(a) The truths of classical identity theory are here engendered by an entity that appropriates for logic the fundamental marks of particularity.

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 not qualify and give a character to anything is obviously impossible. (F.H. Bradley, Appearance and Reality, p. 162, cited in Murphy's 'Substance and Substantive,' U.C. Publications in Philosophy, Vol. 9, 1927, pp. 63-87)

But these correlative features of particularity are irretrievably lost to ontology-free thematizations of the identity-relation. For the point of departure for such thematizations is the supposition that the entities for which individual variables go proxy are devoid of logically relevant structure.²⁴

(b) From individuals and haecceities can be constructed a classical complex, which demonstrably satisfies the usual axioms for identity, including 'x(x = x)', and a non-classical complex, which satisfies the other axioms, together with a weak reflexivity principle²⁵ and '¬x(x = x)'.  But the validity of RLE depends on its applicability to classical and non-classical objects. So RLE is not valid.

For 'bA'--where b is an individual variable of quantification theory--to be valid, 'A' must come out true under every interpretation.²⁶ Consequently,

(1)'x(x = x)' is valid iff 'x = x' is valid.

For A to be satisfiable, A must come out true under at least one interpretation. That is, 'x = x'is valid iff '¬(x = x)' is not satisfiable.

So 'x(x = x)' is valid iff '¬(x = x)' is not satisfiable. But 'A' issatisfiable iff for some domain D, 'A' is satisfiable in D.  Moreover, for any domain D, 'A' is satisfiable in D iff there is a true interpretation of 'A' wherein D is the Universe of Discourse.

Hence, 'x(x = x)' is valid iff for no domain D is there a true interpretation of '¬(x = x)'wherein D is the Universe of Discourse.  But the entities which realize, P-, P-+ and P-- constitute just such domains; and with respect to these domains there is a true interpretation of ¬(x = x). 

So the vaunted "soundness" of predicate logic with identity goes by the board. For the contradictory of RLE is satisfiable by means of a construction from individuals and haecceities, of whose mathematical existence there can be little doubt.²⁷

(c) Set-theoretic restrictions on the interpretation of variables exclude objects which do not satisfy RLE, among them non-self-identical particulars. However, these objects are not only consistently thinkable, but they also play--as I show in sequels to this paper²⁸--an indispensable role in the unification of identity theory. Structured individuals thus challenge set-theoretic restrictions on the interpretation of variables which have held semantics and logic in thrall for half a hundred years.²⁹