Thursday, December 25, 2014

Sexuation & Ideology: A Formal Analysis

A paper I wrote for Prof. Jameson. I think it's pretty good and hope to pursue some of the themes in the future--the formation of universals is incredibly interesting. Think I'll start soon on Badiou's Saint Paul.

Sexuation & Ideology: A Formal Analysis

I. Introduction & Overview
In Seminar XX, Lecture VII, Lacan draws a curious diagram on the board and writes immediately thereafter: “After what I just put on the board, you may think you know everything. Don't” (78). In the top half of this diagram are four propositions arranged into two sets of two—these are the formulas of sexuation, about which no one knows everything. These formulas, already presented with the symbols of formal logic, will in the course of this paper be further formalized. My primary purpose is to trace the relations between the formulas of sexuation and propose a model of their interaction.
Straight away, we must clear up a few things about the scope and aim of my arguments. First, the universals I am referring to are not typical universals like “greenness” or “virtue”, but rather apply to everything or are unbounded. They are not only universal but of universal scope. Second, I pay great attention to formality, to mathemes, because as Zupančič said in her talk “Is the Sexual Political?”, the point is not to positivize sexual difference but to formalize it. Third, there is some question about how fundamental an ideology is versus ideology in general, and how both of these relate to things consciously present to the mind such as philosophical positions. For the purpose of this paper, I will not differentiate too strongly between these three, and I think all of them fit nicely, with only relatively minor differences. This is thanks to the abstractness of the formulas.
I will proceed as follows: first, I will consider the formulas of sexuation from the viewpoint of formal logic, exploring the differences between the masculine and feminine sides; second, I will propose a model of the relation between these formulas and the ideological formation or logical derivation of universals; third, I will deal with some common misreadings of the formulas and clarify some aspects of the phallic function and the fundamental antagonism of sexual difference; fourth, I will present a case study of Meillassoux's primary argument in After Finitude, namely the proposition that everything is contingent except the contingency itself, which is absolutely necessary; finally, I will give a brief conclusion about the antagonism of the formulas and the possibility of going beyond the Phallus.

II. Formulas of Sexuation: Formal Considerations
In this section, I give an overview of Lacan's formulas of sexuation from the standpoint of formal logic, exploring the various contradictions and translations or derivations that can occur between the formulas. I also answer why there are two sides, the masculine and the feminine, and why they are incommensurable while being seemingly formally equivalent.
The masculine formula, in its general form, is as follows: everything is Φ, with the exception of (at least) one thing. We can write this first clause formally as ∀xΦx, “for all x, x is Φ”, where Φ is some function (traditionally the phallic function); likewise, the exception may be formalized as ∃x~Φx, “for some x, x is not Φ”. The feminine formula is the following: not-All is Φ. Formally, this breaks into two propositions: ~∀xΦx, “not all x is Φ”, or perhaps rather “not-All x is Φ”; and ~∃x~Φx, “there is nothing which is not Φ”, in other words “there is no exception to Φ”. In what follows, I will present an abstract, general model of the interrelations of the two formulas and their constitutive propositions before applying it to the problem of the formation of universals.
Now, the universal masculine proposition, ∀xΦx, is in contradiction with the feminine universal proposition, ~∀xΦx; likewise, the masculine existential proposition ∃x~Φx is in contradiction with the feminine existential proposition, ~∃x~Φx. So here we see the complementary/contradictory nature of the sexes expressed in formal terms. These are of course logical contradictions, and this is perhaps to be expected. To avoid triviality, we must therefore deny the principle of explosion, the principle that from a contradiction anything can be derived, and we assume for this task a paraconsistent logic minimally different from classical logic. But this is not far enough. It is not only that we must acknowledge the contradictions between the masculine and feminine formulas—we must also realize the contradictions inherent to each formula.
To do this, we must use the standard logical rules for quantifier negation (henceforth QN) and double negation (DN): from ∀xΦx we can derive ~~∀xΦx by DN, and from there we derive ~∃x~Φx by QN; similarly, from ~∀xΦx we can directly derive ∃x~Φx by QN. So the universal side of the masculine formula leads to ~∃x~Φx, which is the negation of the existential side of the masculine formula, ∃x~Φx, and thus a contradiction. Likewise, one side of the feminine formula, ~∀xΦx, leads to ∃x~Φx, which is the negation of the other side, ~∃x~Φx, which is another contradiction. To summarize, each proposition is in internal contradiction with the other of its sex and with one other of the opposite gender, namely the one with the same quantifier. For example, the masculine universal proposition is in contradiction with the feminine universal proposition, but not with the feminine existential proposition, with which it is formally equivalent. We must now take a closer look at this seemingly innocuous non-contradiction, or more precisely, equivalence: What keeps the two formulas from collapsing into one another, since formally speaking they are equivalent? In other words, what is the real difference between them, if it is seemingly not to be found on the purely formal level?
