Here's one argument for the necessity of identity. Assume a=b. a has the property of being essentially identical to a. By Leibniz's law, b also has this property. So in every world in which b exists, b is identical to a, so there's no world in which a and b are distinct.
Will that convince the anti-essentialist? No, because it relies on an essentialist assumption: that every thing x has the property of being essentially identical to x. (This is not the premise that every thing x has the property of being essentially self-identical, which the anti-essentialist may well grant.)
Can the necessity of identity be established without relying on essentialist assumptions? I don't think so. But there is a tradition of thinking it can be established solely by considerations concerning rigid designation. I thought this tradition had died and been buried, but it has arisen in a recent article in the Philosophical Quarterly by Sören Häggqvist. (Häggqvist, Sören, (2006), ‘Essentialism and Rigidity’, The Philosophical Quarterly 56:223, p275-283.)
He argues as follows. If ‘a’ and ‘b’ are rigid designators then they designate in every possible world that which they designate in the actual world. Since they designate the same thing in the actual world (let us suppose) they therefore designate the same thing in every possible world, and so ‘a=b’ is a necessary truth.
I don't think this argument works, and want to bury it for good. The argument moves from (1) ‘a’ and ‘b’ designate the same thing in the actual world and (2) in any world, both ‘a’ and ‘b’ designate what they designate in the actual world to (3) in any world ‘a’ and ‘b’ designate the same thing. But (3), pretty obviously, only follows from (1) and (2) if the necessity of identity is true, and so to rely on this move is just to beg the question. Suppose ‘a’ and ‘b’ actually co-designate but that there is a world, w, in which ‘a’ designates something distinct from ‘b’. Does it follow that one or other of ‘a’ and ‘b’ designate in w something that they do not designate in the actual world? Not if a and b are identical in the actual world but distinct in w. If a and b are merely contingently identical then of course the rigid designators ‘a’ and ‘b’ will not necessarily co-designate; in a world in which a and b are distinct then, since ‘a’ must still designate a and ‘b’ still designate ‘b’ in this world, ‘a’ and ‘b’ will designate distinct things in this world. And so to conclude that the rigid designators ‘a’ and ‘b’ necessarily co-designate because they actually do begs the question against the contingent-identity theorist.
11 comments:
I just noticed that wo thinks the argument I don't like is better than the one I cite at the start. See his blog below.
http://www.umsu.de/wo/archive/2006/08/09/Kripke_s__Alleged__Argument_for_the_Necessity_of_Identity_Statements#c902
I left a comment there making much the same point I make above. (Actually I made it twice: warning for any potential posters on wo's blog - you have to click submit twice to prove you're not a spammer. This means that those of us, like myself, who do not scan the page in careful detail on the assumption that this was going to happen have to re-write our entire comments.)
oh dear, sorry about the lost comment. I've added a warning to the comments form on my blog so that this shouldn't happen again.
And good point about the rigity argument. Though I still think it is better than the argument from Leibniz' law. It depends a little on what the conclusion is supposed to be: is it the quantified "for any x,y, if x = y, then L(x = y)", or is it the schema "if a = b, then L(a = b)"? A typical proponent of contingent identities from the 1960s might have accepted the former, but rejected the latter. The rigidity argument can then be used to show that if something like the quantified formula is true (and rigidity interpreted in a certain way), then the schema, restricted to rigid designators, is also true. But as you say, against a die-hard anti-essentialists (or, I would add, a counterpart theorists) who even rejects the quantified formula, the argument is useless.
Here's the best argument for the necessity of identity based on rigid designation:
1) ‘a’ and ‘b’ are rigid designators. (Assumption)
2) ‘a’ and ‘b’ co-refer. (Assumption)
3) For all worlds w, ‘a’ refers in w to what it refers to in @. (From 1)
4) ‘a’ refers to a in @. (Analytic)
5) For all worlds w, ‘a’ refers to a in w. (From 3 and 4)
6) For all worlds w, ‘b’ refers in w to what it refers to in @. (From 1)
7) ‘b’ refers to a in @. (From 2 and 4)
8) For all worlds w, ‘b’ refers to a in w. (From 6 and 7)
9) For all worlds w, ‘a’ and ‘b’ co-refer. (From 5 and 8)
10) For all worlds w, ‘a=b’ is true in w. (From 9)
11) Necessarily, a=b. (From 10)
Is this a good argument for the necessity of identity? I don’t think we should be able to derive 8 from 7 and the claim that ‘b’ is a rigid designator. The problem lies with 6. It seems to presuppose that ‘b’ has a unique referent in w, and also that there is a unique thing in w that is identical to the thing ‘b’ actually refers to. But of course, if contingent identity is an option on the table (which it must be at this stage, to avoid begging the question) then these can’t both be true. So it seems to me that to conclude 6 from 1 presupposed the necessity of identity, and so begs the question.
