Logicality, invariance, potential isomorphism

0:00 If you watch French movies, you encourage introspection, but unfortunately it happens that only movies with the word introspection are commercial failures. Very simple to follow when movies are commercial failures. Ok, and the best simple to follow means that this is a correct inference. And if this is correct, probably you're going to say that this is correct. So now you need to tell a story to the right schema. Independently from what the theory is, we do need another term of how we get from one to the other. Is that there is a big difference which favors, on the one hand, the content of tendencies in the argument, which... Yeah, that's too great. Is the conodore illogical or not?

2:30 Here, something like 2, okay? And then, one conclusion is true is true no matter how we interpret it. That is, QIQI is simply true. A list of tasking definitions of logical truth. In addition, we'll use this logical truth. We have this idea of understanding of all possible implications of non-logical symbols.

5:00 Of course, this is the point. All this depends on different sets of logical symbols. These terms are concerning whether a given sentence is a proof or not might be different. So to give an example of this, I should ask you a question which is the following. We know that sometimes if Nessie was a president of the United States, actually, then Nessie was a man. And let's make the somewhat real assumption that if Ben was a political expression, but so far president... And that's it. The only non-analytical expression in the sentence is the autonomous energy. Then, if we apply half this step, we will get that y is a non-equal 2. And indeed, and actually, 2 happens to be 2 for the history of the next step. So we will create y is a non-equal 2. And y is our prediction 1. We will get a function with respect to the difference between the logical and non-legal expressions. And now, so this is bad. and make a nicer distinction between the two kinds of earth equations. So, here's a question I hope to answer as a question. So, how do we manage to give a list that gets, in most cases, the results and so on? Or, is it actually possible to give a principal characterization of a symbol that should count as radical? Do we have a way to count some symbol as radical? When you're just in this formulation of Tarski test, what is the domain of quantification? It just goes like everything, like for any P, what is quantifiable?

7:30 Here I want to be as neutral as I can be with respect to this. So you could take the modern model theoretic formulation for Tarskin's original formulation. And then Tarskin just goes forever from there. Yeah, and this is also the case. There is a very big discussion about it. Yeah. Big papers of Paolo McCall, Michael Goff, who had different interpretations of the domain. At Tarskin, it seems that there is no change in the domain, So here I want to remind you not to follow the text files. The only thing is that there is some engineering consequence, regarding the kind of issues that you mentioned, but no matter how you do this engineering, you do have to come up with a nice notion of what the technical concept is if you want to be sure that the notion will give you good predictions. Tarski was skeptical with respect to the possibility of drawing the line between a logical expression and a non-logical expression in a non-conventional way. He just said, OK, I do this with what people think are logical expressions and it seems that I get the correct result. Which was also used before that by mathematicians and mathematicians like Matovsky, which you will be able to find in a lot of books, and this is made to evaluate them. So this is more or less the standard answer to answer the question. So in the first part of the talk I will present the standard answer and explain why it doesn't work and even if it is the standard answer. Usually people think this is a necessary condition, something to count as logical, but it's not sufficient, so we just review the arguments here. And then, if we have this idea of invariant and random permutation as a possible criterion, but which actually does not really work,

10:00 I would suggest we take one step back and take a picture of a very... There are several criteria that we might choose in the same theory. Maybe there is an invariance under x for a given x respect to what x should be and x should be. We have to view quantifiers as second-order predicates. This is a Phrygian view. You want to say there is an x in a given number f. We interpret this in terms of set, because actually it is equivalent to the fact that the set of objects in the domain we satisfy in class is not empty. In that case, you see the existence of an effect on the material as a property of set. Namely, it gives the value true if an object is applied to a set which is not empty, where the set is providing the interpretation of the formula class in the domain. The second question is about the domain N, the function which takes as argument the subset of a domain and gives the true value. So, for example, the set is true if and only if the set is the domain N.

