Récanati

Lacan

Lacan, it seems, in his 1st ‘seminar’—as it is called—of this year, would have spoken, I’ll give you a thousand guesses, about love, no less! The news spread…[Laughter] It even came back to me, not from very far of course, from a small town in Europe [Amsterdam] where it had been sent as a message[Laughter].

Since it came back to me on my couch, I can’t believe that the person who reported it to me really believed it, given that they know well that what I say about love is precisely that one cannot talk about it. ‘Talk to me about love’ means little songs. I spoke about the love letter, about the declaration of love; that is not the same thing as the word of love.

In any case I think it is clear… even if you have not formulated it for yourselves… it is clear that in this 1st seminar I spoke about stupidity, about the kind that conditions what I have given this year as the title of my seminar and that is called ‘Encore’. You see the risk!

I am telling you this only to tell you what gives weight here, the weight of my presence, is that you enjoy it: my presence alone—at least I dare to believe it—my presence alone in my discourse, my presence alone is my stupidity. I should know that I have better things to do than to be here.

That is precisely why I can simply want it not to be assured to you in any case. Nevertheless it is clear that I cannot put myself in a position of withdrawal, to say ‘how “encore!” and let it last’ is stupidity, since I myself collaborate in it, obviously.

I can place myself only in the field of this Encore. And perhaps by tracing back a certain discourse, which is the analytic discourse, up to what conditions this discourse… namely that truth, the only one that can be incontestable in that it is not: that there is no sexual relationship… this does not in any way allow one to judge what is or is not stupidity.

And yet it cannot be—given experience—that with regard to analytic discourse something is not questioned, namely whether it does not essentially consist in supporting itself on this dimension of stupidity. And why not, why not after all, not ask what the status of this nonetheless very present dimension is. For after all, there was no need for analytic discourse for it—this is the nuance—as a truth to be announced that ‘there is no sexual relationship’.

Do not believe that I hesitate to get my hands dirty. It is not since today that I will speak of Saint Paul; I have already done so. That is not what frightens me—even to compromise myself with people whose status, whose lineage is not, properly speaking, what I frequent. Nevertheless ‘men on one side, women on the other’, that was the consequence of the message; that is what, over the course of ages, has had some repercussions. That did not prevent the world from reproducing itself at your scale. Stupidity holds firm, in any case.

It is not quite like that that analytic discourse is established, what I formulated to you of a and of the S2 that is underneath and of what that questions on the side of the subject. To produce what? It is quite obviously that it gets installed in that, in stupidity—why not?—and that it does not have that distance, that I have not taken either, to say that if it continues it is stupidity. In the name of what would I say it? How to get out of stupidity?

It is nonetheless true that there is something, a status to give to what it is of this new discourse, of its approach to stupidity; something is renewed from it. Surely it goes closer, because in the others that is indeed what one flees. Discourse always aims at the least stupidity, what is called the sublime stupidity, for ‘sublime’ means that: it is the highest point of what is below. Where is, in analytic discourse, the sublime of stupidity? That is why I am at the same time legitimized to put to rest my participation in stupidity insofar as here it envelops us, and to invoke whoever can bring me the reply of what, no doubt in other fields… but no of course, since it is a matter of someone who listens to me here, who by that fact is sufficiently introduced to analytic discourse. How?

That is what already, at the end of last year, I had the good fortune to gather from a mouth that will be the same. That is where, from the beginning of the year, I hear that someone brings me—at their own risk and peril—the reply of what, in a discourse—namely the philosophical—resolves, obliquely, leads its way, clears it by a certain status, with regard to the least stupidity.

I give the floor to François Récanati whom you already know.

François Récanati[Scilicet 5 p. 61]

I thank Doctor Lacan for giving me the floor a second time, because that will introduce me directly to what I am going to speak about, in the sense that it is not unrelated to repetition. But on the other hand, I would also like to warn that this repetition is an infinite repetition, but that what I am going to say, there too, will not be finished in the sense that I will absolutely not have the time to reach the end of what I have prepared.

That is to say that here, in a way, it was truly at the closing of the loop that what, as a preliminary, will bring me there, had to take on meaning. There I will be obliged, because of time and unless I take this up another time, to stick to the preliminaries, that is to say properly not yet to enter head-on into that stupidity of which Dr Lacan spoke.

You remember that what last time I tried to show you is that repetition occurs only on the 3rd stroke, which was the stroke of the interpretant. That means that repetition is the repetition of an operation, in the sense that for there to be a term to repeat, there must be an operation that produces the term; that is to say that what must be repeated must indeed be inscribed, and the inscription of this object cannot itself be done except at the end of something of the order of a repetition.

There is something there that resembles a logical circle, and that is in fact a bit different, rather something of the order of a spiral, in the sense that the arrival term and the departure term, one cannot say that they are the same thing: what is given is that the arrival term is the same as the departure term, but the departure term itself is not already the same; it becomes the same, but only after the fact.

There are therefore two repetitions to consider, asymmetrical: the first, which is the process by which this object that must repeat is given, and one can call that, in a way, the identification of the object, in the sense that it is a matter of the decline of its identity; and one sees very well what that means: when one declines this identity of the object, this identity declines at once. And the initial tautology ‘A is A’—of which we remember Wittgenstein says that it is a coup de force devoid of meaning—is properly what institutes meaning, because something passes in that.

That is to say that in the ‘A is A’, A first presents itself as the entirely potential, undifferentiated support of everything that can happen to it as determination. But as soon as an effective determination is given to it, as soon as it is existence that is at stake and not the whatever of all its possible determinations, then precisely there is a kind of transfer of power. That is to say that what was supposed to function as support—in this case this undetermined A, this potential A—is in a way marked by the fact that there is being all of a sudden that interposes between it and itself; that is to say that it repeats itself, and it repeats itself in the form of a predicate.

That is to say that there is a kind of diminishing, and this diminishing is symbolized by this: that in ‘A is A’, the A that had the function of support suddenly finds itself supported by something of the order of being that supports it, that surpasses it, that envelops it; and it itself is in this relation only what predicates the predication, insofar as the predication is what being supports. I will come back to this.

Lacan—Besides, everyone knows that ‘War is war’ is not a tautology, any more than ‘a penny is a penny’!

François Récanati

Exactly. I will come back to that because it is more or less the nerve of the whole matter and I would like to speak—and it is that which I fear I will not have time to do—about the logic of Port-Royal, because it is a theory of substance, precisely, and it was said last time that we do not refer here to any substance. But I will come to it in a moment. Let it simply be known that repetition indeed, the first one, repeats the initial indetermination of this object that is given as potential, but that in repeating this indetermination, the indetermination suddenly finds itself determined in a certain way.

That is to say that one can indeed posit that the repetition of emptiness or the repetition of the impossible—in short, that this type of repetition of something that is not given and that must therefore be produced in the time in which one would like to repeat it—one can indeed posit that it is the impossible, and that is what more or less everyone says; but it suffices that it be impossible for there to be something assured there, and that this assurance precisely allows a repetition; moreover it is a 2nd repetition.

