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(All parts in English)

I will try to outline for your use some essential, fundamental relations, starting from what constitutes our subject this year. I hope no one will object to abstraction merely on the grounds that the term would be inappropriate.

As you will see, nothing is more concrete than what I am about to put forward, even if the theme does not correspond to the kind of thickness that its connotation implies for many. It is about making you sensitive to a proposition that, until now, I have only presented in the guise of a kind of aphorism, which might have played the role of an axiom at a certain turning point in our discourse, such as this one:

“There is no metalanguage.”

A formula that seems to go precisely against everything that is given, if not in experience, at least in the writings of those who attempt to establish the function of language. At the very least, in many cases, they identify in language some differentiations they find useful as a starting point, beginning for example with an object-language [Russell], on the basis of which they build a certain number of differentiations.

The very act of such an operation seems indeed to imply that to speak about language, one must use something that is not part of it, something that would encompass it in an order other than the one that makes it function. The solution to these apparent contradictions, which manifest in discourse, in what is said, is to be found in a function that it seems essential to me to bring out, at least by means that will allow me to inaugurate it specifically for our subject.

For the logic of fantasy could not, in any way, be articulated without reference to what it concerns, namely something that, at least to name it, I pin down with this term: writing.

Of course, this is not to say that it is what you know under the ordinary connotations of this word. But if I choose it, it is precisely because it must have some relation with what we have to articulate. One point on which we will have to insist today, constantly, is this: that it is not the same thing, after having said something, to write it or to write that one says it. The second operation is essential—it is the function of writing, in the sense in which I want to demonstrate its importance, as far as our most specific references in this year’s subject are concerned.

This from the outset presents paradoxical consequences. After all, why not—to stimulate your attention—start again from something that, by a certain approach, I have already presented before you, and without it being possible to say, I believe, that I am repeating myself.

It is quite in the nature of the things that stir here to emerge… from some angle, some approach, some edge that pierces a surface on which, by the very act of speaking, we are obliged to position ourselves… that they appear at some moment before taking on a function. So here is what I once wrote on the board:

This could have been presented in the form of a little character from whose mouth emerged what in comic strips is called a bubble, in which case you would all have agreed, and I would not have contradicted you, on the number 5. It is clear that from the moment this sentence is written: “The smallest whole number that is not written on this board,” the number 5, by the very fact of being written there, is excluded. You then only have to ask yourselves whether the smallest number being sought might, by chance, be the number 6. But you will encounter the same difficulty again: as soon as you consider the number 6 as “the smallest whole number that is not written on this board,” this number 6 is written there. And so on…

This, like many paradoxes, is only of interest for what we want to make of it. What follows will show you that it may not have been useless to introduce the function of writing by this approach, where it can present a certain enigma. A logically proper enigma: it is not a worse way than another to show you the close relationship between the apparatus of writing and what one can call logic.

From the outset, this also deserves to be recalled… at the moment when most of those present here have an adequate notion of it, and for those who would have none, this may serve as a point of attachment… reminding them that in no way can “new” steps, certainly new in the sense that they are distant, be subsumed within the framework of classical or traditional logic. The new developments in logic are entirely linked to operations of writing.

Let us pose a question here. For a long time, I have been speaking about the function of language. To articulate what concerns the subject of the unconscious, I constructed the graph. I had to build it level by level, with an audience whose reluctance to listen to me, at the very least, was palpable. This graph serves to organize what, in the function of speech, is defined by the field required by the structure of language and demanded by the paths of discourse—or what I called “the corridors of the signifier.”

Somewhere in this graph is inscribed the capital letter A, on the right, on the lower line. If someone can erase this, I could quickly redraw the entire graph for those who do not know it.

This little a [Lapsus of Lacan], which in a sense can be identified with the place of the Other, which is also the place where everything that can be called an enunciation occurs, in the broadest sense of the term—that is, what I have incidentally called “the treasure of the signifier,” which is not limited, in principle, to the words of the dictionary.

Precisely when, correlatively to the construction of this graph, I began to speak of the witticism, approaching things by a path that may have seemed the most surprising and the most difficult to my listeners at the time, but which was precisely necessary to avoid all confusion.

