# SEXUATED TOPOLOGY AND THE

SUSPENSION OF MEANING

## A NON-HERMENEUTICAL PHENOMENOLOGICAL

APPROACH TO TEXTUAL ANALYSIS

WILLIAM J. URBAN

### 6.2 From the Aristotelian to the Lacanian Logical Square

One of the undeniable strengths of Le Gaufey's book Lacan's Notall: Logical Consistency, Clinical Consequences (2006)599 is to provide an account of the debt Lacan owes to Jacques Brunschwig's 1969 article on Aristotle. The discussion begins at the exact mid-point of the second chapter of Le Gaufey's text, and there we learn how this article effectively initiated Lacan's complete revision of the classical logical square. The article in question concerns Brunschwig's analysis of problems encountered by Aristotle when he tries to specify the meaning of particular propositions used in deductive or syllogistic reasoning. The overall contention of Brunschwig is that Aristotle significantly shifts his approach over the course of the Prior Analytics. At first relying on the equivocal quality of 'natural language,' this position is eventually found to compromise his logical system so that by the end of the text Aristotle's quest for definitiveness compels him to reject his initial position for one that is better able to control the troubling equivocations. The two positions Aristotle implicitly takes concur with the two options discussed above regarding the reading of the particular and the shift he undergoes is precisely a move from option two to option one. Brunschwig calls these two the 'maximal' and the 'minimal' readings of the particular, respectively. Restated in simple terms, the particular affirmative 'Some S are P' in its maximal sense concurs most closely to natural language, as in the case when someone says 'Some of Gadamer is Heideggerian.' Very few people, if any at all, would conclude from this statement that 'All of Gadamer is Heideggerian.' Yet a logician having chosen a minimal reading would do just that and accordingly would ask us to consider how this statement is only a particular instantiation of the universal affirmative. The contention of Brunschwig is that in the end Aristotle favors exactly this minimal reading of the particular (whereby 'At least one S is P' does not eliminate the possibility that 'All S are P'), and one might add that logicians since the Organon have even more consistently excluded its maximal reading (whereby 'At least one S is P' implies no more than one S having the property P). However, in retrospect it is evident that its exclusion was not absolute. Indeed the possibility is raised that the exclusion in question concerns an element which haunts any effort to construct a complete and universal logical system.600 It is against this backdrop that Lacan's project of formulizing sexual difference most clearly surfaces. As Le Gaufey writes, 'Lacan is striving to pick up the challenge of what Aristotle, according to Brunschwig, had to drop in order to make his proofs of non-conclusiveness consistent.'601 By the early 1970s the time had finally come to work through the consequences of choosing the maximal particular.

The immediate consequences of this choice are paradoxical. To express this in terms already developed, by choosing to read the particular in the maximal or restrictive sense (i.e., 'some, not all'), Lacan completely undermines the logical relations of the classical logical square to the point which strains common sense. Recalling the above discussion, it is clear how the logical relation of subalternation is replaced by contradiction whenever the truth of a particular is affirmed. That is, 'Some S are P' excludes 'All S are P.' But the former must now be said to be parallel to 'Some S are not P.' In slightly different words, if 'Some (but not All) S are P,' then it is also the case that 'Some (but not All) S are not P.' With the removal of subalternation, the particular affirmative and particular negative are no longer logically related as subcontraries but should rather be thought of as effectively equivalent. It follows from this that since the two universals contradict their opposing particulars and since the latter are equivalent, the universals are equivalent as well. The logical relation of contrariety, just like subcontrariety, is replaced by equivalence. In the end, Lacan has simplified the classical logical square with respect to its logical relationships, reducing the logician's task from thinking with four (contradictoriness, contrariety, subcontrariety and subalternation) to only two (contradictoriness and equivalence). But the claims Lacan effectively makes here seem entirely unreasonable. How can 'All S are P' be considered the equivalent of 'All S are not P,' both of which are, moreover, said to stand in contradiction to 'Some S are P' and 'Some S are not P?' Expressed in these terms, this cannot be defended. But they do become defensible through Lacan's proposed changes to the writing of each of these propositions.

Before turning to examine these changes it should be acknowledged briefly how the choice of the maximal particular reopens the prospect of using the structure of a logical square to better articulate the dynamics of the hermeneutical circle within which meaning stands. Recall how the choice of the minimal particular closed down this potential in the classical logical square. But Lacan's choice of the maximal particular initiates the development of a new logical square in which the objection of the particular to its universal reestablishes a distance between these two levels that otherwise threaten to collapse into each other with the choice of the minimal particular. Such a distance is obviously necessary to effectuate the desired analogy between the universal and the particular of the logical square to the whole and the part of the hermeneutical circle. The possibility of using Lacan's new logical square to articulate (and ultimately to suspend) the hermeneutical circle is a thesis that is returned to time and again below.

