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Sunday, January 01, 2017

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Good post.

The artificial distinction between epistemic and ‘metaphysical’ possibility is precisely Kripke’s move in Naming and Necessity. But let me use another of Kripke’s favourite moves against him. I shall grant him his distinction. I grant that it is epistemically possible, but not really possible or ‘metaphysically possible’, that not (a = b). But then let *F be ‘It is epistemically possible, but not really possible, that a is not identical with ---’. Then clearly *Fa is false, and *Fb is true, even given a=b. Your move.

>>Given only what we know from the story it is possible that a not be identical to b.
OK then let **F be ‘Given only what we know from the story it is possible that a not be identical to --’. Then clearly **Fa is false, and **Fb is true, even given a=b.

>> For any x, y, if x = y, then necessarily, x = y. (Kripke's Necessity of Identity thesis)
It is clearly possible (in some given-the-story sense of possible) that a = b, but possible not a=b, where ‘a’ and ‘b’ are placeholders for singular terms.

>> from a story-immanent perspective to a story-transcendent perspective.
I am not sure there is such a distinction. One of my examples in the book is the man in Mark 14:51-2 following Jesus after his arrest, and the man who the women see when they enter the tomb in 16:5. Many commentators have speculated whether the man in 14:51 was the same as the man in 16:5. But Mark does not tell us. Note the list of speculations in that article. Clearly those who speculated that the man in 14:51 was James the Just were not speculating that the man in 14:51 was self-identical. Yet this is nothing to do with ‘story-immanent perspective’ versus ‘story-transcendent perspective’. If we cannot transcend the story, then we obliterate the distinction between myth and history. If all of history is just a story, then historical reality evaporates.

You may discern a connection between what we are discussing here, and the issues about biblical allegory and literality.

>>Kripke's Necessity of Identity thesis

It doesn't really originate with Kripke. In 'Identity and Necessity' 1971, he gives the following argument:

1 ass. (x)(y) [ x = y -> (Fx -> Fy) ]
2 ass. (x) nec (x=x)
3 subst. (x)(y) (x = y) -> [nec(x=x) -> nec(x=y) ]
4 conc. (x)(y) [ x = y -> nec(x = y) ]

He says ‘This is an argument which has been stated many times in recent philosophy. Its conclusion, however, has often been regarded as highly paradoxical’. Then he mentions that while some philosophers have been discontented with the conclusion, others have advocated it.

So I think it's better to regard Kripke not as the originator of the thesis, but rather as the person who first provided arguments so superficially plausible that there is now almost universal consensus that the conclusion is correct.

Except for me, of course, although my beef is not with the quantified or de re version, but the placeholder or de dicto version.

And in answer to that question (see ‘On a Derivation of the Necessity of Identity’by John Burgess, Synthese May 2014, Volume 191, Issue 7, pp 1567–1585) it was Ruth Barcan Marcus who postulated the necessity of identity thesis (‘Identity of Individuals in a Strict Functional Calculus of Second Order’, Journal of Symbolic Logic 1947 12-15). However her proof was complex and laborious. The simple proof mentioned by Kripke is due to Quine (‘Three Grades of Modal Involvement’ Journal of Symbolic Logic 1953, 168-169). In Kripke’s first publication [‘A Completeness Theorem in Modal Logic’, Journal of Symbolic Logic, 24, 1-14, 1959, bottom p. 9] he cites ‘a theorem of Quine’, namely necessity of identity.

">> from a story-immanent perspective to a story-transcendent perspective.
I am not sure there is such a distinction."

As I see it, your non-admission of this distinction is at the root of your problematic view. It is connected with your denial of real reference and your attempt to reduce all reference to intralinguistic reference such as back-reference.

That a = b is not part of your little story. Will you say it is part of a wider story? So the world is narrative all the way down? Very post-modern! I have always suspected you of being some sort of linguistic idealist. This also explains your nominalism.


>>As I see it, your non-admission of this distinction is at the root of your problematic view.

OK then let's scrub out 'story' and substitute 'history' in my example. We have a witness, a manuscript which seems genuine, and which contains an assertion logically equivalent to "there was someone called 'a', and there was someone called 'b'".

Then it's still your move. Let **F be ‘Given only what we know from the history it is possible that a not be identical to --’. Then clearly **Fa is false, and **Fb is true, even given a=b.

Do you deny substitutivity, as I do, or not? You need to bite the bullet.

'History' is ambiguous as between narrative and events narrated. Which do you mean? (When a feminist speaks of His Story, she means a narrative.)