We need to be more careful in stating 6. What are we entitled to conclude from “‘a’ is a rigid designator”? Here are three options:
(i) In every world w in which ‘a’ refers, ‘a’ has a unique referent, and this referent is identical to the unique referent of ‘a’ in @.
(ii) In every world w in which ‘a’ refers, ‘a’ has a unique referent, and this referent is the thing in w that has the a-haecceity.
(iii) In every world w in which ‘a’ refers, ‘a’ refers to any thing in w that is identical to the thing that is referred to by ‘a’ in @.
Each of these options seems compatible with the contingency of identity.
On the first: ‘a’ and ‘b’ can fail to co-refer at a world, w, and (i) still be true: the thing that ‘a’ refers to in w is identical to its actual referent, and likewise with ‘b’. But since these things are distinct in w, ‘a’ and ‘b’ don’t co-refer. On this option, 6 should be replaced with 6i: For all worlds w, the referent of ‘b’ in w is identical to the referent of ‘b’ in @. But we can’t move from 6i and 7 to 8; because if there are two things in w that are identical to the referent of ‘b’ in @ then 6i can be true and there some thing in w that is not referred to by ‘b’ that is also identical to the referent of ‘b’ in @ (namely, a).
The second option has it that rigid designators are haecceity trackers whereas non-rigid designators refer to things with different haecceities in different worlds. The necessity of identity doesn’t follow on this option, because nothing rules out one thing (a/b) having both the a-haecceity and the b-haecceity in the actual world, but there being another world in which some thing (a) has the a-haecceity and not the b-haecceity (and so is referred to by ‘a’ but not ‘b’) and some thing (b) having the b-haecceity and not the a-haecceity (and so is referred to by ‘b’ but not ‘a’). On this option, 6 should be replaced with 6ii: For all worlds w, ‘b’ refers in w to whatever in w has the b-haecceity. 8 obviously doesn’t follow from 6ii and 7. We would need to add the premise that if a has the b-haecceity then it necessarily has the b-haecceity (that, together with the uncontroversial premise that ‘b’ only actually refers to a if a actually has the b-haecceity, will get us to 8). But this just brings us back to the essentialist argument for the necessity of identity: we are no longer deriving it from principles concerning rigid designation alone.
On option iii it looks like what we should say concerning a world in which a and b are distinct is that ‘a’ and ‘b’ lack unique referents. We should replace 6 with the cumbersome 6iii: For all worlds w, each referent of ‘b’ in w is identical to the unique referent of ‘b’ in @, and every thing in w that is identical to the unique referent of ‘b’ in @ is a referent of ‘b’ in w. 8 follows from 7 and 6iii. 9 follows from 8 if what it means for terms with no unique referent to co-refer is either that one of the referents of one term is identical to one of the referents of the other, or that each of the referents of one term is identical to one of the referents of the other. But if it means that then 10 doesn’t seem to follow from 9. ‘a=b’, on option iii, should not be true but indeterminate at worlds in which there are two things that are identical to the unique actual referent of ‘a’ and ‘b’: it is true on some admissible assignments of referents to ‘a’ and ‘b’ and false on others.
So I don’t think there’s any version of the above argument that works. (None of this should be surprising: it would be amazing if the necessity of identity could be established in this way.) But maybe I’m missing some ways of understanding rigidity. Any other suggestions? The criterion is that the characterisation cannot presuppose the necessity of identity (like I think 6 does).
I'm not convinced by the objection to iii. As it stands it seems to confuse plural and indeterminate reference. If you think of `a' and `b' as plural referring terms relative to other worlds, then shouldn't `a=b' be evaluted just as `the people carrying the piano are the Beatles' would be? And since all the multiple referents of `a' are also the multiple referents of `b', `a=b' should be true.
It looks to me as if at least the letter of nec identity would be sustained on this appraoch (though interestingly, doesn't seem any way to preserve even the letter of the necessity of distinctness). By the way, (iii) doesn't seem a crazy way of extending a rigidity constraint motivated, say, by direct reference considerations, to this metaphysical setting.