Rather than spreading myself out on that, I cite this sentence of Kierkegaard:

‘The only thing that repeats is the impossibility of repetition’.

That shows very well what is at stake, and it makes the joint with what I said last year about the triad that supports all repetition: object-representamen-interpretant. That is to say that between the object and the representamen, one changes in a way of space, or at least there is something like a hole that makes precisely the object and the representamen unapproachable in this relation.

But this hole, insofar as it insists, allows one to found a true repetition in this sense that on the next stroke there is something that will embody this hole, which will be the interpretant, and which will in a way be able to repeat in two ways what was passing between the object and the representamen:
– on the one hand to inscribe it by saying ‘there was hole’, and by allowing that this impossibility or this hole repeats itself.
– but on the other hand it will not only signify it but repeat it because, between the initial impossibility that was passing between the object and the representamen and its signifier which is the interpretant, there is the same impossible relation that there was precisely between the object and the representamen; that is to say that a second interpretant will be needed to take charge of the repetition of this impossibility.

In the interpretant there is something like the carrying out of an impossibility up to then potential, and the impossibility inscribed by the interpretant is, let us say, the first term of this existence of which the potential zero was the bearer, in the sense that in some manner the whole leads to ‘it exists’, and I will return to that as well.

What is important is that the impossibility of the object-representamen relation is given as such for the interpretant. The interpretant says: ‘That is impossible’, but insofar as it is given for the interpretant as such, as soon as the interpretant itself is given for another interpretant, it is then that this impossibility is truly a term, a founding term of a series. That is to say that it allows the new interpretant to assure something solid, as if this solidity were the first interpretant that had founded it from something originally fluid. What was escaping in the object-representamen relation comes to be imprisoned in the interpretant.

But one sees well, and I have already said it,
– that what is imprisoned in the interpretant,
– and what was escaping in the object-representamen relation,
is not exactly the same thing, since precisely what was escaping in the object-representamen relation continues to escape in the relation between that relation and the interpretant.

That is to say that in any case there is the same shift, the same inadequation. And it is indeed the impossibility of repetition, on which I am now going to lean a bit, that produces what happens and that one can observe, that is to say the repetition of impossibility.

What institutes the shift, this shift from which repetition originates, is the impossibility for something to be at once that something and at the same time to inscribe it. That is to say that the existence of something is inscribed only for something else, and consequently it is inscribed only when it is something else that is given.

And if it is a matter of punctual existence, the existence of something is inscribed only at the moment when it precisely declines, from the moment when it is another existence that is in question. This disjunction is more or less what passes between being and predicated being, and I hope to have time to get as far as the logic of Port-Royal, which was theoretically the core of my presentation, but it is doubtful.

You remember that last time Lacan characterized being as ‘section of predicate’. And it is properly about that that it is a matter. And right away I will give some reflections if only on this formula ‘section of predicate’, which immediately makes one feel the recurrence where what is precisely supposed to support every predicate is constructed, that is to say being: what supports predicates beforehand is given after predicates.

And in a certain manner, if there is ‘section of predicate’ to find being, that means that what supports predicates is what is not in predicates. It is precisely what is absent from predicates, what is absent in predication.

It is therefore the absence of being, in a certain manner, that bears predicates, which also implies, and in a somewhat indirect way, that predicates are themselves predicates only of this absence. That the predicate can be cut is as if, in a way, there were already an elementary partition, as if a line were given as dotted, a boundary, and that it suffices to cut out as in certain packages.

Lacan

Articulate well the notion of ‘section of predicate’, since that is what you latched onto in what I left, and I almost just stumbled on that.

François Récanati

The ‘section of predicate’ is properly the core of my presentation. One can imagine it like a vibration; that is to say that it is starting from a kind of halo that I will try, by truly going around, to encircle this core that will appear in all the examples I am going to give.

Section of predicate is therefore as if it could be cut. I do not insist on that, except that it is obvious that it is not by having cut the cut that one will recover the uncuttable, and that the boundary, once one has slashed into it, insists all the more as it manifests itself as a hole.

Let us say that section, to take the senses that come, is also just as much making two of what was One; and if I signal this sense, which is not what is received here, it is because it is the one that Groddeck gives to one of his concepts, which is called precisely the ‘sexion’ [pun: ‘section’ altered with an ‘x’ to evoke ‘sex’], that is to say that it is not without interesting sex, in a certain manner.

And that is Groddeck’s way of referring to Plato, and when I say Plato it is not the ‘Parmenides’ but the ‘Symposium’. You remember that in Aristophanes’ speech the problem is raised of this myth of the original androgyne that would have been cut in two. That would have been the ‘sexion’ with an x.

Now, what I would like to insist on is something that stands out very well from the Symposium, not specifically from Aristophanes’ speech but a bit from all the speeches, even those that are supposed to be contradictory, and I will take only two examples:
– the speech of Diotima on the one hand,
– that of Aristophanes on the other.

And the Symposium is about love. Love, says Diotima, is what, everywhere there is a two, serves as boundary, as middle, as intermediary, that is to say as interpretant.

When I say ‘interpretant’, it is because one can very well translate like that the word Plato uses, which is a word derived from Μαντιχή[mantiké] which means interpretation, and Plato says that this word comes from Μανιχή[maniké] which means delirium. That is what serves as interpretant.

But the only interest of this formula… because after all, no one in the Symposium assembly contests it… is what allows the following to ensue: that love in no case could be beautiful, because what is posited as the object of love, what as a series falls under the stroke of love… love being like a mark that makes pass by, that institutes a kind of corridor where a series of objects will pass, the objects it has marked… love cannot be beautiful because its objects are beautiful.

And it is said that in no case,
– what is the agent of a series,
– the very instance of the series or the ultimate term of series,
– what caps a series,
can have the same characteristics as the objects that are in this serialization; that is to say that if the objects of love are beautiful, love cannot be beautiful.

That is properly a character of this instance of serialization, a character of the interpretant, that no one, among the polemicists present in the Symposium assembly, calls into question.

And one can see fairly easily the relation there is with Aristophanes, even if it seems more distant: it is that when he says that at the origin men had four legs, four arms, two faces and two sexes, they became a bit too arrogant because they no longer really had desire, they lacked not much, so Zeus decided to cut them in two so that they would become humbled.

But what Zeus said is that it does not count, a cut, if there are no effects of the cut; that is to say that if the cut is punctual and after that it continues as before, it is of no use. So what he wanted was that it remain, that there be an effect, and for that he turned the faces… which were then like the sexes on the back, and the place of the cut was properly the belly since there is the navel which is the index of the cut… he decided to turn the faces toward the navel, so that men would remember this cut. And then while he was at it he turned the sexes as well, so that they could try to reattach and so that it would keep them busy.

But the important thing, and that is why I unfolded all that in relation with Diotima’s speech, is that the result of this whole operation, which can appear derisory, is simply that man, his face has been turned; he can no longer look behind him; he no longer sees except forward; he sees only what precedes him.

Does one see well that it is precisely also what Diotima says, that is to say that that is the end of the whole, that is to say the end of the whole insofar as, for every series, the ultimate term of serialization will be missing, the point of view: that from which the serialization is built.