It would help if you tell what you are arguing for. What is your thesis? That there are identity statements that are contingently true?

I believe that the new dean is a lefty. Tom is the new dean. But it doesn't follow that I believe that Tom is a lefty.

'Tom' cannot be substituted for 'new dean' in the first premise.

OK let **F be ‘Given only what we know from the manuscript it is possible that a not be identical to --’. Then clearly **Fa is false, and **Fb is true, even given a=b.

Do you agree or not?

>>'Tom' cannot be substituted for 'new dean' in the first premise.

Right - does this imply you are implicitly conceding my point? It would help if you actually said. I don't know whether you agree but don't want to concede, or whether you are not sure, or whether you disagree.

The point is that Kripke's whole argument depends on substitutivity. If this fails for epistemic modality, why should it succeed for any other kind of modality?

By the way, did Burgess mention Quentin Smith who rather indelicately some years back raised the question of plagiarism of Kripke from Marcus and in so doing raised a stink at the A. P. A.?

>>OK let **F be ‘Given only what we know from the manuscript it is possible that a not be identical to --’. Then clearly **Fa is false, and **Fb is true, even given a=b.

Do you agree or not?<<

A yes or no answer is not possible.

It is clearly impossible that a not be identical to a. This is because it is a formal-logical truth that a = a, a truth that holds whether we are inside a story or outside a story, inside a doxastic network or outside it.

Given that a = b outside the story (in reality), could a not be identical to b outside the story? NO!

Given that a = b outside the story, could inside the story the person named 'a' not be identical to the person named 'b'? YES

>>The point is that Kripke's whole argument depends on substitutivity.<<

Argument for what? Which Kripkean thesis are you opposing?

This seems inconsistent with what you stated in the main post. You said

Suppose 'Axwell' and 'Buswell' are the two names in the story and that both refer to an existing man, the same man. That a = b is no part of the story. Given only what we know from the story it is possible that a not be identical to b.
You are making two statements in that post:

1. a=b
2. **Fb is true, where **F is ‘Given only what we know from the story it is possible that a not be identical to --’.

From your comment above you are also making the third statement

3.**Fa is false

I see no inconsistency provided we distinguish internal and external PsOV.

I still don't know what you are arguing for. What is the point of the little story?

>>Which Kripkean thesis are you opposing?

The thesis that a=b, Fa, and ~Fb are inconsistent whenever the reference/designation of 'a' and 'b' are preserved. He argues that the apparent exception cases are when 'a' has a different designation in 'Fa' than in 'a=b', or likewise 'b' has a different designation.

I oppose that thesis, but per your remarks above, it seems you oppose it too? I.e. you seem to agree that the following 3 statements are consistent:

1. a=b
2. **Fb is true
3.**Fa is false

where **F is ‘Given only what we know from the story it is possible that a not be identical to --’.

Or have I misunderstood you?

What you say implies that the following set of propositions is consistent:

Hesperus is Phosphorus
Hesperus is a planet
Phosphorus is not a planet.

I don't understand. Surely that trio is inconsistent.

But now suppose there is a story in which we read that there is something called 'Hesperus' and something called 'Phosphorus.' Suppose there is no more to the story than that. Obviously, from the story we cannot conclude that H = P or that ~(H = P).

I can say, truly, on the sole basis of the story, that it is possible that the names are coreferential and it is possible that the names are not coreferential. From this I conclude that it is contingent whether the names are coreferential.

But this contingency derives from the indeterminateness of the story. It is not real (extramental, extraliterary) contingency but something analogous to epistemic/doxastic contingency.

You have to distinguish what is really possible from what is possible given what we know. It looks like you are failing to make that distinction.


Our comments crossed.
>> What is the point of the little story?
It is that substitution fails, without reference-shift taking place. Was that not obvious from the start? Perhaps not. I can see one line of defence on your side.

1. a=b
2. within the story a could be a different object from b
3. not(within the story a could be a different object from a)

This violates substitution only if there is no reference shift. Is your point that the ‘within the story’ operator changes the reference of ‘a’ from its reference in ‘a=b’? In this case you could concede 1-3 without conceding failure of substitution. Was that your point all along?

>>What you say implies that the following set of propositions is consistent: Hesperus is Phosphorus Hesperus is a planet Phosphorus is not a planet.
No. I did not claim that substitution always fails, only that it sometimes fails.

>>But now suppose there is a story in which we read that there is something called 'Hesperus' and something called 'Phosphorus.' Suppose there is no more to the story than that. Obviously, from the story we cannot conclude that H = P or that ~(H = P).

And here you have given a perfect example of substitution failure.

