In Lacan’s formulae of sexuation, the masculine side and the feminine side each include two (contradictory) propositions, which are equivalent to one another, in terms of their truth-values: The masculine “all x are f(x)” is equivalent to the feminine “no x is not f(x)”, and the masculine “an x is not f(x)” is equivalent to the feminine “not all x are f(x)”. According to Žižek, this shows the indifference between the sexes:
“there is no sexual relationship” means that there is no direct relationship between the left (masculine) and the right (feminine) side, not even that of contrariness or contradiction; the two sides, set side by side, are equivalent, which means they just coexist in a non-relationship of indifference. (Less Than Nothing, page 760)
Consider another logical expression, that of implication: P ⇒ Q. In terms of truth-values, it is equivalent to say ~P ∨ Q. Both formulas express the same logical implication, but with different logical pathways. P ⇒ Q says “Q must be true whenever P is true”, whereas ~P ∨ Q says “Either P must be false or Q must be true” . Let’s compare these formulas.
To compare the two versions of the logical implication, consider an implication that is well-known in Lacanian psychoanalysis: The Master-Signifier as an exception grounds the universality of knowledge. In other words, S1 ⇒ S2. In terms of truth-values, it is equivalent to say ~S1 ∨ S2. Let’s interpret these two equivalent formulas.
It’s easy to interpret the first formula, S1 ⇒ S2, which is reminiscent of the Master’s discourse. It is the direct and explicit formulation of an implication. Logically, it says “S2 must be true whenever S1 is true”. We can simply interpret it as the grounding of a symbolic chain of signifiers on the exception of a Master-Signifier.
The second formula, ~S1 ∨ S2 is more interesting because it’s an indirect and implicit formulation of the same implication. Logically, it says “Either S1 must be false or S2 must be true”. In this formula, there appear two choices “~S1 or S2?”. How to interpret these two choices?
Remember that the signifier S1 represents the subject $ for the other signifier S2. Thus S1 comes to replace $. Therefore, what negates S1 must be the subject $ itself. In other words, ~S1 signifies the truth of S1, that is, the subject $.
We can now rewrite the second formula as $ ∨ S2. Now it says “Either the subject must be true or the chain of signifiers must be true”. In other words, it’s now the choice between being and thinking. Either “the subject is”, or “it thinks”. If the subject is, then the Master-Signifier is negated and the chain of signifiers is no longer implied. Otherwise, the subject is represented by the Master-Signifier, and the chain of signifiers is implied by the formula.
Of course, the choice between being and thinking is a forced choice: negating the Master-Signifier is the wrong choice that invalidates itself. The subject $ can properly exist only by getting represented by S1. Otherwise, it can only ex-sist in the Real, which amounts to a symbolic non-existence. In this way, the first choice ~S1 can function as a threat.
The formula $ ∨ S2 can also be interpreted as the choice between alienation and separation. Either the subject gets represented and alienated in the chain of signifiers, or it ex-sists in the Real and gets separated from the symbolic order.
The second implication formula is more interesting than the first implication formula, but they are two sides of the same coin.
I would like to call these two formulas, the masculine logical implication (S1 ⇒ S2) and the feminine logical implication (~S1 ∨ S2). Just like the formulae of sexuation, the two formulas are equivalent in terms of their truth-values, “which means they just coexist in a non-relationship of indifference.” This is my little addition to the formulae of sexuation.
The main difference between the two formulas is that the chain of signifiers S2 appears as a requirement/necessity in the masculine implication where S1 explicitly requires S2. But in the feminine implication, it appears as a choice, albeit a forced choice: it’s either $ or S2.
 In this text I use the expression “either … or …” in the sense of “or”, not in the sense of “xor”.