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Acknowledgement | Dedication
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2.       It remains to show how the foregoing analysis can be deployed to account for the breakdown in modal contexts of the principle of the Substitutivity of Identicals (SI). SI embodies the supposition that an exchange of referentially identical singular terms must leave a statement's truth-value unaffected--being an exchange, as is said, salva veritate. Indeed, the principle according to which referentially identical singular terms can be exchanged salva veritate is widely held to constitute an inviolable canon of valid deductive inference, for it is vouchsafed by a framework principle of philosophical semantics: that a particular's properties stand unaffected by its mode of designation. What then to make of (18) and (19), where such an exchange leads from truth to falsehood?
(18) necessarily(Scott = Scott)

(19) necessarily(Scott = the author of Waverley)

The philosophical literature is replete with proposals for preserving SI, in the face of a bewildering variety of "apparent" counterexamples such as (18, 19). To discuss at length any of these proposals falls outside the scope of this paper. Instead I will provide a reason for what I take to be the real breakdown of SI in modal contexts: a breach of the Leibnizean principle of the Indiscernibility of Identicals. If my assessment is correct, what roils the modal waters is not, as is often claimed, confusion concerning scopes, or unclarity about contexts, or a lack of ability to grasp the difference between coincidence between individual concepts and identity between individuals, or anything of the sort.6 On the contrary: what roils these waters is confusion concerning, unclarity about, and a lack of ability to grasp the nature of either formal or material identity.7  For without an understanding of these and how they are related, it is impossible to make out why Scott and the author of W, or nine and the number of planets, behave modally as they do.

The bearing of formal identity on the Indiscernibility of Identicals is not difficult to ascertain. To begin with, the self-identity of x is a condition on x's having any property or relation.

(20) for allxnecessarily(Fx if-then (x = x))
So, necessary self-identity is an indispensable condition for having properties necessarily at all:8
(21) for allx(necessarilyFx if-thennecessarily (x = x))
As a consequence, particulars that are not necessarily self-identical have none of their properties necessarily.
(22) for allx(¬necessarily(x = x) if-then ¬necessarilyFx)
As a result, the only identities in which contingently self-identical particulars feature are contingent ones.
(23) for allx(¬necessarily(x = x) if-then ¬for somey((x = y) & necessarily(x = y)))
Therefore, since self-identity is an indispensable condition for having properties or relations, and necessary self-identity a sine qua non for having properties or relations necessarily, contingently self-identical particulars confer mutual consistency, both upon the LPC=-irreconcilable:9
necessarilyFx
(24) x = y
¬necessarily(Fy)

and upon the LPC=-irreconcilable:

necessarily(x = x)
(25) x = y
¬necessarily(x = y)

For, from (23)* it follows that if--as our pre-theoretical understanding of the difference between Scott (who can not but have been Scott) and the author of W (who need not have been the author of W) would seem to suggest--Scott is self-identical necessarily but the author of W is not, then Scott and the author of W, although identical are not identical necessarily. But it follows from (21)* that if the number of planets is not self-identical necessarily, then--although nine = the number of planets and it is necessary that nine is odd--it is not necessary that the number of planets is odd.

By making sense of the modal sundering of numerical and qualitative identity, the (dark) doctrine I am about to expound restores to logical grace the behavior of 'the author of W' and 'the number of planets' in modal frames. For, despite the panic that their misbehavior sows amongst swillers at the classical trough, and the maladies to which departures from classical norms are variously attributed,10 if the author of W and the number of planets are contingently self-identical, their breach of the Indiscernibility of Identicals, so far from subverting the sovereignty of self-consistent thought betokens fealty thereunto.

Thesis III* posited the unity of x and X as root and source of the self-identity of x. This link, between the unity of x and X and the self-identity of x, is moreover a necessary one: it is no more possible that x should exemplify X and x not equal x than it is possible that x should equal x and x not exemplify X.

