Venn, Boole & les diagrammes entre logique et algebra

0:00 I'm happy to be here because there's a lesson. I want to say that it's a bit of a discussion with Hermes Promet, who was in the CV, that I had heard a little bit about what she was doing, and I found myself quite happy because she was doing everything. So I wanted to make a little bit of a project. So, I have two photos, I don't know what date, I don't know when, but I have done it on the internet. There is one of the most aged ones apparently, but I don't know at all. So Greil is born on April 1834, in the north end of Angleterre, and he is born at Cambridge on April 4th of 1923. So he is born, he had 98 years old. But it's not true, he is born, but I suppose it's not true. You know, he is born, he is born. Yes, he is born. But you, he is born, he is born. He is born, he is born. But Greil, he is born. So he is born from a family where they were in prison. son père était son arrière-grand-père, je ne sais pas, et c'est une famille intéressante dans la mesure où ils étaient assez engagés pour l'époque, ils vivaient contre l'escalade, etc. donc c'est quand même assez intéressant. Sa mère meurt lorsqu'il est encore enfant, donc il a été élevé, ou presque apparemment, par son père. Donc il fait ses études primaires à l'ombre, sur le père, sur une paroisse, and he is going to the college in 1853. He comes to the college in 1857, he is a diatribe in 1858, and he is ordered to be a priest in 1859. And then, until 1862, he is a priest. He is going to be a priest, I don't know about it, I don't know about it. And he is going to be a priest as a lecturer of moral sciences in 1869. And he is not a priest. So it's there, apparently, that he studied the logic, mainly in Boole and in De Morgan, he studied the probability, I think that he is also in the book, and in the psychosophie. He started in 1967 with his wife, his family, and he will have a son, who will be president of Queens College in 1832, in 1932.

2:30 In 1983, he is elected to the Royal Society, and this year, he quitts the religious, abandoning the priest, because he considers that his research work requires a limit of the spirit, which is incompatible with the doctrine empirical imposed by the 31st parties of the form of the law. So it's also pretty nice, I think. Well, I'm going to ask you a question, but for my personal question, is that it's a return of the Darwinism or not? I don't know. According to the testimonies, he was a marcher montagnard, or we would say randonneur. He was interested in the botanique, and he was also quite a linguist. And it's true that in what he cited as an ouvrage, apparently he said, without a problem of latin, it's normal, in English, it's normal, in English, in English, in English, in English, in English, in English, and in English, in English, in English, in English, in English. Well, he was also a double of his hands, because he described in his own book, as I mentioned earlier, the realisation of these diagrams. There is not a plan, but a schéma in which he explains that we can fabricate a mechanisation of these diagrams. And he also used to measure his legs, for example, which is a machine, because it's in English, a machine to launch the ball of clicker. and this machine would have fought the best world australian in 1909 so it was visited in Angleterre so it's something that would be noted in the bibliography Well, when he was at Cambridge, from the 80s he published several articles in different reviews about the idea of the machine which I mentioned earlier and mainly there were three books concerning the logic In 1981, symbolic logic, with a second edition of 1984, which is very interesting, according to the commentators, The Principle of Empirical Logic, in 1889. And from there, there is no logic, they don't care about it. He does all his bibliothèque biologist at the University of Cambridge, and it is considered because in his book, he says that he has read 60 books in the last century. the last one, in all the world he was able to get into the book. There were some who said, I don't have to get into the book, but I don't have to get into the book.

5:00 But there was no internet, for having a book, he was going to command in the library, in the bibliothèque. It had to be complicated. So he gave his bibliothèque a book and he did his story. So, he takes the biographical history of Gombeville and Caius College in 1349-1897, and then he writes the story of his family. He takes, with the help of his son, to publish the biography of the students at Cambridge University. So, the first part is published with 76 millions, which are the students at Cambridge in 1751. the second volume, but it is not yet published, there are only 60 million people in the second volume. So it's a bit of a picture of the character, which is actually interesting. So my exposé will be in five parts. First, a story story to replace Evan in the situation that he has found, or he has found the logic, before he does his work. a part of the introduction of the diagram in general. Then I will talk about the function, but I will not enter into details. Then I will ask the question, is there a new age that is behind these diagrams and what is the relation of these diagrams with the new age? And then I will see if there is an aspect of the finance that we have. Well, in fact, we have to see that it was the first to algebra the logic. And, despite all these defects, this algebra comes practically on the theory of ancient. In fact, the theory of ancient what was? It was the logic of the reality of the Moyen-Âge, like we still found in the logic of Port Royale in 1662. It's the analysis of the discourse, and the analysis of the discourse will guarantee the reasoning just, by looking at whether the discourse is correct, whether the discourse is not correct. And so the rules of the good reasoning are ruled by the discourse. And it's the analysis that gives the right rules to know when we conclude, when we conclude and when we don't go through certain data.

