Has anybody attempted to rewrite the formula of metaphoric substitution to account for psychotic substitution? How does the formula change for an “un-triggered” and/or “triggered” psychosis? In “On a Question…,” Lacan writes two versions. In the first, you have the generic version of metaphoric substitution which moves from the signifier, to the expression of the barred signifier through a signified. This formula is expressed in the following way:

S/$ * $’/x -> S(1/s)

The S’s, we are told, are signifiers. The $’ is the signifier after it has undergone a new meaning by way of the x, which is the unknown signification. The result is that the signifier expressed a new signified, s.

Okay, next, Lacan provides the formula for ordinary neurosis (I choose the title “ordinary” neurosis):

Ndp/Md * ~md/Sd(x) -> Ndp(1/Phi)

So, in this formula the name of the father stands in place of the mother’s desire (md), thereby effacing the desire to make way for an unknown new signifier for that desire, x. The result is that the name of the father is a function through which the phallus comes to dominate the mental life of the individual.

Okay, very well.

But next Lacan does something very interesting, and it still holds up for contemporary Lacanian thought – he insists that the formula is changed because the name of the father (ndp) becomes replaced by a hole.

Perhaps this formula would read as follows:

Zero/MD * MD/MD -> zero (Semblant)

The non-effacement of the mother’s desire [MD/MD rather than ~MD/s(x)] occurs which produces the possibility of a system of semblants. This, then, might be the formula for pure psychosis, if such a thing exists.

However, the ordinary psychotic – which is, only by degree, relatively stable whether problematically or not – may be written as follows:

Zero/MD * MD/MD * Semblant/MD * ~MD/Sinthome -> …

This implies that the name of the father does not efface the mother’s desire, because it is zero, so that, to compensate, a semblant provides the latter corrective. However, what it produces is not a signifier but rather an unknown symptom, a sinthome. I’m not sure what this would mean for the other side of the formula, and this is why I’ve left an ellipses.

The semblant represents the sinthome, which, anyway, is already negative (whereas the phallus requires an inversion of 1/Phi so that it becomes imaginary). Finally, then, it must be:

Zero/MD * MD/MD * Semblant/MD * ~MD/Sinthome -> Semblant(Sinthome)

I want to highlight a part of this formula, the part that I am now placing in square brackets is the compensatory function of the metaphor – what some have referred to as the delusional metaphor:

Zero/MD * MD/MD * [Semblant/MD * ~MD/Sinthome] -> Semblant(Sinthome)

You can see that I have really only redoubled the original metaphor formula. It is precisely the same, in the end. This means that the means by which a psychosis becomes ordinary is precisely the same means by which neurosis is grounded. The redoubling only serves to emphasize that the semblant(sinthome) bond is fragile whereas the bond of the signifier to the signified [S(1/s)] is more tightly bonded.

This demonstrates the inadequacy of the formula. What we require instead is a topology or a knotting. For example, if you demonstrate the way in which neurosis begins in much the same way as ordinary psychosis becomes stabilized then you end up missing something essential. The result is that an ordinary neurosis would be a variation on this formula (rather than psychosis being a variation on the falling of a neurosis) written as follows:

Zero/MD * MD/MD * NDP-as-semblant/MD * ~MD/Signifier(x) -> NDP-as-semblant(sinthome(1/Phi)).

What you miss is the topology of the zero as compared with the topology of the NDP. This is why the return to Frege that I’ve drawn in my paper (not included here) is essential.

Please feel free to interrogate and correct.