Transcript Numerical identity

Identity
Recall Quine’s slogan: “no entity without identity”
What does this mean?
We have to have a way of saying whether this x is
the same as that y.
We have to have a way of saying that this x is the
same x over time.
An Important Distinction
Qualitative identity: two things are qualitative identical iff
they share their properties.
E.g. there is a sense in which you can say that you and I
have the same book (Kim & Sosa’s anthology).
Numerical identity: two things are numerically identical iff
they share all their properties.
According to this definition, a thing is numerically
identical only with itself; that is, numerical identity is the
relation that a thing has with itself and with nothing else.
Numerical identity seems unproblematic. But it actually
involves us in certain deep problems, even in its formulation.
Principle of the Identity of Indiscernibles (PII):
If, for every property F, x has F if and only if y has F, then x is
identical to y.
Or,
(F)(Fx ↔ Fy)  x = y
Principle of the Indiscernibility of Identicals:
If x is identical to y, then for every property F, x has F if and only if y
has F.
Or,
x = y  (F)(Fx ↔ Fy)
Sometimes, the conjunct of these two principles is called “Leibniz’s Law”.
I.e. (F)(Fx ↔ Fy) ↔ x = y
Two distinctions:
(i) intrinsic vs. extrinsic properties
(ii) pure vs. impure properties
Extrinsic properties can be seen as relational properties (though
it’s trickier than that).
Impure properties involve reference to other things.
E.g. “to the left of” is extrinsic; “to the left of Bob” is extrinsic
and impure. (It would seem that all non-relational properties
are pure.)
Strong PII: restrict properties to pure intrinsic properties
Weak PII: restrict properties to pure properties
Worthless (uninteresting PII): allow extrinsic properties.
Some problems with identity
A.
Frege’s Puzzle: If a = b, it would seem that the one term
should be substitutable for the other salva veritate. (That is,
if the terms ‘a’ and ‘b’ refer to the same thing.)
But this is clearly false.
E.g. “I thought that Cicero was a great orator” is true; “I
thought that Tully was a great orator” is false. But Cicero =
Tully! Answer: in intensional contexts, something else is
going on.
B. Kripke’s Puzzle:
1) a is necessarily identical to a
2) a = b
3) a is necessarily identical to b
E.g.
4) The morning star is necessarily identical to the morning star.
5) The morning star is identical to the evening star.
6) The morning star is necessarily identical to the evening star.
But (6) is supposed to be telling us something new and interesting
and contingent. (I.e., it could have been the case that the
morning star wasn’t identical to the evening star.)
What is Black’s argument?
PII is false. (Or, rather, PII is not necessarily true, for there is
a possible world that violates PII.)
Thought-experiment: imagine a universe that contained two
exactly similar spheres. (67b)