Le Gaufey locates the precise date at which Lacan introduces his first writing of the four formulae of sexuation which replace Aristotle's four propositions to compose the new logical square. This occurs during his March 17, 1971 session of his eighteenth seminar. These formulae continue to be rewritten over subsequent sessions and only reach their finalized state a year later on March 3, 1972. The intriguing question of how these changing formulae precisely capture the topological transformations of Lacan's thought during these twelve months must be set aside to instead focus on the final set of formulae which most closely inscribe his position in the early 1970s. Now, even a cursory glance of these formulae confirms how Lacan 'is proposing to rewrite Aristotle... with the function and the quantification invented by Frege.'602 What Le Gaufey is referring to rather unclearly is Lacan's annexation for his own devices of what has been called 'Frege's most remarkable and indisputable achievement,... the revolution that he effected in logic, which for over 2000 years, ever since its origins in Aristotle's Prior Analytics, had been dominated by syllogistic theory... [This real logical breakthrough occurs] in his creation of what we now know as predicate logic, through his invention of quantifier notation.'603 This achievement essentially proceeds from Frege's novel 'mathematical' approach to the analysis of sentences in ordinary language. That is, he replaces the subject and predicate terms of a sentence with the mathematical notions of argument and function, respectively. Applying this approach to the familiar sentence 'Gadamer is Heideggerian,' as was seen Aristotle and traditional grammar would take 'Gadamer' to be the subject (S) and 'is Heideggerian' the predicate (P), so that the sentence takes the form 'S is P.' In contrast, Frege would consider the subject term 'Gadamer' an argument and 'is Heideggerian' a function. Yet this is far from simply an alternative terminology, for this introduces a significant change in the understanding of how sentences are constructed. To see this, suppose one substituted 'Badiou' for 'Gadamer' in the sentence 'Gadamer is Heideggerian.' Doing so proceeds from a claim that is possibly true to one that would be much more contentious in its 'obvious' falseness. In this way sentences can be seen as consisting of two components, one that is constant like '( ) is Heideggerian,' and one that is potentially replaceable like 'Gadamer.' What Frege has done is to make an extension of mathematical terminology by which, for example, 8 is the value of the function × 4 for the argument 2, and 12 is the value of the same function for the argument 3. In this way 'Gadamer is Heideggerian' is the value of the function '( ) is Heideggerian' for the argument 'Gadamer' just as 'Badiou is Heideggerian' is the value of the same function for the argument 'Badiou.' And what Lacan does is apply this Fregean approach to the classical logical square. So what argument(s) and function(s) does Lacan use and how does he write them?

One symbol used in both the argument and function components of each of the four new logical propositions is which is most 'naturally' read as the human being who is to be classified as either 'man' or 'woman' according to these propositions. These four propositions are the four formulae of sexuation after all. But Lacan frustrates this simple understanding precisely by using this symbol which he borrows from algebra where (as seen above) it traditionally operates as a variable ready to be filled in with different object-terms. This alone greatly extends the applicableness of Lacan's propositions to fields other than those which entertain the question of sexual difference or alternatively, extends the notion of sexual difference to more abstract fields like hermeneutics and phenomenology. So this is best considered an 'element' which may (or may not) be grouped together and which may (or may not) exist. These possibilities are determined in part by the quantifiers Lacan chooses to modify . One such quantifier is the universal quantifier, marked by the symbol , and which reads as 'for All' or simply 'All.' Thus the argument can be read 'All .' The other quantifier Lacan uses is the existential quantifier, marked by the symbol , and which reads as 'at least one.' Thus the argument can be read as 'There is at least one .' These two are, respectively, Lacan's versions of 'All S' and 'Some S' in the classical logical square. Lacan also follows the Aristotelian manner of determining the quality of a proposition through affirmation or negation. But he departs from classical practice, which only entertains the affirmation or denial of the specified predication, by further allowing for the direct negation of the argument component of a proposition. The method he chooses is to place a bar over the component of the proposition to be negated. Thus the argument can be read as 'There is no ' and the argument can be read as 'Not-all .' A component which remains unmarked in this manner signifies affirmation. To produce the four propositions, each of these four arguments of , , and are placed in relation to a function which together determines its value. Given Lacan's use of , one might expect that the function will take the traditional mathematical form of , and this is indeed the case, although he drops the parentheses and replaces the with so that the function to which the four arguments are (or are not) submitted to becomes . This is the symbol Lacan has long used for the phallic signifier discussed in Section 5.3 above and when it is graphically combined with it produces which can be read as 'the phallic function.' What allows Lacan to think of the phallic signifier as a function stems from Frege. As implied above, after Frege functions are no longer strictly numerical, for through his mathematical understanding of language the concept itself can now be taken to be a function which maps objects on to truth-values. Generally speaking then, the phallic function can be taken to be the concept of the phallic signifier reconceived as a function. As per Frege, we could then consider in its capacity to submit an under the concept of . More specifically and insofar as the latter is a quasi-transcendental signifier, the signifier of the symbolic order as such, this amounts to saying that has the ability to include this in the symbolic order which simultaneously makes subject to all the limitations of the signifying system, or in psychoanalytical speak, subject to symbolic castration.604 But as was seen, an stands in relation to a function like as per one of the four arguments of , , and , and what makes this even more complex is that Lacan allows for the placement of the bar of negation over this function so that these four arguments can potentially be combined with either or to produce the four propositions. Lacan's final decision on these combinations are reflected in Figure 6.2, the Lacanian logical square.