12:30 Okay. So, these are the kind of things that our logical symbols can be, and we can make it, set it, when using tags, we can apply it also to propositions, connectives, whatever. Our task is physics. Under all terminations, the symbol is the decoder, and it is over all domains. So, we have the subsets of the main. The habitation environment. Even on English, here, the set of values, when you apply it to a set of the image of a subset of the main here, of the image of such a set, one does not have a data set. But what exactly do you mean by permutations? So, we have it in the next slide. So, like in the next slide, there are two examples. So, in this slide, ABC. And permutation is just a bijection of a domain of itself and there are six objects. Let us consider a predicate like weight. Let us imagine that there are two weight objects, three... But permutation, yes, just normal usage of the term, it makes sense even with this total ordering, right? So, you suppose that your domain is like n to the power, right? It's not just a set, so the domain in your sense is not a set, right? It's an antipode.

15:00 Each one of those pi's is a permutation, pi1, pi2, and it's simply a mapping from every object in the domain to some... Yeah, but domain in this sense is just... No, domain in this sense doesn't mean anything. Yeah, without... It's a permutation from a set to a set? Yeah, it's not ordered. A set doesn't have to be ordered. Right? You can imagine the set when it comes to the projection from the second to the third. Okay, okay. Yeah. They're the same set, but they're not the same part, they're not the same permutation. Yeah, yeah. Okay. Very good. Yeah, so we can now call it the 2D empirical project, right? And it is the way you have actually two wave objects, A and B, and of course you can see that The number of types of A, A to the right, is applied to the image of A under the foundation of I-5, for example. The type of A is between two kinds of objects, the red one and the yellow one. So it is sensitive to the identity of the object, so it is not available under the foundation. Now, as I presented before, the question of existential classification is just the nomenclature. You will see that it is going to be invented under the foundation. If we take the subsets A and B, where the image of it on the right side is an empty set, it is very true.

17:30 The image before was also an empty set, near the same value. And of course, this is going to work no matter what the permutation you consider, because if you have an empty set, and you take the image of it on the permutation, it will again be an empty set. And the same thing if you take the image of it on the permutation, it will again be an empty set. In infinite cases, when you have isomorphism on a proper subset, it drops this in a way? No, infinite permutation has bijection onto itself. Yeah, of course, but here, by definition, the permutation, here as it is defined, the permutation is a function which is onto the set, which is subject to account by definition to subset of the unit. So here we see how it works, but I didn't have to convince you as to why the invariant permutation is interesting from a conceptual point of view. One which is a logic proposed by Starsky in his paper, which I shall call the generality of humans, and a different one which was put forward by Scher, for example, in his book. So first we have to start with an intuition of what logic is. The most general theory one can think of. Why? Just because no matter what kind of theory we are investing in, we are going to use the logical principles to derive...

20:00 So, logic is the most general of our theories, just because it is a theory. So now, we need to take some kind of formal grade on this intuition and to get a formal account of the theory. And here, the idea was to use Klein's insights from his Erlanger program about geometry, of analyzing the generality of the theory in terms of the group of transformations that are associated with this theory. So, for example, in the case of geometry, we're going to say that Euclidean geometry is about the notions of space that are invariant under ratios of distance. In this sense, when you have a triangle and a chemical transformation which results in that, it's again going to be a triangle. This is why a triangle is a notion in geometry. On the contrary, for example, topology studies just the notions that are invariant and of continuous mapping of the space of twistor. You can map the triangle of a circle or the axis of a circle of the wall, which is why it is triangular in the notion of topology. Metry, in a sense, speaks about the process of spacing, some of which are not trivial. And it shows that in the groups of transformations that are associated with that, If the transformation of space which preserves the ratio of distance is a continuous mapping, the converse is not true. In that sense, topology is more general because it is associated with a bigger group of information. So, if we have this idea that we can measure generality by means of the size of a group of information, Then, it is natural to think that the group of transformations that we shall use to characterize logic is the biggest one. And of course, the definition of a transformation is just the group of a way to reconstruct Tarski's argument in his paper.

22:30 Sometimes the two are the same thing. What are the different intuitive starting points? Here the idea is that logic deals with formal properties as opposed to empirical ones. We have to take some kind of mathematical break to the idea of formality. So here our proposition is that formal properties are insensitive to the ISP of the subject. An empirical property is something that makes a difference between objects, just like a web within a computer. Whereas a thermal property is something which is not sensitive to the identity of objects, which is not sensitive to arbitrary switching of objects. For example, when you say that a set is not empty, it is not sensitive to replacing some object in the set by other objects. In this case, we are replacing an object by another object, whereas it is precisely by the permutation that we can represent switching of objects by permutation. In this case, won't that be centuries? Things can change places. Yeah, yeah, yeah. I mean, I'm trying to figure out what an arbitrary system of objects is. Because of spatial geometry, you will have relevant symmetries that correspond to the geometry you are using in any case. So you mean you've got an equalized triangle if you move it?

