SEXUATED TOPOLOGY AND THE
SUSPENSION OF MEANING
A NON-HERMENEUTICAL PHENOMENOLOGICAL
APPROACH TO TEXTUAL ANALYSIS
WILLIAM J. URBAN
CHAPTER 6
DERIVATION OF THE FORMULAE OF SEXUATION AND STASIS
I posed the question of what one could call a matheme, positing already that it is the pivotal point of any teaching. In other words that the only teaching is mathematical, the rest is a joke.594
Lacan spoke these words near the end of the year he spent devising his formulae of sexuation, a set of four logical propositions written in his own unique logico-mathematical shorthand. They represent a high point in Lacan's thinking that the matheme conveys something essential and in the case of sexuation this something most obviously has to do with sexual difference. His controversial claims regarding this difference have repeatedly been examined in the decades since his death. But what is often overlooked is how these formulae readily lend themselves to other fields of thought precisely because they are written as logical propositions. Since they bear no preconceived notion regarding this difference – indeed they deal with the very notion of notion, with the notion as such – this difference has fruitfully been used in the past to characterize the fields of philosophy and politics by the likes of Copjec and Žižek. By directly identifying the elements of the hermeneutical circle with the formulae themselves, the present chapter endeavors to begin demonstrating how the field of meaning undergoes a fatal disruption as it too similarly harbors this difference. Even less considered in the literature has been the fact that Lacan did not write his formulae in isolation but developed them from Aristotle by way of an influential contemporary commentary. A notable exception is Le Gaufey. This chapter makes extensive use of his recent work and further argues for a reading of these formulae that prove fruitful for interpretive theory.
Section 6.1 is a schematic presentation of the logical system of Aristotle leading up to a discussion of the classical Aristotelian logical square. Under specific consideration is whether this logical square could legitimately be conceived as a hermeneutical circle. Section 6.2 examines Lacan's decision to read the Aristotelian propositions in a more natural (and less logical) way which completely subverts the classical square and eventually leads to his own logical square – an arrangement effectively equivalent to the formulae of sexuation. Section 6.3 continues to provide a reading of the propositions in both the classical and Lacanian logical squares together with their differing consequences for textually analysis. The thesis throughout this chapter that the Lacanian logical square offers the potential to radically suspend meaning is more fully considered in the following chapter.
6.1 The Aristotelian Hermeneutical Circle?
Aristotle was the first thinker to formulate a logical system. The treatises which comprise this system were later compiled into what is known as the Organon (4th c. BC)595 and broadly speaking, the thought contained therein casted a dominating influence over philosophy from its immediate reception on through the medieval period until the 19th century. It succeeded in tying together all aspects and fields of philosophy because logic for Aristotle and his followers was not viewed as a separate and self-sufficient academic field to be considered in isolation from other fields of inquiry. Indeed that it was seen as a preliminary requirement for the study of every branch of knowledge is reflected in the very term organon [Greek, for tool or instrument] itself which suggests that careful thinking should proceed along its rigorous methodological path. We focus here on the first two treatises which make up Aristotle's logical system, viz., Categories and De interpretatione.
In his Categories, Aristotle describes the world as composed of separate yet unified whole things called substances to which various properties can be ascribed. Substances are of two kinds, primary and secondary. A primary substance is an independent object composed of matter and characterized by form. An example would be an individual man. A secondary substance is the larger group to which these primary substances belong, say, man qua species. Aristotle's logic endeavors to correctly ascribe specific properties to secondary substances, which then indirectly ascribes these same properties to primary substances. Along with the category of substance (or essence), two additional ways to describe what is in the world would be with respect to the categories of quantity and quality. It is with these three categories that Aristotelian logic evaluates arguments in the form of statements composed of words of which the most basic is the proposition. A proposition is a complete sentence that asserts something and is a complex involving two terms: a subject (a word naming a substance), a predicate (a word naming a property) and a copula [Latin, for connection or link]. The generalized logical form of a proposition is thus 'Subject is Predicate' or in traditional symbolic shorthand 'S is P' where S stands for the Subject and P for the Predicate. Following this schema, simple assertions can be analyzed accordingly. Consider the statement 'Gadamer is Heideggerian.' Gadamer is the subject (S), the property of being Heideggerian is the predicate (P) and the verb 'is' (the copula) links Gadamer and this property together in a single affirmation which claims a truth about the world. Of course asserting a truth about the world resounds more deeply if the subject in the proposition is a (secondary) substance to which an essential property is attributed, as in the case 'Triangles have interior angles which sum to 180°.' But Aristotle is flexible enough to consider propositions involving primary substances (as in the above example regarding Gadamer) and even properties of substances in the subject position.
