Yet Another Exchange on the Necessity of Identity

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.

My interpretation:

1. Let ‘a’ rigidly designate a  and ‘b’ rigidly designate b
2. Suppose a=b
3. Then there is a single thing, call it ‘x’, such that x=a and x = b
4. ‘a’ designates x and ‘b’ designates x
5. If designation is rigid, ‘a’ designates x in every possible world, likewise ‘b’
6. If ‘a’ and ‘b’ designate x in any possible world w, and not a=b, then not x=x
7. Therefore a=b in w
8. 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).

BV's interpretation:

A. 'a' and 'b' are rigid designators.
B. 'a' and 'b' designate the same object x in the actual world. 
Therefore
C. 'a' and 'b' designate the same object x in every possible world in which x exists.  (By the df. of 'rigidity')
Therefore
D. There is no possible world in which x exists and it is the case that ~(a = b).
Therefore
E. If  a = b, then necessarily, a = b.

I see no reason for Substitutivity if we are given Rigidity and Coreferentiality.   

Comments

[Astute opponent]

>>I see no reason for Substitutivity if we are given Rigidity and Coreferentiality.
Agree, but now you are assuming what needed to be proved. Clearly if ‘a’ and ‘b’ are co-referential, i.e. if there is a single object that they both designate, and if they rigidly designate, then by definition they designate the same object in every possible world. That is obvious. But earlier you said ‘Apparently, the Opponent wants to know what validates the inference from ‘Hesperus is the same entity as Phosphorus’ to 'Hesperus' and 'Phosphorus' designate the same entity’. Again, how do we get from my 1-3 above to my 4 ‘a’ designates x and ‘b’ designates x’?

Here is Kripke’s full argument (N&N p.104, my sentence numbering).

(1) But we, using the names as we do right now, can say in advance, that if Hesperus and Phosphorus are one and the same, then in no other possible world can they be different. (2) We use 'Hesperus' as the name of a certain body and 'Phosphorus' as the name of a certain body. (3) We use them as names of those bodies in all possible worlds. (4) If, in fact, they are the same body, then in any other possible world we have to use them as a name of that object. (5)And so in any other possible world it will be true that Hesperus is Phosphorus.

(1) States what needs to be proved, i.e. if H=P, then necessarily H=P.
(2) for some x, ‘H’ designates x and for some y ‘P’ designates y.
(3) In every possible world ‘H’ designates x, and ‘P’ designates y.
(4) Suppose x=y, so in every possible world ‘P’ designates x, and so that ‘H’ and ‘P’ are co-referential in every possible world.
(5) Hence in every possible world H=P.

Step 4 is invalid without substitutivity. You cannot infer ‘P’ designates x from ‘P’ designates y without this assumption.

[Astute opponent]

There is a further puzzle about what Kripke thinks his argument is, i.e. a puzzle about exegesis. Right there at the beginning (NN p.3) he says

Waiving fussy considerations deriving from the fact that x need not have necessary existence, it was clear from (x) nec (x = x) and Leibniz’s law that identity is an ‘internal’ relation: (x) (y) (x = y -> nec (x = y)).

My emphasis. He says in a footnote that Leibniz’s law known as ‘the indiscernibility of identicals’ is the principle that identicals have all properties in common; schematically, (x) (y) [ (x = y & Fx) -> Fy). ‘Not to be confused with the identity of indiscernibles’. Yet elsewhere (p.20) he says

Some critics of my doctrines, and some sympathizers, seem to have read them as asserting, or at least implying, a doctrine of the universal substitutivity of proper names.

I don’t see how he can get to the necessity of identity without Leibniz’s law, and in fact the law is not universally valid, as he correctly notes.

[BV]

You are assuming the the Indiscernibility of Identicals and Substitutivity are the same principle. Is that obvious? Let's see you prove it.

The first says nothing about language: For any x, y, if x = y, then whatever is true of x is true of y and conversely.

The second is roughly: coreferential expressions are intersubstitutable salva veritate.

Quine, Word and Object, 142: "When a singular term is used in a sentence purely to specify its object, and the sentence is true of the object, then certainly the sentence will stay true when any other singular term is substituted that designates the same object."

Quine's principle is true, no?