The nonsensical trait… not “senseless” but close to that play which English defines quite well, resonates under the term “non sense”… which there is in the witticism, whose dimension I aimed to bring out and of which I showed the kinship—at least at the level of reception, of the tympanic vibration—the kinship it has with what was, for us, in a time of trial, the personal message. The personal message, that is, any enunciation as such, insofar as it is cut out nonsensically. I alluded to this last time, by recalling the famous “Colourless green ideas,” etc.

The whole set, then, of enunciations—I do not say: of propositions—also belongs to this universe of discourse which is situated in A. The question that arises, and which is properly a structural question, the one that gives meaning to my statement that the unconscious is structured like a language—which is a pleonasm in my enunciation, since I identify structure with “like a language”—concerns the structure, precisely, that I will try today to set into operation before you.

What is it about this universe of discourse insofar as it involves this play of the signifier? Insofar as it defines these two dimensions of metaphor: to the extent that the chain can always be “grafted” with another chain through the operation of substitution. And insofar as, essentially, it signifies that slippage which is due to the fact that no signifier properly belongs to any signification.

Recalling this mobility of the universe of discourse that allows for this sea—m,e,r—of variations in what constitutes meanings, this essentially shifting and transitory order where nothing, as I said at the time, is secured except by the function of what I have metaphorically called “quilting points”: it is this, today—this universe of discourse—that must be questioned starting from this sole “axiom,” the one for which we must determine what, within this universe of discourse, it can specify.

An axiom which is the one I advanced last time: that the signifier… this signifier which we have until now defined by its function of representing a subject for another signifier… this signifier, what does it represent in relation to itself, in its repetition as a signifying unit? This is defined by the “axiom”: that no signifier… even—and very precisely when it is—reduced to its minimal form, that which we call the letter… can signify itself. The mathematical usage, which precisely lies in the fact that when we have somewhere—and not only, as you know, in an algebraic exercise—when we have somewhere posited a capital letter A, we then reuse it as if it were, the second time we use it, still the same.

Do not raise this objection—I am not here today to give you a mathematics lesson—just know that every proper formulation of any use of letters—even precisely in what is closest to us today, for example in the use of a Markov chain—will require of any teacher, and this is what MARKOV himself did, the step that is in some way propedeutic to making clearly felt what there is of impasse, of arbitrariness, of the absolutely unjustifiable, in this use, the second time, of the A (even if only apparently so) to represent the first A as if it were always the same.

This is a difficulty that lies at the very root of the mathematical use of this so-called identity. We are not expressly concerned with it here today, since we are not dealing with mathematics. I simply want to remind you that the foundation that the signifier is not entitled to signify itself is admitted even by those who, on occasion, may make a use of it that is contradictory to this principle—at least in appearance. It would be easy to see by what trick this is possible, but I do not have the time to get lost in that.

I simply want to continue, and without tiring you further, my argument, which is the following:
— What is the consequence, in this universe of discourse, of the principle that “the signifier cannot signify itself”?
— What does this axiom specify in this universe of discourse insofar as it is constituted, all in all, by everything that can be said?
— What kind of specification is it, and does this specification determined by the axiom belong to the universe of discourse?

If it does not belong to it, then that is, certainly for us, a problem. What is specified, I repeat, by the axiomatic statement that “the signifier cannot signify itself” would have as a consequence the specification of something that, as such, would not be part of the universe of discourse—when we have just admitted, precisely, that this universe includes everything that can be said.

Would we then find ourselves in some deduction that would mean this: that what, in this way, cannot be part of the universe of discourse, could not be said in any way? And of course, it is clear that, since we are talking about this thing I am bringing to you, it is obviously not to tell you that it is the ineffable theme of which it is known, purely by coherence and not for that reason because I belong to Mr. WITTGENSTEIN’s school, I consider as: “it is pointless to speak.”

Before reaching such a formulation—which, after all, you can clearly see I do not spare you the prominence or the impasse it constitutes, since we are indeed going to have to return to it—I truly do everything I can to clear the paths for you within what I am trying to have you follow me in… let us first take care to test this: namely, that what is specified by the axiom that “the signifier cannot signify itself” remains part of the universe of discourse. What must we then posit?