Lacan—That is indeed what I was saying just now: that he does not see the encore.

François Récanati

What I have just isolated from two speeches will be found again as two very linked points concerning ordinals. What makes the ordinal—as you have already been told—is something of the order of a name of a name, and we will see more precisely what it is about, in the sense that the ordinal is a name, but if it is a name, the function of this word is to name something that is not precisely its own name.

It is in a way the 2nd name of what precedes, of the name that precedes and that—as name itself—is indeed a name, but serves only to name something that precedes, etc. That is the relation with Aristophanes. I do not insist.

There is a problem that will arise right away, and I will try to approach it: it is that the 1st ordinal is not really a name of a name, because there is no name that precedes it, if it is indeed the 1st. That is why I wrote beside it the ‘name of the name’, because that is what the 1st ordinal is.

And I will even say: if that is what happens at the beginning, it is because of that that afterward there is name of name, because precisely, as soon as one gives a name to what does not have one, it is in identification precisely that something like the decline of identity, in the sense that one says a bit more, and that this more that one says will itself have to be, not so much reabsorbed but identified, given a name, and from there it is the infinite shift. [It is God who carries out Creation, but it is Adam who commits to the naming of what has been created, cf. metaphor and metonymy]

To name, in general, is to make the point of what precedes in the series. But the point, insofar as it itself functions as name, precedes something to come as well; and this something to come, if one considers it absolutely, what is always to come will be what one could call the ‘encore’, which itself precedes nothing that is not itself, that is to say does not hold a name, unnameable as a result.

One sees that from this point of view, what I call the encore is the index of the infinite.

And on the other hand one can say that the infinite is already there: it is given from the start in the homonymy of name and no. That is to say that the name is something like the propagation of the most radical no which, before any naming, in the instant of any naming, is given as something infinite.

One therefore sees something detach as two limits,
– the no on the one hand,
– and the encore,
and ordination is what passes between the two.

That is to say that what will interest me… and one can see the relation of this with the section of predicate… that is to say with this expression and this recurrence… is the relation between the two.

The system of naming in general, you see roughly how one can grasp it: it is the wrapping of an initial impossible, a wrapping which precisely, in this relation to the impossible, is sustained only by the encore as index of this transcendence of the impossible with respect to any wrapping. And if the impossible is what says no… which is not obvious and I regret not having the time to develop this point… it will have to be heard more or less as a radical denial, insofar as denial is something that is already infinite.

That is to say that insofar as it is already infinite, denial does not care much about what comes, in a way, behind it, what it supports, that is to say all the play of predication, all the play of predicative objectification that takes denial, for example, to deny it, by saying no or by saying yes; it never yields any yes; denial remains intact, with little games that happen on its body, one could say. And then it is not even, for the infinity of denial, a tickle.

So this leads us to think—it is a parenthesis—that even if what I called ‘logical manipulation against a background of infinity’ becomes infinite in its turn, that does not mean that we are going to cure infinity by means of infinity and that it is suddenly going to yield the finite, or something like a yes.

On the contrary, it is going to become worse in the sense that what in nomination can become infinite is not the same thing as what is already there as infinite in what I call this initial denial, in the sense that what in the logical manipulation comes as infinite is the nomination of the infinite, and that what is already there as infinite denial is what infinitizes every nomination; it is the infinity of nomination.

Which means that the nomination of the infinite will be a nomination like the others, that is to say that it will equally be subject to this infinitization that is already there, which starts from a source that is at the beginning. That is to say that it is not going to change anything and that one can posit something like omega, the smallest infinite ordinal; it is not going to stop there, it continues in the power set of omega, in the alephs etc.

As soon as the infinite is given in that position, the infinite itself must be infinite, that is to say that we continue these passages from infinity to infinity, etc., that we continue ‘encore’. As if what wants to be reached in this story is precisely the encore itself. The encore to be given as the limit of the extension of this radical no of which I spoke,

And I am now going to speak of the relation between the radical no and the encore, since it is to that that what I am going to return to will introduce me retroactively, that is to say the ‘section of predicate’. The ‘section of predicate’, one sees it immediately, is at once
– what there is after every predication, that is to say once one can say ‘there are no more predicates’,
– and it is equally what, before every predication, supports it.

But what must be understood is that this before and this after are the same thing, that is to say that it is what constitutes, what sustains predication as the wrapping of an impossibility, this impossibility that must be understood as the very impossibility of predication, that is to say the impossibility of providing all the predicates, of putting them together, without at least one detaching as representing, in impossibility, in existence, impossibility, or if one prefers the encore.

More precisely as to ordinals, the ordinal names the name of the one that precedes it. That means two things:
– that an ordinal does not name itself but is named by its successor,
– and that to each ordinal belongs the mechanical summation of all those that precede it. Since an ordinal names its predecessor, its predecessor names its predecessor etc., that is to say that there is, attached to each ordinal, the series of all the ordinals that preceded it.

Now, already these two points imply an essential discordance between the name and the name-of-name, and that is what I will call an effect of flattening.

What comes to identify zero for example, in a definition of zero, as something like the unique element of the set identical to zero, or for the empty set one can very well say: what is the unique element of the set of its parts, or simply that set of its parts of which it is the element that comes to identify it properly, this is given as a predicate of zero. Now, one sees well that in this predicate there is something extra that is given, extra beyond the empty set, extra beyond zero, and it is so tangible.

The proof of it is that precisely the 0 and the 1, which is supposed to be nothing other than the identification of 0, that makes precisely 2. One sees that one changes level, that it has no relation, that it is not situated—there is a shift, one passes from a level to a higher level. But what is remarkable is that this 0 and this 1 that have nothing to do with each other, that are not situated at the same level, one puts them together as the elements of this new set constituted by the ordinal 2. 0 and 1 make 2 precisely in the sense that 0 and 1 are in a way leveled, put on the same plane in 2. And for 2 itself, the operation will repeat in this passage from 2 to 3 etc. The representamen has there with the object no possible relation, and it is always this cursus of the interpretant that intervenes, that is to say that it is embodied by something, and insofar as it is embodied, insofar as the something that escapes is bridled, it resurges as well just after this embodiment.

One can take the formula of an ordinal to see better what is at issue.

Lacan—Give it back to Cantor all the same!

François Récanati

Here is the formula that one can consider as the formula of 4.
0 0 I : 0 I 2
0, { 0 }, { 0, { 0 } } : { 0, { 0 }, { 0, { 0 }}}
0 I 2 : 3

In this formula, what happens?
One knows that it is the ultimate term of this series that counts. One sees that in 4, what is repeated is 3. And one sees that 3 itself repeats 2, which itself repeats 1, which itself repeats 0.

But what is important is that 4
– is not only the parenthesizing, the nomination of 3, which itself parenthesizes and names 2 etc.
– it is not only the exposition, even repetitive, that is to say with additional parentheses, of what was already given in 3.

It is the putting into one and the same set of 3 already as flattening, as ‘ensemblization’, of heterogeneous terms, that is to say the same thing as in 2, the fact that there are 0 and 1 that are put absolutely on the same plane.