[Opponent]

PS I sent you a clever paper by Bostock which is not quite the same as my argument, which cuts off the Kripke argument before it even gets off the ground. Bostock grants that 'a' and 'b' are rigid, that a=b, and that 'a' and 'b' are co-referential in the actual world. (My argument, by contrast, is that it co-referentiality in the actual world does not even follow from a=b). Bostock then considers (my example) the statement ‘if Hesperus had not been Phosphorus then …’. He calls this an impossible situation. Then either (i) the names are not rigid designators in this example. But then Kripke’s argument for rigid designators is collapses into circularity. The antecedent is impossible because the designators are rigid. Yet they are not rigid because the antecedent is impossible!! Or (ii), as intuitively seems correct, they are designating as usual. Then ‘Hesperus’ and ‘Phosphorus’ designate the same planet when they are used in the specification of any counterfactual situation. But it does not follow that Hesperus and Phosphorus will be the same planet in any possible situation. From the premise that the expressions designate the same planet in our specification of the situation, it does not follow that the planets are the same planet in that situation. On the contrary, the antecedent ‘if Hesperus had not been Phosphorus ..’ is deliberately specifying a situation (an impossible world) in which they are different planets! Surely Meinong would approve!!

Generally, the number of entities designated when specifying a counterfactual situation does not have to be the same as the number of entities in the situation so specified. E.g. consider the counterfactual ‘if Nixon’s mother had not had any children ..’. Then the number of entities designated in thus specifying = 2 (Nixon and his mother). But the number of entities in the situation thus specified = 1 (the mother).

In either case I refer to Nixon when specifying the situation, but Nixon is not in the situation I specify.

Oxford 1, Princeton 0.

>>Quine, Word and Object, 142: "When a singular term is used in a sentence purely to specify its object, and the sentence is true of the object, then certainly the sentence will stay true when any other singular term is substituted that designates the same object."

It's not clear what 'purely to specify its object', and 'sentence is true of the object' mean. I will take it to exclude intralinguistic reference statements. I.e. I hold that ‘the name ‘Moses’ refers to Moses’ is true because the token that is used refers back to the token that is mentioned. So the statement is not ‘true of the object’ in that it says nothing about Moses himself. Nor does the name ‘purely specify its object’. Rather, it is picking out a text that the author indicates or suggests we are familiar with. You of course will hate this, but as I pointed out earlier, if the IL theory implies contingent identity, it is equally true that the falsity of contingent identity implies the falsity of the IL theory. We can both see something good in this.

[Opponent]

>>The second is roughly: co-referential expressions are intersubstitutable salva veritate.

Care again. The second is obviously true. The problem is getting from an identity statement which uses the names, to one which mentions them. I.e. from

1. a=b and designates('a', a) and designates('b', b)

to

2. for some x [ designates('a', x) and designates('b', x) ]

[BV]

Part of what complicates this discussion is that your IL thesis plays a role in it, a role that it not clear to me and not set forth explicitly by you.

I did not receive from you the Bostock paper.

Doesn't the burden of proof lie upon you to show that (2) does not follow from (1)?

Suppose I write a piece of fiction in which a purely fictional character bears two names, 'Jake' and 'Mack.' Then, in that story, (1) is satisfied. But it doesn't follow that there exists an x in reality external to the story such that 'Jake' and 'Mack' designate it.

Is this why you think the inference fails?

Or maybe you don't think it fails, you just want to know what justifies it.

Also, how can you presume to refute Kripke using IL when he would never accept such a thesis?

[Astute opponent]

I think you are confusing the two claims I am making. I am saying

(1) Kripke's argument is not valid without substitutivity. I think he would accept that, because he accepts substitutivity.

(2) Kripke's argument fails because substitutivity fails for the 'designates' function. That is a much more radical thesis, which I have not defended here in much detail.

[BV]

You are not answering my question. You say or rather imply @ 7:00 that (2) does not follow from (1). Explain why it doesn't follow.

How can (1) be true and (2) false?

Answer directly.

[Astute opponent]

1. a=b and designates('a', a) and designates('b', b)
to
2. for some x [ designates('a', x) and designates('b', x) ]

It would be false precisely in the case where (e.g.) Hesperus and Phosphorus were one and the same object, where ‘Hesperus’ designates Hesperus, but ‘Phosphorus’ did not designate Hesperus. You are going to leap up and down at this point and yell ‘but how is that possible? It is absolutely certain and obvious that if ‘Hesperus’ designates Hesperus, and if Hesperus is Phosphorus, then ‘Phosphorus’ designates Hesperus. The antecedent cannot be true and the consequent false. And I reply ‘precisely. You have just invoked the principle that Fa and a=b cannot be true, and Fb false. I.e. Fa and a=b implies Fb.

