Undecidable Relation

The "Undecidable Relation" names the structural impasse that obtains between the barred subject ($) and the barred Other (Ⱥ) as they are distributed across the asymmetric sides of sexuation. On the masculine side, the subject can locate woman as the signifier of the barred Other—man constructs an imaginary figure of Woman through the procession of objet petit a objects, making her stand in for the lack at the heart of the Other. On the feminine side, however, woman neither coincides with the symbolic Other nor simply falls outside it: she is "not-all" inscribed within the phallic function, which means she has no determinate, decidable position in relation to the barred Other. This asymmetry produces a zone that is neither affirmable nor deniable within the binary logic of the symbolic—a structural undecidability that is not an epistemological failing but a formal property of the relation itself.

The undecidable relation is therefore the logical consequence of the non-relation between the two terms $·Ⱥ and $·a (the barred subject's relation to the barred Other versus the subject's relation to objet petit a). Because the feminine side generates no totalizing exception—no positive anchor point that could close and decide the set—the relation between the two sides of sexuation cannot be written, only circled. It is this undecidability that grounds Lacan's axiomatic claim that "there is no sexual relation": the two terms are real in their effects but structurally indeterminate in their conjunction, making any imaginary resolution (the fantasy that woman is the Other, that man has the Other through her) a symptomatic covering of the void.

In jacques-lacan-seminar-20-cormac-gallagher, the undecidable relation appears as a precise logical characterization of the impasse at the heart of the formulas of sexuation. It functions as a technical sharpening of several canonical concepts working in tandem: the barred Other (Ⱥ) supplies one pole of the relation—the symbolic order marked by its own foundational incompleteness; the not-all supplies the logical structure of the feminine side that prevents any totalizing decision from being made; lack names the productive void that the undecidability both marks and sustains; and objet petit a names the imaginary object-series through which the masculine subject tries—and fails—to decide the relation by constructing woman as the signifier of what is missing in the Other. The concept thus sits at the intersection of the formula of sexuation and the impossibility of the sexual relation, serving as the formal name for the gap between the two terms that no signifier can bridge.

The undecidable relation is best understood as a specification of the Imaginary's structural limits: the imaginary construction of woman (as objet a, as the signifier of Ⱥ) is the masculine attempt to decide what is structurally undecidable. The concept also extends Feminine Sexuality by giving a logical, quasi-mathematical label to what it means for woman to occupy a position "between centre and absence"—not by defining her positively but by insisting that her relation to the barred Other remains formally open, resistant to the binary closure that the symbolic order imposes on the masculine side. The undecidable relation is, in this sense, what the not-all looks like from the side of the relation rather than from the side of the set.

Seminar XX · Encore: On Feminine Sexuality, the Limits of Love and KnowledgeJacques Lacan · 1972 (p.207)

Between the two terms3».$A. and^.^k , there is the undecidable.

The quote is theoretically loaded precisely because it places "the undecidable" not as an adjective modifying a subject but as a substantive noun occupying the space between two formalized terms—$·Ⱥ (the barred subject in relation to the barred Other) and $·a (the barred subject in relation to objet petit a)—thereby inscribing undecidability as a structural position internal to the formulas of sexuation rather than as an external epistemic limitation.