In 3, it is already a flattening of 0, of 1 and of 2, that is to say that one puts them in one and the same set. And 4, it is here precisely the relating, in one and the same set, of 3 as flattening, as this forced ‘ensemblization’, with the elements that 3 flattened, separated from 3, outside of 3.

That is to say that it is a repetition. One sees that the left part and the right part are the same thing, except that on the right there are additional parentheses.

It is here, between 2 and 3, that there is like a bar of splitting, which allows me to say that one can see in this formula that
– if 3 already is the designation of what happened, of a passage-flattening, between 0 and 1, and of 0 and 1 to 2,
– if 3 is already this flattening, that is to say a way of designating what happened from a rupture before, a rupture that is precisely the passage from 0 to 1, a rupture that is to say a bursting of the parts of what was already given as set,
one sees that what is designated in the formula of 4 is precisely this designation itself, insofar as one can see exposed on the same plane
– on the one hand all the parts of what forms 3,
– and on the other hand 3 itself.

That is to say that the flattening itself, the fact of putting additional parentheses, is not sufficient as a result to keep salient this passage from 0 to its flattening in 1, from 1 to its flattening in 2 etc., 2 or 1 as result no longer expressing this passage.

It is necessary that in the set constituted by 4 there be present at once the terms separated from the different passages and the series of passage-flattenings, so that 4, as nomination of all these impossible but effective passages, takes charge, in its own formula, of the history of the progression that one sees here repeated, that is to say leaves open what is posed as question, as irresolution in this movement, that is to say the insistence in this race of what…
through the different successive limits
that in a way oppose the passage from 0 to 1, from 1 to 2 etc.,
…the insistence through these successive limits of what is given as absolute limit and that would be the encore.

And if 4 as totalitarian flattening…
that is to say as summation of everything that happened before it,
of all the flattenings powerless to finish
…if 4 leaves this question open, it is indeed because it itself, as flattening, responding to this fault that calls for an impossible closure, can in its turn only flatten itself encore, that is to say reproduce the fault, namely in the new formula that includes it as element, and that is to say 5, and that in order to do so confronts it with all the elements that it contains, placed beside it, to make arise between all these elements and their flattening in 1, the impossible identity.

It would therefore suffice to repeat everything that there is here and to put the parentheses back to obtain 5. The impossible identity is what repeats at each new flattening with this that in the sequel, in the confrontation, inside 4, of constituted 3 and of all its elements, it is already the flattenings that flatten themselves encore a little.

Whereas the paradigm of flattening one can find it at the beginning in the passage from 0 to 1, and this flattening must be understood in an entirely concrete way, like that of Icarus, that is to say that there is something that takes flight and that crashes miserably, and that does not crash into the hole that was supposed to be flown over, that crashes on the cliff on the other side, in a way.

One can consider that between one ordinal and another, or rather between the nothing of the empty set and its inscription in 1, there is something like a barrier, a boundary, or else a hole. But this hole, one cannot reach it, exactly in the sense that, as Lacan recalled last time, as in the case of Achilles, one can go beyond it but one cannot reach it.

If once a flattening is given it repeats, it is precisely because what is posed as boundary has not been reached; it is always there, this boundary, existing. One is never in the in-between, between two ordinals, but always in one or in the other,
– the one being the set that takes charge but is not itself counted,
– and the other being what takes the first set but is still not itself counted.

That is to say that the limit of which I speak and that atomizes and fragments into a series of boundaries that one can never reach and that therefore reproduces itself, is posed as absolute limit; it is therefore the whole, the whole that is to say
– the something that sustains itself all alone, that has no need of anything else,
– and that is for philosophy the substance, or again ‘the substance of substances’, that is to say being.

This limit insists as always elsewhere, and the passage that manifests it as hole, between something and its support, this passage not for an instant can be grasped as between two.

One sees it as concerns the passage from the finite to the infinite for example because, as I said, one can posit the smallest infinite ordinal. Nevertheless this does not present itself in a harmonious way as preceded precisely by the greatest finite or preceded by something finite, because this infinite would then be only finite plus 1. Between the two, there is truly this hole that could not be reached, and that repeats itself therefore in the infinitization of infinities.

That said, this insistence of the limit insofar as it is excluded, insofar as it ex-sists more exactly, does not only express that there is a gap between 0 and 1, but it is much rather their flattening in 2 that implies a certain misrecognition of this gap, a refusal truly, something that resembles a disavowal or a denial, that is to say something that participates in these unconscious processes that defines formal logic in a certain way since they put infinity into operation, and that to put infinity into operation is truly to disarm most of the processes of logic.

I cite an example that I read in a recent article on modern mathematics where it was said that in a school class, when one asks for an example of an infinite set, it is never answered by something like ‘the integers’, it is never answered numerically, but always by a finite set, a large finite set like ‘the pebbles of the earth’ or something like that.

That shows well that as concerns number precisely, there is something that makes one believe that it can stop, and at the same time it is very right, because it does not stop stopping. But if I say ‘it does not stop stopping’, that is indeed it, that is to say that it will never stop stopping. The limit of which I spoke, one can conceive it in analogy with death, with silence, and I regret not having much time to develop it, but in general it is what discourse converges toward, that is to say that repetition is the representamen of death.

And I would like to show, by taking a minimum of examples, that in the dream for example, as has already been said, there is something that manifests itself as equation of desire =0, but this equation of desire, it is extra, it is held back. It is the one who interprets the dream who says: ‘it is the equation of desire’ who manages to make 0. The dream itself, it is in 0, that is to say that it balances itself.

At the same time ‘equation of desire =0’, that obviously does not stop there. It cannot stop there, because the dream precisely continues to produce statements, it continues to speak. And of course, it would very much like to be equal to 0, but for that it would have to fall silent, which is not the case. Now 0, if it is inserted in this equation, equation of desire =0, that signifies that it is supported, that it is designated by the equation that produces it as that at which it arrives.

Now the fact that it is designated, that it is supported, is properly the transformation already of this 0 into 1. 0, when one puts braces around it, it becomes 1. Now, it is precisely the task of interpretation to make sensitive in this 0 the 1 of which it is the bearer, the 1 of which, insofar as 0 manifests itself, insofar as it is designated, it is then that it is produced from 1.

And one can understand how it happens that interpretation is like an added wagon to an equation already given: it is that precisely the dream itself is the ultimate term of the series, it is for example 1.

But when one is in 1, 1 bears the whole, it is focused entirely on this 0 that it inscribes, and if it itself makes 1, it is for something else, that is to say for the coming of something else that arrives in interpretation.

What is given as resistance to the interpretation of the dream in an analysis, this kind of boredom at speaking of a dream, as if it were already not bad as it is, as if as it is it is good, and as if one must add nothing to it, that has to do with the resistant bar to signification that is supposed to separate the signifier from the signified.

If one lets oneself be guided, insofar as interpretation is in question, by Pierce rather…
if there is an opposition between them
…than by Saussure, one must remember well that the signified of which one speaks is nothing other than signifier, but in a series, in the sense that precisely there are functions in this series, roles that exchange, and that one can say that effectively there is a role of signified with respect to a role of signifier.