What is at stake, what is specified by the relation I stated in the form that “the signifier cannot signify itself”… let us arbitrarily adopt the use of a small sign employed in this logic that is grounded in writing, this “W” which you will recognize in form—these games are perhaps not purely accidental—as my punch mark, whose cap, so to speak, has been tipped over, opened like a small box, and which is used, this W, to designate exclusion in set theory. In other words, what the Latin “or” designates, expressed as “aut”: one or the other.

… the signifier in its repeated presentation functions only as functioning either the first time or functioning the second: between the one and the other, there is a radical gap. This is what is meant by the signifier cannot signify itself: (S) W (S).

We suppose, as we have said, that what this axiom determines as a specification in the universe of discourse, which we will designate by a signifier B, an essential signifier—which you will notice can be appropriately aligned with what the axiom states: that it cannot, in a certain relation and from a certain relation, engender any signification. B is precisely that signifier of which nothing objects that it be specified by this: that it marks, if I may say, this sterility.

The signifier in itself being precisely characterized by this: that there is nothing obligatory, that it is far from being spontaneously, from the outset, generative of a signification. This is what gives me the right to symbolize with the signifier B this trait: that the relation of the signifier to itself engenders no signification: B◊B.

But let us begin with what, after all, seems to impose itself: namely, that something I am stating to you belongs to the universe of discourse—let us see what results from this.

This is why I momentarily use—because after all it doesn’t seem inappropriate to me—my little punch mark to say that B is part of A, that it has with it relations whose richness I will certainly have to bring into play for you throughout this year, and whose complexity I indicated to you last time by breaking down this little symbol in all the binary ways in which it can be rendered: B < A

The question is then to know whether some contradiction results from this, namely whether…
by the very fact that we have written that the signifier cannot signify itself…
we will be able to write that this B, not that it signifies itself, but, being part of the universe of discourse, can be considered as something which, in the mode that characterizes what we have called a specification, can be written: B is part of itself.
It is clear that the question arises: “Is B part of itself?”

In other words, what roots the notion of specification—namely, what we have learned to distinguish in several logical varieties—I mean that I hope there are enough people here who know that the functioning of a set is not strictly superimposable on that of a class, but that all this, at its origin, must be rooted in this principle of a specification.

Here, we are faced with something whose kinship must surely still resonate in your ears from what I recalled last time, namely Russell’s paradox, insofar as I state that here, in the terms that concern us, the function of sets… insofar as it does something that I, myself, have not yet done, since I am not here to introduce it, but rather to keep you within a field which, logically, is anterior; but introduce something it does—this is the opportunity, on this subject, to try to grasp: namely, what grounds the deployment of the apparatus called set theory, which today presents itself as entirely foundational, undoubtedly, to any mathematical statement, and for which logic is nothing other than what mathematical symbolism can grasp… this function of sets will also be the principle—and this is what I question—of any foundation of logic.

If there is a logic of fantasy, it is precisely because it is more principled with respect to all logic that fits into the formalizing channels where it has, as I said, proven itself so fruitful in the modern era. So let us try to see what Russell’s paradox means when it covers something that is not far from what is there on the board. Simply put, it promotes as entirely encompassing this fact of a type of signifier, which it considers moreover as a class. Strange mistake!

To say, for example, that the word “obsolete” represents a class in which it would be included itself, on the pretext that the word “obsolete” is obsolete, is certainly a little sleight of hand, which has strictly no interest except in founding as a class the signifiers that do not signify themselves. Whereas we precisely posit as an axiom here that in no case “can the signifier signify itself,” and it is from there that we must begin, from there that we must find our way—if only to realize that it must be explained differently how the word “obsolete” can be qualified as obsolete. It is absolutely essential to bring into it what the division of the subject introduces.

But let us leave aside “obsolete” and start from the opposition that RUSSELL introduces to point out something that would be a contradiction in the formula that would be stated thus: B < A / (S) W (S). Of a subset B whose status would be impossible to determine, based on the fact that it would be specified in another set A by a characteristic such that an element of A would not contain itself.

Is there any subset defined by this proposition of the existence of elements that do not contain themselves?

It is certainly easy, under this condition, to demonstrate the contradiction that exists in this, since we only have to take an element y as being part of B, as an element of B: y ∈ B [∈: belongs to…, ∉: does not belong to…] to realize the consequences that follow from making it, as such, simultaneously: — part of A, as an element: y ∈ B, y ∈ A, — and not being an element of itself: y ∉ y.