Before a more detailed discussion of these propositions is begun, there are a number of general points which should be made. The most obvious is that Lacan does not present a logical square in any of his seminars or published writings. But as Le Gaufey suggests, nothing prevents its construction.605 In fact, its construction is rather straightforward given that these four propositions appear without modification in the upper portion of the table of sexuation presented at the beginning of the March 13, 1973 session of Lacan's well-known twentieth seminar Encore (1972–3).606 These four propositions form what is usually referred to as the formulae of sexuation. Grouped together in two pairs, one pair is said to define 'man' and the other 'woman.' More specifically, the universal and particular affirmative propositions occupy the man side of the table while the universal and particular negative propositions occupy the woman side of the table. But in this table Lacan does not strictly observe the vertical ordering of the two pairs of propositions of the classical logical square from which they are originally derived. As can be seen in Figure 6.1 or 6.2, the universal propositions appear on top of the particulars. With Lacan's arrangement however, while the two 'feminine' formulae do indeed follow this format, he does reverse the ordering with the 'masculine' formulae so that the particular appears on top. So constructing a Lacanian logical square, which allows for a direct comparison to the classical Aristotelian logical square, merely involves reversing the ordering of the formulae on the masculine side. A further point to be noted is that Le Gaufey constructs his Lacanian logical square so as to preserve the horizontal ordering found in both Lacan and Aristotle whereby the affirmative propositions (or masculine formulae) are placed on the left side of the square and the negative propositions (or feminine formulae) are placed on the right. Yet Figures 6.1 and 6.2 reverse this ordering. As has already been noted, this is only a minor change in presentation. The main reason for doing so is rather strategic: to better reflect the fact that the feminine formulae are more primordial than the masculine formulae. Thus, to serve as a reminder of this logical ordering the negative propositions have been placed on the left side where the reader's eye first takes them up before moving onto the affirmative propositions which occupy the right side; this approach thus harmonizes with the customary reading practice which moves across the written page from left to right. Yet this logical ordering whereby negation rightly precedes affirmation conflicts with the ordered sequence of numbers their quadrants have been assigned. That is, the symbols of A, I, E, O traditionally used to concisely refer to the propositions of the classical Aristotelian logical square of Figure 6.1 have been substituted by , , and , respectively, in Figure 6.2. This has been done to capture the fact that despite the logical precedence of the elements inscribed by the propositions occupying quadrants and , they nevertheless paradoxically come after those of and . One of the endeavors of Part II is to explicate these and other similar claims.

Setting aside the ramifications of substituting these specific arguments and single function for subject and predicate for the moment, a cursory comparison of Figures 6.2 and 6.1 shows that Lacan apparently makes a change to the writing of each proposition in some respect except for the universal affirmative. At least in terms of quantity and quality, it appears to be the case that does not represent any change over 'All S are P.' But Lacan clearly makes changes to the writing of the other three propositions by modifying their quantity or quality (or both). Beginning with the classical proposition 'Some S are P,' it is now seen to be written as . In both squares these are considered particular affirmative propositions yet Lacan's writing negates the phallic function. So while both squares have an 'official' negative left side or deixis,607 negation does appear on the affirmative right side of the Lacanian square in stark contrast to the classical square. This captures Lacan's claim that the particular affirmative stands in contradiction to its universal. Moreover, contradiction occurs within the other deixis, for this same relation is to be had between the universal and the particular negative propositions. That is to say, where the classical square articulates 'All S are not P,' Lacan counters with his own universal negative proposition which he formulates with a double negation: . Very broadly speaking (and indeed at this point only summary observations are being made at the level of the apparent differences in the writing of the propositions in each of the two squares) a double negative returns the proposition to a qualitative status effectively equivalent to the universal affirmative. This double negative thus places it in contradiction to its particular which, for its own part, retains a single negation of its argument (and not of the phallic function which would be the case had Lacan strictly followed the format of 'Some S are not P' in the classical square): . Here is the proposition said to retain an equivalency to the similarly singly-negated particular affirmative proposition. The negative deixis also has a further curiosity in that the quantifiers of their arguments are reversed from those of the affirmative deixis, the latter of which seem to better accord to those of the classical square.

A high-level comparison between the two squares thus reveals how the general layout of the classical square is retained in the new square: both operate with the same two (universal and the particular) levels and the same two (affirmative and negative) sides. But set aside these formal similarities between the two logical squares and the differences rapidly begin to mount, both at the level of the individual propositions and by extension to the relationships between them. The question which now must be asked is why Lacan choose these specific argument-function combinations. Or more simply said, what does each proposition say?

It was said above that does not represent any change over 'All S are P.' This is true enough at the level of its two-component format, but by considering its writing especially in terms of argument and function this assessment alters considerably. As Le Gaufey reminds us, by early 1971 the name of Peirce begins to frequently appear in Lacan's seminars608 whose own work in logic casts a formidable influence on how Lacan reads the quantifier . The issue at stake takes up a problem known for centuries which concerns the existential assumptions of the classical logical square. Suppose that the S of 'All S are P' is an empty term. By subalternation 'Some S are P' is also false which makes 'All S are not P' true by contradiction. But by subalternation again 'Some S are not P' is also true, which must be wrong since there are no S terms in existence. The traditional solution to this problem was to insist either that the logical interrelations of the square unobjectionably hold even for empty S terms, which would then make 'All S are not P' (vacuously) true but not false, or to not even consider empty terms at all. Most traditional logicians opted for the former solution, which rigorously upholds the logical relationships; they could thus easily hold the truth of a universal affirmative statement like 'All unicorns have one horn' as leading to the truth of its subaltern 'Some unicorns have one horn' despite the fictional status of the S term unicorn. However, modern logicians like Peirce refocused the problem onto a question of the existential import of particular propositions. Surely by saying 'Some man is white' is to imply that at least one thing is a man who has to be white if this proposition is to be true; likewise with 'Some man is not white.' Since the classical logical square requires that one of these propositions is necessarily true as both cannot be false and since both imply that some thing is a man, it therefore necessarily follows that men exist. Such reasoning has lead to the modern assumption that for a particular proposition to be true there must be at least one case in which the subject term exists. More broadly speaking, the modern assumption which Lacan thoroughly follows is to take universal claims referencing non-existent objects like 'All unicorns have one horn' to be true even if there are no unicorns, while the particular claims about them like 'Some unicorns have one horn' are seen as false since this claim requires that at least one unicorn exists for it to be true. Generalizing these assumptions, the truth of does not imply the existence of a term to which refers, so it can be true even if there is no (i.e., even if is empty, an element of the empty set: ∈ ∅); yet for to be true, at least one must exist. Hence the reason for Lacan's choice of the existential quantifier .

