As a value for a variable, logicians posit a particular whose constitution is irrelevant to its logical relations:  nothing about this particular circumscribes its relation to itself and its properties.  No matter how complex it may otherwise be, the logicians' posit is LOGICALLY SIMPLE: a 'that' but not a 'what', an existence but not a content, an individual but not a universal.

Into such a particular, change can only be introduced from without--as something extrinsic, involving the properties of the particular but not the particular itself.  To such change, the particular is superfluous. The particular thus disappears into the assemblage of its properties. Therewith, permanence without change passes into change without permanence. For the minutest change, in a particular that is the assemblage of its properties, suffices to make the particular other than what it is. Along with the necessity of its identity, the logicians' particular thus encloses within itself twin impossible worlds: a world of things without change, and a world of change without things. To merge these into one, a world of things that change, it is necessary to reject the logicians' particular for one fashioned after things as they are.

Such a particular will be ontologically differentiated and logically complex, with essential qualities and relations.  Its nature will be to HAVE a nature, to be a 'that' and a 'what', an existence and a content, an individual and a universal.  Two-sided and multi-faceted, this particular will embrace the interconnection of 'that' and 'what' which enters into the identity of every existing thing.

In relating such a particular to itself, the identity-relation will relate the particular to each of its aspects, and each of its aspects to the others. The identity-relation will thus relate a particular to both a 'that' and a 'what', for a particular IS both a 'that' and a 'what'; and so will relate a 'that' to a 'that' and a 'what', and a 'what' to a 'that' and a 'what'. 

The relation in which such a particular stands to itself will thus posit the unity of the 'that' and 'what' which constitute it. But this unity will not present itself en bloc. It will present itself instead as the sum total of manifold distinct unities, each encompassing a different facet of the relation in which a particular stands to what it is.

Two facets of one such relation are the relation of Scott to Scott, and the author of Waverley to the author of Waverley.  Upon analysis, the relation of Scott to Scott thus discloses the unity of what-it-is-to-be Scott and that-which-is Scott, and this unity is essential to Scott. If what-it-is-to-be Scott were to be extruded from that-which-is Scott, Scott would surely not perdure. The relation of Scott to Scott is thus a logically necessary one, for whatever else he might have been, Scott cannot but have been Scott. 

In contrast: the unity of what-it-is-to-be the author of W and that-which-is the author of W is not essential to the author of W. Had he never come to be the author of W, the author of W would have remained, although he would not then have BEEN the author of W. The relation of the author of W to the author of W is thus not a logically necessary one; unlike Scott, the author of W is contingently self-identical.²

Against the notion that identity is a relation that holds necessarily between a logically simple particular and itself, I am thus propounding a theory of contingent identity based on five Theses about structured particulars and the identity relation these support:

Thesis I:

Every particular is a this-what.

Concerning what it is to be real, F. H. Bradley writes:

If we take up anything considered real, no matter what it is, we find in it two aspects.  There are always two things we can say about it; and if we cannot say both we have not got reality.  There is a ‘what' and a ‘that', an existence and a content, and the two are inseparable.  That anything should be, and should yet be nothing in particular, or that a quality should qualify and not give a character to anything is obviously impossible.(Appearance and Reality, p. 162. Cited in Murphy's ‘Substance and Substantive,' UC Publications in  Philosophy 9 (1927), pp. 63-87).

Thesis II:

The self-identity of a this-what posits a series of new relations internal to the whole.

Modulo the interchangeability of x and x.X, (1) underwrites (2), so that in relating x to itself, identity relates x to each of its sides.

(1) x = xx = x

(2) x = xx = x.X

(3) (x = x.X)(R₁(x, x) & R₂(x, X))

Let R₁ and R₂ hold when their respective terms are one in substance. Call R₁ 'containment' and R₂ 'embodiment'. Transcribe (3) as (4).

(4) (x = x.X)(x cont x & x emb X)

(5) (x cont x)   (R₃(x, X) & R₄(x, x))

(6) (x emb X)  (R₃(x, X) & R₄(X, X))

Let R₃ and R₄ hold when their respective terms are one in substance. Call R₃ 'exemplification'. Call R₄ 'identity'.  Transcribe (5, 6) as (7, 8).

(7) (x cont x)  ((x ex X) & (x = x))

(8) (x emb X)  ((x ex X) & (X = X))
 

The self-identity of a this-what amounts to the unity of the individual and haecceity which constitute it.

(2,4,7,8) prove (9), and (9) proves (10)*.

(9) (x = x)((x ex X) & (x = x) & (X = X))

(10) (x = x)  (x ex X) 

Exemplification relates individuals to haecceities--and identity, individuals to individuals³--when these are one in substance:⁴

(11) xy(x ex Yz(z cont x & z emb Y))

(12) xy(x = yz(z cont x & z cont y))

(13) (x = x)z(z cont x & z cont x)

(14) z(z cont x & z emb X)

(15) z(z cont x & z cont x))

(16) x = x

On the other hand, haecceities are identical when there is some particular to which these belong:

  (17) xy(X = Yz(z emb X & z emb Y))

To sum up so far: the foregoing analysis of self-identity--to cop a phrase from then neo-Hegelian Bertrand Russell's The Foundations of Geometry--is one which:

[...] n'est ni purement synthétique ni purement analytique; car il est d'un côté la détermination plus complète d'un certain tout, et donc de ce point de vue analytique, et d'autre côté il implique l'émergence de nouvelles relations à l'intérieur du tout, et de ce point de vue il est synthétique.

[...] is neither purely synthetic nor purely analytic; for on the one hand, it is the most complete determination of a certain whole, and from this viewpoint, analytic--but on the other hand it implies the emergence of new relations internal to the whole, and from this viewpoint is synthetic.  (From a book by Pierre Jacob, for which I no longer have a reference.)