But the signified is a signifier plunged into interpretation in the sense of Pierce, and that finds itself in a way flattened, minimized, diminished, singularized, in the surge of another signifier, surge of another[signifier] that allows, by this confrontation…
which is the same one sees here
…to understand that one is dealing with units of another set, with elements of a larger set.

And this flattening takes place without what makes hole between the two, in the surge of this new signifier between the two signifiers, being properly produced, but it is in the repetition of this phenomenon, in its infinite character, that something like the limit of interpretation is given.

And the limit of interpretation or of signification for Pierce is the gaping of the potential, that is to say something that must be put in relation with the subject and, if one is to put it in relation with something, one can also see whether it is connected with what one calls ‘the set of all sets’. Because ‘the set of all sets’ perhaps, precisely, is this infinitely silent potential of which Pierce speaks and that is found at the beginning and at the end of every series.

To say that it does not exist is equally to say that it exists as limit of every inscription, and equally as a grain of sand in the machinery of every equation that wants to equal zero, for in the time of this ‘equal to 0’ the 0 is produced as this term, and from then on it can be confronted with something else that one would take in the equation that gave birth to it and that would singularize it in another more general set where it figured as an element.

If I say that it is because I heard, not long ago, an analyst declare that most of the time, future analysands come to see him for a preliminary interview as soon as something has happened, that is to say as soon as a grain of sand, a small something of nothing at all has come to jam, has come to make unbearable an economy up to then very well tolerated.

Now this grain of sand is nothing other than this 1 of which I spoke, that is to say that it is constituted
– of the global taking-into-account of this equation,
– of this very satisfactory economy in their extreme singularity which is not nothing,
that is to say in opposition to something else, something that one can possibly take within this equation, and singularize, that is to say posit as currently in front of the entire equation.

It suffices that a single trait of the equation be produced in isolation for it to break the equilibrium of the equation itself, which was an equilibrium of folding back on itself, and for it to function as grain of sand.

It suffices of a slight shift…
I cannot here cite examples and that is a pity because it shows extremely well
…of an entirely derisory change of level, that is to say of a carrying, a carrying of what is given as equation into something else, where other elements are at play, for this equation satisfied with itself, this closed set, to become suddenly something else, that is to say for one to realize that it can equally function as an element of another set, as part of another set which can precisely be the set of its parts as one sees here, that is to say as an element of a set where the whole of the previous equation figures beside anything at all, beside any trait at all and on the same footing as the empty set for example.

There is no ‘whole’ that cannot be brought down, be burst to the rank of elementary singularity in something that is given as a larger set, that is to say the set of its parts. And this singularity, as soon as it is given, precisely in an instant of floating, calls equally for flattening, leveling in a new set, which guarantees to it, to it, this new singularity, a place of its own, a function, something like an employment.

The passage from a set to the set of its parts is therefore the rout of ‘whole’. But this rout takes singular forms, as soon as it takes place, as soon as dispersal occurs only to reform a new ‘whole’, only to re-flatten immediately into a new ‘whole’, that is to say so that what disperses re-consolidates, but in a way that does not return to the point of departure but following a progression, consolidates into something else that this time forms a compact set.

Perhaps in the end victory goes to dispersal in the sense that if the impossibility of repetition can repeat itself, the impossibility of totalization cannot, it, totalize itself.

Since if one takes the set of all these ‘wholes’ whose totalization is broken by their fractioning in the set of their parts, if truly this set is constituted of all these ‘wholes’ as of its parts, then it undergoes the same destiny, that is to say that it itself can be fractioned, which implies that never will all these ‘wholes’ be able to totalize, otherwise it would be something other than the set of its parts, something other than what one knows of a possible totalization or flattening.

One sees that the ruptures of sets lead to the constitution of new sets, to flattening, and these new sets tend, they too, toward rupture, which allows one to say that in the end…
and I will not insist on it although it is important
…everything is a question of rhythms.

At a somewhat general level there is no system except of rupture…
and I also regret not being able to spread out a bit on that
…but it was one of the errors of contemporary linguicism to posit something like an intra-systemic regulation in a set, without positing it as function of something that participates in an order, function of an excluded limit.

Lacan—Function of a…?

Function of an excluded limit. Something like Pierce’s interpretation was perceived in linguistics as only a part of what for Pierce is interpretation, that is to say the possibility for example in a system of passing from one signifier to another, whereas what this elementary operation has as background is a more essential semiotic work…
I only mention it
…which is precisely, for one and the same signifier or for one and the same set of signifiers, the passage from one system to another of a different type.

There is there something like the torsion, the flattening of the signifier, and moreover it suffices to look at the dream to notice what that can mean. That is to say that overdetermination must be understood not only as semantic overdetermination in a system, but more properly
– as semiotic overdetermination,
– as possibility of a passage for one and the same signifier from one system to another,
– as flattening of the signifier.

The remark of such a process—linked to something else that is interesting, which I will say—
one finds it in Bacon who, from his reflections on language, founded a cryptography procedure. This procedure consists in passing from an inner letter to an outer letter and in making the path in both directions, that is to say in jumping a boundary that this passage brings into relief.

I am not going to insist on in what there is a change of system in Bacon, but I give the example to see
– something that is properly what already insisted in this example here,
– something that one finds at all crossroads, which is namely something like the omission of parentheses, and that precisely allows the passage of the boundary,
– something that has to do with the possibility of a substitution of two terms, that is to say that in the substitution of two terms, everything is function of parentheses, and if I allowed myself to ignore the parentheses or to change the place of the parentheses or of the braces, at that moment everything is possible.

It is moreover what Frege reproached Leibniz for, what he reproached him for having done, and it is what one finds again in Bacon in his cryptographic procedure of which I give you the example. To each letter of the alphabet—Latin in this case, that is to say of 24 letters—one makes correspond a group of 5 letters.

And this group is formed only of a and b, according to one of the 32 possible combinations. That is the 1st stage: it is a simple interpretation.

In the 2nd stage it is the message that one is going to transform by means of this transposition. The message that is only in a and in b will be transformed back into the Latin alphabet according to another interpretation, according to another law of transformation.

A B C
(aaaaa) (aaaab) (aaaba)

Lacan—…?

François Récanati

The first operation is therefore this. Now the essential phenomenon of the change of system…
although I am not pointing out that it is precisely a change of system,
but what makes there be interpretation
…is that once one has a message formed only in a and in b by the transcription from each of the letters in this table, one is going to transcribe again into the original Latin alphabet, by taking not each group of 5 a or of 5 b, because that would properly re-effect this cutting-up that one is trying to mask, one is going to take each a and each b separately, and to each a and each b…
since these are the only two letters of which the intermediate message is formed, the boundary message
…there can correspond to each an enormous number of letters of the Latin alphabet.

If one takes a complicated Latin alphabet of capitals and italics, each letter appearing in capital and capital italic, lowercase and lowercase italic, one will have 4 times 24 letters, and the a and the b will each have half of these letters as possible translation. That is to say that the only thing that will count will be the order of the letters of the message, insofar as the decoder knows that one must cut the message into portions of 5.