7:30 In fact, there is still a part in the logical analysis that exists when we look at the status of the previous one. And that's where it is. Well, it is evident that this thing is not going to be indifferent to the construction the development of the scientific development of the 17th or 18th century, and the retombées are present in the 19th century. So it's there that it's in the sense, that it's in the sense, until we disappear. In fact, we can still look at the logic that it exists classically, or even the other. But if you look at the logic of the Fort Royale, you will see that there is a lot of logic, but there is also a lot of grammar. Well, it's the most important thing. And this symbolical logic, which emerged in the 19th century, was in fact elaborated at the beginning of Great Britain by people like Hamilton, Morgan, Benjamin, Ben, Carole. Carole, he was interpretive. And then, this symbolical logic, it was spread and reponsed ailleurs, in Germany in particular, by Schroeder, Frege, Well, so, l'école algerique anglaise, in fact, dont Morgan et Boole sont les représentants, de seconde generation, avait mis en avant l'importance du symbolisme dans l'expression et maintenant des concepts. Pourquoi ? Simplement, parce que pour cette école algerique anglaise, la stagnation de la pensée, qui était figée jusqu'au début du monde du siècle, It came from an archaic conception, and it was founded on an archaic conception, like Peacock and all the savants, like Huell, Schell, etc. They had a conception of the algebra, in fact, as science of discourse, and it was intended to give a cadre unified to all the development of the knowledge, which was, in fact, universel, and which allowed to eliminate the conditions locales. It is then finally the abstract symbol which would port the noix of the general, and not the notion of grandeur and amplification which would be classified. The utility of the representation is not the same. And it is not in the geometry that the mathematics will attach to its legitimacy, it is in the noix of the calculus. In these conditions, obviously, we understand that

10:00 So that he would be able to do this graphically. He would be completely outside the pensée. He would be an adept of this tradition that he was created in Grand Bretagne. And yet, a trentaine of years later, since the pensée is 1854, a trentaine of years later, since we are in 1884, he would use the diagram which is part of the system. I would like to say, It's not an illustration, it's not something that is above it, it's something that is beyond. It's part of the system. And, to my opinion, it was quite a bit of a digestation. It was digested. L'avancée de boule a been amendée when it was insufficient to its major défauts. And at this moment, we see, at this moment, there is a system in which the diagram has their place because they are in the system but the other part they do not have a representation or a legitimation of the type genetically. The first apparition of the diagram is in an article of 1880. You can see . Oh, I'll show you! It's a little suspense. I'll show you the diagram of Venn of 1880. So these diagrams are a little complicated. I'm not photographing the thing with a certain scope. I know that everyone is able to do this. So I haven't done it. These diagrams are in the article of 1880 which is the title of the representation of the diagram and mechanics of the propositions and the raisonnements in Philosophical Magazine. The next one, Marcant, who is a logicist, is he intervened on the same subject in the same review. And then, on the basis of his article, Bell develops the first edition of Symbolic Logical in 1880. And the second edition, which is the of 1994, because in the edition of 81, there are things that... I said 94. It's 94. 94, the second edition, because that is definitive and the things that were in the edition of 81 were reviewed by Ben

12:30 who said that he was not agree with what he did, that it was not very good, that he had improved the things, etc. So, we can consider it that it is the thought of the thought of Ben because it is the same in this edition which is, of course, that I managed to find in a text simulant, but I don't know from that time, it was 61. There is a text simulant from 61 that I found in my book. So, here we go. So, here's what is the cadre in which I'm supposed to be. So, on the diagram, there are some generalizations that you need to know about the diagram. because, to see the work that we have in relation to that, it is to understand that a diagram is a representation of the schematic of the situation. And there is a double aspect. One part, it can allow you to give a particular vision in occupying other aspects of the situation. But, on the other hand, it can consider, it can give you a sort of synthesis, in recapitulant globalement, at a moment where the knowledge is acquired at the end of the month. Well, the algorithm can be efficient, it's his but, if he is not efficient, it's not the same. And his efficacy is conditioned in a big part by his equation, more or less than his own logic. He demands a double translation, a translation of effectuation and of codage, a representation of an expression, and a second translation, in another sense, an interpretation of an intuitive. And we need to be able to do these two senses. and in the two sense, he has obviously a guarantee of fiability and simplicity because otherwise, it's rare. So, his role is often static in fact. It's a diagram where it takes into account the situation at a moment, and when we see something, it can simplify things, and make an eye on the situation. Well, but sometimes it can also be dynamic, and dynamic as support to a reasoning, in cumulant different stages, and eventually in organizing these stages of the reasoning. When they play this last role, and it's the last one in the frame of the frame, it must be understood that there are two ways to do it. So by adding an instruction on the diagram, or then, and this is not easy with a paper frame, And so, with a graph and a tableau, it's more easy. With a graph and a graph, it's more easy.