The contradiction reveals itself when we substitute B in place of y: B ∈ B, B ∈ A, B ∉ B, and we see that the formula plays out in the following way: every time we make B an element of B, it results, due to the interdependence of the formula, that since B is part of A, it must not be part of itself; and if on the other hand—B being substituted in the place of this y—if, on the other hand, it is not part of itself, thus satisfying the right-hand parenthesis of the formula, it then becomes part of itself, being one of those y that are elements of B.

Such is the contradiction before which Russell’s paradox places us.

The question is whether, in our own register, we can stop here, even if only in passing to realize what the contradiction highlighted in set theory signifies—which may allow us to say in what way set theory is specified within logic, namely what step it constitutes in relation to the one we are attempting to institute here—a more radical one.

The contradiction in question, at the level where Russell’s paradox is articulated, lies precisely—as the mere use of words delivers it to us—in the fact that I say it. For if I do not say it, nothing prevents this formula—very precisely the second one—from standing as such, written, and nothing says that its usage would stop there.

What I say here is by no means a play on words, for set theory as such has absolutely no other support except that I write it as such: that everything that can be said of a difference between elements is excluded from the game. To write, to manipulate the literal play that constitutes set theory, consists in writing—as such—what I state here, namely that the first set may be formed simultaneously of:

— the pleasant person who today, for the first time, is typing my lecture, — the condensation on that windowpane, — and an idea that at this very moment is passing through my mind,

…that this constitutes a set by virtue of this: that I expressly say that no other difference exists than that which is constituted by the fact that I can apply to these three objects, which I have just named, and whose heterogeneity is evident to you, a unary trait on each, and nothing else.

So then, this is why, since we are not on the level of such a specification, since what I bring into play is the universe of discourse, my question does not encounter Russell’s paradox—namely, that no impasse, no impossibility follows from the fact that B, which I do not yet know, but which I have begun to suppose might be part of the universe of discourse—indeed it…

even though formed by the specification that: “the signifier cannot signify itself”

…may perhaps have with itself that kind of relation which escapes Russell’s paradox—namely, may demonstrate to us something which might be its own dimension, and in relation to which we will see under what status it does or does not form part of the universe of discourse.

Indeed, if I took care to remind you of the existence of Russell’s paradox, it is probably because I am going to be able to use it to make you feel something. I will make you feel it first in the simplest way, and then in a way that is a bit richer.

I make you feel it in the simplest way because I have, for some time now, been prepared for all concessions [Laughter]. People want me to say simple things—well then, I will say simple things! You are already sufficiently trained in this, thanks to my efforts, to know that understanding is not such a direct path. Perhaps, even if what I say to you appears simple, you will still retain a certain mistrust…

A catalogue of catalogues: that is indeed, at first glance, a matter of the signifier. Why should we be surprised that it does not contain itself? Of course, for us, this seems required from the outset. Nevertheless, nothing would prevent the catalogue of all catalogues that do not contain themselves from printing itself inside itself! In truth, nothing would prevent it—not even the contradiction that Lord RUSSELL would deduce from it!

But let us consider precisely the possibility that, in order not to contradict itself, it does not include itself.

Let us take the first catalogue: so far, there are only four catalogues that do not contain themselves: A, B, C, D. Suppose another catalogue appears that does not contain itself—we add it: E.

What is inconceivable about thinking that there is a first catalogue that contains A, B, C, D, and a second that contains B, C, D, E, and not being surprised that each lacks that letter which would precisely designate itself? But from the moment you generate this succession, you only need to arrange it around the edge of a disc and realize that it is not because each catalogue will be missing one—perhaps even more than one—that the circle of these catalogues does not form something that corresponds exactly to the catalogue of all catalogues that do not contain themselves.

Simply, what this chain will constitute will have this property of being a surplus signifier that is constituted from the closure of the chain. An uncountable signifier and one that, precisely because of this, can be designated by a signifier. For being nowhere, there is no inconvenience in a signifier emerging that designates it as the surplus signifier: the one that is not grasped within the chain.

I take another example: catalogues are not made, first and foremost, to catalogue catalogues—they catalogue objects that are there in some respect: the word “title” having its full importance here.