I claim that there are actually three (hidden) formal differences in these formulas. First, there is the banal point that to show the equivalence of any two propositions requires the passage from one to the other, so the move from masculine to feminine in terms of quantifiers is precisely the opposite of the move from feminine to masculine. Put another way, the two formulas express the same things but in different ways, and that difference hinges on a difference in quantifiers used. Second, each quantifier on the masculine side is positive, having no negations before it, whereas each feminine quantifier is already negated. For example, the masculine must add two negations in the passage from ∀xΦx to ~∃x~Φx, whereas the feminine must remove two negations while making the opposite move. The feminine existential proposition has negation spontaneously present, whereas the masculine must posit that negation. Third, both QN and DN are needed to derive ~∃x~Φx from ∀xΦx and vice versa, but only QN is needed to derive ∃x~Φx from ~∀xΦx and vice versa. The former thus requires two separate steps: the positing of the equivalence of ∀xΦx and its double negation ~~∀xΦx by DN, and the application of QN. In sum, really there are three possible derivational asymmetries: the quantifier to be derived; the relationship to negation; and the order of application of DN and QN. These delineate the formal differences between the two formulas, that which keeps them separate, though “really” there be no difference between them. Nevertheless, there is a certain labor, particular to each formula, involved in passing from one to the other. Now each of these asymmetries must be interpreted and their existence and relevance justified.
The first asymmetry is relatively straightforward: both formulas express the same things in immediately different ways, this difference being due primarily to the quantifiers used. If we are metaphysical realists about universals, which both Lacan and Žižek are (Žižek, For They Know Not... 105), then the formulas' immediate differences mean we must maintain some sort of gap between the universal and existential quantifiers. In other words, the universal quantifier is not just the empirical summation of a mass of particular things: “Everything is subject to the Phallus” is in some sense “more” than the statement “A is subject to the Phallus, B is subject to the Phallus, C is... (and so on perhaps ad infinitum)”. This difference closely parallels several others: necessity and contingency, essence and appearance, transcendental and empirical. However, it would be a mistake to interpret this difference as being directly any of these oppositions, and even more of a mistake to assume that its character as a formality is irrelevant. In fact, it is precisely the formality, the marked inability to express the difference in any informal categories, that is of the utmost importance. The crux of the matter is the purely formal shift which is constitutive of the application of QN. Because of this, strictly speaking ∀xΦx and ~∃x~Φx are not exactly equivalent but formally entail one another. Said another way, they have the same truth conditions, reference the same things, but yet still there is a gap between them that is not reducible to these conditions, to these referred-to things.
The second asymmetry is the relationship to negation: the masculine propositions have positive or un-negated quantifiers, and the feminine propositions both begin with a negated quantifier. Of course, intuitively this is because the feminine and masculine positions are the exact opposite of each other. But this is not all, because negation itself has a positive content. It is a formality with decidedly non-formal implications. For example, affirming a negative statement and denying a positive one are different, though they may have the same content or truth-conditions. In other words, there is no way to formally “get around” the fact that the male propositions are non-negated; they are affirmative whereas the feminine propositions are negative.
The third asymmetry is related to the use of DN, which is required for one move but not for the other. It may seem quite artificial, if not an awful violence, to resort to such a degree of formality. Is DN really relevant to the discussion? Isn't it just a totally empty formality, used only in logical proofs? Doesn't ~∃x~Φx directly entail ∀xΦx and vice versa in one single mental act, without the need for DN, which is not simply a formality but an outright technicality, something superfluous introduced from the outside, a contingency of a particular logical system? Actually, it is important that the move between ∀xΦx and ~∃x~Φx has a difference of two negations (this is what DN does, add or remove negations), whereas the passage between ~∀xΦx and ∃x~Φx simply moves a negation without either adding or removing any (this is what QN does). This means that in the passage between the universal and the negation of the exception, there is a labor of the negative that is far greater than that between the other two propositions, which have no difference in number of negations. Since negation has positive content, and along with the model proposed in the next section, the earlier claim that the universal has something over and above is corroborated—now we see it is, at least to some degree, the forced expulsion of negations.

