Introduction to Philosophy & Logic (1999)
Lecture 39: Validity and Invalidity in First Figure Syllogisms

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Is that statement here obvious or is it in need of proof? It seems to be obvious. Yeah. Notice, huh? If B is said of something else, right, and that is not an A, right, then that B, because B is said of it, it is B, will not be an A, right? And that kind of gets such a given here, right? So if A is said of all B, then A will be said of whatever B is said. If B is said of all C, then A will be said of all C. If B is said of some C, then A will be said of some C. Now you can state this another way, dramatically, it's more the way we're supposed to do it now, but you can say, if every B is A, then whatever is a B is also an A. If you understand what it means to say, every B is an A, right, that's in the form of a universal affirmative statement, right? There's no exception, right? Every B is an A. If every B is an A, right, if every B is an A, then whatever is a B, an A, right? Again, let's state the same principle here, but a little differently, dramatically. Again, is that Aadis? Yeah. You can see how to deny that would be to Aadis, right? Okay? Because if something is a B, and it's not an A, then not every B is an A, right? Okay? So, this is one of the two beginnings, principles upon which the socialism is based, right? Okay? If the set of law is found in the arrangement of the terms, right, then something falls necessarily, right? Okay? And you've got to recognize it right away. I mean, if someone has, for example, every B is A, and every C is B, it should be clear right away, that every, what, C is an A, right? That was involved in this principle here, right? If every B is an A, and you told that in the first sentence, lay it down, right? Then whatever is a B, will have to be an A. Since you told all of the Cs, every C is a B, right? Then every one of those Cs will have to be an A, right? Okay? Now, if you were told only that some C is B, then you would conclude necessarily only what? Some Cs. Some C is A, right? That's rather kind of obvious, right? Okay? Now, in the first figure, when we study that, you'll find out that if it is a syllogism, the set of all, or its twin principle, we're going to talk about it, the set of none, will be really obvious the way it, what? It stands, huh? Okay? In the second and the third figure, it will not be obvious the way it stands. And therefore, our style calls those syllogisms imperfect in a sense and not clear. And you have to, even to get them, to see what files, you have to turn statements around. That's why we have to study conversion before we can do the second and the third figure. But in the first figure, if it's valid, we see the set of all, or the twin principle, the set of none in it, as it stands. And if either set of all or set of none applies to what it stands, you can guess it's not going to be a syllogism. But then we're going to prove this now, syllogism, by examples, right? It will show that when the premises are true, nothing is always so, and therefore nothing is necessarily so, right? Let's take the set of none, which is the other principle. If A is said out of none of B, then A is denied. Right? Of whatever B is set of, huh? Okay? So, if, um, cat is set of none of the dogs, right? Then cat will be denied of whatever dog is set of, right? If dog is set of all cocker spagnols, then it will be denied of none. Now, it might be better to state this way around, because that's the form we're using more. So, if no B is A, then whatever is a B is not an A, right? Okay? If no B is A, then whatever is a B is not an A. Again, is that obvious? So, um, in the first figure, if it says none applies, you'll see it right away. You'll have a statement like, no B is A, and you will have, um, either every C is B, in which case none of the C's are A's, right? Or you'll have some C is B, in which case some C is not A. Okay? Now, um, notice, huh, in all of these forms here, where you have a set of all instead of none, you have a universal statement, right? And then you have an affirmative statement, right? Placing something under the subject of that universal statement. Now, if you don't, um, have any universal statement, you can't have the set of all instead of none, right? So, in all of those arrangements in the first figure, where the two premises are particular, you would guess right off the bat, right? That they're going to not be syllogisms, if the syllogism, in fact, depends upon the set of all instead of none. Because there's no universality, right? Okay? Likewise, if you have two negative statements, can you have the set of all instead of none? No. Because even the set of none requires an affirmative statement, placing something under the subject of universal negativity. So, right away, you might guess, huh, that the two negative statements, or there are two particular statements, it will not be a syllogism, right? But we won't just guess that, we'll use examples to show that it is not so, right, huh? We'll talk about what you need in the way of examples, huh? Now, um, what do you need in the way of examples? We need two, we need three terms that will make the premises true. Let's take an example of the first figure, okay? Let's take this example here. Every B is A, and no C is D. Now, can you draw any conclusion with C as a subject and A as a predicative question, or ask it, right? I customarily call the predicative conclusion the major term, huh, because the first figure tends to be more universal. And the conclusion, the minor term, and the term that's found twice but not in the conclusion is called the middle term, okay? And this is where the middle term is in the first figure. Now, um, I'll call this first statement here the major premise, because it has the major term in it, and I'll call this the minor. Now, what you want to show is that there's nothing you can say, necessarily, about C as a subject and A as a predicative, when these statements are so, right? In other words, you can't say, necessarily, that every C is A. You can't say, necessarily, you can't say, necessarily, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can't say, you can You can't say necessarily that some C is A. You can't say necessarily that no C is A. And you can't say necessarily that some C is not A, right? So there's nothing affirmative or negative, right? Universal or particular, right? That you can say, right? So you've got to find examples that show that none of these is always true when these two are true, right? Okay? So you've got to find examples for A, B, and C that, one, satisfy the condition and place them in this form, right? The statements are true, right? And one example where this is false, one example where this is false, one example where this is false, and one example where that is false. But now we simplify less, right? Instead of finding four sets of examples, right? If I find one example where no C is A is true, right? Then these two would both be what? False. Yeah. If I find one example where every C is A, these two would be false then, right? Okay? So what I have to