15:00 So, we add things to the diagram and we enlève. And the two elements can be understood. So, what does Venn for a diagram? He talks about page 2. I don't know, that's not possible. But before, maybe I should... Well, it's not possible. The situation Venn is introduced, in fact, in the logics of three periods. He talks about the view ordinary or predicative. It's the story of the logics classique. The view predicative or ordinary, it's liable to the predicates. We do the reasoning on the predicates It's not a good logic. A second period in the logic, it's the view of the classes inclusives and exclusives. It's a logic based on the principle of inclusion and exclusion. And he attribue the merit to Euler. He says that it's the second phase of the logic. And finally, the third phase, in which he sits at it, it's the phase of the view compartimental or existential. The symbol of the Fiboulet, in fact. it depends on the problem by diagram. So, at the level of diagram, what is happening? Well, obviously, it's in the 3rd period in the 2nd period, in the 2nd period, to justify the 3rd period. And, for him, he is very clear, at the beginning, that it is not to solve the problem, and to solve the problem, it is to enlarge the logic the language of the pure symbols which are already with success in mathematics. So it concerns one chapter of the diagram, which is chapter 5. So that means that before, there are four chapters, in which, after the introduction of the day, there are two functions of the system. In fact, it's the system that I'm talking about. But in fact, the diagrams are explained simply in the fifth chapter. Or, from the page 7, there is a diagram. If he doesn't even talk about it, he does a diagram. The diagram, it's two spheres of the sun. I don't know how to describe it. So it also means that if he did it before, it's still important. If it was simply an ajout to his system, he would not do it from the page 7 without having to talk about it. So at that moment he did that? It's when he looked at the schémas of the air. The schémas of the air, I'll show you. It's still a little bit.

17:30 What I'll show you is practically in the air. The Schémas Euler, it's the Schémas that exist in the letters by the princesses of Germany in 1748, in which Euler did the logic. There are 20 letters in which he did the logic. And at a moment, he presents the syllogism classically and he illustrates it by the five, which concern, in fact, two, a predicate and a subject. In fact, it's complicated when we have the syllogism, of course, but it's a great debate. critique fondamentale, bon il faudrait étudier ça mais ça a été plus ou moins clair, j'ai fait un travail ici là-dessus mais bon c'est pas important pour nous, elle trouve que finalement c'est pas symétrique, c'est pas clair, c'est pas flexible, c'est pas général, il fait plein de critiques. Le seul crédit qu'il aurait attribué apparemment c'est qu'il y a une symétrisation entre le sujet et les critiques. After all, it's not symétrique. I didn't understand why it was not symétrique. But it was still symétrique for the subject and the predicate. And I remember that the subject and the predicate are not on the same plan, while the logic, obviously, is not the difference between the subject and the predicate. I'll read that later. Well, Venn, it's not for the opposite of his diagram of the lens that he has established in fact the properties that he has in the diagram. For him, it must provide an air for his eyes, it must be clear, transparent, it must avoid a curve of calculating, It needs to be done with the pensée, it needs to be done with time, it needs to be protected against errors and errors, it needs to be symétrique, and it needs to be élément in itself. The aesthetic of the thing is not innocent. Euler, well, I will cite Euler, because it's kind of a joli Euler.

20:00 Euler, who was kind of Swiss, not Swiss-Français, was kind of a French from the 18th century. These figures, or rather, these spaces, are very propres to a possibility of an effect on this matter, and they discover all the mystery in which we see in the logic, and that we demonstrate with the pain, while by the means of these symbols, D'abord, on envoie donc des espaces formés à plaisir pour représenter chaque notion générale et on marque le sujet d'une proposition par un espace contenant A et le prédicat par un autre espace contenant B. Bon, en fait, ces figures traduisent la situation extensive d'une relation de sujets prédicats. Par exemple, quand vous avez ce match-là, ça veut dire que A est bien de toute taille. Bon, alors, évidemment, quand on regarde les quatre types d'insertions reconnues par la logique classique, il y a cinq positions respectives de Neussert qui se dit que ça ne va pas marcher. Cinq positions respectives de Neussert, quatre figures d'insertions dans la logique classique, il y a un truc qui va pas. Donc c'est pas l'objectif, effectivement, mais pourquoi ce serait l'objectif, finalement ? If you look at, for example, when he tries to treat it in a way a bit more modern, in 1817, the aristotélician logic, it's not that much, it's not that much, it's not that much, it's not that much. But the situation becomes almost inexplicable, in fact, when you have more of two subjects and two articles, when you have three terms, you can't get out of it. Well, simply because, that is Venn, that is the translation, it is the translation, in logique commune, of the 5th century, and it says that it does not do anything, it does not do anything. Well, obviously, in logique quantifiée, that's to say in a new logique, where we can quantify the predicament, we'll talk a little bit more about it, because we see immediately that there is no more way to do the same thing. But there, to my opinion, it takes a little bit to the ficelle. And then, he gives it to his traduction to him. That, it's the traduction of Venn.

22:30 that is the logic logic, that is correct. Well, if you say, of course, that is not exactly the same thing. Well, for Open, in fact, the objective, is to obtain a method which allows us to treat problems logically, which will intervene in various variables. It is not like for O'Leary to take into account a situation after a coup and to represent a situation. It's about to régler the problems, and so it's about to have something to be evolved. The Diabarne-Dobain is simply a retombing of the logic Boolean. In fact, it explains that there are two types of symbols. There are symbols of operation and the symbol of class. Well, it's like you, we're used to, I don't know how to explain it. The elements, when you describe the situation of the two classes of objects, which you call x and y, there are elements that are common, the objects that possesses the X and Y, which are the X, Y, which is a note by the juxtaposition, which is a multiplication. So these are the elements common. There are objects that are not the X, which are the non-X, So it is noted by the surlignement, so we apply it to that. And then when we are going to take an object that are X or Y, we obtain a disjonction, and this disjonction is noted by the addition. But this addition, Venn the prend non-exclusive, so that Venn, in his first edition, the prend exclusive. And then in the second edition, Symbol Ethology, it says that it is an error because it works well and that they will have all interest to take an addition, which is intuitive. So, these are the limitations of the language that we have now, I don't know how to say that. But Venn doesn't know what the language is, because it's a concept of inventing. So it's actually called obligations when we want to represent that. So what is important for him is to have the compartments of the classes which are, I repeat at least 10 times in his book, mutually exclusive and collectively exhaustive. It's to say that he has to be joined