III. Ideology & The Formation of Universals
We now turn to the relation between the formulas of sexuation and ideology, namely to the formation of universals. Reality only takes on its particular contours through the imposition of, to use Kantian terminology, the categories. However, it is not only Kant's categories that intervene in and structure reality—there are also at least some of Žižek's master-signifiers, in particular the Phallus or Phallic function. Whatever the particular differences between these may be, they are all universals, in the sense not that they are abstract, but, as above, that they apply to everything. But are these universals which structure reality always there, or are they produced according to certain mechanisms, (a)temporal or retroactive though those mechanisms may be? I claim that the formation of these universals follows a definite process that can be outlined with the help of the above formal analysis. In particular, this process has to do with the derivation of the masculine formulas—the universal and its constitutive exception are at the heart of the structure of reality, though perhaps not directly identical thereto.
I will now attempt to address the problem of the formation of universals by establishing three things: the primacy of the feminine formula, the meanings of the various particular propositions, and the ordering of the two derivations.
The feminine non-All might be thought of as primary, prior to the masculine formula, since a universal must be created from the non-All. Universals are never pre-given but must be abstracted, posited, constituted from the feminine proposition ~∃x~Φx. The universal's intervention into reality must be just that—an intervention. This action, however, need not happen “in time”, and in fact it follows a Hegelian logical temporality. It is not necessarily as if the feminine not-All exists “out there” and then the masculine principle, some moments later, intrudes and structures it. Rather, as in Žižek's discussion of the tautology A = A, the temporality here is a “minimal temporality” or “minimal delay” (For They Know Not... 36).
In some sense, then, we might say that ~∃x~Φx is the universal's worldly counterpart, in that its truth conditions involve an impossibly-long process of investigation, where each element is tested to see if Φ applies. Its counterpart, ∀xΦx, is the same thing said with that extra something the universal brings. Similarly, any particular exception, ∃x~Φx, relies on the not-All's multiplicity; the feminine proposition ~∀xΦx is the transcendental precondition for the masculine ∃x~Φx. The feminine as an unordered or inconsistent multiplicity, as a not-All, is the prerequisite for the concrete existence of any single exception to Φ. We might say that ~∀xΦx is the transcendental formula of difference, something along the lines of “the One is not”. In short, the universal and its constitutive exception are never given to us but must be produced (by the revolutionary subject, the mind, the master-signifier, or whatever).
From these considerations, I propose a model for the formation of universals containing two steps:
(1) ~∀xΦx → ∃x~Φx
(2) ~∃x~Φx → ∀xΦx
In explicating this, I will need to address several concerns: why the derivations are ordered as they are, what each move's specific mechanisms are (for there are several different mechanisms), and what the results and implications of such a model might be.
The proposed logical (not temporal) ordering of the derivations is based primarily on the fact that what is posited in the forming of a consistent totality is the exception. To form the totality, the exception must first be posited:
This, then, is the basic paradox of the Lacanian logic of “non-all” [pas-tout]: in order to transform a collection of particular elements into a consistent totality, one has to add (or to subtract, which amounts to the same thing: to posit as an exception) a paradoxical element which, in its very particularity, embodies the universality of the genus in the form of its opposite. (Žižek, For They Know Not... 44)
The action is here clearly on the side of the exception. The second move, the formation of the universal, must follow this positing.
The first mechanism is thus positing an exception, which we will now examine in more detail. This is done simply and completely in order to found a universal, a teleological operation. As Žižek describes the process above, positing the exception is always done “in order to.” For many things, this takes the form of the effect effacing its own cause or of retroactive positing. In any case, this extra effort, we might say the startup cost, of the first derivation explains why there are far less universals than there are conceivable non-Alls, ideologically or as philosophical positions.
