Lacan’s formulae of sexuation are well-known to Žižekians. Žižek frequently refers to the exception that grounds the universal and the logic of non-All where this does not work. But I think the formulae still remain quite enigmatic. Let me resolve this enigma through computer science.
A signifier represents the subject to another signifier. In fact, the subject is represented by a unique signifier (called Master-Signifier or S1) to the rest of the signifiers (called S2). This representation by the S1 happens at a concluding moment that retroactively determines the preceding chain of signifiers.
Let’s describe the process: At first, there is a need to represent the subject for a signifier Sa, then this need to represent is displaced to Sb, then it’s displaced to Sc, then to Sd… etc. At some point, the subject is actually represented by S1 for the preceding chain of signifiers (Sa, Sb, … etc.), and at that moment, we call the preceding chain of signifiers S2. Thus, there are two logical times in the process: (1) The time when a signifier needs to represent the subject to another signifier. This signifier is repeatedly displaced, until finally it’s (2) the time when the representing signifier S1 is found, and it represents the subject to S2. Therefore, S1 stops the continuous displacement of signifiers.
This process resembles what we call a recursive function in computer science. A recursive function has two parts: (1) If the termination condition is not yet met, the function calls itself. (2) If the termination condition is met, the function returns without calling itself further. An example is factorial(x). In the first part, factorial(x) is defined as x*factorial(x-1) for a positive integer x. In the second part, for the condition that x becomes zero, it’s defined as factorial(0) = 1. Here’s how its code looks like:,
If we express signification as a recursive function, what would the termination condition be? Imagine that we have a function called “represent” and that we call “represent(x)” for an x to be represented. When should the function call itself, and when shouldn’t it?
If x is inside language, represent(x) should call itself: in language, every representation refers to another. If x is outside language, represent(x) shouldn’t call itself: x here becomes the exception to the language. Therefore the termination condition of representation is that x gets out of the field of language, that is, it becomes an exception.
As we know, to be inside language is expressed by the phallic function Φ(x) in Lacan’s formulae of sexuation. We also know that when represent(x) terminates, it should return S1. We can write the function represent(x) based on these ideas.
As you can see in the code, represent(x) calls itself when Φ(x) is true, and returns S1 when Φ(x) is false. Each time it calls itself, it displaces x, until x gets out of language and Φ(x) becomes false, when it returns S1.
Now we can clearly formulate Lacan’s formulae of sexuation in terms of this function.
In the masculine side, “Some x are not Φ(x)” means that the representation function actually terminates because x gets out of language after its displacements. “All x are Φ(x)” means that all of the x before the exception that terminates the function were Φ(x).
In the feminine side, “There’s no x that’s not Φ(x)” means that x never gets out of language in its displacements, and that the representation function never terminates. “Non-all x are Φ(x)” means that x continues to get displaced and it can never be completed and totalized, because the termination condition is never met. The displacements go on and on, and we call it the ex-sistence of the woman.