For example, one gives oneself an ordered series in a very simple way of a and b, in order, and one then makes correspond the alphabet to each a and to each b, which means that each time one has an a, one can put whatever one wants that corresponds to it, and each time one has a b, it will be the same thing.

The essential thing will be the position of italics and the general order of the letters.
a b a b a b a b
A . a . B . b .

What happened between the two is precisely that one made these parentheses that grouped the groups of 5 fall away. One made them fall away, and that is the essential point. That said, I regret not having the time to develop this point.

What allows the rupture and the bursting of which I spoke is therefore the open structure of ordination. It is moreover this fact that the term, the agent of the series…
that is what I said at the beginning
…is absent from the series that it arranges, that is to say that it will be present there only on the next stroke. From that, from this absence, is born the possibility of the shift that is the re-objectivation of the entire series.

It is very palpable in a case narrative that the grain of sand of which we spoke, if it manifests a change of level, it is that what was properly the totalizing agent of the preceding formation, that is to say what was the last parentheses, in a way, of the formation preceding the grain of sand, that becomes an element, that is counted in the series for a new totalizing agent.

That is to say that it is clear that the vanishing point or the falling point of a formation in general, of an unconscious formation for example, this point is absent from the formation at the level of the designated, at the level of what it designates, of what it manifests and of what it stages. That is to say that it is a matter, from the designated, of making this ascent, of bringing to light these parentheses, in a way, that are there, but that are absent.

Let us take a single example which is that of this dream, where then really it goes without saying, commented on by Freud at the time when he was looking everywhere for wish fulfillments, where precisely there is a patient who brings him on a tray a dream where there is no apparent desire. One can rack one’s brain; one will not find desire, one will not find an equation of desire, but wish fulfillment.

But Freud, who understood this process very well, says:

‘Precisely, her desire is that there be no desire in the dream, that is to say that I be wrong’.

Which shows well that what in the dream is present is 0, no desire, no equation, etc. But all this 0, it is encircled in parentheses, it is inserted in the more general set, as a part of this set that represents desire in its generality. That is to say that it is supported by a desire, and desire, insofar as it has there the function of support, is absent from the designated. And it is for interpretation to make this 1 arise that was in a potential state in this 0.

There is something in the rupture that does not want to finish, what I called misrecognition, and that leads to successive flattenings. And flattening, it, cannot finish; it cannot be complete.

But what the process tends toward, since I have already spoken of it a bit, is flattening. The flattening of everything that can happen, that is to say of all ruptures, a complete flattening that would delimit and would complete the totality of possible ruptures.

‘The set of all sets’ is the set of everything that can produce, by rupture, a new set. And if it is said that every set, by rupture, gives birth to a new set, then ‘The set of all sets’ defines itself as a possible.

Now precisely what is impossible is to encircle a rupture, to put it in a box. For as soon as from a rupture a new set is produced, it is to push back, to shift the rupture that, from the new set, will make encore another.

The rupture is never in the set, even if the set holds only from wanting to encircle the rupture, and ‘The set of all sets’, the one that would encompass the rupture, is impossible.

After these preliminaries, one can say that what passes…
since I return to my starting point which was the question of the ‘A is A’
…between a subject and the operation that objectifies it, defines it or limits it in predication, has close ties with the category of what supports itself.

Now since what supports something is supported only by something else, we have just seen it, the category of what supports itself, it seems that it is impossible. But if it is impossible, this very impossibility can have effects on predication, which is nothing other than an encircling supported by what wants to be encircled. And it goes without saying on looking that something supports its predicate, but that the predicate at the same time is going to try to encircle that, to bind what supports it.

What is real in these effects could appear a bit anywhere. It would no doubt have been more attractive to see what appears of it for example in Proust’s work, but in any case I took the logic of Port-Royal because it is precisely a theory of substance, a theory of what supports itself, and such a theory can function only—I think—on what we have just seen, even if it is in order to reproduce endlessly a misrecognition.

What brought me to the logic of Port-Royal, where one finds an interlacing of interesting themes like the sign, predication, substance and being, is what was said of a section of predicate characterizing being, for in the logic of Port-Royal, the elementary predication ‘man is’ is considered there as the empty form of all predication, as if the predicate were in this case ‘no predicate’, unpredicable.

There is in the logic of Port-Royal a series of objects that predicate themselves precisely not to predicate themselves, and that participated both in their Jansenist preoccupations on the one hand and Cartesian on the other. I develop a bit this question of predicate and substance to show that if one pushes a bit to the end these concepts that are found in a theory of substance, one obtains something that is more or less what I said before.

A predicate is something in the set that is supported by a thing, a substance, substance being what supports itself. Substance is what one conceives as subsisting by itself and as the subject of everything that one conceives in it. The predicate is what, being conceived in the thing and as not being able to subsist without it, determines it to be in a certain way and makes it be named such.

These are two definitions one finds at the beginning. Now, already from there there is something that is going to fail; there is going to be a stumbling point that will be in a way produced by ordinary language. In logic, it is said that a name of substance is quite naturally a substantive or absolute, whereas a name of predicate is an adjective or connotative.

The problem that arises is that there are substantives that have nothing to do with substances apparently, that are not things, substances like earth, the sun, fire, spirit, which are the examples given of substances in the logic of Port-Royal.

That is to say that apart from these substantives of which I have just spoken, there are also names that express connotative qualities, that is to say names that participate in predication. For example roundness. It is said on the one hand:

‘the idea that I have of roundness represents to me a manner of being or a mode that I do not conceive can naturally subsist without the substance of which it is mode’.

And immediately after, it is said:

‘The names that signify first and directly the modes because in that they have some relation with substance, are also called substantives and absolutes, like hardness, heat, justice, prudence’.

In other words, it is from a point of detail rather derisory that one can conceive—and it unfolds in the logic of Port-Royal—that what was at first mode, or in discourse predicate, after having first and directly been such, it suffices of a certain shift for it to become in its turn substance, substance being what supports itself. Now this shift, it will be necessary to try to pin down, and you will see that it has to do with the set of parts of a set. It is the passage for example in discourse from a predicate ‘round’ to the substantive ‘roundness’. Now roundness is participated in by all the objects that can be predicated round.

That is to say that roundness, to use another expression, is the extension of the predicate ‘round’. And the extension of the predicate is not a predicate; it is a substance. Which means that from an extension of predicate, one obtains a substance—and I am going to dig into this matter—you see well that a substance like earth, sun etc., that is to say a collection of predicates, is an object to which a multiplicity of possible predications relate. Whereas an extension of predicate is properly a predicate that is sustained by being able to be referred to a series of possible objects that are then in the position of predicated of predicate, which means that from an extension of predicate one obtains a substance; that has something to do with the set of parts of a set.