The second mechanism, which founds the universal, follows a purely logical development. After positing the exception, ∃x~Φx, we are left with two formulas: that exception and its negation, ~∃x~Φx. These formulas are in immediate contradiction, of the form P and ~P. And as the Hegelian wisdom goes, thought cannot stop to rest on a contradiction, or at least an immediate one: “some logical moves (precisely the right ones) can be made only after other (erroneous) moves have been done” (Žižek, Less Than Nothing 629). This immediate contradiction, I claim, is what forces the move from ~∃x~Φx to ∀xΦx. Once that occurs, we have not an immediate contradiction but an ideological contradiction between our two formulas, the universal and its constitutive exception. Ideological contradiction is not immediately obvious and often manifests as a kind of disavowal: “Everything is subject to the Phallus, except women”. This sentence could be fleshed out: “Everything is subject to the Phallus without exception, and yet there is one exception—woman”. They are still contradictory, of course, but now the contradiction is subsumed in some regard, hidden and not immediately present. We could dub this logical restructuring the labor of the negative, as discussed above with regard to the “more” of the universal.

IV. Sexual Difference
We must now turn to some aspects of the model that might be misunderstood. Because much has been written about the formulas as they relate to sex (and not as they relate to the intervention of ideological universals), I will address these concerns with a greater focus on sex than in the previous sections. First, I will provide arguments against the anticipated charge that I fall prey to one or both of the misreadings explicated by Žižek in his paper “Woman is One of the Names-of-the-Father, or How Not to Misread Lacan's Formulas of Sexuation.” Second, I will investigate the Phallic function, and show that it is not masculine but neuter. Unfortunately, I will not have space to address formally the bottom half of Lacan's famous diagram from Seminar XX, Lecture VII, though I will touch the content of those mathemes tangentially.
The first misreading is collapsing both formulas into the masculine position, such that the feminine formulas are the exception to the masculine universal. It might seem that the derivations of my model are doing just that, constructing a masculine reality wherein the feminine is somehow “left out” or even a leftover. This is not the case because the two feminine formulas are not done away with but are rather transformed—the not-all does not become an ontological leftover because it is, as above, the transcendental formula of difference; the negation of the exception may always follow from the masculine universal, because with regard to whatever function one is working with it is the universal which is “more” than the collection of non-totalized particulars. Those particulars are, of course, “more” than the universal, but not insofar as the function applies to them. Further, the meaning of the formulas is not to propose a mythical pre-symbolic mother not subject to castration. The feminine in my model is thus not akin to a pre-existing Ur-mother but something that exists along with the masculine—they are mutual preconditions, and the feminine is not “outside” reality as structured by the masculine, and neither is it simply “inside” as the masculine's exception.:
However, if we are to assume fully the true paradox of Lacan's formulas of sexuation, one has to read them far more literally: woman undermines the universality of the phallic function by the very fact that there is no exception in her, nothing that resists it. (Žižek, “Woman is One...”)
What this means is that Woman is not the exception to the masculine, but the lack of the exception. Instead, the opposition between the two sets of propositions has the status of a fundamental ontological rift.
The second misreading is introducing a distinction between the different uses of the quantifiers and the different scopes attributed to the 'x's to which the function is applied. For example, one might say that the masculine universal “everything is subject to the Phallus” refers to every singular object, and that the feminine not-all “not everything is subject to the Phallus” refers to there being a part of things that is not subject to the phallic function (the feminine part), such that there is something inside woman that resists the phallic function, though woman and everything else be subjected to it. On the contrary, no such distinction between 'x's can be made—it is fundamental and ubiquitous. This problem has been avoided in the present essay by studying the propositions formally—this is perhaps the best way to avoid the subtle semantic distinctions leading to the second misreading, because I make no semantic distinctions but treat the propositions primarily syntactically.
But what is the significance of the phallic function, as opposed to the other sorts of universals, e.g. Communism (the communist worldview) as ideological universal, or a category like space? The phallic function is unique in that it is the pure universal, whose content exactly coincides with its formal aspects as analyzed above:
In other words, the paradox of the phallic function resides in a kind of short-circuit between the function and its meta-function: the phallic function coincides with its own self-limitation, with the setting up of a non-phallic exception. (Žižek, “Woman is one of the Names-of-the-Father”)
The phallic function has precedence because it is directly the emptiness of its own form-and-content. The phallic function posits its own exception and thus founds itself as universal, and is in itself nothing but this positing, nothing but the set of propositions described by that positing, their empty forms.