25:00 Yeah. That's arbitrary switching? Yeah. Yeah, that's the switching. I mean that's just the space. But here, optimization is just any way of doing that without reflecting the structure of the space. I think you can completely destruct the triangle you wish, which is why permutations, all permutations will not be as near to geometry as they should be. I think this is just what you get to generalize the idea of space transformation, if you completely generalize it. But I mean, right, that approach actually works if we have just one domain, right? We have no types. If we have types, then we have much more things, actually. Yeah, yeah, thank you. If we have a finite domain is there, can we generate a list of which are the logical notions? I mean I can, given your definition, the variance in the permutations of some and all and the most, what about at least two? Yeah, yeah. At least three, at least four, right? So those are all logical. Yeah. Now what about when you have more than one property, like as many Fs as Gs are? Is it defined? Is it defined when it's not just at least three NEMs, where there's a single problem? Yeah, you can extend the definition, but it applies to non-monadic... Right. Modifiers take more than one property in an argument. And then such a quantifier, I mean, in your example it would be logical because actually... Since you take invariant logarithmic notation as the basis, all quadratic properties, which is the number of objects, return algebrities. And I actually believe in heuristics, but of course, people like Schindler and his studies back in the frontier dimension do say that our things are consistent. Well, some of these intuitions seem to you to kind of presuppose that language is in some important way finite.

27:30 Maybe, because if you allow the language, it may be invariant. So it could be infinite and like the far left and then things can change a lot. I could uncalibrate all strings and then someone puts them. Yeah. Okay. I'm not going to say. You're the booster. Yeah, yeah, yeah. I think that, I mean, if you're using... I'm very proud to say from Google Syntax, a really cool symbol, that I... If an equal set of formulas looks like that, then you can take, for example, a conjunction or a subjunction or arbitrary if-an-equal-set of sentences. You can guess that your expression, in that case, let's say, a conjunction, maps the assignment functions that satisfy the formula in the set, and yields the set of assignments that satisfy the formula in the set. We did that to me too, I guess. The question was the next slide.

30:00 So, by the way, in the case of compositional logic, we have this idea that this is the logic of truth. Okay, so we can ask for functional completeness. With mediation and convention, we have all the truth functions. Usually, in the case of personal logic, there is no functional completeness here, just because there is no agreement about what personal logic is about. But precisely, the idea of all this is to give us an idea of what personal logic in general is about. So here we say it is about operations, turn-invariance, or permutation, and now we can ask for functional-competence theorem, we can ask in what kind of a grid is powerful enough to give us all these operations, turn-invariance, and permutation. So usually this is a theorem, like it might be, we say that if you fix the domain, then the... There are many different kinds of foundations, even the latest is available in the biggest engineering language of the Spanish-speaking community, and you can take assumptions about arbitrarily big sets of formulas and you can quantify in one shot other infinite sets of variables. So of course, for people who think that technology is a white logic, as a shark you mean, it is of course much more than that. First said, it might not be an objection, and again, a partial criterion, but there are some worrying consequences, namely the following. As I am going to give you a question, all cardinality ones. But for example, you can have a contrail which says that there are uncountably many objects, such as blah blah blah.

32:30 Inclusively, it is a self-theoretic notion, because it belongs to a self-theorist who tells us what exactly f1 is, at least f1. And we sense that this is something which is more complicated because there are issues with what LF1 is, and that it should not be settled just on the basis of logic, what we mean with it. So the fact that all the countries, in particular all the countries of infinite solidarity, is not really a welcome feature of this framework. Here we are working in a first-order setting. The differences between first-order and third-order logics collapse because, as has been shown by Mr. Scheffermann, in LHC 360, other domain can simulate third-order quantification. So we are not really working with first-order logics, but with second-order logics, with strong quantification. In logical theory, a given mathematical structure opts for a randomism, so if you can give a theory of a natural number such that all its models are isomorphic to n, becomes trivial because in one shot actually you can define a point failure which just tells you a number of natural numbers and you close that under isomorphism, you get something that is less of a limitation. Just because you made a logic superpower fall right at the beginning. So, I think you would agree that this is way too much and that we are starting quite a bit over the night. So the idea is that this is not the framework where we should have simple domain for every...

