Everything is identical to itself, and nothing is identical to
anything except itself. No two things are ever identical. If A and B
are identical then "they" are one, not two.
These are platitudes about identity, or rather about a
somewhat technical use of "identity" common in mathematics and
philosophy.
No doubt there are other uses. For instance, "identity" and its
cognates are often used to express sameness of kind, as in "this
record is the same Jones bought last week". Sometimes, "identity" is
used as a singular term for a thing's characteristc properties or
individual essence, as in "the festival has lost its identity". The conceptual platitudes
above do not apply to these other uses.
Humeans distinguish between how things are in themselves and how they are related to other things. The latter, they say, is always a contingent matter: Even though this cup of tea is about 20m away from a book and stands on a table, it could very well not be 20m away from the book and not stand on the table. In slogan form, there are no necessary connections between distinct entities.
Understood literally, this leads to a position one might call strong humeanism:
Meinongians say that some things do not exist. In other words, existence is a property that befalls only some of the things there are. It follows that by 'existence' these Meinongians do not mean the trivial property that every thing whatever has. What else do they mean? Maybe they mean by 'existence' being in space or time, as Meinong sometimes does. Or maybe they mean an alleged primitive property of certain things. At any rate, I have no objection to this except that I'd rather not use the word 'existence' for that. But I can't really say that ordinary usage is on my side, given that a) ordinary quantification is almost always restricted (though not always in the same way), and b) there is hardly an ordinary usage of 'existence' at all. So far, Meinongianism is utterly trivial. It merely holds that some objects lack a certain property.
Yesterday, I said that it doesn't really matter whether we regard identity simpliciter as identity-at-our world -- individuationg referents extensionally -- or as identity-at-every-world -- individuating referents intensionally. Suppose we want to do the latter, so that the referent of "the amazon" determines a function from worlds to world-bound individuals, that is, an intension. So on the present account, we identify the amazon with something that completely determines the intension of "the amazon". The intension? What if, as two-dimensionalists argue, "the amazon" has two intensions? Which one is the one we want extensions to determine?
So there are several ways to make sense of restricted identities. Which is the right one? Maybe there is no fact of the matter.
The difference depends on which contexts are regarded as referentially transparent and which as opaque. And that in turn depends on how the referents are individuated. For instance, (de re) ascriptions of modal properties will be transparent iff the referents of singular terms are such that they determine the truth value of all such ascriptions, perhaps because they (the referents) are fusions of world-bound individuals with their counterparts, or because they are Carnapian individual concepts, or because they simply contain some hidden tag that determinately settles all their modal properties. At any rate, for de re modal contexts to be referentially transparent, the referents have to provide us with a function from worlds to world-bound individuals, as that's what we need to determine the the truth value of those ascriptions. Alternatively, if we hold that those contexts are referentially opaque, we decide that the referents do not contain that information. Instead, we put the information into another aspect of meaning, which we call the terms' intension. Is the difference really more than just a relabeling of semantic vocabulary?
Now restricted identities threaten to violate
Leibniz's Law: If R1 is identical with R2, then how can they differ in
their courses? If AD1 is AD2, how can they differ in their history?
If A1 is A2, how can they differ in their modal properties?
They can't. So either R1 and R2 (and AD1 and AD2, and A1 and A2) are
not really identical, or the don't really differ. Let's look at the
first option first. It says that R1 and R2 are not really
identical. Hence "R1 = R2" is false, even though
If you follow the Rhine upstream, you'll reach Reichenau in Switzerland, where its two tributaries, the Vorderrhein and the Hinterrhein, meet. As far as I know, it is undefined which of them, if any, is the Rhine. Obviously that's not a mystery but just a matter of stipulation. So let's stipulate that 'R1' is to denote the continuation of the Rhine through the Vorderrhein, and 'R2' its continuation through the Hinterrhein.
What's the difference between substitutional and objectual quantification? I'll use the old-fashioned round brackets for objectual quantifiers and square brackets for substitutional quantifiers. The standard interpretations are
OB) (x)A is true under an interpretation I iff for some new constant t, A(x/t) is true under all interpretations I' that differ from I at most in what they assign to t.
SUB) [x]A is true under an interpretation I iff for all constants t, A(x/t) is true under I.
Assume that predication (and the truth functors) is interpreted in one of the usual ways, for instance by ruling that Ft is true under I iff I(t) is in I(F).
Then if (x)A is true under any interpretation, [x]A is also true under that interpretation. The converse holds iff every interpretation assigns every object in the domain to some constant.
Let K be a class of sets such that whenever x is in K and x is a subset of y, then y is also in K. It follows that if the empty set is in K, then every set is in K. Let's rule this out by stipulating that some set is not in K. Thus every set that is in K is not empty. So instead of saying outright that some set is not empty we can instead say that it is in K, which sounds less controversial but really comes down to the same thing.
I think this is the trick in Gödel's ontological proof of god. His class K is the class of 'positive' properties, where properties are individuated intensionally. Gödel claims 1) that whenever some property Q is necessarily implied by a positive property P, then property Q is also positive (which is just the closure principle above), and 2) that not all properties are positive. On these assumptions saying that a property is positive means saying that it is not empty, that is, not necessarily uninstantiated. Hence when Gödel says that 3) necessary existence is a positive property he in effect says that a necessary being possibly exists, which in turn means that a necessary being actually exists.
The fallacy is to assume that there is any class of ('positive') properties satisfying (1)-(3).
Sometimes we wonder whether some thing A is identical with some thing B: Is the man in the brown hat (the same as) my neighbour? Is the table in the mirror over there (the same as) the one here in front of me? Is the square root of 841 (equal to) 29?
What determines whether A really is identical with B? According to a view I find very irritating it's the identity conditions of A and B. The idea is that all things fall under kinds, and every kind comes with an associated identity condition. Different kinds may be associated with the same identity condition, but there is never more than one identity condition for a kind. So to find out whether A = B, we first have to find the relevant identity conditions of A and B. A good way to find those is to find the relevant kinds and look for their associated identity conditions. If the identity conditions differ, A is not identical with B. If they are the same, they tell us under which conditions A and B is identical, and we only have to find out whether these conditions obtain.