But we might ask what the statuses of the different sets of formulas are—is the phallus then on the side of the masculine? On the contrary, the feminine is described in terms of that function as well. I conclude from this that the phallic function is neither masculine nor feminine, and the two sets of propositions are both directly the definition of the phallic function. I would propose that it is not the simple positing of the exception or the formulas themselves but also the matrix of relations between the masculine and feminine. Thus the phallic function manifests immediately as many contradictory things, and the masculine logic of the exception is only one of those.

IV. Meillassoux's Masculine Logic
I will now apply my model of universal formation to Quentin Meillassoux's argument for the necessity of contingency as found in After Finitude. First, I will present the derivation of the necessity of contingency. Then, I will show how it fits into my model of universal formation but also differs from it in an important aspect. Then, I will examine Žižek's critique of Meillassoux as found in the Less Than Nothing, Interlude 5, titled “Correlationism and Its Discontents.” I aim to show that this critique is in many ways correct, but that it raises some important questions that must be further developed.
In order to establish his “speculative realist” (henceforth SR) position, Meillassoux begins from the standpoint of “strong correlationism” (henceforth SC). SC is marked by a philosophical agnosticism, such that every positive statement about objective reality as it exists “in-itself” or independent of a subject or observer, in his terminology independent of the correlate, is subject to the rebuttal “That might be the case, but it could also be different.” For example, when presented with the proposition that there is life after death, SC responds with “There might be life after death, but there might not be as well.”
SR undoes this agnosticism from within in the following manner: in maintaining absolute agnosticism while avoiding dogmatism, SC must claim that one cannot know what is outside the correlate but that there might be something outside the correlate all the same. SR replies by stating that this “there might be” is actually not an epistemological limit but a positive ontological one—it is not that “we don't know” whether there is a world outside the correlate, but that world is facticity itself, the “there might be.” Facticity is thus not a veil hiding the in-itself but is directly the in-itself. Note that this conversion is precisely logically warranted: to avoid absolute idealism, SC must posit the real possibility of things outside the correlate—that possibility cannot itself be inscribed within thought without falling into absolute idealism. If one at the position of SC admits that there might be this real possibility, then one actually moves to the position of SR.
Now we can rephrase this argument in terms of necessity and contingency. SC can be summarized as the position “Everything is factual.” SR then adds “except for facticity itself, which cannot be factual.” This is because, if facticity is factual, then it ceases to be absolutized, it ceases to apply to everything, and this is the aim of both SC and SR. If facticity were factual, it would not be able to apply to everything. Thus, Meillassoux (through the mediation of Ray Brassier's translation) proposes the term “factiality,” the principle that only facticity cannot itself be factual (80). From here it is a short step to translate the “intra-correlate” principle of facticity into contingency, which does not heed the limits of the correlate (see the bit about “real possibility” above): “For if contingency consists in knowing that worldly things could be otherwise, facticity just consists in not knowing why the correlational structure has to be thus” (Meillassoux 39). So the crux of the argument turns out to be the proposition that everything is contingent, except contingency (which is necessary).
It should be relatively clear how this relates to the formulas of sexuation, in particular the masculine propositions. To be truly universal, an exception to contingency must be posited—this exception is contingency itself. If contingency were not necessary but contingent, it could not apply to everything because it could be different—contingently contingent things are not absolutely contingent, only contingently so. The “it could be different” of contingency must be absolute and therefore necessary. So contingency is a universal with a constitutive exception, pure and simple.
Žižek's analysis of Meillassoux in Less Than Nothing is long and has many different sub-arguments, but I will focus on (and quote at some length) his argument that “...when Meillassoux asserts contingency as the only necessity, his mistake is to conceive this assertion according to the masculine side of Lacan's formulae of sexuation...” (636). What is left out, then, is the feminine side, the not-all and the negation of the exception:
[B]ut what about the non-All of contingency: there is nothing which is not contingent, which is why not-All is contingent? Simultaneously, there is the non-All of necessity: there is nothing which is not necessary, which is why not-All is necessary. (Žižek, Less Than Nothing 636)
And finally, the real clincher: “How do these two non-Alls relate? Since reality is contingent, we should begin with the non-All of contingency: it is out of contingencies that, contingently, necessities arise” (Žižek, Less Than Nothing 636).
Unfortunately, Žižek does not go on to elaborate these points. However, we have all the tools necessary to explicate them. I will proceed with several claims: first, that Meillassoux does indeed omit the feminine non-All; second, that Meillassoux would reject the need to worry about the non-All of necessity; third, that Žižek's position is feminine and includes an explicit rejection of Meillassoux's masculine propositions; and finally, that this feminine position is actually much more complicated than it at first appears, implying a sort of higher-order masculinity even as it provides avenues for escape.