1. From the story we cannot conclude that H = P
2. From the story we can conclude that H = H

This is true even if H in fact is P. Perhaps there is a confusion here about what ‘substitution failure’ means? ‘Substitution’ is sometimes known as the indiscernability of identicals, and it states that Fa, ~Fb and a=b are not consistent. I did point this out in an early comment. Philosophers before Kripke thought that it sometimes fails, Kripke’s contribution to the subject was his argument that the failure is due to reference shift.

>> You have to distinguish what is really possible from what is possible given what we know. It looks like you are failing to make that distinction.

This is irrelevant to my argument. My argument is that sometimes there is substitution failure. You reply ‘that is because we must distinguish what is really possible from what is possible given what we know’. I object ‘I know that, but there is still substitution failure’. And so it goes on. I thought I had made it clear from the outset that substitution failure. For example, at the end of my story example which you posted, I said ‘This is the case even if a, in fact, is identical with b’.

Here is another example:

1. Given what we know, it is possible that a is not identical with b
2. Given what we know, it is not possible that a is not identical with a

Both are true even if a=b (assuming a=b is not part of 'given what we know').

'Someone called "Al"' is a general term whereas 'Al' is a singular term.

In your original formulation you slide from 'there was someone called "a"' to 'a.'

I am wondering if this is a source of trouble.

It is not clear what you mean by 'substitution,' but the Indiscernibility of Identicals (InId) is very clear: Nec, for any x, y, if x = y, then whatever is true of x is true of y, and vice versa, in which case >> Fa, ~Fb and a=b are not consistent<< as you say.

So what are you claiming? That there are counterexamples to InId?

I think that is what you are trying to show. You think that there is a predicate F that is true of a but is not true of b, despite the known fact that a = b. The predicate is: 'The story is consistent with a not being identical with ___'.

Your argument, then, is this:

1. a is such that the story allows its non-identity with b
2. b is not such that the story allows its non-identity with b
3. a = b
Therefore
4. InId is false.

Now is that your argument?

>>In your original formulation you slide from 'there was someone called "a"' to 'a.' I am wondering if this is a source of trouble.

I don’t think so. Consider

A man called ‘Barabbas’ was in prison with the insurrectionists who had committed murder in the uprising. Wanting to satisfy the crowd, Pilate released Barabbas to them.
That’s pretty much all we have about Barabbas. But wait! I just said ‘That’s pretty much all we have about Barabbas.’ Am I referring to Barabbas or not? I don’t see a problem with introducing a character by a general term like ‘man called ‘N’’, then going on to use the name ‘N’ to refer to that man. This can happen either in the source document itself (the clay tablet) or I can use back reference to the document to refer myself, as I have just done.

In fact, all historical characters are introduced to us in some such way, yet we can refer to them. Consider poor Nabu:

[Regarding] 1.5 minas (0.75 kg) of gold, the property of Nabu-sharrussu-ukin, the chief eunuch, which he sent via Arad-Banitu the eunuch to [the temple] Esangila: Arad-Banitu has delivered [it] to Esangila. In the presence of Bel-usat, son of Alpaya, the royal bodyguard, [and of] Nadin, son of Marduk-zer-ibni. Month XI, day 18, year 10 [of] Nebuchadnezzar, king of Babylon.
Can I not refer to Nabu-sharrussu-ukin? Or to Arad-Banitu? Indeed, have I not just done so?

Jeremiah 39:3 states that a man called Nebo-Sarsekim a chief officer, took a seat at the middle gate of Jerusalem after Nebuchadnezzar laid siege to it. We can now speculate whether he was the same person as the chief eunuch Nabu-sharrussu-ukin. However:

1. It can be established from logical principles alone that Nebo-Sarsekim is identical to Nebo-Sarsekim.
2. It cannot be established from logical principles alone that Nebo-Sarsekim is identical to Nabu-sharrussu-ukin.
3. This is so even if Nebo-Sarsekim is identical to Nabu-sharrussu-ukin.

Substitution failure again. And if that is correct (it surely is) Kripke’s whole theory collapses. His theory depends on the assumption that the principle of substitution is sure and certain, and that all the apparent exceptions are caused by reference shifts. Now it is true that his exception cases do involve reference shifting. But the cases I am highlighting do not.

>>It is not clear what you mean by 'substitution,'
Strictly speaking, I have been talking about what the literature calls the ‘Principle of Substitutivity’. Ricardh Cartwright (‘Identity and Substitutivity’) formulates it as:

for all expressions alpha and beta, {alpha - beta} expresses a true proposition only if substitution of beta for alpha is truth preserving
An exception to the Principle will occur any time we can find examples where alpha = beta is true, but where there is no truth preserving, such as the many examples I have given above. I call these examples of ‘substitution failure’.
>>Now is that your argument?