35:00 If you fix the domain, you fix the language, then for every second order sentence in this language, there is a sentence of extremely equivalent with a small twist on how you represent a logical operation. So instead of thinking in terms of function, you can clarify this and think in terms of class of structures. So you can say that the interpretation of there exists is just the class of structures based on the existing domain of the project extension, but the project extension is not. And in that case, invariance is not by injection. Well, what you get is just the standard notion of closure under isomorphism, which means that the class of structures you want to say are consistent is closed under isomorphism, so you get something familiar. But, so it shows that implicitly, Tarski's criterion relies on the standard notion of isomorphism as being a function of what it means for two structures to be together. Does it make sense that it's got to be reflexive and symmetrical but not kind of? No, no, similar but in the sense that, I mean, the idea of that, the idea of that is that if two structures are, I mean, in this setting,

37:30 I'm going to tell you the class of structure, okay, and if two structures are formally similar, you don't want the contributor to separate them. And here you have to come up with the, yeah. We have an idea about what it means to be formally similar, and I don't want to spend too much time on this. So, any notion of similarity might be a candidate. Now there is a question about whether you should have special properties with similarity variations. And I hope you really want this to be an equivalent position on that. I just remember a paper by Hannes Leitgeber where he was trying to work out some ideas in Karajan's logical structure of the world and he used similarity relations where that's a technical notion for a relation that's It's reflexive and symmetrical but not transitive. It's not an equivalence relation. Okay, okay, okay. But do you hear something as strong as any equivalence relation in the candidate? Yeah, I mean, potentially any equivalence relation, even relations which are not symmetry. So here, imagine that you have a symmetry relation, which tells you when you put pictures on the set. The idea of invariance about S is that the variation O is S-invariant if and only if one of those constructions are related by S, then one belongs to O, if and only if we base our S as an origin, then we have the derivative of O. We have this abstract view. What we take from Tarski is the idea of defining logic in terms of invariance. But we have to provide an answer to the question what it means for two structures to be logically similar, that is, to pick an S, and then we come up with a notion of what, and then we come up with the event aeration as a logical aeration.

40:00 As I said, a partial idea was to take for S isomorphism, but we have this low-generation problem, so the question is, can we pick a different S? Do we have some good conceptual ways to reason to pick a different S and solve this problem? For example, Plekermann, who studied logic, logic and logicism, proposed to drop the injectivity requirement, so you can collapse a set of different sizes, and to shift from n1 to 2. He hoped in this way to solve the regeneration problem. In order to know whether you did solve a problem, you have to ask the question about the function of a computer, what logic you get. And what I did in my dissertation was to prove that if you take a common criterion, what you get should make sure that it works. So it's the same with history of physics, it's what you call history of physics. So there's no big progress, you still have a strong... And then in this paper I recommend an argument for some syntactic restriction on the kind of operators who want to apply this, actually wanted to apply this only to monadic operations, so that you would not get the override only by lambda fraction, only by lambda defined ability, and then you would get it right, but I mean, in fact, agree with this, so you can... And so now there was a paper in the GPL last year on how it makes the classification clear, but... But then the thing is that you get a strongly under-generated sense that full cosmological logic is not a logical and contextual base, and that the notion of a logical operation is not chosen by the nation.

