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The Aristotelian Roots of Lacan's Formulas of Sexuation

On Tuesday, March 13, 1973, the psychoanalyst Jacques Lacan walked into his seminar and wrote two pairs of logical propositions on the blackboard:

masculine and feminine universal and particular propositions

He then began lecturing that day, referring to the propositions on the left as 'masculine' while dubbing those to the right 'feminine.'

That Lacan formulized the psychoanalytic notion of sexual difference in this fashion is known to anyone who has read the official transcripts of Seminar XX (a seminar first published in French in 1975, and in English in 1998). Given that most of his seminars remain unpublished, readers might be forgiven for thinking Lacan came up these propositions on the fly, as he was occasionally wont to do. But in actual fact he constructed them over the course of an entire year, from their initial introduction on March 17, 1971 to their finalized form on March 3, 1972.

A lesser known fact is how Lacan's march through sexuation proceeded from Aristotle by way of Frege, and a 1969 article published by Jacques Brunschwig entitled "Aristotle on Particular Propositions and Proofs of Inconclusiveness."1 We have Guy Le Gaufey to thank for having extensively traced through these developments in his 2006 book Lacan's Notall: Logical Consistency, Clinical Consequences.2

What follows is a high-level comparison of the Lacanian and Aristotelian logical systems.

Aristotle's Logical Square

Aristotle was the first thinker to formulate a logical system. The treatises which comprise his system were eventually compiled into the Organon (4th c. BC). In one of these treatises, Categories, Aristotle describes the world as composed of separate yet unified whole things to which various properties can be ascribed. There are two kinds of substances, primary and secondary. A primary substance is an independent object composed of matter and characterized by form. A secondary substance is the group to which primary substances belong.

For example, an individual man is a primary substance, while man qua species would be a secondary substance. Aristotle endeavors to ascribe specific properties to secondary substances, which indirectly ascribes these same properties to primary substances: if man qua species is rational, an individual man is rational as well.

Along with the category of substance, Aristotle describes what is in the world with respect to the categories of quantity and quality.

It is with these three categories that Aristotelian logic evaluates arguments. These arguments take the form of statements composed of words, the most basic of which is the proposition. A proposition is a complete sentence that asserts something. It is a complex involving two terms: a subject (a word naming a substance), and a predicate (a word naming a property), linked together by a copula.

The generalized logical form of a proposition is thus 'Subject is Predicate,' or simply 'S is P.' This form can be used to assert a truth about the world.

After fixing the proper logical form of a proposition, Aristotle in his treatise De Interpretatione makes two further distinctions. One of these is between particular and universal terms. The particular term refers to individual things, while the universal term refers to groups of things and so is applicable to all members of a group. The second distinction 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, denoted by the adjectives 'all' or 'some' (where 'some' is 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, which is determined by an affirmed or a negated verb.

This allows the logician to reduce every categorical proposition to one of four logical forms:

  1. The Universal Affirmative (the A statement) takes the form '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 then proceeds to discuss how these four categorical propositions are related to one another. Ever since, his propositions have been arranged in the following fashion:

The classical Aristotelian logical square (aka square of opposition), with 4 propositions: All S are P, Some S are P, All S are not P, Some S are not P; with arrows expressing the logical relationships of contradictoriness, contrariety, subcontrariety, and subalternation; accompanied by the traditional A, E, I, O shorthand

But it is important to note that Aristotle explicitly articulates just two of the four relations in the logical square which bears his name, or the 'square of opposition' as it is more generically known. These relationships are contradictoriness and contrariety.

Contradictoriness: Aristotle reasoned that universal affirmative propositions stand in contradiction to propositions in the particular negative logical form.

That is, if A is true, then O must be false. As well, if O is true, then A must be false. The same logical relation holds between E and I propositions. Generalizing, two propositions stand in contradictory relationship if (and only if) they cannot both be true and both be false.

