1/0 is not very convincing, because (in normal arithmetic) there is nothing that 1/0 names, so an expression of the form 1/0=1/0 is neither true nor false, but ill-formed. It's like asking "Does three men?" It doesn't mean anything.

Now, there are metaphysical theories that everything exists that we can try to describe, even something like 1/0 and round squares and Sherlock Holmes. This is called Meinongianism after Alexius Meinong. Meinongian writers generally make some sort of distinction between existence and being real or actual, and some of them countenance different sorts of predicates applying to the non-real objects, or those predicates applying in a different way. In these theories, a=a might not apply even to some things that exist.

UPDATE: Reading this article, I learned that my terminology was wrong with respect to original Meinongianism. What I was describing was the work I've read (mostly Zalta and critics of Meinongianism), which is apparently discussing a modified form of the theory. The original work (which I haven't read) distinguishes between a thing's being and the thing's existence. Meinong would say that there is a flying horse named Pegasus (Pegasus has being) but the flying horse named Pegasus doesn't exist.

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You can view the question in several ways. For example, you can consider 1/0 to be an expression, which clearly does exist as I have just typed it, in which case you can say that 1/0 is the same as 1/0, meaning that the expression is equivalent to itself. Then you can consider this particular instance of the typed expression 1/0 and say that clearly it is identical with itself, just as the inscription on Karl Marx' gravestone is identical with itself. The fact that 1/0 is undefined is in some ways, at least, no different to the undefined nature of the string kspltvrymuuub, yet I suspect you would have no problem in accepting that kspltvrymuuub is the same as kspltvrymuuub, and that this particular string kspltvrymuuub is identical to itself.

In what I have said so far, I have evaded your point, somewhat, by adopting the tactic of making an unreal thing real by equating it with its typed label- typed characters being definitely real. I think you can argue that concepts, such as the concept that the typed expression 1/0 represents, have mental counterparts as well as typed counterparts, so that when you think of 1/0 there is some mental state associated with it, and if you believe mental states exist then you can apply the principle identity to a mental state (albeit with some qualifications).

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