After having fixed the proper logical form of a proposition, Aristotle in De Interpretatione makes two distinctions which have allowed logicians to classify his propositions into four different kinds. One distinction is between particular and universal terms. The particular terms refers to individual things, like the name Gadamer, while the universal term refers to groups of things (e.g., man) and so is said to be universally applicable to all members of a group. But as his own examples make clear, Aristotle does treat propositions with an individual subject as universal propositions. The other distinction made follows from Aristotle's suggestion that all propositions must either be an affirmation or a negation. With these two distinctions each different categorical proposition asserts a relationship between two categories along two lines. On the one hand a categorical proposition possesses a quantity insofar as it represents a universal or a particular predication which is denoted by the adjectives 'all' or 'some' (where 'some' is to be understood as 'at least one'), respectively. On the other hand a categorical proposition possesses a quality insofar as it affirms or denies the specified predication and this is determined by an affirmed or a negated verb. This allows one to reduce every categorical proposition to one of four logical forms:
(1) The Universal Affirmative (the so-called 'A' statement) takes the form of 'All S are P.'
(2) The Universal Negative (E) is 'All S are not P.'
(3) The Particular Affirmative (I) is 'Some S are P.'
(4) The Particular Negative (O) is 'Some S are not P.'
Aristotle discusses the way these four categorical propositions are related to one another. As he writes,
'I call an affirmation and a negation contradictory opposites when what one signifies universally the other signifies not universally, e.g. every man is white–not every man is white, no man is white–some man is white. But I call the universal affirmation and the universal negation contrary opposites, e.g. every man is just–no man is just. So these cannot be true together, but their opposites may both be true with respect to the same thing, e.g. not every man is white–some man is white.'596
The claims Aristotle makes here (that A and O are contradictories, that E and I are contradictories and that A and E are contraries) are captured in Figure 6.1. This figure reproduces the classical Aristotelian logical square but inverts its traditional presentation which places the universal and particular affirmatives on the left side with their negative counterparts to the right. The reasoning for this modification (a cosmetic change only which alters nothing substantial) is explained further below.
Aristotle's own examples are often confusing and differing translations, particularly with the choice of words like 'every' and 'no' to capture the two universal propositions, only compound the matter. In the present discussion the endeavor is thus made to use 'All' and 'Some' as per Figure 6.1, although with the license to omit the term 'All' when is it clearly implied in its opposition to 'Some.'
The two logical relationships Aristotle explicitly expresses as existing among the four logical forms can easily be illustrated by returning to the proposition 'Gadamer is Heideggerian.' As said above, although Gadamer is an individual man occupying the subject position, such a proposition is to be treated as a universal proposition. Reflecting on the standard practice in academia that the invocation of a scholar's name is often tacitly understood to refer to his body of work and not to his actual existence as a man should make this practice in Aristotelian logic equally acceptable. Thus 'Gadamer is Heideggerian' effectively says something like 'All of Gadamer's texts are Heideggerian' and does not refer to that man whose remarkably long life touched on three different centuries. It is also clear that this universal proposition is in the affirmative form as the verb affirms the predicate. Now, in considering Aristotle's first claim that such universal affirmative propositions stand in contradiction to propositions in the particular negative logical form, one finds that this is clearly the case. For if the (A) proposition 'All of Gadamer is Heideggerian' is true, then the (O) proposition 'Some of Gadamer is not Heideggerian' must be false. The relation of contradiction equally holds for the reverse case, for if it is true that 'Some of Gadamer is not Heideggerian, then 'Gadamer is Heideggerian' is obviously false. The same contradictory relationship holds as well between (E) propositions like 'Gadamer is not Heideggerian' and (I) propositions like 'Some of Gadamer is Heideggerian.' Generally speaking, two propositions stand in contradictory relationship if (and only if) they cannot both be true and both be false. Aristotle's second claim is that the universal affirmative and the universal negative propositions are related as contraries. Contrary propositions cannot both be true. For example, it cannot both be true that 'Gadamer is Heideggerian' and 'Gadamer is not Heideggerian,' for the truth of either of these contrary propositions excludes the truth of the other. However, contrary propositions are not contradictories because it is possible that both of them may be false. This would arise when it is indeed the case that 'Some of Gadamer is Heideggerian' and 'Some of Gadamer is not Heideggerian.' Generally speaking, two propositions are contraries if (and only if) they cannot both be true but can both be false.
