De Morgan's Formula

De Morgan's Formula, as deployed in Seminar 14, is Lacan's appropriation of a classical logical law — the equivalence between the negation of an intersection and the union of the negations — as a formal instrument for articulating the structure of alienation. In set-theoretic terms: ¬(A ∩ B) = ¬A ∪ ¬B. Lacan seizes on this formula not as a piece of mathematical curiosity but as a precise description of the forced choice that governs the subject's relation to the two fields of thinking and being. If the intersection of "thinking" and "being" is impossible (that is, the subject cannot simultaneously occupy full being and full meaning), then what is available is only the union of their respective negations — non-thinking or non-being. This is the exact logical skeleton of the vel of alienation: the subject can be (without thinking) or can think (without being), but cannot have both. De Morgan's Formula thereby formalizes the asymmetric disjunction at the heart of the cogito and extends it toward Freud: the Id and the unconscious are readable as the "negation of A and negation of B" that the union makes visible once the impossible intersection has been negated.

The formula also bears on the empty set as the logical place of the stating subject. If the intersection of the two propositions — "I am thinking" and "I am" — yields nothing (is empty), then De Morgan's equivalence distributes this nullity across both terms in their negated forms. The empty set is not a failure of logic but its proper result: it stands in for the subject who enunciates but cannot be pinned down within what is enunciated. Freud's discoveries are thus not a departure from Descartes but an inhabitation of the gap Descartes himself refused to close — the gap that De Morgan's Formula renders with algebraic precision.

This concept appears once, in jacques-lacan-seminar-14-1 (p. 67), situated within Lacan's sustained re-reading of the Cartesian cogito as a structural scene of alienation rather than a founding certainty. Its immediate neighbors in that argument are the Empty Set as Stating Subject — which occupies the place vacated by the impossible intersection — and the vel of alienation that De Morgan's Formula logically underwrites. As an extension of the canonical concept of Alienation, De Morgan's Formula provides the set-theoretic backbone for what the definition of Alienation describes as the "forced choice structured by set-theoretic joining (union)": the formula makes explicit precisely why the union of negations is what survives once the intersection is cancelled. It thus translates the existential drama of Alienation into a manipulable logical notation.

In relation to Negation and Logical Time, De Morgan's Formula introduces a second-order negation — not merely the negation of each term but the negation of their conjunction — that maps onto Lacan's account of how the unconscious is produced as the non-thinkable remainder of a subject who cannot be. Its link to Desire is more medial: by formalizing the constitutive impossibility of full presence (being + thinking), the formula lays the ground for the lack that is desire's condition of possibility. The Duality Principle named as a cross-reference is itself a meta-logical counterpart to De Morgan's law (which is one of its canonical instances), suggesting that Lacan treats De Morgan not as an isolated trick but as a specimen of a broader logical symmetry that structures the subject's field.

Seminar XIV · The Logic of Phantasy (alt. translation)Jacques Lacan · 1966 (p.67)

de Morgan's statement is expressed as follows: that in the set formed by these two fields here covered by the two propositions in question, the negation of intersection … is represented by the union of the negation of A … and the negation of B

The quote is theoretically loaded because it places "negation of intersection" — the impossibility of the subject holding together thinking and being simultaneously — in direct equivalence with "the union of the negation of A and the negation of B," which formally maps the only livable options onto negated, partial terms; this is the logical skeleton of the vel of alienation, where both available positions are positions of loss rather than plenitude.