42:30 And I shall go for potential hypnotism, actually, but I just want to introduce some... And then I will go back to the concept tool argument in favor of Starsky's criterion and try to show where they are prone, why they are prone, and why when we mutilate them, we actually come up with differential algorithms instead of algorithms. Yeah, sometimes it's known as partial algorithm. Here I shall use a partial algorithm. Here, this is like a function of a partial algorithm just in case if an algorithm is being substructured. So, you have structures in your head which are located similar. So, somewhere in your two structures you can link two subsets or ends. There are parts of structures which are similar. So, because it means that they are square, the structures can be very different. So, you cannot just say that two structures are similar if they are partially independent. It is much weaker, it just means that... What domain, co-domain, you say? Those we have, you also know, it's sub-structure. Yeah, so... So you just require that two sub-structures are isomorphic. Ah, even two sub-structures, okay. Yeah. It's like local. Oh, yeah. Yeah, but there's nothing between the sub-tropics. No, that's not what that is. That is a partial isomorphic. There are substructures C and D. C is a substructure of A and D is a substructure of B, such that there is an isomorphism between C and D.

45:00 A manifold and topology. We have all set of partial mappings. But the good question there is, is there any over-position that the mixture of the multiple partial mappings go together properly? Yeah, I will answer the question later on in the talk. I mean, the question makes sense and then I can stop. And so, now we will say that two structures are potentially isomorphic, if and only if there is a partial isomorphism, you can extend it in an arbitrary manner, so that you start with a major isomorphism between two structures, somewhere, you want this local similarity to extend in the structure, that if you have your partial isomorphism here, And you take three main objects A in the first structure, which was not always in the domain of the first structure of S, then you can find a number which extends S and which has A in the next structure of A in the domain of the first structure of S. Even though we are adding a lot of similarities, let's look for a notion.

47:30 Now, if you have two structures that are countable and potentially atomorphic, you can actually collect and get data from the media, and in that case, they can be potentially atomorphic and not atomorphic. It's easier to be potentially atomorphic, structurally atomorphic. Do you have a toy example of this? Yeah, so... Yeah, I would need the... Oh, really? The predicate feature of P, which is unary, and consider two models of that, one in which there are one of size LF1 and the other of size LF0, respectively. On the closed domain, there are LF1 objects which are P, LF1 which are not, and on the other domain there are LF0 objects which are P and LF0 objects which are not. So these two structures are not isomorphic. But if you just pick one object which is P in the first one and one object which is P in the second one, this is going to be a very partial, a very small partial algorithm, but it can be extended as long as you want because you have infinitely many objects which are P on the left and on the right as well, so even though... The size of the set of objects, which are two, the first structure and the second structure, is not the same. You do have a different structure in the long run.

50:00 If you take, say, a differential manifold would be an example. They are locally all Euclidean, so you can just locally map everywhere, but still they are not, say, the same. Yeah, yeah, yeah. Excuse me one second. Here's what threw me about this definition. You've introduced the symbol I, but it seems to me it should occur after the word non-indicent. A potential isomorphism between A and B is a non-indicent I, a partial isomorphism. Because when I first parsed that, I was thinking non-indicent some other thing, G or P or whatever. So Ives has got to be over there. And then for every F member of Ives, F is the partial isomorphism? Yeah. Ives is the partial isomorphism. Okay. And what is the potential isomorphism between them? Because if you are working in a quantable model of LFC, you could have a potential isomorphism between two structures, then there is a forcing expansion in which there are isomorphisms. So now we want to go back to the generality argument and explain what's wrong and what we did. So, coming to the generality argument, we should pick invariant-derived logarithm because it is the most general notion of invariant non-transformation that we can think of. By contradistinction, with what happens in the case of geometry, we do not require to preserve some extra structure, like a structure, a quotidian structure, and so on. And this is the first parameter for generalist theory, that you want to get rid of all the extra structure which is implicitly there in your domain, in your space.