Contrariety: Aristotle also reasoned that universal affirmative and universal negative propositions cannot both be true. Clearly, the truth of A excludes the truth of E, and vice versa. However, they are not contradictories because it is possible that both of them may be false (i.e., when I and O are true).

Generalizing, two propositions are contraries if (and only if) they cannot both be true but can both be false.

Medieval Supplementations to Aristotle

Logicians since the medieval period have traditionally extended the number of logical relationships to be had between these four prepositions by two others. It is usually argued that these are wholly in keeping with Aristotle's original analysis. These relationships are subcontrariety and subalternation.

Subcontrariety: The two particular propositions at the bottom of Aristotle's logical square are subcontraries. That is, they cannot both be false. To demonstrate this, suppose that I is false. Then its contradictory E is true. This makes A, the contrary of the latter, false, and its contradictory O true. This rules out the possibility that I and O are both false. Generalizing, two propositions are subcontraries if (and only if) they cannot both be false but can both be true.

Subalternation: This logical relation adheres between a universal proposition (the superaltern) and a particular proposition (the subaltern) of the same quality such that the latter is implied by the former. So if A is true, then its contrary E must be false. But then the contradictory of the latter, I, must be true. Thus, if A is true, I is also true. Generalizing, 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.

For most logicians, the logical conclusion of subalternation can proceed without argument since it is readily inferred from universal propositions. That is, if all elements of an existent group possess (or do not possess) a specific property, it follows that any smaller subset must also possess (or not possess) that specific property.

But it is important to recognize that the logical relation of subalternation only proceeds from the universal to the particular proposition and does not work in the opposite direction (note how the arrow denoting subalternation in Aristotle's logical square is uni-directional, unlike the others). So if I is true, it need not follow that A is true. However, if I is false, it does follow that A is also false.

Upsetting Aristotelian Definitiveness

The supplementation of the latter two logical relationships appears to complete the fragment of the logical square Aristotle initially provided. Whereas Aristotle only explicitly discussed contradictoriness and contrariety, with the addition of subcontrariety and subalternation each of the four propositions can be seen as standing minimally opposed to each of the others. This certainly gives the impression that his logical square is a rigorously defined network of relations. It seems that once you enter the square, you are compelled to follow a well-defined and restrictive trajectory.

Nevertheless, 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.

Illustrating with the particular affirmative 'Some S are P,' there are three possibilities.

The truth of this particular proposition could, of course, contradict the falsity of the universal negative 'All S are not P.' But this conclusion is rather inconsequential.

The two remaining possibilities are more significant in that they provide a source of this truth by way of their own truth.

  1. By subalternation, the truth of 'Some S are P' could be sourced to the universal affirmative 'All S are P.' For if the latter is true, then, a fortiori, the former is also true.
  2. By subcontrariety, the truth of 'Some S are P' could be sourced to the truth of the particular negative proposition 'Some S are not P.' But in this case, the truth of the universal affirmative is set aside since the negative proposition stands in contradiction to it.

Generally speaking, subalternation and subcontrariety are thoroughly incompatible from the perspective of the particular, in the sense that the propositions they connect cannot be affirmed together. A choice must therefore be made as to how particular propositions should be read.

If option one is chosen, the meaning of the particular will be in agreement with the universal of the same quality.

If option two is chosen, the meaning of the particular will exclude the truth of the universal of the same quality by way of the truth of the particular of the opposite quality.

The importance of this choice cannot be overstated. (Here's a political example: Aristotle, Trump and Mexicans)

Recall how the logical relation of subalternation only proceeds in one direction, from the universal to its particular proposition. Yet if affirming 'some' simultaneously affirms 'all,' it would proceed in both directions. Not only would the truth of the universal imply its particular, but the truth of the particular would imply its universal. The uni-directional arrow in Aristotle's logical square would then become double-headed. This would shatter the logical square's pretense to definitiveness since the particular and universal levels could no longer be distinguished.