Hi Ross, when considering (i) you say:
On the first: ‘a’ and ‘b’ can fail to co-refer at a world, w, and (i) still be true: the thing that ‘a’ refers to in w is identical to its actual referent, and likewise with ‘b’. But since these things are distinct in w, ‘a’ and ‘b’ don’t co-refer.
But I'd have thought the advocate of the rigid designation argument would say that this violates the transitivity of identity: if x = y and y = z then x = z. In your case the actual referent of 'a' and 'b' (call this 'y') is identical to both the referent of 'a' in w (call this 'x') and the referent of 'b' in w (call this 'z'). So we have x = y and y = z, but allegedly not x = z.
I suppose that the transitivity of identity might be doubted, but I don't see that it's a particularly essentialist assumption. In particular the dialectic starts to look funny if the defender of the contingency of identity accuses his opponent of begging the question by presupposing transitivity. That looks like it falls on the wrong side of the line separating a legitimate charge of begging the question from a decision to take an argument as a reductio of its premises. What am I missing (hope it isn't too obvious)?
If I were a Lewisian realist with overlap I might be convinced to hold the necessity of identity by these transitivity considerations (actually it's that identity is Euclidean that does the work, but that is entailed by transitivity and symmetry - sorry, totally pedantic!).
But I'm not sure anyone else should be convinced. The relation between y and x and y and z isn't identity after all - no one thinks that, apart from the (fictional) modal realist with overlap. Everyone on all sides is agreed that x simply represents y as existing and that z represents x as existing. And nothing in the logic of 'represents' suggests that x can't represent y as existing but not z, and z represent y as existing but not x.
Is that satisfying?
Ross: why do you think that it's only the modal realist with overlap that thinks transworld identity is identity? Why doesn't, e.g. Williamson think this too? (And maybe some abstractionists). Am I missing something here?
Even if you think that the transworld identity isn't identity simpliciter, I can imagine theorist's making it a theoretical constraint on anything deserving the name "transworld identity" that it satisfy enough of the properties of strict identity, e.g. transitivity.
I can think of motivations for insisting on that which (as Daniel suggests) are certainly motivated independently of essentialism. (Whether they're motivated independently of the Humphry objection is another matter).
Ross, as well as concurring with what Robbie says, I'd add two things. When you say that we're not really talking about identity here, you're shifting the goalposts, because (i) is no longer an option. The new situation is that given (i) as an interpretation of rigidity the argument for the necessity of identity goes through, whereas given (ii) or (iii) it doesn't (perhaps). But since the advocates of the argument were presumably assuming (i), that's where the action is - you need to make your objections to (i), which you hint at in your last comment, more explicit and more central. (I'm not demanding that you do that here. But I think you should if you want to publish it...)
Also I think there are various anti-realisms about possible worlds which have trans-world identity. For example, if you think that possible worlds are projections, they should be projections involving actual objects. That seems attractive because we want modal talk to be about actual objects. In my view the realism/anti-realism issue should be about whether possible worlds exist, not whether they can contain the same objects as the actual world. That's the framework I was coming from, not Lewisian realism.
anti-realism about worlds? get thee behind me satan! Cambridge herisy!
(anti-realism about which worlds are possible however, is perfectly ok in our neck of the woods. Curious.)
Hi Andrew,
Sorry, I was over-simplifying quite a lot. I do want to allow for it to be possible that objects other than the actual ones exist, and that the actual objects don't exist. But at least sometimes we do want to know what the possibilities are for actual objects. And that's at least a prima facie reason for allowing trans-world identity. I get why modal realists like Lewis might have metaphysical reasons to reject trans-world identity; but those reasons don't apply to anti-realist views. In that sense trans-world identity looks like a default position, even or especially for anti-realists; in which case it's fair enough to assume it in an argument for the necessity of identity.
Also, although I do think of anti-realism about worlds as more natural than mere anti-realism about possibilia (what's a world if not the fusion of the objects in it?), what I really mean by modal anti-realism is anti-realism about modal discourse, i.e. thinking that modal discourse isn't descriptive, at least initially (or that truth in modal discourse needs to be earnt etc. etc. standard Cambridge heresy). But it's not as if I have a well-worked-out view, so I won't distract attention from Ross's argument any further by going on and on.
Hi guys,
Loads of useful stuff for me to think about - thanks!
I'm not going to get a chance to think much about this for a while, but I see I need to think a lot more about the transitivity point. I grant that some ersatz accounts will have strict identity - say worlds as sets of Russellian propositions - and so it may be that whether or not the transitivity argument is suasive depends on your metaphysics of worlds. I'm not too happy with that, but there you go.
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