Brute necessity is hard to accept, much harder than brute possibility. If someone claims that necessarily there are no purple cows, I expect an explanation. Perhaps he knows what kind of DNA is essential for cowhood and also that this kind of DNA can never produce purple beings, and he also believes that the laws of nature are necessary. This would make his claim understandable. But suppose he had no such explanation. Suppose in fact that we all know that only a minor mutation would be required to produce purple cows, a mutation perfectly compossible with the laws of nature. And still he claims that there could not be any purple cows. This would seem bizarre.
According to the epistemic account of vagueness, there aren't really any vague statements: When we're uncertain whether to call somebody bald that's not because he is a borderline case of baldness. There are no borderline cases. The border between being bald and not being bald is perfectly precise. It's only that we don't quite know were it runs.
Not many people believe in this account. That's surprising, because many people do believe that there are rigid designators -- terms denoting the same thing in every possible world --, and this seems to imply something that looks to me just like (an application of) the epistemic account of vagueness.
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.