Wolfgang Schwarz :: Slingshooting against the necessity of identity

Slingshooting against the necessity of identity

Posted on Monday, 17 Apr 2006.

The set $$m[1]$ is the empty set if p is false, otherwise it is the set of all numbers. Hence $$m[1]$ iff either p and q are both false or p and q are both true. So

1) $$m[1]$

Moreover, (1) isn't contingent. It doesn't just happen to be the case at our world that $$m[1]$ iff p and q have the same truth value. Such truths of logic and set theory hold with necessity:

2) $$m[1]$

Now assume the necessity of identity:

3) $$m[1]$

From (1) and (3) it follows that

4) $$m[1]$

And from (2) and (4) it follows that

5) $$m[1]$

But (5) is absurd. (In fact, (5), together with the innocent assumption that there is any necessary truth, entails that all truths are necessary truths: let p be the necessary truth and q any other truth.)

So what's wrong?

Comments

# Jim on 17 April 2006, 15:41

(2) is true (in the right-to-left direction) only if "{x:x in N & p}" etc are construed as non-rigid designators--i.e., as picking out, wrt a world w, the set of all x such that in w: (x is in N and p), rather than the set of all x such that in the actual world: (x is in N and p).

The rest of the derivation fails if they are so construed.

If we construe "{x:x in N & p}" in the way described, then from 1 and 3 it only follows that:

(4*) (p < - > q) -> (w)(z)(w={x:x in N & p} & z={x:x in N & q} -> NEC:(w=z))

and from 2 and 4*, 5 does not follow.

# wo on 17 April 2006, 18:00

Hi, yes, that sounds exactly right.

So this means that universal instantiation ((x)Phi(x) -> Phi(a)) must be restricted to *rigid* terms (to prevent the derivation of (4) from (3) and (1), with the terms in (1) understood so as to make (2) true), whereas necessitation must be restricted to sentences with *non-rigid* terms (to prevent the derivation of (2) with the terms understood so as to make it false -- I take it that (1) remains a logical truth on this reading). Interesting!

Thinking about it, it seems clear that necessitation must be given up once we have rigidification devices: "if something is F, something is actually F" should be a logical truth (counting "actually" as logical), but it is surely contingent.

# Shawn on 18 April 2006, 07:44

I believe this is the same basic argument used by Quine in Word and Object to argue that the modalities collapse. Dagfinn Follesdal discusses it in detail in his Referntial Opacity and Modal Logic. He comes to the conclusion that it is too strong because there is nothing in the argument that relies on the modality being necessity rather than something else. The premise he rejects is the same one that the first poster mentioned: the treatment of names as descriptions rather than rigid designators. It might be worth checking out in more detail.

# wo on 20 April 2006, 10:18

Thanks Shawn, will have a look!

Wolfgang Schwarz

Slingshooting against the necessity of identity

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