The universal and particular levels precariously held apart by subalternation is now replaced by a logical relation which more radically maintains their opposition. Žižek's own discussion of the universal and particular affirmative propositions is illustrated by a citation from Lacan's ninth seminar,609 a fact which allows one to see how Lacan was already aware of the relation between these two propositions despite not yet formalizing this relation at a logical level – an achievement which only comes a full ten years later. The citation in question concerns how every existing father stands as an exception to the universal notion of father captured by the proposition 'All fathers are ' as Žižek generically expresses it. That is, despite how the universal paternal function determines all fathers, this in no way implies that there exists a particular individual which exemplifies its truth. For each individual is either deficient or overbearing as a father such that the only father who fully exists at the level of his notion is the mythical 'primordial father' standing precisely as a particular exception to all the other fathers who waver between a too little and a too much. Le Gaufey's own illustration entails what is arguably the most elemental universal affirmative: 'All men are mortal.'610 If this proposition does not imply existence, to aspire to belong to it as a man simultaneously coincides with an abstraction from it. In other words, a particular man who says 'All men are mortal' arrogantly assumes his ability to draw a line over his own existence while he is alive. With these illustrations it begins to be seen that while the two propositions contradict each other, the existential declaration of has the potential to undermine from within the universal notion . In terms of textual exegesis, if the notion is initially held that 'All of Gadamer is Heideggerian,' this in no way presumes that there exist particular passages of Gadamer which exemplify this truth. What will actually be found are passages which fall short of Heidegger or alternatively, which parody him to ridiculous excess. These existing passages problematize the notion they were intended to illustrate and thus call for its revision which in turn will re-initiate the search for its own (non-existent) exemplification. In this way the hermeneutical circle turns for the individual interpreter while a larger circle operates at the level of the secondary literature itself where the following strategy is quite often employed: scholar A effectively critiques scholar B by targeting what the former views as the latter's 'poor choice' of supportive textual material. But what Lacan's two propositions articulate is how this is really a foregone conclusion, for any use of existing textual material cannot but undermine the interpretive thesis it is meant to illustrate.

Such a conclusion ultimately stems from Lacan's privileging of the maximal form of the particular which objects to the universal. He affirms that if , then one cannot conclude that . In terms of the above examples, if at least one possesses some property like the paternal function or mortality or falls under the concept 'Heideggerian,' it is wrong to conclude All do, that is, that all fathers, all men and all of Gadamer's texts submit to their respective function. Rather, what must be concluded is that Not-all do, or . In this sense the Not-all is an affirmation of the setting aside of by and thus can be seen as mediating between these two propositions. In doing so its own status is highly undecidable, for it operates as a sort of index to the discord between the universal and existence: to the which opens up the field of existence without implying anything regarding the existence of the it universally affirms, it links which affirms the existence of at least one able to sustain the negation brought to bear on the phallic function. That stands between the universal and existence (and thus troubles a simple opposition between existence and inexistence) can already be seen in Lacan's choice to use the negation of the universal quantifier to express the negative of a proposition dwelling at the particular level assumed to carry existential weight. But if by affirming one concludes that , this still leaves open a potential for misunderstanding. If some do not possess the property and simultaneously Not-all possess it, traditional logical would conclude that their conjunction brings back the whole of the universal such that All may be said to possess it. To prevent this from happening (which at once shows his concerted effort to undermine the universal), Lacan recognizes how the particular negative proposition cannot stand alone but must be supplemented by the universal negative . The relation between these two propositions occupying the negative deixis of the Lacanian logical square can be classically read from top to bottom (so that is a necessary condition of ) or from bottom to top (so that the latter is the sufficient condition of the former): if Not-all possess it, then this implies that there is no that does not possess it. But the difficulty of understanding this is immense. For how exactly can it be that while there is no that does not satisfy , these nevertheless do not totalize themselves into an All set?