Evidence suggests that while Aristotle wavered between these two options, his quest for definitiveness ultimately drove him to settle on the first option. Classical logicians have largely followed suit in their own attempts to construct an unequivocal logical system which could resolve the ambiguous status of the particular proposition once and for all. In contrast, Lacan chooses the second option which excludes the universal. For he recognizes how no amount of work can ever remove the equivocation which roots itself at the formal level of any articulated system of logic. Accordingly, he sets out to construct a system that inscribes that very equivocation itself into its set of logical propositions. What results is the Lacanian logical square, a set of four propositions effectively equivalent to his formulae of sexuation.

Lacan Meets Frege

In many respects, option one is the most natural choice. Few would conclude from 'Some S are P' that 'All S are P.' Yet a logician favoring option two would do just that, arguing how the former statement is only a particular instantiation of the universal affirmative.

But the immediate consequences of Lacan's choice are paradoxical. By choosing to read the particular as 'some, not all,' he completely undermines the logical relations of the classical logical square to the point which strains common sense.

Consider how 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 is now parallel to 'Some S are not P.' In other words, with the removal of subalternation, the particular affirmative and particular negative are no longer logically related as subcontraries but must be thought of as effectively equivalent.

It follows 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 this seems entirely unreasonable. How can 'All S is P' be considered equivalent to '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 (Aristotelian) terms, Lacan's claims cannot be defended. But they do become defensible through his proposed changes to the writing of these propositions. As a cursory glance of his formulae will confirm, Lacan rewrites Aristotle using Frege's predicate logic – a novel 'mathematical' approach to the analysis of sentences in ordinary language. Using Fregean logic, the subject and predicate terms of a sentence are to be replaced with the mathematical notions of argument and function, respectively.

For example, 8 is the value of the function algebraic expression: x times 4 for the argument 2, just as 12 is the value of the same function for the argument 3. In a likewise fashion, the function '( ) is P' takes on different values according to which arguments 'S' are used.

Far from a simple alternative terminology, this introduces a significant change in the understanding of how sentences are constructed. And again, 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?

The Mathemes of Sexuation

One symbol used in both the argument and function components of each of his new logical propositions is algebraic symbol x. Given that his four propositions are the four formulae of sexuation, this could be read as the human being who is to be classified as 'man' or 'woman.'

But Lacan frustrates this simple reading precisely by using the symbol algebraic symbol x which he borrows from algebra where it traditionally operates as a variable able to take on different object-terms. So this algebraic symbol x is better read as 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 algebraic symbol x.

One of these quantifiers is the universal quantifier, which is marked by the symbol logical symbol for 'All' and can be read as 'All.' Thus, the argument logical symbol for 'All x' reads as 'All algebraic symbol x.'

The other quantifier is the existential quantifier, which is marked by the symbol logical symbol for 'at least one' and can be read as 'at least one.' Thus, the argument logical symbol for 'There is at least one x' reads as 'There is at least one algebraic symbol x.'

These two are, respectively, the Lacanian versions of 'All S' and 'Some S' in the Aristotelian logical square.

Lacan also follows Aristotle in determining the quality of a proposition through affirmation or negation. But he does depart from classical practice which only entertains the affirmation or negation of the specified predication, by further allowing for the direct negation of the argument component of a proposition. His method is to place a bar over the component of the proposition to be negated.

Hence, the argument logical symbol for 'There is no x' can be read as 'There is no algebraic symbol x,' and the argument logical symbol for 'Not-all x' can be read as 'Not-all algebraic symbol x.' A component that remains unmarked in this manner signifies affirmation.

To write the four propositions, each of the four arguments logical symbol for 'All x', logical symbol for 'There is at least one x', logical symbol for 'There is no x', and logical symbol for 'Not-all x' are placed in relation to a function which together determines its value. Given Lacan's use of algebraic symbol x, one might expect the function to take the traditional mathematical form of algebraic symbol f(x). This is indeed the case, although he drops the parentheses and replaces the algebraic symbol f with Lacanian symbol for the phallic signifier so that the function to which the four arguments are (or are not) submitted to becomes Lacanian symbol for the phallic function. This Lacanian symbol for the phallic signifier is the symbol Lacan uses for the phallic signifier, and when combined with algebraic symbol x it produces Lacanian symbol for the phallic function, which can be read as 'the phallic function.'

