The Opponent comments in black; my responses are in blue:
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) 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 necessarily, H = P, because necessarily, H = H. 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.
BV: This seems correct. There is no immediate valid inference from the principle of identity to the necessity of identity. The inference would seem to be valid only in the presence of auxiliary 'mediating' premises.
But let me play the role of advocatus diaboli. We know empirically that H = P. And we know a priori about the identity relation. We know that it is an equivalence relation (reflexive, symmetric, transitive). We also know that it is governed by the Indiscernibility of Identicals (InId) which states that for any x, y, if x = y, then whatever is true of x is true of y and vice versa. InId is not a principle external to the notion of (numerical) identity, but part of what we mean by 'identity.' Obviously, if two putatively distinct items are one item, i.e., are identical, then whatever is true of the one is true of the other, and vice versa. We would never apply the concept of identity to any thing or thing that violated InId.
So if we know that H = P, then we know that in reality (i.e., extralinguistically, and extramentally) there is just one thing where H and P are. Call this one thing 'V.' We know from the principle of identity that necessarily, V = V. Now suppose, for reductio, that it is not the case that necessarily, H = P. Suppose, in other words that possibly, ~(H = P). One would then be supposing that the identity of H and P is contingent. But that is to suppose that the identity of V with itself is contingent, which is absurd. Therefore, the necessity of identity holds.
So it appears that I have validated the inference from the the principle of identity to the necessity of identity by adducing premises that are well-nigh self-evident. One of my supplementary premises is that we know some such truths as that H = P. I also assumed that if x = y, then there are not two things denoted by 'x' and 'y,' but one thing. I also assumed that when we use terms like 'H' and 'P' we are referring to things in reality with all their properties and relations and not to items like sense data or Husserlian noemata or Castanedan guises or any sort of incomplete object or epistemic deputy. I am assuming that our thought and talk about planets and such reaches right up to the thing itself and does not stop short at some epistemic/doxastic intermediary.
And now back to the Opponent:
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.
BV: That's right. The names do not have the same Fregean sense (Sinn). This is why 'H = H' and 'H = P' do not have the same Fregean cognitive value (Erkenntniswert). To know one is to know an instance of the principle of identity. It is to know a logical truth. To know the other is to know a non-logical truth, one that is synthetic a posteriori in Kant's sense.
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.
BV: it is true that there are counterexamples to the principle of substitutivity in the 'wide open' formulation that the Opponent provides. Sam can believe that Hesperus is a planet, not a star, without believing that Phosphorus is a planet, not a star, despite the fact that Hesperus = Phosphorus. So the following is a non sequitur:
Hesperus has the property of being believed by Sam to be a planet.
Hesperus = Phosphorus.
Phosphorus has the property of being believed by Sam to be a planet.
This example is also a counterexample to the Indiscernibility of Identicals which is presumably equivalent to the substitutivity principle. I think that should worry us a bit.
To appreciate the dialectical lay of the land it may help to set forth the problem as an aporetic tetrad:
A. InId: For any x, y, if x = y, then whatever is true of x is true of y and conversely.
B. Hesperus = Phosphorus.
C. It is true of Hesperus that it is believed by Sam to be a planet.
D. It is not true of Phosphorus that it is believed by Sam to be a planet.
The tetrad is inconsistent: any three limbs entail the negation of the fourth. One could solve the problem by rejecting InId in its wide-open or unrestricted formulation. What speaks against this solution is that InId in its unrestricted formulation is part and parcel of what we mean by '=.' If you were trying to explain to a student what relation '=' stands for, you couldn't just say that it stands for an equivalence relation since not every such relation is picked out by '=.' You would have to bring in InId.
A second way to solve the tetrad is by denying (B). It can be true that H is the same as P without it being the case that H = P. Note that '=' is not a bit of ordinary language; it is a terminus technicus. One can't just assume that the only type of sameness is the sameness denoted by '=.' Suppose we distinguish between formal identity statements of the form a = a and material identity statements of the form a =* b. While both are equivalence relations, the former are necessary while the latter are contingent. We can then say that H and P are materially identical and thus contingently the same. Because they are contingently the same, they are not one and the same. H and P are together in reality but are nonetheless distinct items. If so, (C) and (D) can both be true in the presence of InId/Substitutivity.
At this point I ask the Opponent whether his denial of the necessity of identity amounts to an affirmation of the contingency of the relation picked out by '=,' or whether it amounts to a rejection of the relation picked out by '=.' It seems to me that if you admit that there is a relation picked out by '=,' then you must also admit that it holds noncontingently in every case in which it holds.
One could hold the following view. There is a relation picked out by '=.' Call it formal identity. It holds of everything. But no synthetic identity statement is noncontingently true if true. No such statement is reducible to the form a = a. All are contingently true if true. So 'Hesperus is Phosphorus' is contingently true, and what the names refer to are distinct items. They refer directly to these items. But these items are something like Castaneda's ontological guises or Butchvarov's objects.
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.
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.
BV: It seems to me I gave a reductio-type demonstration in my first comment. The paradigm cases of the relation picked out by '=' are the cases of the form a = a. Now if 'H' and 'P' designate one and the same entity, then what appears to be of the form a = b, reduces to the form a = a. Clearly, if a = a, then necessarily a = a. The assumption that the identity of H and P is contingent entails the absurdity that a thing is distinct from itself. Therefore the relation denoted by '=' holds necessarily in every case in which it holds. Q. E. D.
Note that I didn't use Substitutivity/Inid or Semantic Identity in this reductio. But I did assume that there is a relation picked out by '=' -- which is not obvious! -- and that it is this relation that the 'is' expresses in the synthetic truth 'H is P.' Which is also not obvious!