First, Žižek claims that actually Meillassoux should deal not just with the masculine formulas of contingency, but also with the non-All of contingency. We might say that Žižek claims Meillassoux forgets about the feminine proposition(s) and goes straight to the masculine propositions. But here we must keep in mind Meillassoux's starting point, SC. What is the claim of SC, expressed formally? It is hermaphroditic: ~∃x~Γx and ∀xΓx, where “Γ” is the predicate for contingency. These statements are equivalent, but the latter cannot express a true universal along with the explicit denial of its exception. This is important because it distinguishes the philosophical position of SC from operations such as the phallic function or ideological intervention. Here, for symmetry's sake, we should also look to the inverse of SC, the hermaphroditic position holding the formulas ~∀xΓx and ∃x~Γx. This position, as is to be expected, is what Meillassoux calls metaphysics. In this context, we can translate ~∀xΓx as “not everything is contingent” and ∃x~Γx as “there is a necessary thing (for example God).” Alternative derivations beginning with the metaphysical position are also be possible, but this must for our purposes remain only a vague proposal. In any case, what neither SC nor metaphysics can allow is the constitution of the masculine or feminine.
What Meillassoux does is deny the negation of the exception and thus posit that exception, repudiating the sterile hermaphroditic SC for a vigorous masculine position. Interestingly, since Meillassoux begins not with the non-All but with SC, he begins already with the universal and then posits the exception, rather than positing the exception and thereby deriving the universal. Perhaps more importantly, there is not a legitimate logical relation between the exception and its negation—so where does Meillassoux get the material to derive the exception? One could make the move from ~∀xΓx to ∃x~Γx, but does Meillassoux do this? It is in order to constitute the universal that Meillassoux posits the exception, but this is formally suspect without the not-All. I believe this is Žižek's point, somewhat refined.
Second, Žižek proposes that Meillassoux busy himself not just with the non-All of contingency, but also with the non-All of necessity. But why should Meillassoux care about the non-All of necessity? There are perhaps an infinite number of non-Alls one could propose, but why any particular one of them is relevant would be uncertain. Here, aside from proposing a rather strange logical relation (concluding ~∀xΓx from ~∃x~Γx and likewise for necessity), Žižek does not give us much with which to work. In any case, Meillassoux would likely not accept that “there is nothing which is not necessary” (Zizek, Less Than Nothing 636). In other words, he might say that not all non-Alls are true, and certainly not the universals one would form with regard to them. In fact, if Meillassoux constructed a universal from the non-All of necessity, he would produce statements immediately contradictory to his main thesis. There are rational or external considerations which determine the acceptance or rejection of various functions, and Žižek does not here explicitly provide us with any of these. However, there is an implicit reason for Žižek's proposal, which I will outline in the conclusion.
Third, while Meillassoux obviously subscribes to a masculine logic, it seems rather unclear what Žižek's stance is. For starters, he proposes that Meillassoux begin not with SC but with the robust feminine position, namely the non-All of contingency. Immediately, this corroborates my model which gives (logical) primacy to the non-All—it is the thing with which one begins. But does this proposal indicate Žižek's rejection of Meillassoux's masculine propositions? Or is it more along the lines of “Meillassoux is not directly incorrect but he left out something (the non-All)”? Or is there a tricky retroactivity at work here?
Žižek's subsequent argumentation leads me to believe that he is outright rejecting conceiving contingency from the masculine perspective. He calls the masculine perspective “[Meillassoux's] mistake” and then proposes the two non-Alls, asking not how they relate to Meillassoux's masculine propositions but how they relate to each other (Less Than Nothing 636). If Žižek were proposing some sort of “neutral” position that would finally take both the masculine and feminine propositions with regard to contingency into account, his criticism of Meillassoux's ignorance of the non-All would likely focus on the derivation between the non-All and the exception (and Meillassoux's lack thereof).
