Les sexuations, le féminin et la logique intuitionniste

Abstract

Objectives

We propose a logical reflection on the feminine. In the Freudian corpus, this enigma finds various answers in which feminine desire can be interpreted from the phallic register's side. But, as a letter written to Marie Bonaparte testifies, Sigmund Freud belatedly concedes that he neglected a grey area of feminine desire, which caused him to fail to understand what women want. He calls this lack the dark continent. In this respect, Jacques Lacan took this reflexion a step further and, while considering the phallic dimension, he proposed additional theoretical elaborations such as the formulas of sexuation, which allowed him to theoretically extend Freudian ideas.

Method

In the Encore seminar, Lacan proposes the table of sexuation as an attempt to formalize the question of the difference of the sexes. This difference is questioned here at a symbolic level and through a logical approach. Lacan's formulas of sexuation thus overturn a simplistic and anatomical conception of sexual difference, by subverting a binary logic of the excluded third. In this way, these formulas meet the intuitionistic logic suggested by Luitzen Egbertus Jan Brouwer. Thus, based on a review of the Lacanian and Brouwerian literature, we question the interest of relying on intuitionism to apprehend the logic that operates on the feminine side of the formulas of sexuation.

Results

At the beginning of the 20th century, Brouwer, a Dutch mathematician, initiated his work on intuitionist logic. In 1908, in “That Logical Principles Cannot be Trusted,” Brouwer partially rejected the classical axiom of the excluded third, thereby opposing the Aristotelian foundations of logic. He refused to extend the logic of the excluded third to the treatment of infinite sets, for which, according to him, no a priori can be postulated. A double logical treatment is suggested in intuitionism: the infinite is treated logically differently than the finite, and this double treatment allows us to apprehend certain specificities of the feminine position. Lacan specifies that for feminine subjects, an undecidable relation is at work between the not-all formula and a double negation ¬∃x.¬Φx.

Discussion

The logical double treatment of intuitionism seems to illustrate the discordant logic of the not-all. The not-all subject is in a discordant relation between the phallic function Φx and S (Ⱥ). The formula S (Ⱥ) expresses a certain relation to the lack in the Other that we associate with a castration carried out on the Other. Similarly, the different interpretations (algorithmic, semantic) that have been made of intuitionism also seem to point in the direction of a discordance in this logic. We are interested in the importance of the interpretation of the Real and its consequences on the way we consider the different Lacanian dimensions (Real, Symbolic, and Imaginary). Indeed, from a psychoanalytical point of view, the sustained interpretation seems to have consequences on the way one can conceive the articulation between the dimensions and the jouissances. This reflection also allows us to question the impact of considering the temporal dimension within an interpretation.

Conclusions

Based upon the formulas of sexuation, we propose a reflection on the interest of intuitionist logic for approaching the logic of the feminine position. This position reflects a logic other than that of the excluded third in its relationship to castration. On the feminine side, Lacan puts forward an undecidable relation to castration, which leads us to question the meaning he gives to this term. In order to understand it, we have drawn on the two definitions of Verleugnung proposed by Freud, in which this negation corresponds either to the way of denying the difference between the sexes in the perverse structure, or to a singular negation of the neurosis that can be associated with the end of the cure. In both cases, the logic of these negations diverges from that of the excluded third, because in perversion, it relates to an affirmation of castration that is secondarily denied (yes but no) and in the case of the end of the cure, it is rather a concomitant consideration between affirmation and negation of symbolic castration (yes and no).