Jacques Lacan's Sexuation Formulas
(in progress)
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(the One)
(the Other)
In Le Savoir du Psychanalyste, Lacan glosses this matheme as "There exists an x determined by its saying no to the function of castration." This x corresponds to the necessary, to "that which does not cease to write itself." In this respect, it is equivalent to the symptom, or Sinthome. It is a lackless real that never ceases to repeat itself qua impossible.
Lacan equates this at-least-one (au-moins-un) who says no to castration with the mythic father-jouisseur of Totem and Taboo. He emphasizes here that from the start "the question of existence" is "tied to something of which we cannot misrecognize that it is a saying (dire)," yet reminds us that in the myth of Abraham this father is sacrificed in the form of a ram (Gilson 167). Lacan adds that, in Judaic tradition, "as in all human lines that respect themselves, its mythic ancestry (descendance) is animal" (7/1/72). In effect, for this exception to fulfil its totemic function, it must be something non-human; it must not be a speaking subject, which by definition would be constitutively divided--castrated--by the signifier by the signifier--reduced to what is represented by a signifier for another signifier.
But for this exceptional existence, or ex-sistence, to "be something other than a myth," we must conceive of it terms of a structural logic in which it serves as "the inclusive function: . . . this existence plays the role . . . of the complement, or to speak more mathematically, of the edge" to the contradictory universality of the possible. In relation to the impossible, this exception is equivalent to the empty set (7/1/72).

1. Pierre Skriabine, "Clinique et Topologie," La Cause freudienne #23, 1993, p. 124. This article is now available in an English translation by Ellie Ragland and Veronica Voruz titled "Clinic and Topology: The Flaw in the Universe," in Lacan: Topologically Speaking, eds. Ellie Ragland and Dragan Milovanovic (New York: Other Press, 2004), pp. 73-97. Back.

2. Jean-Paul Gilson, La Topologie de Lacan: Une articulation de la cure psychanalytique (Montréal: Les Editions Balzac, 1994). Back.

"All are subject to the law of castration." This all corresponds to the possible, to "that which ceases to write itself."
"No x exists which is determined as subject by the saying-no (dire-non) to the phallic function" (Gilson 167). This "no x" corresponds to the impossible, to "that which does not cease to not write itself." savoir
"Not all are subject to the law of castration." This not-all corresponds to the contingent, to "that which ceases to not write itself."
The divided subject (subject of lack).
The Phallus.
The object a. Pierre Scriabine describes the object a as both "agalma and refuse (déchet)." It is "what, in the fantasy, sutures the subject's lack in a fallacious completeness that misrecognizes its division." It "is also what splits the subject, causing it, beyond the fantasy." But still, "as a correlate of the failure of the Other, [the object a] is the logical consistency that completes the inconsistency of the Other" (1).
In his Seminar at Barcelona, Jacques-Alain Miller states, "The object a is only the elaborated part of jouissance, it is the fantasmatic or semantic part of jouissance, the part of jouissance already drawn into the fantasy . . . Object a is a false real." In his later teachings, Lacan situates it as a point at the center of the Borromean knot, as in this diagram. This suggests that the objet a partakes of all three of the orders knotted together by the symptom (the real, the symbolic, and the imaginary) in an extimate relation to sense and to the two jouissances (phallic jouissance and the jouissance of the barred Other).
The signifier of the barred Other.
"The woman does not exist."