25:00 and all the classes that he has established will recover the totality of the situation. That's exactly what he said. No, no, he insisted. He said a moment, and I was talking about it a little earlier, but I don't want to confuse the places and the compartments. The places should be vides, because the xy can't exist, but the compartments are not vides. I don't do the distinction between the places and the compartments. The compartments, it's a representation and the classes, it's the treatment the situation and the classes are inserted into a group of people. So it's a mutual exclusive and then you have to do the rest. Collectively exhaustive. Collectively exhaustive. C'est la partition. C'est la partition. Voilà. Alors, il s'agit en fait, pour lui, et il le dit, il s'agit simplement de faire procéder à la dichotomie. Puisqu'à chaque fois qu'il y a une classe, il est coupé en nous. C'est un travail de dichotomie. Well, it's not the dichotomy in the sense aristotélicien-cutter, where there are animals, animals, etc. It's inspired by that in a certain way, but through a médiation that you will see all of a sudden, and that you will see all of a sudden. So, in fact, we need to divide the universe of things in all these unique subdivisions in prenant for each term in prenant for each term of his contract it is to say that every time he has a term he considers the term and his contract and for all the terms he has to do that and he obtains at this moment there are classes and who are by dichotomy and who are exclusively in the collective and it is from this that he works so he can assign each term has a place, and by this démarche, in fact, they find a place in terms, and they fix a point of ancrage for the diagram, because the diagram will turn, in fact, around the representation of the R. Now, concerning the proposition of the number of terms, it's Y, the figure is the number of terms, it's grand, so it's... Well, it insists on the fact that it doesn't represent a proposition. Now, contrairement to the circle of R, it represents the propositions. For example, this situation in the circle of R, it's not a BBA, or not a BBA.

27:30 For him, in fact, it doesn't represent a situation, but simply a table in which the situation is simple. It's to say that for R, it's all these situations with two things. This is what you know well. It's the circle. And this, it's the compartments in which the place should enter. There are five things that correspond to different situations. For LR, it's the same thing. Just after that, it's not the same thing. So it's just a case in which the propositions come out. It's what it explains. It doesn't have to do with the same two. It's another one. Evidently, there's no difficulty. So I'll come back to what I said earlier. There's no difficulty when we rest with three propositions. The situation of the proposition would be synonymous? It's the same thing. It's the same thing. It's the same thing. It's the proposition of the situation. The existence of the proposition. The existence of the proposition. Okay. So, here's what it is. I'm going to read it now. It's a long time. It's a long time. It's a long time. So the first thing in his article, his article is for the situation of 4 classes. You can see, there are 3 classes, there are 3 certs. And then the fourth, obviously, we are obligated to make a figure which coupe all the places preceded. It's not me, it's not me, it's not me. So, he says, we can think about things in H4. So, that's the thing in H4. Then you have five classes, five variables, it's more embêtant. So in his article 280, he proposed that. So it's not idiot like this. Because in the formal logic, in symbolic logic, he gives this. For three variables, he gives the three CRs. for 4, for 4 years it's a problem, so he does 4. Well, and there he has a note, because it's an example, he gives a note in saying that this case is a note of what it is. And then for 5, there is a problem, because it's an art form, it's easy. So he does this thing, he takes the diagram for 4, and then you see this thing, this bizarre thing with a trou.

30:00 Then he explains that the Z, it's all that except the trou. It's not just that, it's not just that, it's not just that. And he says that at this moment, it's a bit more embêtant. He also says that for simple, you can do two diagrams like this. You can do two diagrams like this. One for the Z, one for the N, one for the Z. And he says that, well... He says that you can do a little bit of an achievement in form of pen. In form of pen. If it's a matter of fact, then after I have a copy of a photo, I have a copy of a photo that I have done with a symbol of magic. Ah, non, non, non, il va pas jusque là. Non, il est pas pour lui parce qu'il écrit là des deux côtés de la feuille. Allons-y en même temps... ... j'en ai bien compris pendant que ça marchait. Voilà, donc, ils proposent ça. Alors, ils disent de toutes façons que... Non, ils concluent pas là-dessus, alors le truc intéressant, je vais vous mettre, le truc intéressant c'est que je vais parler de Marc-en tout à l'heure. Well, Mr. Marcant, what's he does? Mr. Marcant, he has a solution in the article, in the year after the next year. No, it's good, it's good. So, you're going to say that, when he solves a problem, he does this. I'll leave it like this. That is marquant. That is marquant. I have maybe heard this kind of show. That is marquant to the six variables. You have the A and the non-A because the marquant doesn't make the negations with a bar, but it makes the minuscule. So here you have, all these parts are the A, all these parts are the non-A. Now if you have a linear variable B, the A is shared in two and then the things that are A and B. And in the same way, the no A, you have the A who are B, and then the no A who are B. And then you have C, you know what's going on. And then from the other side, now you have the possibility of doing that. And here, there's a little problem because it explains that here, it's about the no A, which are B, which are non C, which are non D, which are E and which are F. it's the nuclear exclusive and the collective exclusive.