Having established that Žižek's position is that of the feminine non-All as against Meillassoux's masculine totality, where does that leave us? Žižek here provides us with one explicit argument: “Since reality is contingent, we should begin with the non-All of contingency: it is out of contingencies that, contingently, necessities arise” (Žižek, Less Than Nothing 636). Does this statement apply also to the necessity of contingency? Is there a primordial contingency that could give rise to its own necessity, which giving-rise would thus be retroactively determined? In another context Žižek writes, “...the pre-symbolic 'eternally Feminine' is a retroactive patriarchal fantasy—that is, the exception which grounds the reign of the Phallus” (The Žižek Reader 140). Here we see something rather strange: Žižek, in proposing the primacy of the non-All of contingency and its ability to give rise to necessities, has taken part in just such a “retroactive patriarchal fantasy.” Here, the exception is on a higher level, no longer the simple ∃x~Γx but now the feminine itself. This is precisely the first misreading discussed in the previous section. In taking the feminine position Žižek has shown himself more masculine even than Meillassoux!

VI. Conclusion
What are we to conclude from this bizarre state of affairs? Žižek's critique of Meillassoux provides us with some generally relevant conclusions. Importantly, it shows that one cannot read the formulas in a neutral way—Žižek provides us with some ways “not to misread” Lacan's formulas, but how can one read them correctly? Since “Woman does not exist,” there does not seem to be a way to occupy a feminine position that is not also masculine—but this is just what we would expect, since She is a mask or screen, coincident with the patriarchal order though with a formal shift from the for-other to the in-itself; in-Herself She is for-other. Another way to conceive this shift would be the difference between the masculine and feminine formulas—they 'say the same thing,' but are formally different in the ways outlined in Section II of this paper.
As Žižek writes, the two sexes “...involve a different modality of the very antagonist relationship between these [stereotypical male-vs-female] opposites” (“Otto Weininger...” 144). So Žižek's modality differs from Meillassoux's, but how are we to decide between them, to pronounce a judgment? Žižek, though involved in a patriarchal fantasy, is at least able to speak the antagonism, even if from the side of the masculine. It is by taking the side of the feminine that he is absolutely masculine, and yet this partiality allows him to address the ontological rift of sexual difference. This precisely parallels Žižek's later point that “the very epistemological failure (to reach reality) is an indication and effect of our being part of reality, of our inclusion within it” (Less Than Nothing 636). Thus, it is as partial that we can relate to or interact with reality, because that reality itself is partial, subject to sexual antagonism.
Finally, there remains the question of the exclusively feminine possibility of going “beyond the Phallus.” In the context of sex this means having non-phallic enjoyment, but what does it mean for our model? I believe the answer is to be found in Žižek's proposal of the non-All of necessity—going beyond contingency here means an agnosticism of sorts. Žižek occupies the feminine (and therefore also masculine, as shown above) position insofar as he decides on the non-All of contingency, but he goes “beyond” insofar as he considers other non-Alls. And further, this agnosticism is due to the feminine: it is because of the inconsistent nature of the non-All that alternatives are possible. From within the masculine, Meillassoux cannot do this. Truly holding a feminine position, then, may involve any number of non-Alls, but these are precluded by an exclusively masculine position.
Sexual difference, then, is definitively shown to be fundamentally present. The model could, with minor tweaks, be utilized for everything from ideology and Kantian categories to philosophical positions and of course sexual relations, wherever there are universals. This formal work, though incomplete, is important precisely because sexual difference itself is “the failure a symbolization”, itself formal (Žižek, “Otto Weininger...” 145).

Works Cited
Lacan, Jacques. The Seminar of Jacques Lacan, Book XX: On Feminine Sexuality, The Limits of Love and Knowledge, 1972-1973. Ed. Jacques-Alain Miller. Trans. Bruce Fink. New York: W. W. Norton & Company, Inc, 1999. Print.
Meillassoux, Quentin. After Finitude: An Essay on the Necessity of Contingency. Trans. Ray Brassier. New York: Bloomsbury Academic, 2009. Print.
Žižek, Slavoj. Less than Nothing: Hegel and the Shadow of Dialectical Materialism. New York: Verso, 2013. Print.
Žižek, Slavoj. “Otto Weininger, or 'Woman doesn't Exist.'” The Žižek Reader. Eds. Elizabeth Wright and Edmond Wright. Malden: Blackwell Publishing, 1999. Print.
Žižek, Slavoj. “Woman is One of the Names-of-the-Father, or How Not to Misread Lacan's Formulas of Sexuation.” lacanian ink 10 (1995): n. pag. Web. 5 December 2014.