Addressing this difficulty provides an opportunity to raise awareness of the fact that these formulae make use of set theory, which is that field of mathematics lying at its very foundation. Discussing how set theory relates to the Lacanian logical square is important, if for no other reason than to lay the groundwork for understanding Lacan's further usage of a branch of set theory called topology to better articulate how these formulae interrelate. Topology is a topic that is discussed in Chapter 7 below. Basically, the Lacanian logical square involves set theory precisely at the level of the universal proposition where the quantifier becomes operational by referring the that follows it to some thing which is then written as belonging to a determined set. Above it was implied that was a variable ready to be filled in with any one of an array of different object-terms which have a determinate status. But since the universal affirmative does not imply existence, strictly speaking this is inaccurate. As Le Gaufey suggests, Lacan is following Frege's notion of how the 'indeterminacy' of 'is not a descriptive epitaph of "number," it is rather an adverb modifying 'indicate." We will not say that [] designates an indeterminate number, but that it indicates numbers in an indeterminate manner.'611 This notion raises the question of sets. More specifically, the use of is effectively the hypothesis that a set which covers the range of values of truly exists such that an element can be extracted from it given the employment of the proper function the satisfies. In terms of the distinction Russell draws between the 'intensional' and 'extensional' approaches to the set,612 Lacan here seems to opt for the former. For the proposition collects corresponding to , which is consistent with the intensional or rule-governed understanding of a set as a collection of objects corresponding to a predicate or concept, an approach which presumes that the concept takes logical priority over its application. But it should be carefully noted how intensional set theory is strictly applicable only to the right deixis of the Lacanian logical square, for the left deixis undeniably inscribes the extensional or combinatorial understanding of a set which eschews proceeding along such strictly defined lines. In contrast to the intensional conception, any set that results from the extensional approach is built from the bottom up by a simple bundling together of elements and what solidifies the priority this approach enjoys over its counterpart is that it avoids the fatal limitation of holding to the rule-governance theory of set formation. The limitation in question is none other than the one exposed by Russell to Frege's great disappointment. Frege had defended the intensional approach, but Russell's famous paradox pointed out that the former's system allows for the distinction between properties that include and do not include themselves within their intensional grasp and so leads to sets which include themselves and others which do not. This is perfectly acceptable, but it further suggests that the formation of a set of all sets that are not members of themselves is possible. Yet it is easily shown how the attempted formation of such a set leads to a paradox, for if it is a member of itself then it is not a member of itself and if it is not a member of itself, then it is a member of itself – a paradox because it conflicts with Frege's sensible view that any coherent condition determines a set. Simply said, this set of all sets that are not members of themselves does not exist. Restating Russell's paradox in concise existential terms: if it exists, it is a member of itself if (and only if) it is not a member of itself; this is a contradiction and so it does not exist.613 Given that Frege's system leads to such a paradox, the conclusion must be that it cannot be logically sound. So the extensional approach must be seen as taking logical priority to the intensional approach and what makes the Lacanian logical square that much more 'complete' is that it inscribes both these approaches. In contrast to the right deixis where sets exist, the left deixis has priority through its inscription of the 'deeper' truth that some sets do not exist.

To relate these results back to the propositions of the logical square, to think that there is no exception to the satisfaction of a function () does not reunite those which do satisfy this function under the aegis of a universal set () from which one could indeterminately draw an element and inscribe it into the existential order of the particular (); rather, a domain results () which refuses to collectivize into a set. Žižek privileges 'politics' from the series of other terms he has employed throughout his career to helpfully illustrate this paradoxical domain. As he recently writes: 'Politics which occurs in this in-between space is non-All: its formula is not "everything is political," but "there is nothing which is not political," which means that "not-all is political." The field of the political cannot be totalized... there is no meta-language in which we can "objectively" describe the whole political field, every such description is already partial.'614 Reading these words closely, one sees how Žižek effectively makes reference to the logics of each of the four propositions. Thus we are admonished not to move too quickly to declare how 'everything is political' []. Such a declaration seems rather naïve given the series of examples that immediately spring to mind which surely countermand its assertion []. Sex perhaps is such an exception, providing a meta-language position from which an objective assessment of the field of the political might be possible. But then again as feminist discourse endeavors to prove, a closer examination reveals how 'there is nothing (not even sex) which is not political' []. That is, taking each of these contradictory examples one by one will reveal that they do not hold any exceptional status to the function in question. This, however, does not re-submit them to the universal All but rather demonstrates how the political field can only be described in a partial manner. Here one must avoid thinking that this is so because this field is simply too vast to grasp all at once and thus only permits a piecemeal approach. For although the results of such partial analysis can be gathered together, their sum must be conceived as a loosely grouped and non-totalized domain (rather than a determinate set) such that it is only permissible to say 'not-all is political' []. This illustration by Žižek makes it more clear why Lacan has paradoxically chosen to write the universal negative proposition by negating the existential quantifier () which in the right deixis affirms the existence of an exception to the universal All; while the particular negative proposition is written by negating the universal quantifier (), which stands as a reminder not to conceive it as 'some' or 'at least one' with a determinate and essential existence. The two propositions in the left deixis thus work in unison just as those of the right. If we begin by claiming Not-all satisfy the function, this of course rules out that All should do so, but also that there should be one that does not since the universal negative compliments the Not-all claim by affirming that there is no one who does not satisfy the function.

Now that the propositional components, their specific combinations and the relations held between the two propositions in each pair have been introduced, a greater appreciation for how each proposition relates to all the others is needed. The example from Žižek above already suggests how the two pairs of propositions do not stand in simple opposition to each other but do seem to enter into an interplay of mutual relations just as do the propositions of the Aristotelian logical square. So while it is perfectly legitimate to speak of elements in the left or the right deixis of the Lacanian logical square, care must always be taken not to slip into thinking that this reflects a bi-partition which somehow occurs 'within' the universal. To do so is to precisely take up the meta-position offered by the particular affirmative (which the universal negative denies). In contrast, by continually approaching the global consistency of these four propositions which respects their interplay, the desire to comprehend them 'all at once' is frustrated. It also becomes clear that what is at stake in each deixis is the conception of the exception, affirmed on one side () and denied on the other ().

