Non-identity to Itself

Non-identity to itself is a concept Leclaire (reporting within Lacan's Seminar 12) takes from the logic of the Russell–Frege tradition, where the concept of "non-identity to itself" — the class of all things that are not identical to themselves — is assigned the extension zero (the empty set), precisely because nothing can coherently fail to be identical to itself. That assignment of zero is what Leclaire calls "suture": the moment logical discourse closes itself, plugging the void it had opened with the empty signifier, and so producing a coherent, self-consistent system. The concept of non-identity to itself thus names the logical crack or gap in discourse — a place where something threatens to fall out of identity, out of self-coincidence — which formal logic immediately neutralizes by declaring its extension null.

In the Lacanian frame, the theoretical weight of the concept is transferred from logic to the analytics of the subject and the analyst. The subject, constitutively split by the signifier ($), is precisely a being that does not coincide with itself; it is the living instance of non-identity to itself — never fully present in any signifier that represents it, always falling between signifiers. The analyst, in Leclaire's argument, is one who does not suture this gap: where logical discourse assigns zero and closes, the analyst holds open the space of radical difference — sexual difference, castration, death — refusing to occupy any fixed place in the symbolic topology. Non-identity to itself is therefore the logical correlate of the split subject, the formal index of what suture covers over when discourse constitutes itself as complete.

The concept appears once, in jacques-lacan-seminar-12 (p. 259), within Leclaire's intervention on the logic of suture and the analyst's distinctive position. It sits at the intersection of several cross-referenced canonical concepts. Most directly, it is the logical underside of the Splitting of the Subject: the split subject ($) is precisely the being who fails to coincide with itself, who is non-identical to itself in the very movement of being represented by a signifier. Where logic forecloses this non-identity by assigning it zero, the subject of the unconscious inhabits it. The concept also articulates with Castration: castration is the structural operation by which the speaking being loses self-coincident fullness on entry into the symbolic — non-identity to itself is, in formal-logical terms, the notation of that constitutive loss. The analyst's refusal to suture connects further to Desire and Objet petit a: the analyst occupies the position of the cause of desire (a), a void that must not be filled; to suture would be to replace that void with a positive term, destroying the transferential and clinical space. The reference to Death and Castration as what the analyst remains open to echoes the Death Drive — the insistence of a repetitive gap that no symbolic closure fully masters. Finally, the contrast with Psychosis is implicit: where psychosis involves the failure of the Name-of-the-Father to quilt the signifying chain, suture is logical discourse's own quilting operation — non-identity to itself names what both logical and psychotic discourse handle in structurally opposite but analogously symptomatic ways.

Seminar XII · Crucial Problems for PsychoanalysisJacques Lacan · 1964 (p.259)

It is in the decisive statement that the number assigned to the concept of non-identity to itself is zero that logical discourse is sutured.

The phrase "decisive statement" marks the suture as an act — a performative intervention in logical discourse rather than a neutral description — while "the concept of non-identity to itself is zero" names the precise maneuver: assigning an empty extension to the structurally destabilizing concept so that discourse can close around it. The juxtaposition of "non-identity to itself" (radical self-difference, the formal echo of the split subject) with "zero" (foreclosure, nullification, suture) concentrates the entire argument about what analytic practice must refuse to do.