But while Aristotle only explicitly articulates the two logical relationships of contradictoriness and contrariety (indicated in Figure 6.1 by the solid arrows), logicians since the medieval period have traditionally extended the discussion of the possible logical relationships to be had between these four categorical propositions by two others. It is usually noted that these two additional logical relationships (indicated by dotted arrows), which thereby 'complete' the logical square, are wholly in keeping with Aristotle's original analysis. One of these relationships exists between the two particular propositions occupying the lower half of the logical square. As subcontrary propositions, they cannot both be false. To demonstrate this, suppose that (I) 'Some of Gadamer is Heideggerian' is false. Then its contradictory, (E) 'Gadamer is not Heideggerian,' is true. This makes (E)'s contrary, (A) 'Gadamer is Heideggerian,' false. So (A)'s contradictory, (O) 'Some of Gadamer is not Heideggerian,' is true. This refutes the possibility that the propositions 'Some of Gadamer is Heideggerian' and 'Some of Gadamer is not Heideggerian' are both false. Said in another way, since every proposition has a contradictory opposite and since a contradictory is true when its opposite is false, it follows that while the opposites of contraries can both be true, they cannot both be false. Generally speaking, two propositions are subcontraries if (and only if) they cannot both be false but can both be true.
The other logical relationship not explicitly mentioned by Aristotle but that traditional logicians find implicit in his treatise is subalternation. As seen in Figure 6.1, subalternation is a logical relation between a particular proposition (the subaltern) and a universal proposition (the superaltern) of the same quality such that the former is implied by the latter. To demonstrate this, suppose that (A) 'Gadamer is Heideggerian' is true. Then its contrary (E) 'Gadamer is not Heideggerian' must be false. But then (E)'s contradictory, (I) 'Some of Gadamer is Heideggerian,' must be true. Thus if (A) 'Gadamer is Heideggerian' is true, then (I) 'Some of Gadamer is Heideggerian' is also true. A parallel argument establishes subalternation from (E) 'Gadamer is not Heideggerian' to (O) 'Some of Gadamer is not Heideggerian.' Generally speaking, a proposition is a subaltern of another if (and only if) it must be true if its superaltern is true, and the superaltern must be false if the subaltern is false. Traditional logicians after Aristotle have also noted that drawing the logical conclusion of subalternation can proceed without argument as it is immediately inferred from universal propositions. This does seem to be the case, for if all elements of an existent group possess (or do not possess) a specific property, it must follow that any smaller subset of that group must possess (or not possess) that specific property. So if it is true that (A) 'All of Gadamer's texts are Heideggerian,' this immediately implies that (I) 'At least one of Gadamer's texts is Heideggerian' must be true. Reformulating as a negation, the truth of (E) 'Gadamer is not Heideggerian' entails the truth of (O) 'Some of Gadamer's texts are not Heideggerian.' But it is important to recognize that subalternation is a logical relation which only proceeds from the universal to the particular proposition and does not work in the opposite direction (as indicated in Figure 6.1 where the subaltern arrow is distinct from the others by its uni-directionality). So if it is the case that 'Some of Gadamer is Heideggerian,' clearly it need not follow that 'All of Gadamer is Heideggerian.' However, note that if 'Some of Gadamer is Heideggerian' is false, it does follow that 'All of Gadamer is Heideggerian' is also false. These immediate inferences proceed in a likewise manner with the negations of these propositions.597
Traditional logicians have of course drawn other immediate inferences from these propositions as well as extending Aristotle's analyses in many other directions. But for present purposes it is enough to reflect on how the supplementation of subcontrariety and subalternation completes the fragment of the logical square Aristotle initially provided. Whereas Aristotle only explicitly commented on the logical relationships of contradictoriness and contrariety which exist among the four propositions, with the subsequent supplementation each of the four propositions can be seen as standing minimally opposed to each of the others. Taken as a whole, this gives the impression that the logical square is a rigorously defined network of relations. This network of oppositional relations between its four corners is a crucial characteristic of the 'square of opposition,' as the classical logical square is more commonly known. As has been