Following Frege, Lacan considers the phallic function Lacanian symbol for the phallic function a concept of the phallic signifier Lacanian symbol for the phallic signifier reconceived as a function. That is, it has the capacity to submit an algebraic symbol x under the concept Lacanian symbol for the phallic signifier.

But as seen, an algebraic symbol x stands in relation to a function like Lacanian symbol for the phallic function as per one of the four arguments of logical symbol for 'All x', logical symbol for 'There is at least one x', logical symbol for 'There is no x', and logical symbol for 'Not-all x'. And what makes this even more complex is that Lacan allows for the placement of the bar of negation over this function so that the four arguments can potentially be combined with either Lacanian symbol for the phallic function or Lacanian symbol for the negated phallic function to produce his four propositions.

Lacan's Logical Square

Lacan's final decision on these combinations is reflected in the following figure, the Lacanian logical square:

The Lacanian logical square, with 4 propositions written with Lacan's mathemes of sexuation with accompanying text: All x are submitted to the phallic function, There is (at least) one x which is not submitted to the phallic function, There is no x which is not submitted to the phallic function, Not-all x are submitted to the phallic function; with arrows expressing the logical relationships of contradictoriness and equivalence

It should be noted that Lacan does not present a logical square in any of his seminars or published writings. But its construction is rather straightforward given that these four propositions appear without modification in the upper portion of the table of sexuation Lacan formally presents on March 13, 1973. While he does invert the vertical ordering on the left side of this table, this is merely a cosmetic difference which changes nothing substantially. A direct comparison of Lacan's and Aristotle's logical systems is thus made possible by simply reversing this ordering.

Such a comparison 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 that Lacanian sexuated formula 'All x are submitted to the phallic function' does not represent a change over 'All S are P.'

But Lacan clearly departs from the writing of the other three propositions by modifying their quantity or quality (or both).

Where classical logicians write 'Some S are P,' Lacan now writes Lacanian sexuated formula 'There is (at least) one x which is not submitted to the phallic function'. In both logical squares these are particular affirmative propositions. Yet Lacan's writing negates the phallic function. So while both squares have a negative right side, Lacan complicates matters by having negation appear on the left side as well. This captures his claim that the particular affirmative stands in contradiction to its universal.

Contradiction also occurs within the other side, for this same logical relation is to be had between the universal and the particular negative propositions. That is, 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: Lacanian sexuated formula 'There is no x which is not submitted to the phallic function'.

Broadly speaking, a double negative returns the proposition to a qualitative status effectively equivalent to the universal affirmative. It 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 have been the case had Lacan strictly followed the format in the classical square where 'Some S are not P'): Lacanian sexuated formula 'Not-all x are submitted to the phallic function'. Apparently, this proposition retains an equivalency to the similarly singly-negated particular affirmative proposition.

The negative side of Lacan's logical square has a further curiosity in that the quantifiers of their arguments are reversed from those of the affirmative side, the latter of which seem to better accord with those of Aristotle's square.

A high-level comparison of the two logical 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 particular) levels, and the same two (affirmative and negative) sides. But once these formal similarities are set aside, the differences rapidly begin to mount, both at the level of the individual propositions, and by extension, to the relationships between them.

The intriguing question of why and to what end Lacan chose these specific argument-function combinations is beyond the scope of the present discussion. For a further discussion of these matters in the context of interpretive theory and textual analysis, interested readers are directed to William J. Urban's 2015 book Lacan and Meaning: Sexuation, Discourse Theory, and Topology in the Age of Hermeneutics available for purchase or as a PDF download free of charge.

1 A synopsis of "La proposition particulière et les preuves de non-concluance chez Aristote" can be found here.
2 An unofficial translation of Le Pastout de Lacan: consistance logique, conséquences cliniques has been made by Cormac Gallagher.

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