You cannot argue that substitution is a logical truth, for there are clear instances where it fails. E.g. ‘It is analytically true that H=H, therefore it is analytically true that H=P’ is not valid.

[BV]

It is interesting that a principle you call 'substitution' you express in Carnapian material mode. The principle you cite is the Indiscernibility of Identicals.

As I suggested earlier, InId is logical or analytic fallout from the concept of identity, where identity is what is picked out by '=' in standard logic books. Quine calls InId an "axiom of identity." (Methods of Logic, 213)

So it seems to me that I do not need to invoke the principle that Fa and a=b implies Fb if that means to add it as an auxiliary premise to validate the inference from (1) to (2). (2) follows immediately from (1). THis is because indiscernibility is analytically contained in the notion of identity.

Does InId have counterexamples? Suppose Hesperus has the property of being believed by Sam to be a planet, but Phosphorus does not have this property. Does this show that Hesperus is not Phosphorus? Not obviously. It can be taken to show that the 'property' in question is not an admissible property. Intuitively, the identity and difference of things in the real world does not depend on what ignorant people believe.

On the other hand, I cannot substitute 'Phosphorus' for 'Hesperus' in 'Sam believes Hesperus to be a planet' salva veritate. There are clearly cases where substitutivity fails to preserve truth.

So I suggest we distinguish between InId and substituivity. Seems we ought to in any case since what we substitute are linguistic items, words and phrases, and what we substitute them into are linguistic items, i.e., sentences. Phosphorus itself cannot substituted into any sentence. No planet is a part of speech.

[Astute opponent]

Will this help? First order logic with identity adds three axiom schemata to standard first order logic. The third is the principle of substitutivity.

x = y → (φ → φ')

You ask when in the inference above the antecedent would be true and the consequent false. This would precisely be a case when the axiom above had failed. You said a while ago that ‘you don’t need substitutivity’. But why would it be an axiom if we didn’t need it? I am truly perplexed by this discussion!

(I appreciate you are perplexed too. I think you are denying something blatantly obvious, you think the same of me! I think it's blatantly obvious we need the axiom, you think it's blatantly obvious it is not required).

[Opponent]

To be very clear, I am talking about substitutivity alone, i.e. the rule that allows you to replace one term (a linguistic item) with another term (another linguistic item) in a way that is truth preserving. As you say, no planet is a part of speech. Nor do planets enter into logic, which is about the rules governing the transition from premises (linguistic items) to conclusion (a linguistic item).

Kripke formulates a particular thesis in his book Naming and Necessity. This is a book, containing words, written almost entirely in ordinary English (there are a few things he expresses in the language of predicate calculus, but that is a language also). The thesis is that if the ordinary language sentence ‘Hesperus and Phosphorus are one and the same’ is true, then the sentence ‘Hesperus and Phosphorus are necessarily one and the same’ is also true. He wants to persuade us of this logically, i.e. using the discipline of logic, and not rhetoric or emotional persuasion or appeal to authority etc.,

>>On the other hand, I cannot substitute 'Phosphorus' for 'Hesperus' in 'Sam believes Hesperus to be a planet' salva veritate. There are clearly cases where substitutivity fails to preserve truth.

Correct. So my question is why substitution succeeds for the ‘designation’ relation, but fails for the ‘believe’ relation. The two consequences below are of exactly the same logical form:

1. ‘H’ designates H and H=P, therefore ‘H’ designates P
2. S believes that H is a planet and H=P, therefore S believes that P is a planet

You seem to accept that the first one is valid, because ‘indiscernibility is analytically contained in the notion of identity’. But at the same time you concede that the second is not valid, given that the antecedent could be true and the conclusion false. If indiscernibility is analytically contained in the notion of identity, why aren’t both valid?

>> can be taken to show that the 'property' in question is not an admissible property.
Right, so why do you claim that ‘‘H’ designates ---’ is an admissible property, but ‘S believes that --- is F’ is not an admissible property? This is the crux of it. You say you do not need to invoke the principle that Fa and a=b implies Fb, but that is because you are (covertly?) appealing to another principle, namely that F is an ‘admissible property’. So to validate the inference ‘Hesperus is the same entity as Phosphorus, therefore ‘Hesperus’ and ‘Phosphorus’ designate the same entity’, we need something like this:

Hesperus is designated by ‘Hesperus’
Hesperus is the same entity as Phosphorus
‘being designated by’ is an admissible property
Phosphorus is designated by ‘Hesperus’

So my point stands that we need some kind of principle to justify the inference in question, given your concession that not all properties (or relations) are admissible.