Making use of Le Gaufey's occasional practice of generically referencing the propositions in a convenient shorthand (whereby = all say yes; = one says no; = no one says no; = not-all say yes), the logical relations between the four propositions can be concisely articulated. If all say yes, then it is false that one says no and that not-all say yes (as seen in Figure 6.2, enters into contradiction with both and ) and if no one says no, then it is false that one says no and that not-all say yes ( enters into contradiction with both and ). In addition to contradiction, there is also the relation of equivalency. For if all say yes, no contradiction ensues with the fact that no one says no (the two universals and imply each other and are effectively equivalent), as is the case with the two particulars, for if one says no, this does not contradict with the fact that not-all say yes (an equivalency between and ).

Concerning the question of equivalency it will be noticed how the last sentence of the previous paragraph seems to stand in tension to the conclusions drawn in the immediately foregoing paragraphs. For if those conclusions forewarn us not to conceive of the Lacanian logical square as neatly partitioned into two sides, how are we to make sense of the equivalent relations between the universals and the particulars which cannot but give the sense of a perfect symmetry between the two sides? Le Gaufey would suggest that the answer lay with Lacan's conception of the universal negative, which is 'undeniably... the high point of his invention, much more than on the side of the "not-all"'615 The general contention that Lacan wants to preserve the equivocation of the maximal particular is here specifically translated into ensuring that the universal negative vacillates so that if the two universals are said to be equivalent, this 'equivalence' is nevertheless conceived in a way which harbors an internal rift. To understand this, consider how one might begin the movement of the writing of the four propositions with the universal affirmative . As seen above, Lacan's choice of the maximal particular immediately excludes this universal, which he affirms through the particular negative in a writing both denying the universal operator while sustaining the existence of Not-all. But if not-all illustrate the universal, on the right deixis the particular affirmative is obligated to say that some do not illustrate it. The movement just traced out proceeds as → → , and at this point Le Gaufey notes that it becomes clear how Lacan has taken his departure from the Aristotelian conception of the maximal particular. This latter conception would read Not-all as 'some' and so the maximal particular of 'some, but not all' in quadrant would read as as per the logician's choice of reading each proposition as per the relations of the logical square. In contrast, Lacan transforms the negation of the universal quantifier into the existential quantifier since he has chosen to write each proposition at its place.616 These places are made available for writing because of the universal negative, which logicians for their part might well read as But to ensure the universal is well evacuated from the left deixis Lacan instead writes the universal quantifier by negating the existential quantifier, that is, he substitutes for . In doing so, Lacan captures an essential ambiguity which complements the ambiguity attending the claim that the negation of universality produces existence. More specifically, is as ambiguous ('if not some, then all' but it is equally sustainable that 'if not some, then no one') as the ambiguity of ('if notall' then either 'some' or 'no one'). If Lacan's propositions are an ambiguous read, it is with good reason for his intention all along has been to capture the equivocation of language in a logical manner and his writing here does just that. For while the 'not some' of is equivalent to 'all,' it must simultaneously be read as 'no one' or 'none' or 'the place of all.' To speak more broadly so as to capture its relation to all the other propositions, the universal negative can be conceived not only classically as the set of elements not satisfying the function, but also as a locus or space devoid of any element. In Le Gaufey's words, 'with this writing [Lacan] secures a sort of bolting down of his battery of formulae which, otherwise, would go down the tubes.'617 Yet as strange as it sounds, it is the 'nothing' inscribed by that does the bolting down. An understanding of this might be gained by conceiving of this proposition as a subjective gesture, precisely the one described above as taking elements 'one by one.' That is, the subjective activity of submitting each for closer analysis is simultaneously the very operation which 'clears the space' in each of the other quadrants so that a place is made available for the determinate writing of their propositions. The complete movement is thus → → → and the paradox is that what clears the space for the propositions to be written into their places comes 'after' the propositions themselves have actually been inscribed.618

So in contrast to the logician's choice to read the Not-all of the particular negative as a partitive 'some' so that if 'some say yes' the right deixis balances things out with a 'some say no,' Lacan's writing of the universal negative inscribes his desire to break with such a neat symmetry: there, where no one says no, not-all say yes. Thus this Not-all should not be seen as partitive but as affirming that those elements present in the left deixis (and having been taken one by one) are each subject to the same function but without becoming elements of an All. Each element in the left deixis has an existence under the regime of the function but in a way which cannot be made into a determinate set. In this manner the existence of these elements remains unattached to any essence within which they would be subsumed. Because of this one must speak of the asymmetry of the Lacanian logical square. This must always be kept in mind when utilizing Figure 6.2, for despite the equivalencies running horizontally across the square which suggest symmetry there is an internal rift in this relation which is even more troubling than the formally acknowledged contradictory relation running vertically between the universal and the particular in each deixis. Moreover, Lacan's writing has effectively redoubled the logical rift, supplementing the one running between propositions by the one now discernible within the writing of the propositions due to the inversion of the universal and existential quantifiers in the left deixis. This not only leaves the classical Aristotelian logical square far behind but even takes its distance from the logician's reading of the maximal particular, for the latter can only register such rifts through the unfolding of the relations internal to the logical square. In contrast Lacan's writing thoroughly ensures that the left deixis is not treated as a mirror image of the right which would symmetrically oppose each to the other. For now each deixis must be taken as contributing reciprocally to one another as both obstacle and support.