The argument is simply that Substitutivity fails in certain contexts. My ultimate target is the Identity of Necessity principle. You are familiar with that?

To prove Identity of Necessity you need the Principle of Identity, nec(a=a), plus Substitutivity, plus the assumption that every token of the same type has the same reference.

Marcus (1975. ‘Does the Principle of Substitutivity Rest on a Mistake?’ In The Logical Enterprise, ed. A. Anderson, R. B. Marcus and R. Martin, 31-38. New Haven: Yale University Press, p.108) has a more precise formulation:

For all proper names a and b (indexed to preserve univocality), a = b expresses a true proposition just in case, for all sentences P, S and S*, if S is a restatement of P in logical form and if S* is like S save for containing an occurrence of b where S contains an occurrence of a, then S expresses a true proposition only if S* does.

A suggestion:

Substitution succeeds when the notation Fx can be understood as application of the function F to object x. That’s how it’s understood in mathematical logic and mathematics in general. This has to work else we lose virtually the whole of maths!

So if in Astute's examples substitution fails it must be because Fx cannot be seen as function application over objects. Can we see why not? I think so. We have to get a truth value out of somehow applying the open sentence ‘From the story we cannot conclude that a = --‘ to the object x. The only way we can do this is to obtain a name for the object x, textually substitute it into the open sentence, and then form a judgement as to the truth of the resulting closed sentence. No problem. Except…

…objects can have multiple names. This wouldn’t be a problem if for every object every name gave the same truth value. An object's names would then be equivalent and we could factor out the ambiguity. But…

…co-referential names don’t always return the same truth value. Using name ‘a’ for object a returns false but using its alias ‘b’ (we know ‘a’ and ‘b’ co-refer because we have a=b) returns true. That’s exactly substitution failure. It makes F multi-valued and hence not a function.

So we seem to have a criterion for predicting when substitution fails. It’s when Fx cannot be interpreted as the application of some function to some object.

>>So we seem to have a criterion for predicting when substitution fails. It’s when Fx cannot be interpreted as the application of some function to some object.

This is essentially Cartwright's approach, who distinguishes the substitution rule from Leibniz’s Law, i.e. if a = b then every property of a is a property of b. However, Graeme Forbes has objected that 'it is perfectly fine to attribute to Giorgione the property of being so-called because of his size (that token of the predicate is not defective qua expressing a property); equally, it is unproblematically true of Giorgione that he is so-called because of his size. Moreover, Giorgione is Barbarelli. Yet Barbarelli does not have the property of being so-called because of his size; equally, it is not true of Barbarelli that he is so-called because of his size'.

Graeme Forbes Metaphysics of Modality p.64:

But even the treatment of constants as rigid designators is in fact not sufficient to guarantee the validity of [a=b -> nec(a=b)], for if we could make sense of the idea that one object could have been two, then at some world which realizes this possibility, ‘a’ could perhaps attach to one of the two objects and ‘b’ to the other, so that a = b, though true at w*, is false at a world in which the single object of w* is two.
Even I find that a bit weird.

I'm glad you find it weird.

If Hesperus is Phosphorus, then in reality there is just one item, call it Venus. Nec, Venus = Venus. So isn't it blindingly obvious that if H = P, then nec, H = P? If you deny that, then you are saying that it is possible that Venus be two and not one.

Is it not blindingly obvious that a thing cannot be contingently identical to itself? Of course Venus contingently exists; but in every world in which it exists it is self-identical.

So, Ed, please explain why you are not committed to saying that there are things that are possibly self-diverse.

You and I know that Eric Blair = George Orwell. But Joe down at the pub, grousing over the new beer tax, doesn't know this. For all he knows the two names both could be and could not be co-referential. But this epistemic possibility does not induce real world contingency.

This is a complex matter and worthy of a separate discussion. Briefly:

>>Is it not blindingly obvious that a thing cannot be contingently identical to itself? Of course Venus contingently exists; but in every world in which it exists it is self-identical.<<

Concerning Nabu (the man mentioned in the Babylonian clay tablet), I agree that (a) necessarily Nabu is self-identical (b) necessarily Nabu is Nabu, i.e. Nabu could not be other than Nabu (c) necessarily (and for the same reason) Nabu is one thing, not two. For to say that Nabu is two is to say that Nabu is one thing and another thing, but we agree that Nabu cannot be different from himself.