52:30 But the thing is that there is another structure that is preserved. In the case of invariance and or anthropomorphism, you are very general with respect to a class parameter. But, in terms of structural similarity, being anthropomorphic is a very demanding notion, of course. So, both are very similar subjects, you want your structures to be exactly the same. And being semi-hororific, you want your design to be less demanding, so it is more general in that sense. In both ways, anthropomorphic is less demanding, so it is more general in that sense. Just to add to that, it's absolutely clear what you're saying also in case of geometry, because even this Rolandian program doesn't actually work. I mean, you cannot define topological space as through topological group, right? You can, given topological space, you can... You are doing like homology theory, really. You cannot just say reduce the notion of topological structure, for example, right, or whatever, even if leading structure to the notion of group. So it's not sufficient to describe that. If you go to category, it's sufficient. It's more or less the same that you're saying here. So here, in the Tarski-Klein framework, to speak, we have this idea of families of transformation, and in order to get the good one for logic, we have to look at the bottom one, which is the most difficult one, and the idea was to speak in length and in lengthism. But now, when we have this idea that there is another parameter in reality, in terms of the real structure of iteration, what happens if we look at the bottom queue? So, if you want to be very liberal about degree of structure preservation, you can just pick the universal relation which relates any two structures and say that the logical notion, well, it has a notion which are very very invariant, which are invariant, you know, the relation, which says that the relation is the same. And then, you can ask whether you get the logical relation, and the answer is that you get the version of the gradient. So, yeah, I had a function-functionalism profile, which is really interesting. I have just a constant function.

55:00 So, what does this mean? This might mean that in this generalized framework, generality does not make sense, because if we look at what's at the bottom, well, everything is at the bottom, so nothing with the invariant above it. So, I do not take it to show that generality does not make sense to define, to get the good invariant criterion for logical notions, but rather I think that it shows that generality is... We want to use the most general notion of similarity which abides by a certain constraint that captures the idea that logic does think about something. It should be this constraint to fit the machine with something. Here I should say that it is reasonable to think that the most basic The most basic properties of sets, like the difference between being empty and non-empty, the most basic properties of sets should be preserved. At the end of the day, we want existential conjugation to be logical and we don't want to be able to collapse empty sets with non-empty sets. That is, if something is logical, if something can be defined by logical means, then it should be, again, logical. For example, if you think that the future consecutive is logical, that equality is logical, then you should think that there is at least two, which is logical as well,

57:30 because it can be defined in terms of there is and there isn't. Here, I get this as a requirement of non-triviality, and the idea is that if you have two structures, Say that they are different just by looking at one object in these two structures, they should not count as really senior. If you have two structures, you pick an object which is P in one of them, and you see that you cannot do the same thing with the other one, then they are not senior. In this case, they are quite a bit senior. The second constraint I have created is that here you want to get the logical notions as notions which are invariant. And if you have the idea that logical notions are close to non-definitivity, then you want your invariant to be close to non-definitivity. That means everything which is disabled from invariant to variation is again invariant. Why are you doing that? Sometimes that's so basic that it's difficult. If you think that being logical means being free from empirical content, from mathematical content, I don't know what. If you have just logical expressions and you define something just with that, then where could the logical ignorance come from? You have to look at the precise way I use the method of definition here, but, I mean, the idea is just, in the case of, you can define, and then you have the notion of how to compute.

1:00:00 The notion of particular notions there is the first kind, in the sense that it should not matter how you get it. Okay, okay, I agree with that. These two constraints proved to be quite substantial when you put them together. Because, that's the main theorem. If you have, so you have Carsten's idea of picking isomorphism. And then, if you want to look at what's under that, you see that if you take at the bottom the universal relation that's going too far down, so you want to be somewhere up, and you access these two requirements of push-on-the-stick NLPT and non-triviality, and you want to pick up the most general measure of transformation which satisfies these two. And here you can prove that there is a unique one, and that is the notion of being potentially small. So, if you want to be very low world but still preserve its true, you can have more below being isomorphic, but not that much below because you have to stop at being potentially small.

1:02:30 Now, you'd have to have the attention to write some kind of argument to show that you're dead the way you thought you were dead the way you thought you were dead the way you thought you were dead the way you thought you were dead the way you thought you were dead the way you thought you were dead the way you thought you were dead If I discover that they do converge, it is going to be a good sign, obviously, that things are going on in this sense and so on. Here, if I can get, with a different intention, the same kind of thing, which is a co-extension with that, starting from a different intuition, I will be happy, just in the case of a solution. Okay, then one of the things you have to do is to get the attention. You're saying there's some complication because of some converters? Yeah. Right. Alright, but there may be even more phenomena. Otherwise, I don't think that that gets much attention. But the establishment that you've got to connect some or all ends up being logical under your definition.