(26) for allxnecessarily((x = x)equivalent(x ex X))
On the other hand, from (26) it follows that the necessary self-identity of x amounts to the necessary unity of x and X.
(27) for allx(necessarily(x = x) equivalent necessarily(x ex X))
Accordingly, x is contingently self-identical just in case x exemplifies X contingently:
(28) for allx(((x = x) & ¬necessarily(x = x))equivalent((x ex X) & ¬necessarily(x ex X)))
I will now show x exemplifies X contingently just in case X is accidental11 to x:
(29) for allx[(x ex X & ¬necessarily(x ex X)) equivalent (x ex X & ¬necessarily(x = x if-then x ex X))]
(x exemplifies X contingently just in case X is accidental to x.)
To establish (29) from right to left assume, contrary to (29), that X is accidental to x but x exemplifies X necessarily:
(30) [(x ex X & ¬necessarily(x = x if-then x ex X)) & necessarily(x ex X)]
(X is accidental to x but x exemplifies X necessarily.)
Now if x exemplifies X necessarily, X is essential to x.
(31) necessarily(x ex X) if-then necessarily(x = x if-then x ex X)
On the other hand, (30) says x exemplifies X necessarily, so it follows from (30) that X is essential to x.  But (30) also says X is accidental to x.  So (30) is false12 and (29)* follows from right to left.

To establish (29) from left to right assume, once again contrary to (29), that x exemplifies X contingently but X is essential to x:

(32) [(x ex X) & ¬necessarily(x ex X) & necessarily((x = x) if-then (x ex X))]
(x exemplifies X contingently but X is essential to x.)
To refute (32) I require (A-C) as premises:
(A) for allxfor allyfor someznecessarily(x ex Yif-then (z cont x & z emb Y))
(Necessarily, if an i exemplifies an h, these are one in substance.)

(B) for allxfor allyfor allznecessarily((z cont x & z cont y) if-then x = y))
(Necessarily, if i's are one in substance, they are identical.)

(C) for allx(for somey(x ex Y) if-then for someznecessarily(x ex Z))
(If an i exemplifies an h, there is some h it exemplifies necessarily.)

From (32, A, B) we have:
(33) (x = x) & ¬necessarily(x = x)13
On the other hand, from (32, A, B, C) we have:
(34) (x = x) & necessarily(x=x)14
Hence (A, B, C) refute (32) and (29)* is secured*. And so we come to Thesis IV.

Thesis IV:

The unity of an individual and a haecceity is sometimes accidental.

Some haecceities, for example, the property of being the author of W, are accidental to the individuals that exemplify them.  Thus, what-it-is-to-be the author of W is no more essential to that-which-is-the author of W than it is essential to any other such:

(35) ¬necessarily(aw = aw if-then aw ex AW)15
On the other hand, from (26, 29, 35) we have (36):  the author of W is not self-identical necessarily*.
(36) ¬necessarily(aw = aw)
So, granted that Scott and the author of W are identical, from (23, 36) it follows that Scott and the author of W are contingently identical*.
(37) (s = aw) & ¬necessarily(s = aw)
By parity of reasoning:  if, as the circumstance that the number of planets need not have been the number of planets suggests, NP belongs to np accidentally,
(38) ¬necessarily(np = np if-then np ex NP)
then, granted that the number of planets is odd, from (22)* it follows that the number of planets is contingently odd:
(39) ODD(np) & ¬necessarily(ODD(np))
Therefore, granted that nine is the number of planets and that nine is odd necessarily, it follows that these identicals are discernible in respect of necessary oddness.
(40) nine = np & necessarily(ODD(nine)) & ¬necessarily(ODD(np))
Whereupon is secured for Logic* the capstone of that Dark Doctrine which it has here been my pleasure to propound.

Thesis V:

Some identicals differ in their modal properties and relations.16

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Copyright© 1999, William J. Greenberg, all rights reserved.