Lacan and Modality

By the time of his 19th seminar, Lacan started to associate the four places of discourse with four logical quantifications of "'the phallic function," as can be seen below:

For instance, the place of agency in discourse structure is here associated with the expression "there is one that is not subject to the phallic function" or "there is one that says 'no' to castration." Below it the place of truth is associated with the fact that "all are subject to the law of castration." In the place of production in the bottom-right corner "not all are subject to the law of castration," and above it "there is not one who says 'no' to castration." I won't go into detail on these expressions at this point.

Lacan associates the four positions further with the four traditional modal categories of the necessary, the possible, the impossible and the contingent. These are carved out through the function of "the written" (l'ecrit): the necessary marks that which "doesn't stop writing itself," the impossible that which "doesn't stop not writing itself," the contingent that which "stops not writing itself," and the possible that which "stops writing itself." The specific modalities linked to each of the four corners can be seen in Lacan's schema below:

Readers of Lacan will be most familiar with the category of the impossible as applied to the real: "the real is the impossible." This expression must be taken in a technical sense, as in for example the logical conjunction of a term with its denial. That is only one corner, however. I won't elaborate, as for now I am just laying groundwork. I only wish to note at this point that the superimposition of these four modal categories with the permutations of discourse will yield some very interesting results.