Likewise for Nebo, the man mentioned Jeremiah 39:3. All these things are blindingly obvious.

However I don’t agree that necessarily Nabu is Nebo. I challenge you to give a proof of this that does not depend on the Principle of Substitutivity. So in what sense is it ‘blindingly obvious’?

Presumably you agree that ‘man mentioned in the Babylonian tablet’ and ‘man mentioned in Jeremiah 39:3’ could be true of one person, or two people? Do you agree it is contingent whether they are true of one person or two?

PS I can't stress too much that the 'epistemic' vs 'real' distinction is a complete red herring. 'Epistemic' concerns stuff we know. Suppose we have two sources A and B (documents, inscriptions, writing of some kind). Then our possession of these constitutes knowledge about the past. If we got hold of more sources, that would be more knowledge, I agree. But the sources we possess are irrelevant. Suppose what is said by source A and by source B is consistent with there being two people, or one person. Then that is a fact whether or not we in fact possess another source C which when combined with A and B, makes it inconsistent with there being two people. For example:

Source A: 'there is a person called Nebo'
Source B: 'there is a person called Nabu'
Source C: 'Nebo is the same person as Nabu'

Then if sources A-C are all true, there is just one person. If source C is lost in the depths of the British Museum, then we do not know that there is just one person. That is an epistemic matter. Or suppose an archeologist has just found C, and has pieced it together with A and B, and concluded that there is one person. That is also an epistemic matter. But that is irrelevant. Either way, what is said or stated by source A and by source B is consistent with there being two people, or one person. That is a semantic, not an epistemic matter.

Here is the puzzle: how can we establish the necessity of identity without appealing to principles which are either insufficient, or which are not universally valid. The principle of identity (necessarily a = a, or a = itself) is not sufficient. We agree that necessarily Hesperus is identical with Hesperus. That planet could not be numerically different from itself in any circumstance. But the question is whether necessarily Hesperus is identical with Phosphorus. You will object that if H=P, then H necessarily is P, because H necessarily is H. I reply: this begs the question. Under what law of logic or reasoning does nec(H=H) imply nec(H=P)? The principle of identity is insufficient on its own to establish necessity of identity.

What if ‘Hesperus’ means exactly the same thing as ‘Phosphorus’? This is the principle of Semantic Identity. Then it certainly follows that nec(H=H) implies nec(H=P), because both statements mean exactly the same thing. But does ‘Hesperus’ mean exactly the same thing as ‘Phosphorus’? Surely not. When the names were given, when those planets were dubbed, people understood the meaning of both names perfectly. But while they understood that H=H, they did not understand that H=P. The names cannot have meant the same. So the assumption of semantic identity does not hold.

Finally, let’s try the principle of substitutivity, which states that Fa and a=b implies that Fb. Then let F be ‘nec(a= --)’. The principle of identity says that nec(a=a), i.e. Fa. Then if a=b, the principle of substitutivity says that Fb, i.e. nec(a=b). This is valid, but is the principle of substitutivity valid? There are many counterexamples to this, so we cannot assume it is valid.You will object that the principle of substitutivity may be invalid for a type of necessity known as ‘epistemic necessity’, but valid for a type of necessity known as ‘metaphysical necessity’. I reply: under what assumption or principle can you justify that substitutivity is valid for metaphysical necessity, when it is clearly not valid for other types of necessity. You object: we shall define metaphysical necessity as that type of necessity for which substitutivity is valid. I reply: how do you know that anything whatsoever fits that definition? You need to establish that the principle of substitutivity holds for some kind of necessity, without assuming the principle of substitutivity itself. But of course you can’t. If this were possible, Marcus and Quine would have been able to prove the necessity of identity without having to assume substitutivity. But they couldn’t.

My problem is therefore that we cannot establish the identity of necessity without appealing to principles which are either insufficient (the principle of identity) or which are not universally valid (the principles of semantic identity and substitutivity). We could of course assume it as a sort of bedrock, a truth which is obviously true in its own right, a per se nota principle which requires no further demonstration. But I am not sure it is such a truth. It’s not obvious to me, for a start. I mean, I find Graeme Forbes’ claim above pretty weird. But ‘weird’ is not ‘necessarily false’. Furthermore, the weirdness of his claim rests upon an implicit denial of the principle of identity, nec(a=a) which is not in question.

So my challenge to Bill and others is to demonstrate necessity of identity by appeal to principles of reasoning which are stronger than the ones given above, or by demonstrating its self-evidence. Neither will work, in my view.

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