32:30 After the diagram of the Oscar, yes, it's a implication. Well, but it's a great question. Well, it's a great question. Well, I'll just say in a few words, and then it's a little bit, I'll explain a little bit the function of it. Well, of course, the interest of the diagram is that when it's done by Z, And it explains how we traverse one frontier, there is only one that changes the value, which is 1.000 or not 1.000. If you look at the non-x, if you look at the non-x, if you look at the frontier, you look at the x. The problem is that it doesn't work like that. You can't do this type of reasoning with the diagram of Markov. But it's not very interesting if you make a decision and interest that it doesn't work well. Carole, Carole also has made a gap, so you can see that. But that I'll put it on, because I'm going to give it to you. Carole, it's another thing. But it's not... it's not bad. Carole... Well, it's simple. Well. Well, that's when there are two objects, three, four, etc. Well, here I have... It's in the picture of Michel's face. No, it's not in this one. Because in this one, it's all in English. That's the French language. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. It's not bad, but it's not bad because there's nothing in it, there's nothing in it. But on the other hand, what's interesting is the post-phase of the coups, which is interesting. Well, I'm German, then. Yes, I'm German. Yes, I'm German. But the problem is that everything is not translated. So Carole, you can see how it works. It's the same genre, and it's the dichotomy, it's exactly the same thing. It's just the presentation. Well, Carole, he presents his thing in saying that, in his case, the things are much more rigoureuses because in the liberal world of Mr. Venn, the universe of the discourse could be set up to infinity. He could do nothing to do. Here, it's the non-existence. And there's no limit. It's the paper. Venn said that it's the paper. The limit, it's the material. For Carole, he marked it very strongly, but not a problem, not a problem, not a problem.

35:00 He called it the universe of history. Yes, yes, it is the universe of history. He marked it very strongly the universe of history, so that, finally, it was a constant, but without explicity, considering the universe of history and the fact that he did. Well, in his book, Gendrick Jekyll, I'll show you this because it's a lot better, Carole, the plastic that had been invented at this point, and Carole made a series of envelopes, in which there is this, it's in a carton. There is an anthology, it's a book that has been made for the children, for the initiates of the modern logic. And so there's this stuff in carton, with these buttons. And these buttons are designed to be used to solve problems. There were three buttons, I suppose, which would be, I don't know, the colors, the blue, the blue, or something like that. And so it was to make the diagram of the map, but in a unique way. So the magic would be a new one. So it's not the thing to do that, but that's the thing to do with Carole. I'm going to put my diagram here, because I'm going to talk about these diagrams, which is this thing here, in which there are some interesting things. I'll put it in my hand. So, the first example that we have to solve the problem in this case 8x5 with the diagram, well, this is the first example that we have to solve the problem by diagram, it's not a diagram of the problem, it's a diagram of Markov. However, there is no necessity, there is a multi-variable, or 8. Well, using a diagram of Markov, it was perfectly correct. But here you can see that he has 4 variables. So 4 variables, he could do a lot more simple than that. And he takes a diagram of the markup. I don't know if he's going to get rid of it, but for him it's not important. It's the same thing. So, what is it? I'm going to explain. It's not very complicated. On a, in fact, there are classes that can be filled. For example, if you want to be a little, it's y. There are x, y, y, y, y, y, y, y, y, y, y, y, y, y, y, y, y, y, y y a a classé y.

37:30 Donc les classes y, y achurent, ils l'enlèvent. C'est comme quand on fait des résolutions d'équations dans les classes et lycées, le truc qui nous intéresse pas, on l'achure, on l'enlève. Comme j'ai constaté depuis quelques temps qu'ils ont tendance à achurer ce qu'ils gardent. Alors c'est évidemment non comme concurrent, je sais pas pourquoi. Les quatre trucs qui achurent, alors ils le montrent et ce qui achure est quatre fois. So, what is assured, it's what is visible, at costum. Now, obviously, there are cases that are not assured. These cases, we don't know. We don't know. They correspond to the compartments in which the classes can't live. They can't live. And then, there are cases that are not visible. This is for, I'm going to talk about the composition, and at this point, we propose to put a special symbol, for example a plus, notant that this class cannot be fine. Here, obviously, there is a problem contradictor, because there are compartments, there are compartments, and there are compartments that correspond to the classes that cannot be fine. So the problem has no solution. Here, by the way, there is another example that he does. He says that, in the place of putting the plus in the cases that cannot be fine, that they cannot be fine because they exist in the elements, We can simply put the number of the proposition, because we have all the points in the previous proposition. We can put in the case the number of the proposition which means that this case is not valid. For example, here we know that this case is not valid. Here it is because of the proposition 1-3. So there are three things in the diagram of Bell. When we have the compartments, on a chute les compartiments qu'il croate dans les classes vies, il fait bien la distinction entre les compartiments. Les compartiments qui ont un symbole spécial, soit le plus, soit un numéro, sont non vides à coup sûr, et les autres, on ne peut rien en dire. Enfin, ils sont peut-être vies, ils ne sont peut-être pas vides, mais on ne peut pas conclure, il n'y a pas de conclusion. Voilà. Alors il dit même qu'à part le fait qu'il décrit une petite machine, on peut... So this machine is like this, if you want to, where everything is covered in the wall. And all the small pieces that are there, the compartments, are mobile. It's to say that there are some small pieces. It's referenced, for example, if you don't know why. And when you take a small piece of the compartments, you can see it.