Now that a certain 'global' understanding of the Lacanian logical square has been reached, a momentary pause in the analysis is warranted to take stock of the fact that Lacan is not simply a skilled logician. It is of course undeniable that Lacan thoroughly acts the part, masterfully commandeering both classical and modern logic to develop his own logical square. This general truth the present section has made plain and more specifically the immediate foregoing analysis has shown that Lacan retains in the left deixis the central lesson of Russell's paradox, viz., that there are well defined sets which do not exist, as opposed to the right where this paradox is not taken into account. However, these logical moves are made by Lacan the psychoanalyst, whose predominant interests lay outside the strict realm of logic. In the present case Lacan has turned to logic to search for a way to best articulate the difference between the sexes. One might say that his goal is to emulate the accomplishment of Russell. Where the latter's paradox caused a crisis at the level of the once certain foundation of mathematics, Lacan hopes to similarly articulate a radical break with the traditional binary logic of the sexes, which places man and woman on opposite sides of the sexual divide as per some pertinent universal feature. Recalling that each deixis of the Lacanian logical square is also a 'sexual side' whereby the left represents woman and the right man, the foregoing analysis already demonstrates certain successes of his project. For instance, Lacan's infamous claim that 'Woman does not exist' is reflective of the fact that in the left deixis, while individual women certainly exist, they nevertheless resist forming themselves into the set Woman. In a word, Woman is Not-all. This alone ensures that the irreducible asymmetry of the sexes remains entirely independent of any feature one might presume to impose on their relation in an attempt to bridge – or even define – the sexual rift which separates them. Moreover, this sexual rift thoroughly maps onto the logical rift(s) articulated above. As Le Gaufey correctly stresses, for Lacan sex and logic thoroughly intertwine.619

One further point should be made. The operating assumption of the present chapter has been that Lacan's formulae, written as they are in logical script, are flexible enough to be adapted to other realms. Thus both the Aristotelian and Lacanian logical squares have proven useful in discussing and articulating the hermeneutical circle. But it will be noted that those demonstrations largely proceeded from the affirmative deixis. This was not simply to keep the discussion relatively free of negating terms which might otherwise have added an avoidable level of complexity; rather, this anticipated the asymmetry of the Lacanian logical square whereby the negative deixis stands to the 'subversive left' of the affirmative deixis. Such a square suggests that if the hermeneutical circle is confined to the space of the right, the space of the left is where the suspension of meaning is accomplished. In these terms the current chapter has been laying the groundwork for Chapter 7, which seeks to more expressly demonstrate this possibility.

To gain a deeper appreciation of what has been accomplished in the left deixis, one might turn to the right to enquire how Lacan understands the status of the 'exception,' a term that has hitherto been used without complication. Above it was said that the elements to the left do not collectivize into a determinate set since their existence is without essence. Lacan's concern to maintain a distinction between existence and essence is not unrelated to the 'at least one' element to the right, that is, the particular exception to the universal affirmative. Consider that Lacan takes the universal quantifier as an ontological marker and so one directed toward being and essence as opposed to existence, just as do Aristotelian logicians. For instance, by saying 'All of Gadamer' it is understood that these signifiers have a denotative status which awaits effectuation, for they do not establish any existence by themselves but merely produce a being which could be qualified and thus produce an essence. Inversely, the existential quantifier directly asserts the existence of the element that it writes, as in the case of asserting 'There is (at least) one text of Gadamer.' Here is where Lacan begins to take his departure from Aristotelian logic. For the latter has it that this existence corresponds to an essence and so always relates to the supposed being of the universal. That existence is nothing more than the particular actualization of a being always universal in its category is reflected in the classical choice of the minimal particular. The approach of the minimal particular would, moreover, maintain a consistent universal which would unproblematically span the entire logical square so as to develop an essence for each deixis. But this is precisely what Lacan wants to deny, preferring instead 'to make existence prevail over essence'620 so as to contravene the Aristotelian derivation of existence from essence and make the universal vacillate.

To this end Lacan would have us conceive how the universal finds its true foundation in the existence of the exception which opposes it. If this is the case it certainly lends support to Lacan's decision to include negation in the right deixis so that its particular enters into contradiction with its universal just as is the case in the left deixis. Now, if existence is placed with the two particulars and existence trumps the essence to be had in the universals, for Lacan the truth would lie with the particulars, which in turn would make the two universals necessarily false. But while the conception of a universal thus grounded certainly distinguishes itself from the Aristotelian conception, one is hard pressed to understand such a claim. Le Gaufey quotes Lacan, who suggests the proverb 'the exception proves the rule' illustrates the claim, but one would have to agree with his assessment that this sheds little light on how the exception provides support for the universal.621 The further suggestion is made to look to the other deixis and generally speaking this is good advice when difficulties arise in comprehending aspects of a specific deixis; for if the logical square comprises both classical logic and its subversion, indeed there is no other gesture to make than a tautological one. In the present case it is thus pertinent to ask what would occur to the universal if there was no exception. This in effect is the claim of the universal negative , which negates the exception and it has been seen that Not-all results or , precisely the proposition which negates the universal quantifier. Again, while the logic of the right deixis places elements into a set, the logic of the left concerns only a domain of elements whose existence is indeed affirmed (recall how acts as an existential quantifier) but nevertheless cannot be collectivized into a universal set precisely because no element escapes it. Through the contradictory writings of the two propositions in the left deixis, the existing domain of elements thus also escapes having an essence produced of them.622 Here something exists which does not belong to a universal and so is devoid of essence yet at the same time cannot be conceived as an exception. So to the right deixis where an All is founded on the existence of the exception of at least one, the left deixis professes that insofar as no exception exists, those that do exist do not form an All. In contrast to the classical logical square, running across the top of the Lacanian square are propositions which deeply damage the universal, for in both partitions the universal simply cannot collectivize all the elements which would give rise to a homogeneous unity without exception. Le Gaufey is quite right that one must strive to hold together the two types of contradiction which run within and between each deixis or else the entire project easily reverts to a symmetrical square reflecting traditional bipolar thinking subsumed under a grand universal. For his part the key to doing so is to specify the relationship that the universal maintains with the exception better than Lacan himself has done.623 Following the results of his analysis is valuable for its introduction of the term 'limitation' into the vocabulary of the Lacanian logical square.