And namely it is said in the logic of Port-Royal that abstraction is what consists in considering parts independently of the whole of which they are part. And it is said that it is thus that one can conceive the attribute, that is to say the predicate, independently of the singular substance that currently supports it. One starts from a set, a thing as set of predicates, to which belong, but inessentially, these predicates. One separates the parts, the predicates, from the thing, and from there, in a way magical, one can consider a new substance which is that by which singular predicates can have relation to unity, independently of any current relation to a singular substance.

There is therefore a process which, starting from the breaking up of a unity, leads to another unity. One must understand that what is given at the beginning as substance, that is to say as the object to which a series of possible predicates can relate, is the same thing as the first A of the ‘A is A’. It is something potential, that is to say that it is given as the support of everything that can happen as predication, potential support, that is to say that it functions at the level of the whole, at the level of whatever.

But as soon as something is given, as soon as predicate exists, the potential support goes up in smoke; that is to say that as soon as an actual word is given, the support ceases to be subject: it is referred to its actual predicate, as if it itself were only an object pertinent for this predicate, this predicate raising itself into extension of predicate, that is to say into intrinsic value.

And it is the predicate that becomes support, substance in the extension, that is to say that there is a reversal of roles. The extension of predicate is a set of objects referred to a predicate; the objects predicate the predicate. Whereas in potential substance, it was all the possible predicates that were referred to the object.

Now, what passes between these two types of substance, potential collection of predicates and extension of predicates, is of the order of what we saw concerning ordinals—I would very much like that to appear on its own. The potential substance is a set of predicates, and the extension of predicate is a set of objects. One makes come out of the potential substance a predicate that it contains, that it is supposed to contain.

And one puts the substance and this actual predicate in relation, one facing the other, in a new set, as there one put in relation 3 as enclosure of parts that one finds right beside itself, all that in one and the same set.

This actual predicate in a new set, put beside the potential substance, that is to say the designation of the designation that was carried out in the first putting-together, that is to say in the first substance, that is what gives the extension of predicate.

Now, if the predicates abstracted from the first substance manage to make One nonetheless, it is thanks to the singularity of what raises itself as first substance, of what takes over, that is to say the extension of predicate. If one pushes back a bit further the difference that founds the One, one can very well ask, considering extensions independently of predicates: what supports the extension?

That is to say that if the extension is the interpretant that supports predicates in their current relation of potential substance, what supports extensions, what is their interpretant, in their relation to this relation itself?
One sees that insofar as, in the passage from the potential collection of predicates to the extension of predicate, there is a reversal of roles, from a formal point of view the two substances are the same thing: it is that there is something that supports and something that is supported, even if in one case it is the contrary of what it is in the other.

But if one adds to that the properly historical or ordinal dimension, the one that I tried to point to at the beginning, one obtains that in the constitution of a set there is something like the substantification of a predicate and that is correlative of the predication of a substance. And that is exactly what we recognized as rupture-flattening in interpretation.

Now it is possible that the play of collection—or one can say comprehension—and of extension in the logic of Port-Royal covers the dialectic of rupture and flattening. And if that is the case, it is quite obviously in a very particular sense that one will have to understand this property of substance of supporting itself.

Because this autonomy of substance, from then on, is entirely relative, that is to say that it holds in the dyadic relation that opposes it to what predicates it, to its predicate, that is to say that one supports and the other is supported; but if of substance there is predication and of predicate there is substantification, that signifies that one must envisage a triadic relation where something like a shifted reciprocity, a discordant reciprocity, is established.

If the predicate becomes substance to support in the extension objects that on the previous stroke supported—in the collection—predicates, this merry-go-round can equally continue encore a bit, such that the extension in its turn is supported by something therefore of which it is only the predicate. The substance-predicate relation presents itself as that of the singular multiple, I said it, and it is the same thing in one direction and in the other.

After collection and extension, there can be something of the order of a collection of extensions, that is to say a set whose elements are precisely these new substances that are the extensions, but de-substantified, taken as predicates of a higher substance that supports them.

Now, that is properly the category of ‘supreme sets’, because in the logic of Port-Royal, everything has an end, and there one touches something that has to do with Being. The extension of predicate as substance is what makes hold together a subject and a predicate in an actual relation.

That is to say that if in the dyadic relation the subject supports the predicate, in the triadic relation it is the extension of predicate that supports the dyadic relation. Extension as substance thus has the function of the interpretant, I have already said it.

So what is the new interpretant—I repeat this question—that supports the dyadic relation between the first dyadic relation and the extension as interpretant? If it is the case that the ultimate term of a serial relation represents it entirely, minus itself…
and you have no doubt noticed that one does not stop working within this hypothesis
…then, in the same way that the set of object-predicate relations, that is to say extension, ‘takes the place of…’ and interprets these relations, it will be the set of all extensions that will be the interpretant of extension.

That is to say that if one repeats the process, the substantified extension of the predicate will de-substantify and be related as predicate to what supports every extension, Being. Being is the only thing that is said to truly support itself, that is to say that it is the predicate of nothing.

Once Being is produced as term of the series, one can return, one can regress down to substances such as ‘extension’, ‘thought’, and found them. It is including from Being that one is perhaps going to grasp more acutely what predication represents, for one has seen that, step by step, it is finally on Being that the predicative relation rests.

Of Being, in the logic of Port-Royal, it is said that it is part of those things that can in no case be predicated for the obvious reason that, if it is predicable, this predicate that one would give it, if one substantifies it, it will be something more vast than Being, and Being will itself be related as predicate to this new substance which will be the extension of this predicate. Now Being cannot be a predicate, therefore Being has no predicate. I cite ‘The Logic…’ concerning Being and thought:

‘We must not ask that we explain these terms because they are among those that are so well understood by everyone that one would obscure them by wanting to explain them’.

That is generally what one says as soon as it is a matter of things like that. To speak of Being is to reduce it to the least being, just as to speak of thought, since if thought is the set of all that one can think and of all that one can say of it, it is necessarily something extra beyond all that one will be able to say of it at the same time, by the very fact that Being cannot be predicated and that, for this other reason, Being is the support of all predication.

There is something like a disjunction between this Being that supports nothing because it cannot be separated from anything, and this whole that cannot be conceived except as supported by Being. But this is disjunction only if one considers at a first stage Being on the one hand and predicates on the other. We will see that this conception is false.

And if Being is properly this nothing in discourse, it is the set of all discourse, that is to say what escapes discourse, what constitutes it. What escapes discourse is discourse itself, from this point of view, since there is discourse as putting-together, as flattening, only in order to catch up with what precisely escapes it.

Thus Being, one will certainly have to situate it as much at the beginning of discourse, in the radical ‘no’, as at the end in the encore. Now, the difference that we isolated between potential substance as possibility of a predication and every actual predication that brings down substance to the rank of predicate become substance, this difference allows us to understand what being is. It is not nothing that a set as closed totality, for example 3 that you see on the board, is different from the set of what one can enumerate as parts of this set.

Substance as support, collection of predicates, includes in a potential way the series of predicates that belong to it, but independently of any actualization of the predicate, for as soon as one actualizes a predicate, as soon as a predicate exists, on the contrary it is a matter of the expulsion out of substance of a predicate. It is a rupture, the rupture that by dismemberment puts substance in relation with everything that it supports.

