# Sexuation, Formulas of

# SEXUATION, FORMULAS OF

According to Jacques Lacan, *sexuation*, as distinct from biological sexuality, designates the way in which the subject is inscribed in the difference between the sexes, specifically in terms of the unconscious and castration, that is, as "inhabiting language" (Lacan, 1998, p. 80).

Lacan presented the complete table of the formulas of sexuation on March 13, 1973, during one of the lectures of his seminar *Encore* (1972-1973), but as early as 1971 he began to use his own symbols for the logical quantifiers and the function Φx (figure 1).

Lacan's choice of the term sexuation, and not sexuality, indicates that being recognized as a man or woman is a matter of the signifier.

The phallus is situated as a symbol, the signifier of castration and thus also of desire. The Law that is transmitted by the father and that states the prohibition against incest is also the foundation of desire. And this is the Law of castration, which Lacan designated in his graph as the phallic function, Φx.

To construct these formulas, Lacan relied on the Aristotelian logic according to which propositions are categorized in four classes: the universal affirmative, the universal negative, the particular affirmative, and the particular negative. But Lacan adopted modern symbols for these categories, which are based on the universal quantifier, ∀, and the existential quantifier, ∃.

On the left side of the table, there appears the formula ∀xΦx, for all x Φ of x (all men are submitted to the phallic function, that is, castration). But modern logic has demonstrated the necessity of a particular negative, ∃xΦx (there exists at least one that is not submitted to the phallic function), in order to found the universal affirmative. This is the hypothesis that Sigmund Freud developed in his myth of the primal father in *Totem and Taboo* (1912-1913a), and also in his argument that Moses was not a Jew in *Moses and Monotheism* (1939a): there always exists one who is an exception. This is how man is inscribed: by the phallic function, but on the condition that this function "is limited due to the existence of an x by which the function Φx is negated" (Lacan, p. 79). This is the function of the father.

The other side of the table concerns the "woman portion of speaking beings" (p. 80). The upper line is read as follows: there does not exist any x that does not fall under the phallic function. In other words, castration functions for all women. But on the lower line Lacan introduced a negation marked by the barring of the universal quantifier, which is quite inconceivable from the perspective of formal logic. Lacan proposed that it be read as "not-whole."

The woman's side of the table "will not allow for any universality" (p. 80). Woman is not wholly within the phallic function. On this side there is no exception that could serve as the basis for a set of women. It is from this fact that Lacan derived the formula, "Woman doesn't exist." This formula leaves no room for any idea of an "essence" of femininity.

Below the table of formulas, there is a "scanded indication of what is in question" (p. 80). On the masculine side, there is the barred subject "and the F that props him up as signifier" (p. 80). For the male is only able to reach his partner, the Other, through castration and the mediation of the object *a* as its effect. This is indicated by the arrow that crosses from the male side to the female side, which also reproduces the Lacanian formula of fantasy. On the feminine side, woman is doubled: she has a relation with F, insofar as a man incarnates it for her. But she is not wholly in that relation. She also has a relation to the signifier of A, the signifier that the Other would need if a set of women were going to be formed. Woman's jouissance is thus divided between phallic jouissance, linked to castration and appearing on the graph as F, and an Other jouissance that is unique to her. Thus there is neither symmetry between the two sides of the table, nor any complementarity between the sexes.

Alan Vanier

*See also:* Graph of Desire; Jouissance (Lacan); Matheme; Phallus; Sex differences.

## Bibliography

Lacan, Jacques. (1970-1971). *Le séminaire-livre XVIII, d* '*un discours qui ne serait pas du semblant*. [On a discourse that might not be a semblance] (unpublished seminar).

——. (1971-1972). *Le séminaire-livre XIX, . . . ou pire [. . . or worse]*.(unpublished seminar).

——. (1998). *The seminar of Jacques Lacan, book XX, on feminine sexuality: The limits of love and knowledge, encore*. (Bruce Fink, Trans.) New York: Norton. (Original work published 1972-1973)

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