The Opponent by e-mail:
Still puzzling over this. I think Kripke believes we can get to N of I directly, via rigidity of designation.
If names are rigid designators, then there can be no question about identities being necessary, because ‘a’ and ‘b’ will be rigid designators of a certain man or thing x. Then even in every possible world, ‘a’ and ‘b’ will both refer to this same object x, and to no other, and so there will be no situation in which a might not have been b. That would have to be a situation in which the object which we are also now calling ‘x’ would not have been identical with itself. Then one could not possibly have a situation in which Cicero would not have been Tully or Hesperus would not have been Phosphorus. (‘Identity and Necessity’ p. 154, there is a similar argument in N&N p.104).
BV's comment: The great Kripke is being a little sloppy above inasmuch as a rigid designator does not designate the same object in every possible world, but the same object in every possible world in which the object exists. Socrates, to coin an example, is a contingent being: he exists in some but not all metaphysically possible worlds. If names are rigid designators, then 'Socrates' picks out Socrates in every world in which the philosopher exists, but not in every world, and this for the simple reason that he does not exist in every world. 'Socrates' if rigid is known in the trade as weakly rigid. 'God,' by contrast, if a name, and if a rigid designator, is strongly rigid since God exists in every possible world.
But I don't think this caveat affects the the main bone of contention.
- Let ‘a’ rigidly designate a and ‘b’ rigidly designate b
- Suppose a=b
- Then there is a single thing, call it ‘x’, such that x=a and x = b
- ‘a’ designates x and ‘b’ designates x
- If designation is rigid, ‘a’ designates x in every possible world, likewise ‘b’
- If ‘a’ and ‘b’ designate x in any possible world w, and not a=b, then not x=x
- Therefore a=b in w
- But w was any possible world. Therefore, necessarily a=b.
I claim that all the steps are valid, except 4, which requires substitutivity. But Kripke does not assume, or endorse, substitutivity (neither do I).
A. 'a' and 'b' are rigid designators.
B. 'a' and 'b' designate the same object x in the actual world.
C. 'a' and 'b' designate the same object x in every possible world in which x exists. (By the df. of 'rigidity')
D. There is no possible world in which x exists and it is the case that ~(a = b).
E. If a = b, then necessarily, a = b.
I see no reason for Substitutivity if we are given Rigidity and Coreferentiality.