40:00 You can see it. I don't know what it is. But apparently it was like this. It was also like this. for the diagram of 3, 4, 5 elements, to avoid losing time. Well, now we have to look at why it works. Why it works? Well, the methodologies, in fact, which are in the middle of 2020, are the methods that we have put in place, after all a work that has been done, in particular, on the subject and the predicate. Because, as long as it was to combine in fact, the assertions of the type of subject and predicate, we have no relation to inclusion and exclusion of the type of subject and predicate, the subject and predicate. The mises on the same plan of the two elements, subject and predicate, are under a symétrie. And this gesture, in fact, is autorizing a global assertion which does not exist in the traditional class. There are two terms that are related to the predicate, but these two terms have the same status. And this symbolization is impossible if the verb is not even affected by that, support de la phrase. Et c'est De Morgane qui a fait ce double travail. Le premier travail c'est changer le rôle de la copule. Alors pour De Morgane en fait, dans ses écrits en d'octilogisme qui sont de 1850, il a écrit quatre articles, donc il est repris dans un seul ouvrage en 66 à l'an. Il y a quatre articles qui s'appellent en d'octilogisme. Il dit que finalement pour lui la copule elle doit avoir, elle doit être abstraite et procéder deux propriétés abstraites. which is, for him, transitingness and convertibility. For us, it means transitivity, and then convertibility, it's simply the symmetry. It's a different language where we have the composition. In fact, for him, this cobweb abstraite is simply something that is transitive and symmetrically. Reflexive, he doesn't talk about it, because it's Saraswa. and so he envisage, in fact, the treatment of a copule which is a relation to the theory. It's maybe for that why we said that Morgan was the one who had begun to learn the logic of the relation, but in fact, it was in fact, but at the beginning,

42:30 there was this position, to my opinion, which is definitive, and the sign of equality which is used by Morgan to make sense of its propriety, and it's a propriety of equivalence, is the sign that, from there, will not give up the logic. In fact, the sign of equality, here, is the absolute identity of two classes. It's the sign of identity. And it's the only predicament that is envisaged. In fact, because everything will turn into this predicament. And at this point, the algebraicization can be done with the negation. The problem of the logic aristotelic was the prohibition of non-negative. The prohibition of non-negative, because it has no sense. We can't talk about the non-hommes. What is it, a non-hommes? We can't describe it in the aristotelic system. So, the non-negative system doesn't exist. Or, the universe of the discourse that Perlmuk has put in place to Morgan, conjointment with the negative, allows this thing. The thing is very simple, Morgan, as I said to my students, he has a great idea, I think. He has realized that when you have, for example, a troupeau of chevaux, you have selected the chevaux blancs, it's like you want to take the chevaux blancs, When we selected the Churons Blancs, we also selected the Churons Non-Blancs, which remained at the same time. They were not collectivized. And so, De Morgan, he came to realize that when we make a choice, we make a partage. There is always a partage, even if we don't have anything. So it's his vision of the Churons. And at that moment, he said, if we introduce this new technique, the technique, he is well aware that he does something important. He invented a new technique called the universe of the proposition. Then he generalizes the universe of the disco when we call it the proposition. And he says simply that when we choose something, we choose automatically the negation of the same way, and they are exactly on the same page. And so, those who are the same for comptes, comptent the other. There is no reason that this is not so. And from that moment, we have the symétrisation of the subject. When we have the possibility of non-negative, we have the possibility, on this basis, to have a dichotomy system between the x and the non-x.

45:00 And if there are y and non-y, the x can be xy or x non-y, and the x can be separated from the criteria, etc. And so, it's all the basis of the system. Well, it is possible that we have, on this principle, two components and compartments for one category of objects. Now, it remains to consider why every problem of logic will be resolved, at a certain moment, to the non-actuality of certain compartments. It's not all about having a compartments that can be used to mobile, etc. It means that everything is going to be done with X and Y and Y. But we need to see that all these propositions will be able to enter into this case. We need to reflect and reprendre all the classical logic under this form, which is a new one, and visualize the solution completely in the form of a diagram. Well, the problem is that he adopted a point of view that is quite radical, about the existence. It is important to know that at this time, in the 1860, 1880, there was in the review Mind, which was the great review of the university, on all sorts of subjects, such as the philosophy, the economics, the economics, the botanical, etc. There was a big discussion recurrente about the existence of things, in particular. Well, Venn said, finally, about all X et Y, he took an example very simple to explain, The affirmative universe is all x and y. So, all x and y, it says that we can interpret it in two ways. The first one is negative and absolutely. It is to say that in all x and y, there are no things in the same way as x and y. All x and y, there are no things in the same way as x and y. Or, we can interpret it positively but conditionally. if there are x, then these x are in. And he says that in between the two positions, he chooses the absolute position,