already seen in the foregoing discussion of the logical relations themselves, the path one follows is in large part dependent on where one chooses to enter this logical network and with what purpose. So on the one hand the path once entered does indeed necessitate following a well-defined and restrictive trajectory, yet on the other the network does retain an element of contingency regarding the choice of an entry point. This fact alone may give one pause to consider how the corners of the logical square may not be as sharp as one might initially suspect so that even in such a tightly structured system there is room yet for an element of subjectivity. However, the analysis made thus far is incapable of demonstrating the inscription of the subject into such a logical structure. For the moment one must be content with contemplating a more modest possibility. Namely, the analogy to be had between the relation of the universal and particular propositional levels of Aristotle's classical logical square and the relation between the whole and the part of the classical hermeneutical circle. So articulated, it becomes evident how the discipline of the humanities generally manages its own internal rift between its traditional ground in the classical thought of the ancients and its strong hermeneutical roots in Romanticism. Today it seems to every self-assured declaration grounded in Aristotelian logic, the tendency is to quickly counter with 'Yes, but there is meaning there!' This recourse to meaning, caught up as it is in the supportive structure of language, is what begins to round out the sharp corners of any logical square into a welcoming circle.
To illustrate the legitimacy of thinking of the classical logical square as a hermeneutical circle, consider the options available to the academic who decides to make a study of Gadamer with no prior knowledge of his thought. One strategy would be to first read secondary texts which provide introductions to Gadamer's theory, or else to inquire with colleagues who have already engaged with his work.598 In both these cases he may very well initially approach the set of primary texts having learned that 'Gadamer is Heideggerian' and initially encounter particular passages which do nothing but confirm this expectation. In hermeneutical terms, the meaning he extracts from these particular passages are seen through the lens of the whole which is here taken as a proposition universally applicable to all of Gadamer's work. By the logical relationship of subalternation this is really a foregone conclusion. For if it is true that 'Gadamer is Heideggerian' as the academic assumes, then it is certainly the case that 'Some of Gadamer is Heideggerian.' But in retrospect these particular passages which confirm the universal may have just been a fortuitous encounter, for in continuing his reading of Gadamer the academic may encounter a passage in which Gadamer comes across much like the empirically minded Hirsch who has very little in common with Heidegger. This, however, does not give the academic license to now say 'Gadamer is Hirschian' (by subalternation again). Rather, he may only make the particular claim that 'Some of Gadamer is not Heideggerian,' which, of course, contradicts the claim that 'Gadamer is Heideggerian.' Related as subcontraries, both these particular (affirmative and negative) passages can be true, but both cannot be false, which further leads to the falsity of the two universal propositions of 'Gadamer is Heideggerian' and 'Gadamer is not Heideggerian' (as per the logical relationship of contrariety). Either way the academic is obligated to formulate a new universal proposition which, on the basis of these particulars, is perhaps something like 'Gadamer is Heideggerian subject to proviso X.' In hermeneutical terms, the meaning of Gadamer's work as a whole is conditioned through the approach of some of its textual parts. With this new universal affirmative in mind the reading may continue untroubled or at least until the time that enough textual evidence has mounted to again trigger this process which ends in the reformulation of a new universal; this new universal in turn places passages to come and those already read into yet another light. It is obvious this process can continue ad infinitum, ever fine-tuning the universal through which nuanced meaning is continuously unearthed from its particular passages. It should also be clear that the complex web of logical relationships travelled, viz., from subalternation and subcontrariety through contradictoriness and contrariety, is nothing more than a rigorously articulated version of the interdependent relations which exist between the whole and the part in the hermeneutical circle, that circle which hermeneuts have expressed in various fashions from Ast and Schleiermacher onward.