It would be easy to pursue this path to reopen the dialectic of the catalogue of all catalogues, but I will take a more lively route, since I must leave you some exercises for your own imagination: the book. We enter with the book, apparently, into the universe of discourse.

Yet, insofar as the book has some referent and insofar as it too may be a book that has to cover a certain surface, in the register of some title, the book will include a bibliography. Which means something that properly illustrates for us what results from the fact that catalogues may or may not live within the universe of discourse: if I make a catalogue of all books that contain a bibliography, naturally, I am not making a catalogue of bibliographies!

Nevertheless, by cataloguing these books—insofar as in their bibliographies they refer to one another—I may very well end up covering the totality of all bibliographies. It is precisely there that the fantasy may be located—the properly poetic fantasy par excellence, the one that obsessed MALLARMÉ: that of the Absolute Book.

It is at this level where things are tied together at the level not of the use of a pure signifier, but of the purified signifier, insofar as I say—and as I write that I say—that the signifier is here articulated as distinct from any signified, and I then see emerge the possibility of this Absolute Book, whose peculiarity would be that it encompasses the entire signifying chain—precisely in this: that it may no longer signify anything.

In this, then, there is something that proves to be grounded in existence at the level of the universe of discourse, but whose existence we must suspend to the specific logic that can constitute that of fantasy, for it is also the only one that can tell us in what way this region belongs to the universe of discourse. Certainly, it is not excluded that it enters into it. But on the other hand, it is quite certain that it is specified within it, not by that purification I spoke of earlier, for purification is not possible for what is essential to the universe of discourse—namely, signification.

And were I to speak to you for four more hours about this Absolute Book, it would nonetheless remain the case that everything I say to you has meaning.
What characterizes the structure of this B…
insofar as we do not know where to situate it in the universe of discourse: inside or outside
…is precisely this feature I pointed out to you earlier, by drawing the circle of just A, B, C, D, E,
insofar as simply closing the chain results in each group of four being able to easily leave out the foreign signifier that can serve to designate the group, for the sole reason that it is not represented in it, and yet the total chain will come to constitute the set of all these signifiers, causing to emerge this one more unit, uncountable as such, which is essential to an entire series of structures,
which are precisely those upon which I based, from the year 1960, my entire operatory of Identification [Seminar 1961–62].

Namely: what you will find, for example, in the structure of the torus, it being evident that by looping a certain number of turns on the torus, by performing a series of complete turns with a cut and doing as many as you please—
naturally the more there are, the more satisfying it is, but the more obscure—
it suffices to do two in order to immediately see appear the third, necessary so that these two loops can close upon themselves and, so to speak, so that the line bites its own tail: this will be that third loop, which is ensured by the looping around the central hole, through which it is impossible not to pass for the first two loops to intersect.

If I am not drawing this on the board today, it is because, in truth—in saying it—I say enough for you to hear me, and yet too little to show you that there are at least two paths, originally, through which this can be effected, and that the result is not at all the same regarding the emergence of that one more which I am speaking to you about.

This merely suggestive indication does not in any way exhaust the richness provided by even the slightest topological study.
What needs to be indicated today is simply that the specificity of this world of writing is precisely to distinguish itself from discourse by the fact that it can close upon itself. And, closing upon itself, it is precisely from there that emerges this possibility of a 1 that has an entirely different status than that of the one which unifies and encompasses.

But of this “1” that already, from simple closure—without needing to enter into the status of repetition, though it is closely linked to it—just from its closure, brings forth that which has the status of the one more, insofar as it is sustained only by writing and yet is open, in its possibility, to the universe of discourse, since it suffices, as I pointed out to you, that I write—
but it is necessary that this writing take place—what I say about the exclusion of this “1”, this suffices to generate that other plane in which the entire function of logic properly unfolds.

This is sufficiently indicated to us by the stimulation logic has received from submitting itself to the sole play of writing,
except that it always lacks the memory that this rests only on the function of a lack within that very thing which is written and which constitutes the status, as such, of the function of writing.

Today I tell you simple things, and perhaps this very fact may cause this discourse to appear disappointing to you.
Yet you would be mistaken not to see that this fits into a register of questions that henceforth give to the function of writing something that can only reverberate to the deepest level of any possible conception of structure.

