Paradox, from the Greek « para », is commonly translated as “against”, and “doxa” as the true opinion. In his seminar of May 10, 1977, Lacan wonders if it would be possible to represent the paradox.
In order to go a little further on this point, let’s recall the paradox of the liar, which, in the sixth century BCE, raises the question in Epiminedes the Cretan’s phrase: “All Cretans are liars”. How could Epiminedes, in so far as he was Cretan, say that about Cretans? Was he lying then, being Cretan himself? And if he lies, does he not then tell the truth? It is undecidable in terms of logic.
The paradox deals with what is undecidable. Lacan confirms this in his seminar “The Knowledge of the Psychoanalyst” when, on the side of woman, he introduces non-existence on one slope, and the not-all on the other. Between the undecidable on the side of woman, and the contradiction that castration imprints on existence on the side of man, Lacan puts into circulation lack, fault, desire and the object a. As a consequence, he defines castration as “leaving something to be desired” and then affirms that it is because of the fact that it circulates and leaves something to be desired that we are in relation with the object a.
Now, the paradoxes of desire appear from this point: “The One dialogues all alone since it receives its own message in an inverted form”. Because the One dialogues all alone, the object a—which appears because of the circulation between the undecidable and the contradiction—is not only the object which causes desire, but is also the object of jouissance—a jouissance that is desexualized in the Freudian sense of the term in that it does not refer to the phallus.
Lacan noted this already when he constructed the fantasy in obsessional neurosis differently from that in hysteria: if in the former, the object is always metaphorized in reference to the phallus that veils it, in the latter, it is metonymized … . In the first case, the subject knows of the lack inscribed in the Other and does not want to see it in order not to be confronted with the undecidable; whereas in the second, “to try to abolish the difficulty that I designate under the name of the parasitism of the signifier in the subject”, the obsessional, if he aims at the degradation of the Other, does so in order “to restore primacy to desire”. In both cases, it is the possibility by way of what Freud called transference neurosis that can support the wager of being able do without the Other, in opening ways for the appearance of the paradoxes of desire. But in both cases it is also clear that these paradoxes can only be unveiled at the moment when we recognize that what is parasitized [paratisé] by the signifier is, in reality, a Borromean knot which articulates RSI and involves the undecidable in which desire and jouissance are linked.
Translation from Portuguese into French: Elisabete Thamer
Translation from French into English: Susan Schwartz
 « Les paradoxes sont-ils représentables ? […] Δόξα [dóxa], c’est l’opinion vraie. Il n’y a pas la moindre opinion vraie, puisqu’il y a des paradoxes ». “Are paradoxes representable? Δόξα [dóxa], is the true opinion. There is not the slightest true opinion, since there are paradoxes”.
 Lesson of June 1, 1972.
 Lesson of May 10, 1977.
 J. Lacan, Le Séminaire, livre VIII, Le transfert [The Transference], Paris, Seuil, 2001, p. 295
« Ce corps-de, est parasité par le signifiant ; car le signifiant, s’il fait partie du Réel, si c’est bien là que j’ai raison de situer le Symbolique, il faut penser à ceci, c’est que cette corps-de, nous pourrions bien n’y avoir affaire que dans le noir. Comment reconnaîtrions-nous, dans le noir, que c’est un noeud borroméen ? C’est de cela qu’il s’agit dans la Passe ». J. Lacan, Le Séminaire, livre XXIV, L’insu que sait de l’une-bévue s’aile à mourre), leçon du 15 février 1977
“This corps-de [body of] is parasited on by the signifier, for the signifier through it forms part of the Real, it is indeed there that I am right to situate the Symbolic, one must think of the following, which is that we might have dealings with the corps-de only in the dark. How could we recognize in the dark that it is the Borromean knot? That is what is at stake in the Passe”. J. Lacan, The Seminar, Book XXIV, L’insu que sait de l’une-bévue s’aile à mourre), lesson of February 15, 1977. Trans. Cormac Gallagher, unpublished.