A number of possibilities are excluded from Lacan’s formulae of sexuation. He provides:
Masculine = Ex~Phi(x) & AxPhi(x)
Feminine = ~Ex~Phi(x) & ~AxPhi(x)
He does not include:
Interestingly, the feminine logic has close allies. For example: ~Ex~Phi(x) corresponds closely with ~Ax~Phi(x), and ~AxPhi(x) corresponds closely with ~ExPhi(x).
What I can say is that the feminine logic corresponds with the aforementioned absent formulae only because of the result and not because of the operation. The result of ~Ex~Phi(x) is the same as ~Ax~Phi(x), but their processes are different. The result of ~AxPhi(x) and ~ExPhi(x) are the same but their processes are different. The process switches either from the particular to the universal or from the universal to the particular, but achieves the same result.
However, the masculine logic, seems to push toward an extreme on the one hand, even while remaining the same on the other hand. The best we can do is match the remaining two formulae and try to force a correspondence: AxPhi(x) would be the direct opposite of Ax~Phi(x), and Ex~Phi(x) would be the same as ExPhi(x). Thus, if we completely switch the formula, we end up with a dramatic shift such that the negative function becomes universalized, Ax~Phi(x), or the existential function itself becomes negativized with its argument positivized, ~ExPhi(x).
So, in the end, why did Lacan not choose these absent formula?
I think, in a sense, he did choose them. He chose them as absent determinations which all the more demonstrate the importance of his principal selections.