1:05:00 No, that's not true. That's converters. I just wanted to go through with the general computer arguments, show that they both point at the same notion, and then I will figure out what we get. Ok, good. So now, as I said, I want to go back to the formal literature. And here I think that, ok, the problem we have, in terms of generation, has it that we somehow collapse logic and perception. But, it would be a surprise that we come with a notion that collapses the two, because if we start with the equation that logic is for all, then okay, logic is for all, but so are mathematics. So the fact that at the end of the day you get something which collapses logic and mathematics would be true. So here I think that the engineering of a criteria is interrogation, is acting in a marginal way. But in the case of a criteria argument, I think that a problem does not really happen in the world. If you catch it, I think it's hard to formalize it in mathematics, but the problem is in the intuition you are working with, just because even with logic use formals, it's not clear that this is a sufficient condition for some people to come to this course. Okay, so can we do better along with logic use formals and some... So, I don't know if we can go back to our program where we have quantifiers...

1:07:30 One thing is that it heavily depends on the grasp of the self-theoretic universe. Even if it is loaded with the self-theoretic concepts, it means that empirical concepts must project the concepts. We are trying to say that logic is content-free, in some sense, and to do content-free is not enough to be free from empirical content, we want also logic to be free from problematic, in some sense, satirical content. In particular, we would like our notion of invariance not to depend on which set exists, in particular about the satirical universe. But by formal, do you call still now this idea of invariance and permutation, or is that a different thing? And the income will be a characterization of something which can be construed as a characterization of independence from the basic features of the model of space. The idea is that being competitive to defend prevailing on what was said, in the sense that many locate that in the case of state theory,

1:10:00 If you shift to bigger models of theory, they become more complex because in fact, that bigger model can be indeed made more complex. So actually, that's how... Now you say that the formula of theory is persistent if I'm on an image, and it preserves through value when you shift to a bigger model. In particular, we can apply this to our idea of invariance, which has the idea that we are testing the notion of logical similarity, which is absolute, which doesn't depend on which state. We want, being exomorphic, if not, we would like a thing which is closest to that, but which respects this idea of not being free from sexuality, a thing which is definable in terms of being an absolute child. Here, it's just actually a re-creating of a known result from Barbeau-Wein. We have to see that in this theorem, we need that being potentially anthropomorphic, the biggest notion of similarity, which we find on being an iceberg. We claim that the equation is just equal to the units of potential anthropomorphism.

1:12:30 We want to pick the smallest similarity, closer to the... If you have the potential to have as much as they can, then you want to get the biggest thing, the closest to the item of truth, which is here. So here, we have this convergence in both cases. Here I claim that the argument was a game. What's nice, I think, is that you recover the convergence. You replace the informal by linear. To recast generalities in the way that they seem, Seagate has a tool. More ways for you for that, I shall teach.

1:15:00 We can learn again that proportion is a direct consequence. Potentialism gives you a logic relation to Hegel, in which you can take infinite conjunctions and injunctions with only finite conjugations. So we are beyond topology, but certainly not as far as... In particular, it seems that the quantum materials are indeed very important for the choice of quantum geometry, because, in fact, in the case of existential quantification, if two structures are potentially isomorphic, it means that you can, that somehow, there is a powerful environment which can be extended. So, two structures for which, which are separated by a global sphere, are not collaterals of each other. I'm sorry, I just missed probably the previous slide. What do you think about the if-and-only-with part, why do you think it is sufficient?

1:17:30 If you start with the notion that logic is free from problematic content, then you will say that everything which is free from problematic content needs to be solved. And you can say probably a little different aspect, but you would say it's what you are trying to characterize as a set, just something which is, say, space, would be set with a structure, right, which restricts you as a morphism. And now you just take a kind of bare set, you have all those things. The problem was that approach, that it doesn't work only with groups. To characterize this way, you just should look, say, all morphism during some category theory. But now you improve, so before you get that it doesn't work, it's too general and so on, now you improved on that, but still I don't see really reasons to say that it's efficient, because again we might have... There are no obvious control examples. You speak logical and you control the logical. Then you might hope that you can make a decision, but you cannot. And just to hear from you what you can count on, I would just like you to give some more data so you can get some grip on what the proposal is.

1:20:00 And here we are happy because it's not going to be logical. The general contract, the monism, collapsed all the functions of mathematics. So, you can... Which may... You can...