47:30 that is to say that he does not exist of x, which is not his problem. So he takes the position médiative, and he says that x is y, So x et y, it's like that he uses his system, it's for that it works, x et y. He is translated by x, so it's not the subject or the predicament that is in the game, but the non-existence of all subjects who doesn't have the problem. So they work through the negation, and they say that it's absolute. There's no way to do the supposition if they exist or not, it's like that, and even if there's no X, it's like that. And if there's no X, it's equal to 0, it's not that. So, when they look at the problem of the hypothyroidism in which it was quite a bit, there are several chapters on the hypothyroidism in which it is quite a bit, Well, he says, if P, then Q, if P, then Q, that means that there is no object, P, that means that there is no Q. So, he translated it as Q. Here we go... Then the usual rate of change... If it's a Q, it means that when we don't have a Q, it's a Q variable. And then it's a Q variable. So, in fact, we see here, in a way that's impressive, that he found a good thing because, finally, there is a system that allows us to account of two distinct objects with the same formalization and the same function. the propositions of the type of x and y etc. Is this a question, does it make a difference, and is the difference between, like that, many of them are q, and, from the other side,

50:00 that p and non q are false? Well, for him... No, there is no problem, he doesn't pose this question. For him, he wants to represent, he wants to represent, he wants to represent, he wants to represent these problems, he doesn't pose a question theoretical, he wants to solve these problems, because there is no moment to take a look at it, or if they think it's even though they're not wrong with it. They only want to solve the problems in a simple way, and that they are not fatigable, that they don't have an error, etc. Because the calculs of Boulogne, it actually gives an error. It's in the habit of being trompés, especially at this time, you don't have to know anything, it's not clear. And so, their problem is simply to solve the problems in a simple way, clear, and without fatigue. So, all this can work well. Is this the application? Is this the application? Is this the application? No, no, no. He's like this and says it's in the system. You can do this or this, it's the same. The system that I have put in place is the way to do it. This is not the application. No, no, no, not at all. He said I'm going to do it in a moment, and he's going to do it in a little bit. Yes, he said it because he knows, he's reading, he's reading, etc. But... Well, I have a problem. There is a problem with the quantitative term, which is to say the somme, the inn, etc. When you try to do things with the quantitative terms, you say that... but the language of the saint-command is to put the idea of putting the word somme or innée on the same symbolical that the other substantives or objectives, in their assignations by a class. And, if it's not a class, if they're not a class, because they don't know what it is, then they don't want to calculate the class. And there, he is very embêté with that. He says that there's no information there. He's obviously on the quantification existentielle, which is incapable of tradu, because his system is not quite clear. It's not simple. All right. All of this, finally, to say x and y x and y impede the class of x, y to be vile. So, we call x and y. x and y.

52:30 x and y means that the class of x, y is not vile. There are x and y. x, y, n'est pas vide. Alors, comment on peut dire de ça dans les diagrammes ? Eh bien, il dit qu'il faut prendre la négation. La négation, c'est la négation de... Alors, x, y n'est pas vide. La négation, c'est quoi ? Eh bien, c'est x, y qui est strictement positif. Il existe des x, y. Il y a quelque chose dans la classe x, y. Et c'est ça qui note dans ces petits machins-là, ce qu'on met dans le numéro de la proposition qui fait qu'il y a quelque chose dans la classe en question, it would be a little bit more. Because you know, this class is invasive. Because of that. But then, if he tries to enter that in his system with his calculations which are the calculations that's not going to work. And he said, if you have seen it earlier, in a particular situation with that, he made it, he made it, he doesn't solve the problem with that. It's not the same. It's not the same. It's not the same. It's always the system to pass to the negation and say, that the x and y will be positive. But it's not the system of an equation or something like that. And it says that, the particular propositions are quite rare in general and the same, which are, finally, and the problem is particularly particular, it's the domain of probabilities. So that's what I'm interested in. Well, so, I don't know, but that's not fair. Ah, yes, on the O, also. It's also, at a moment, on the dissonction O. So, in one of these chapters, he does the demonstration that, actually, there is a language that doesn't exist, because there are things where we use the O, in the phrases of the language that we can very well use, or sign up, the sense of the O, which is different. So, the ambiguity of the language, in fact, cause of big problems. And so, it's there that there is no problem of interpretation in logic, so he puts the barrières and he says that the language of the symbolical is much more interesting than the long-courant because, for example, an object of x, y, z is equal to zero which is an expression variable, and he gives an example in one of his chapters, he translates to 18 different ways in the long-courant

