I believe it makes perfect sense to say that some things xx are identical to one thing y. The table is identical to the four legs and table-top; my two legs, two arms, head, and torso are identical to me; etc. I wonder what your immediate gut reactions are to such claims.
-Einar
PS: I also wonder about your reactions to my last entry. Come on, let's bring this blog back to life!
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3 comments:
My immediate reaction is, "That's crazy talk." Any relation worthy of being called identity is gonna have to respect the indiscernability of identicals (perhaps relativised to times or something). But the table and the four-legs-and-table-top have different adicity, insofar as one is a single object and the other is (are?) a collection of objects.
Of course, they also might differ in other respects -- the legs and table top might have different modal properties than the table -- but that's a bit more contentious.
So, what do you think? Do you think that you can have many-one identity without violating Leibniz's Law? Or do you think this version of LL is over-rated?
Barak,
Consider the distinction between the following two versions of Leibniz'z Law:
SLL: if x=y then (Fx iff Fy)
OLL: if x=y then for any X (Xx iff Xy)
SLL says that whenever two things are identical any predicate applicable to one is applicable to the other, and vice versa. That is, if x=y, we can substitute x for y (and vice versa) in any wffs of first-order predicate logic.
OLL says that whenever two things are identical they share all properties.
The two principles come apart because presumably there is no one-one correspondence between our predicates and the properties in the world.
Many-one identity must be understood as a generalization of OLL, not SLL. In addition we of course need a distinction between sparse and abundant properties and relations such that the quantifier in OLL is concerned with the former somehow, not the latter. The latter are mere constructions from the former (or something alike). So, as an answer to your question, I do think we can have many-one identity without violating OLL, but not without violating SLL. But that should be no problem because presumably SLL has nothing to do with ontology anyway.
The tricky part is to correctly formulate the generalized version of OLL.
What do you think? (Pretty cool, huh?)
-Einar
This is great info to know.
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