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