The foregoing analysis demonstrates how the circular path in the classical logical square is initialized with the universal affirmative proposition. But instead of repeating the exercise to demonstrate how the circular path is equally set in motion with the three remaining logical forms, we do better to note how the classical logical square harbors a certain equivocation which potentially upsets its definitiveness. The problem concerns the status of the particular proposition and can be raised by inquiring into the source of its truth whenever such a claim is asserted. Illustrating this problem with the particular affirmative, note how this logical form is connected to all the others in one of three different logical relationships (as is the case with each of the four propositions). This is readily seen in Figure 6.1 where three arrows emanate to the proposition 'Some S are P'. Now, which of the three other propositions may be said to account for the truth of the particular affirmative? More generally, what are the implications of affirming the particular proposition? To use the ongoing example, say the academic eschews a preliminary investigation of Gadamer through secondary sources to instead immediately dive into his primary texts; moreover, suppose he there finds evidence to affirm the proposition 'Some of Gadamer is Heideggerian.' The truth of this particular proposition could, of course, contradict the falsity of the universal negative 'Gadamer is not Heideggerian,' but this conclusion is rather inconsequential. The two remaining possibilities are much more significant as they provide a source of this truth by way of their own truth. On the one hand, by subalternation the source of the truth of 'Some of Gadamer is Heideggerian' could be the universal affirmative 'Gadamer is Heideggerian,' for if all of Gadamer's texts are Heideggerian, then it is also true, a fortiori, about some of his texts. On the other hand, by subcontrariety the source of its truth could be the situation in which the particular negative proposition is also true. In this case the propositions 'Some of Gadamer is Heideggerian' and 'Some of Gadamer is not Heideggerian' are both true and this simultaneously sets aside the truth of the universal affirmative. Why? Because the truth of the particular negative stands in contradiction to the universal affirmative. Generally speaking, the two relations of subalternation and subcontrariety are thoroughly incompatible from the perspective of the particular in the sense that the propositions which they connect cannot be affirmed together. Clearly a choice must be made as to how particular propositions should be interpreted: if one chooses to proceed as per the first case, the meaning of the particular will be in agreement with the universal of the same quality. But if one makes the alternate choice, the meaning of the particular will exclude the truth of the universal of the same quality through the affirmation of the particular of the opposite quality.
The importance of this choice should not be underestimated, for it has crucial consequences in terms of textual exegesis. For instance, overly-eager scholarly work which rushes to cast universal judgment on a text or author on the basis of scant readings of particular passages is no doubt symptomatic of having settled with the former option. But the theoretical consequences of this choice are potentially far graver. Recall in the discussion above how the logical relation of subalternation only proceeds in one direction, from the universal to its particular proposition. Yet if the affirmation of 'some' simultaneously affirms 'all,' the logical relation now proceeds in both directions. That is, it is no longer just the case that the truth of the universal implies its particular, for the truth of the particular implies the universal as well. The uni-directional arrow in Figure 6.1 now becomes a doubled arrow to indicate how the logical relation between the universal and particular levels of the same quality is no longer one of subalternation. Their once rigorously defined opposition has now been relaxed and one becomes hard pressed to distinguish the two levels. Most importantly, if the particular and the universal levels risk collapsing into one another, the analogy to be had between the classical logical square and the hermeneutical circle is no longer operative, just as would be the case if it was shown how there is effectively no difference between the whole and the part.
But did Aristotle choose this option? Ultimately, yes. Section 6.2 below further discusses these matters within the context of Lacan's decision to follow the alternative option, which excludes the universal. It also begins to explore some of the consequences of this decision, which is one way of characterizing the undertaking of Part II. But before turning to this discussion, it should be repeated how Aristotle only explicitly articulated contradictoriness and contrariety whereas it was later logicians who 'implicitly' found in his treatises additional logical relations like subcontrariety and subalternation to produce the classical logical square which often bears his name. This raises the possibility that Aristotle was himself unsure of the proper course to follow which may have in turn set these later logicians to work to construct a complete and unequivocal logical system to resolve the ambiguity they found harboring in Aristotle's original texts. But after the work of Lacan and those Aristotelian scholars he followed in the 1960s, it is now known that Aristotle's uncertainty was for good reason. For no amount of work can ever remove the equivocation which roots itself at the formal level of any articulated system of logic. Such a system could, however, theoretically inscribe that very equivocation itself into its set of logical formulae. This is precisely what Lacan endeavored to do with his formulae of sexuation.
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