For if the writing I speak of is supported only by the return, the looping back upon itself, of a cut, as I illustrated through the function of the torus, we are brought to this: that the most fundamental studies, precisely those tied to advances in mathematical analysis, have enabled us to isolate the function of the edge. Now, as soon as we speak of edge, there is nothing that could lead us to substantialize this function, insofar as you might unduly deduce from it that this function of writing is to delimit the shifting realm I spoke to you about earlier as being that of our thoughts or of the universe of discourse. Far from it! If there is something that is structured as edge, what it itself delimits is in a position to, in turn, enter into the bordering function. And it is precisely this that we are going to confront. Or else—and this is the other side upon which I intend to conclude—it is the reminder of what has always been known about this function of the unary trait.

I will end by evoking Verse 25 of a book I have already used, at one time, to begin to convey what is at stake in the function of the signifier: the Book of Daniel, and with regard to a story of a pair of zouave trousers, referred to there by a word that remains what is called a hapax, and which it is impossible to translate, unless it be the clogs worn by the characters in question.

In the Book of Daniel you already have the theory—which is the one I am presenting to you—of the subject, and specifically as it emerges at the limit of that universe of discourse. It is the famous story of the dramatic feast, of which, moreover, no trace is found in any chronicle, but no matter! “Menê, menê”—for so reads Verse 25 [Ch. 5]—“Menê, menê, tekêl, oufarsin.” …
וּדְנָה כְתָבָא, דִּי רְשִׁים: מְנֵא מְנֵא, תְּקֵל וּפַרְסִין. which is usually transcribed into the well-known: “Mene, tekel, phares.”

It seems to me not without use to note that “Mene,” which means “counted”…
as DANIEL points out when interpreting it for the anxious prince…
is expressed twice, as if to show the simplest repetition of what constitutes counting: it suffices to count to two for everything at stake in this one more—which is the true root of the function of repetition in FREUD—to come into play and be marked in this: that except that, unlike in set theory, it is not stated.

It is not stated thus: that what repetition seeks to repeat is precisely what escapes, by the very function of the mark, insofar as the mark is original in the function of repetition. That is why repetition functions from the fact that the mark is repeated, but that for the mark to provoke the repetition sought, it is necessary that, upon what is sought in what the mark marks the first time, this very mark be erased at the level of what it marked, and that this is why what is sought in repetition, by its very nature, eludes, causes this to be lost: that the mark cannot be doubled, except by erasing, from what is to be repeated, the initial mark—that is, by letting it slip out of reach.

Menê, menê… Something, in what is found, lacks weight: Tekêl. The prophet DANIEL interprets it; he interprets it by telling the prince that he was indeed weighed, but that something is lacking, which is called “Oufarsin.” This radical lack, this original lack that derives from the very function of counting as such, this one more that cannot be counted, this is precisely what constitutes that lack to which we must assign its logical function, so that it secures what is at stake in the terminal “phares,” the one that precisely makes explode what is of the universe of discourse, of the bubble, of the empire in question, of the self-sufficiency of what closes itself within the image of the imaginary whole.

This is exactly the path through which emerges the effect of the entry of what structures discourse at its most radical point, which is assuredly…
as I have always said and emphasized, even using the most vulgar images…
the letter in question. But the letter inasmuch as it is excluded, as it is missing. It is indeed this…

which also, since today I am again breaking into this Jewish tradition…
on which, to tell the truth, I had prepared so much, even to the point of having undertaken a small exercise in learning masoretic reading, all work that was in some sense sheathed away by the fact that I was not able to deliver to you the thematic I had intended to develop around the Name of the Father, and yet from all this something remains, and namely, that at the level of the story of Creation:
[בְּרֵאשִׁית בָּרָא אֱלֹהִים אֵת הַשָּׁמַיִם וְאֵת הָאָרֶץ] “Bereshit bara Elohim”—the Book begins, that is to say, with a beth.

And it is said that this very letter we have used today, the capital A—that is, the aleph—was not, at the origin, among those from which all of creation issued. This indeed shows us—though in a way folded back upon itself—that it is insofar as one of these letters is absent that the others function, and that no doubt it is in its very lack that the full fecundity of the operation resides.