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Lacan stresses the importance of the signifying structure in fantasy. He takes as his starting point the matheme <>a, which is the logical articulation of fantasy. The matheme was already introduced in Les formations de l'inconscient, in the graphs of desire, and was later developed in 1960 in "The subversion of the subject and the dialectic of desire in the Freudian unconscious" (Écrits: A Selection) as the first topology of the subject.
represents the division of the subject barred by the signifier that constitutes him. The sign [[Image:lacansem1b1.gif]] <> enunciates the relation either of inclusion/implication, or of exclusion between the two terms. It's a binary system where the verb as such disappears to leave room for the algebraic sign of a pure relation. Definitions of objet a will vary over the years; to understand it here, one should go back to the part object of La relation d'objet et les structures freudiennes, and then address its analysis in L'angoisse and L'objet de la psychanalyse. In 1960, however, Lacan mentions the fascination of the fantasy in which "the subject becomes the cut that makes shine in its inexpressible oscillation".
The objet a would be the primal object, forever lost, the remainder or the product, which cannot be assimilated because it is real, of the cut operated by the primal signifier engendering the subject when it repeats itself in absolute difference (L'identification). "If a is the frame of the subject, this frame falls at the level of the most fundamental act of life, the act in which the subject as such is engendered, i.e the repetition of the signifier." This is the symbolic paternal mark or the phallic mark since there is no signifier of sexual difference: "The phallus alone is the sex-unity." The objet a creates a hole constantly filled, in the partial frives, by the different objets a, the breast, feces, the penis, the gaze or the voice, objects that are in themselves caught in imaginary substitutions. To understand fantasy, one should try to determine the logical status of objet a, which can only be accomplished by way of a topology dealing with gemetrical figures. Is objet a situated on the side of the drive or of desire of which it is the cause? Is it born out of the separation from the placenta as a part of the body proper or from the division from oneself from the signifier, the cost that the speaking being has to pay to become a subject? Is there really an alternative? Lacan talks of a surface where "desire and reality" are "the right and the wrong sides"; however, the passage from one side to the other is unnoticeable, as if there were only one side, because "the relation of texture does not entail any break." Might the fantasy allow oneself to go from the drive to desire and from desire to the drive, to link them or to disjoint them?
Lacan oscillates between exaltation and bouts of anxiety: "The logic of fantasy is the most fundamental principle of any logic that deals with formalizing defiles," and at the same time defers his presentation of "alienation in terms logical calculation" because its formulation is not yet ready. The reason might have been that "truth is related to desire," which "creates difficulties for handling it like logicians do." His aim is to define "a logic that is not a logic, an entirely new logic that I have not named yet, for it needs to be instituted first." Using the character of Diotima from The Symposium, he mentions academic Penia (the lack) before psychoanalytic Poros (male resource) and wonders "up to what point, between the two, he could let the obscurity go."