55:00 so it's always x, y, z is equal to zero and he says that a lot of advantage compared to the language. But you have to pay attention. When you translate, you don't have to make mistakes. Because at this point, of course, you have to make mistakes. Now, I'm going to ask... Is this a new algebra? You see, it's about generalizing the ordinary logic. It's a more scientific and more performance, in particular, to solve problems. That's what I wanted to do. But it's always logic. And for him, it doesn't have to be careful, because he says, I don't know, that it's always logic. He doesn't do algebra, he doesn't do algebra, he doesn't do algebra, he doesn't do algebra, he doesn't do algebra, he doesn't do algebra. The stenographies are algebra, but it's not for that the dissolution of the algebra and absorption of the reasoning by the calculus formal. In fact, he is using to put in the form of the data logics, because in mathematics it works. And you were still a mathematician, and you knew why you were talking about it. And to build solutions, we have now a method proprietary properties which are performative. But, you know, and this is what you said, we are not trying to do math, we are not trying to do math, we are trying to work on the logics. So the algebraic logic, as we said quickly, is something that simplifies the problem. It's not algebraic logic, it's giving the logic a form that allows to use the function of the algebraic logic. Now, there is a problem that is supposed to be because the algebraic logic which is used, are chosen to be a more precise signification intuitive and the habit of calculating, In particular, the equalities, for example. Or the product to show the conjunction of two classes, or the fact that it is equal to zero, if there is no object that benefits to this probability, etc. But these symbols are those of the calculus numeric, and they port in them the habit of the calculus numeric. And as they are in other things, which are these classes, these are the combinatory classes which are expressed through symbols which manifest the calculus numeric. And we arrive at the 11th, ininterprétable, in the sense of the habit of the universe. In fact, the symbol, at this moment-là, is in a sort of the point of application,

57:30 a sort of torsion, in fact, where the nouveau emerges from the ancient, but the nouveau n'arrive pas à émerger. Because the fax se manifest clearly, and the system can be saved that by the recognition of the nouveauté. But the nouveauté, in fact, it demands a médiation, because the symbol shows the existence of the rupture, but it cannot give you a reference precise, and the new references are in order to be constructed. So there is a no man's land in which, as usual, like every time we have a new structure which is trying to put in place at the same time, this kind of no man's land gives things which are at the same time. In particular, I will give you an example with an interpretable. It's the story of the division of the classes. the division of UDN is this. It represents exactly the things that we do. This is the page 267. The use of the multiplication sign for example, which is the x, y equals 0, or z equals x, y. We arrive at z equals x, y, which is what we call. You have the chevaux blancs, you have the chevaux, you have the blancs, and the chevaux blancs, well, it's z equals x, y. Well, obviously, when we do that, well, who is y? Well, y, what is y? Well, y, it's z sur x. What does that mean? Diviser the class z by the class x. Well, very simply, we have not multiplied the class X by the class Y. We have a logical operation which consists of the elements which are at the same time, the property X and the property Y. So when we write Y equals Z on X, we don't divide the class Z by the class X, we look for a class which is multiplied by X. It's to say that for him, in fact, In fact, the symbol is not porting in him-mêmes of possibilities of calculus. What is porting is the symbol. The symbol, in fact, is at the service of the symbol. And when he writes y equal z over x, he says that there are conditions. And the conditions, it is that, obviously, it is necessary that there is a difference between the other. So, here it is what? Z equal x, y. In fact, x is equal to z. So, we need to be equal to 0.

1:00:00 We can never separate that from the fact that X, Z bar is equal to 0. That's the condition of existence of us. That's what we call it also. Yes, yes. It's the defense. It's the defense. Yes, yes, yes. What you say is that it's the interpretation of us. So it's exactly the same. We call it the condition analog to the universe. Yes, that's it. Yes, that's it. It's the sense of the problem. Yes, it's the same. So I said, you do it, you do it, you do it, you do it, you do it, you do it. It's there, the difference. Now, what happens is this page, for example, where, you know, when you want to apply this to this thing, it's the development of the B1 with the xy, xy, xy, etc. and he doesn't do the form of the Taylor or the zero, but only by the dichotomy on the classes. There he doesn't cause a problem, it's not a calculation, it's a result of the classes. And he says that, in any way, we obtain xy equal 1 sur 1 xy, because f of 1 is 1 sur 1, and then f of 0 xy, which is 1 sur 0 xy bar, plus 0 sur 1 x bar y, plus 0 sur 0 x bar y bar. And this result is obtained by a purely logical conclusion. In the third chapter, the theory of the theory, it's simply that he does this and he obtains this. Is this a purely logical conclusion that it's a good idea? No, he doesn't need to. He doesn't need to. He doesn't need to. Yes, he doesn't need to. He explains it. He explains it. He's taking the theory, he develops it. He doesn't need to. He says, when he takes a class, whether it's this class or whatever, there's a piece of xy, a piece of xy bar, a piece of xy bar, a piece of xy bar, and a piece of xy bar. And so he has the coefficients. Now he explains how he found it, etc. And he says, we can do this as a rule, because it's fine, it's fine. In fact, he says, it's written axy plus bxy bar, etc. Because he makes his classes mutually exclusive, and then global, So, he does this. And then he does this. And when he does this, he does this.