This is where the knot of the matter is, for if there is a difference between:
– on the one hand the relating, in the actual predicative mode, of substance to the predicates that define it,
– and on the other hand substance itself insofar as it is supposed to be nothing other than its relation to predicates, the fact of supporting them,
then one will have to conclude that substance is something other than a support of predicate, something other than that to which predicates relate.

Nevertheless, there is nothing other in substance than predicates together, and that is said. And yet, if one relates substance, as set of predicates, to these predicates of which it is the set, one finds oneself facing not a simple redundancy but properly a difference. And what there is more in substance, what makes this difference, the fact that predicates are together, is not only a simple additional determination of predicates, for it is said in The Logic that the whole substance holds in this difference between the fact for predicates to be together or not to be.

That is to say that if one suppresses the possibility of this difference, there can no longer be substance, that is to say that there remains a level of predicates, an undifferentiated universe—what Peirce calls the universe of ‘perhaps’, which is equally absolute nothingness, insofar as it is said in The Logic that without substance, predicates do not hold, they are nothing anymore. Substance is what makes something hold, what allows relations, that is to say what is ‘in addition’ when predicates are together.

Now, at the same time, we have not ceased to observe that this ‘in addition’ holds in that a set of predicates becomes a singular term, makes One, and that this singular term is not part of that of which it is the set at the moment when it designates that of which it is the set. Thus substance is what, when a set is given, constitutes it and is lacking to it, and that at the same time. In other words, what is lacking in a set is what constitutes it: substance.

Now if one looks at what is explicitly lacking in ‘The logic of Port-Royal’, because it is said that something is lacking, one will realize unfortunately or not that it is not substance, precisely. What is lacking is of the set: what, when there is nothing other than what is lacking, is equivalent to nothing. It is a definition like another.

And it is said in The Logic that if, from this ‘whole’ formed by substance and predicates, one removes substance, then nothing remains, for this: that predicates and attributes exist only because there is substance.

There, one is truly embarked in a logical corridor from which one cannot get out, a series of propositions that carry us along. Substance is nothing other than predicates plus something. This ‘plus’ is defined as lacking. And predicates are what alone is nothing, but that is produced when substance is given. That is to say, predicates are nothing without something: substance, which is nothing other than the addition to these predicates, supposed contradictorily already given, of what in any case, in the sum, will be lacking.

Substance supports predicates, but also in a certain manner predicates support substance, as this nothing encore from which by substantification the singularity of a difference will be born. Predicates are only 0. Substance is what is added to 0 to make 1, but in this constituted 1 there are only predicates, that is to say 0, that appear, for what makes 1 precisely in the inscription of 0 is absent from what 1 inscribes, that is to say from the designated content of 1, that is to say 0.

These contradictions therefore, which I have pointed out by these few formulas, seem able to be re-ordered starting from the reintroduction of the ordinal point of view that preceded at the beginning of this taking-in-view of the logic of Port-Royal, that is to say the opposition between collection and extension. It is understood like this: substance supports the predicate which, defined, bears on substance.

Now we are going to take all the contradictory propositions one by one and accept only one at a time; that is the best solution. After that, everything will work. Substance being what is lacking, predicate is an effect of lack, what bears on a lack, the wrapping of lack. But on the other hand, predicate is nothing without substance, and it is impossible to differentiate substance from the actual predicate as manifestation of missing substance.

However, since it is said that predicate is nothing without substance, and since it is said that there is no substance, that it is lacking, then since there is predicate, one is forced to deduce that predicate is substance. Since without substance there is no substance, predicate should be nothing, now it yields 1, which implies that this 1 of predicate is not predicate but properly substance.

Now that can be understood only from this ordinal point of view which is the question of the substantification of predicate. The predicate that is supposed to be nothing without substance, if it manifests as something, this something as other than the nothing of predicate is necessarily substance. That is to say that in the extension of predicate, predicate is substantified, that is to say that predicate in the extension will take the place of substance in a punctual way, for something that will take the place of predicate, that is to say the objects of the extension.

And at the same time, now there is substance, there is substance, now it is supposed to be lacking; at the same time, as soon as the second class of predicates is produced, the operation repeats, and what in the first stage took the place of substance will be lacking as substance, since, by the operation I pointed out, it will apply as predicate to the new term that appears as a provisional substance.

And this to infinity, that is to say that, as soon as substance is given, it is inscribed by actualizing itself through the predicates that apply to it, but as soon as predicates actualize, substance is related to these predicates that acquire a substantial value that is extension, that is to say that it is impossible for substance to be at once given and inscribed at the same time.

Substance can therefore very well be defined as what is lacking and as what makes the set. On the one hand a predicate rests on the first predicate taking the place of substance, to define it, to identify it, to predicate it. And on the other hand the first predicate-substance related in this relation to the second which acquires an extension disappears as substance, to become only an element in the extension of the second predicate and to confer on it the relay of this function of substance. Substance is a function that this one will transmit to a third predicate etc. One sees that the first substance, the one that is supposed to be at the beginning, the potential substance, is entirely mythical.

What counts is this relay play; it is the actual relation of predication which, made possible by potential substance, inscribes it and transforms it into a term, into predicate in a relation, it being understood that the ultimate term of the relation in its turn plays the role of substance, that is to say lack in the relation, and is inscribed only by becoming something other than substance, that is to say predicate. The successive substances are therefore the series of transitory incarnations of what is lacking and that supports every pseudo-substance as wrapping of lack.

Being is indeed what supports all discourse insofar as discourse; it is what is produced on the edge of the hole that it constitutes. Being is therefore at once what is before discourse, that bears discourse, and what is after the end of all discourse, its point of convergence, its limit. In the logic of Port-Royal—I would like to resituate things—it is not such a theory of discourse that one can find; it is the contrary.

But insofar as it is the contrary, there is something like this theory that insists within the very discourse that is held, whereas the initial project of Port-Royal was to construct a metalanguage and that is said namely; it is on the contrary that something insists in Port-Royal—despite Port-Royal—that is to say that takes its effects from this: that as soon as Being is presented
– as what cannot be predicated,
– as set of all that can be attributed.

This unpredication of being is presented in an already eloquent formula. It is said: ‘Being is unpredicable’, now precisely ‘unpredicable’, it is perhaps there this first predicate which, in this attempt to signify the impossible, does nothing but repeat it by the fact of exposing its own emptiness and which thereby traces at once the limit of what is possible and what is not.

In this sense, the possible, the potential, is what is impossible to carry out, it is what cannot be given without transforming itself and changing function.

Whereas the impossible is the only thing that can be realized while leaving open what founds this impossibility, that is to say this gaping, for the type of realization of the impossible leaves the impossibility gaping, this for example which is the predication of the unpredicable.

I end on something that would bring us a bit farther, but I do not want to conclude, that is to say to close this discourse which was only a preliminary: language is what represents Being for speech, that is to say that speech is in the position of the interpretant, between tree and bark, just as the finite is what is woven between two infinities.

[Applause]

LACAN

– I will conclude with these words: with time, it comes out!