Not finding what he needs from Lacan's seminars, Le Gaufey turns instead to L'étourdit [1972]624 to find a passage suggesting that the exception be conceived as a limit. There Lacan writes of the universal affirmative that for all , is satisfied. But with regard to this very same proposition he further writes how 'there is by exception the case, familiar in mathematics (the argument = 0 in the exponential [sic] function ), the case where there exists an for which , the function, is not satisfied, namely, by not functioning, is in effect excluded.'625 The exception he speaks of is the particular affirmative and Lacan's point is that it operates precisely like how = 0 creates a limit for the function () = , which here stands in for the universal affirmative. This is visually confirmed by examining Figure 6.3 which presents the graph of this hyperbolic function.

The table determines several points (, ) on the graph of . Reading from left to right, it can be seen that for < 0, increases towards infinity as decreases towards 0; while for > 0, decreases towards 0 as increases towards infinity. However, presents no value if = 0. Mathematically speaking, = 0 is not in the domain of . That is, it is impossible to divide a real number like 1 by 0, for dividing 1 by 0 is to effectively ask 'Which number multiplied by 0 gives 1?' There is no such number because the product of any real number like 1 and 0 is 0. In a word, division by 0 is undefined. This is visually confirmed by the sloping curves of the graph which will never cross its asymptotes. The asymptotes are the lines that the branches of the hyperbola approach as they recede farther and farther from the origin; that is, as the absolute value of becomes greater and greater. For such an (equilateral) hyperbolic function as () = , the coordinate axes (i.e., = 0 and = 0 ) act as its asymptotes and it is precisely at this locus or place of where Lacan situates . This graph quite nicely provides a visual representation of how the universal takes its support on the exception, but as seen, this foundation is anything but solid. There is an undecidability here: if the submission of All under the is conditioned by the fact that at least one escapes it, should this escaped = 0 be counted amongst the All or not?

While noting how this example is quite useful in illustrating this undecidability, Le Gaufey is right to suggest how the graph tempts one to favor a 'transcendent' reading of the exception, one that is wholly in line with St. Augustine's God qua superior element to the worldly series.626 That is, since there is a stark disparity observable between the graph and its asymptotes, the temptation is to read the function as founded on a perfect exteriority. This reading is to be rejected because if accepted, it becomes legitimate to ask the subject from where he has made such an assessment. To the believer such a reading is authorized by faith, but for the non-believer well-versed in the impossibility of taking up a meta-position or bird's eye view from which a neutral assessment of the function and its limit could be made (i.e., there is no meta-language), this assessment must be abandoned. The analysis here is perfectly sound and Le Gaufey strengthens the argument by noting how no meta-position is available in the left deixis as well, although in this case the assessment would concern the status of the absence of the exception. In the end he proposes to think the exception as effectively starting from the series of of , which would allow one to resist thinking of both the series and its exception or limit as occupying a common space. This is thoroughly in line with the impossibility of taking up a meta-position. But the problem with this solution lies with its very endeavor to provide the 'correct' way of reading the relationship between the universal and its exception. In contrast, one should instead approach the right deixis in a way which preserves its undecidability and, moreover, allows one to 'wrongly' choose the meta-position reading of the relationship in a first move. For this 'incorrect' reading is part and parcel of what Lacan intends to capture with the two propositions in the right deixis. More precisely the exception is the meta-position, the point from which man audaciously claims to be able to assess it All. To clarify Le Gaufey's failing on this issue along different lines, certainly his reading of the right deixis and the status of the existing exception is ultimately correct, but what is overlooked is how this reading is done from the vantage point of the left deixis which stands logically prior to the right. Here is the reason for his recourse to the left deixis when articulating his solution and no wonder, since it is this very deixis which inscribes the impossibility of taking up a meta-position. Hence his further specific advisement that one tarry along with the series of (i.e., one by one) in order to break the spell of the meta-position. His failing here also explains his misreading of Lacan on this score. Had Le Gaufey consciously recognized how the spell of the right deixis is only broken by reading it from the more subversive left, he would not doubt that Lacan intends to capture 'a wish, religious in style, clearly worthy of... hope for an eternal life' with the exception.627 His denial that this is the case leads to a general confusion over what exactly is being inscribed in the right deixis. Broadly speaking, what is missed is how Lacan places the secondary and more phantasmatical dimension of subjectivity (inclusive of the field of meaning, as is seen below) in the right deixis while reserving the space to the left for its subversion.

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