do then is find examples for A, B, and C such that one, when I place them in this arrangement, the statements are true, right? And one of the examples where every C is A, which makes both negatives false ones, right? And one where no C is A is true, which makes both what? Affirmatives false ones. Do you see that? So you have to kind of forward with that many times when it becomes a second agent for you, okay? So, notice, when I look at this form, every B is A, no C is B, the first figure I say to myself, do you have the set of all or the set of none? Well, obviously you don't have the set of all, because you have one A distinctly there, right? And the set of all involves one. Do you not have the set of none? Because the set of none says you have a universal negative statement, and something comes under the subject of a universal negative statement. You've got a universal negative statement here, but nothing is said to be a C either. So neither the set of all nor the set of none applies to this, huh? So now I suspect that it's not a what? It's syllogism, right? So now I'm going to try to find examples for A, B, and C, such that this will be, what, true, these two statements when I substitute my examples in. And I want one example where every C is A and one where no C is A, right? I think it's something obvious where every B is A. You know, there's a kind of second nature right here. Every dog is an animal. No problem, right? Okay. Okay. Animal for A, and dog for B. Every dog is an animal. That's true, right? Okay. Now, I've got to think of a C such that every C is an animal, but no C is a, what, dog. Can you think of something that is always an animal, but never a dog? Man, yeah. For a cat, for a horse, right? Take a cat, right? Now, can you think of an example of something that is also never a dog, but never an animal either? It's a problem, right? Okay. Now, I tell the students to get an idea. I say, you think you've found examples of the test of the two conditions, right? Now, check it out. You say, every B is A. Every dog is an animal. Yep, that's true. No C is B. No cat is a dog. Yep, that's true. No stone is a dog. Yep, that's true, right? I have satisfied, what, condition number one, right? With my examples, these premises are both true. When both of these are true, sometimes every C is A, right? Which means that sometimes both negatives are what? Not so, right? And sometimes no C is A. Not so, right? Not so, right? Not so, right? Not so, right? Not so, right? Not so, right? Which means both affirmatives are what? False ones, right? So every possible thing, right, is not so always, right? Now, if nothing is so always, then nothing is so what? Necessarily. And if nothing is so necessarily, you don't have any syllogism, right? Notice the way I'm reasoning it, I'm saying, if it's a syllogism, something's necessarily so. If something's necessarily so, it is always so. My examples show that nothing is always so. Therefore, nothing is necessarily so. Therefore, there's no syllogism, right? So this is what we're going to do in the 16 cases of the first figure, right? We're going to look at all 16 of them, and we'll ask ourselves, does the set of all or the set of none apply to them as they stand, right? If the set of all or the set of none apply, it should be obvious right away what follows necessarily, and if something does follow necessarily. If the set of all or the set of none does not apply to them, then you suspect they're invalid, right? And then like in this example here, you look for examples for A, B, and C that fulfill these two conditions. When you substitute them in, the statements are true. In one case, every C is A. In the other case, no C is A, right? Now, when I, you know, give exams to students on these things, and I'll say to them, I'm going to check your examples to see if they fulfill those two conditions, right? Okay. But in a separate question, I'll ask you, why do those two conditions, right, why do they suffice to show it's not a syllogism, right? And then you have to bring other things in, right? You have to point out that the universal affirmative being so excludes the two negatives, right? And the universal negative being so at some time means universal, the two affirmatives are never always so, right? And then you have to see a connection between necessary and always, right? Between necessary and syllogism, right? So I ask them then a separate question to see if they understand why examples fulfilling those conditions are sufficient to show this is not a syllogism. And then you have to go back and say, hey, if it is a syllogism, something would have to be necessarily so. And if something necessarily so would have to be always so, right? And then these conditions that I say these examples have to have show that nothing is always so, right? The fact that universal affirmative is sometimes so shows that both negatives are not always so, right? And the universal negative is sometimes true, right? It shows that the two affirmatives are not always so, right? So there's nothing with C as a subject and E as a predicate so you can say it's always so when these are true, right? There's nothing there for that is necessarily so when they are so. You see that? Now let's look at the 16 cases in the first figure. Now the most important ones are the universal cases, huh? So let's put those in the fourth case. Every B is A Every C is B Now notice this is the arrangement in the first figure in the middle terms of slant form. No B is A Every is B Every B is A No C B is A C is B Now I ask myself in which case does the set of all or the set of none apply as it stands, right? And notice for the set of all or the set of none to apply you have to have one, a universal statement and then secondly an affirmative statement putting something under the subject of the universal statement. Thank you. So, let's look at this first one. Every B is an A, and every C is a B. You have a universal affirmative statement, every B is A, and is anything said to come under the Bs? Yeah, every C is B, right? So, obviously, the set of all applies here, right? Every B is an A, so whatever is a B must be an A. We're told that every C is a B. So, necessarily, then, every C is A. So, this is by the set of all, right? Recognize the set of all involved in, right? Okay? Clear? Now, over here, no B is A, every C is B. Well, now you're going to have to try it for the set of none. Do you have a universal negative statement? Yes, no B is A. Does something come under the subject of a universal negative statement? Yeah, we're told that every C comes under the Bs. Well, now, it follows necessarily that none of the B is. None of the Cs are A. See that? Instead of none, we're saying, Hi, kitty. Want some logic? Okay? So, if no B is A, right? Whatever comes under the Bs cannot be an A, right? You're told that every C comes under the Bs. So, what follows necessarily is that no C is A, right? And this is by the set of none. Now, every B is A, and no C is B is a form that is apt to deceive people, right? Okay? But do you have the set of all or the set of none as it stands? Obviously, you don't have the set of all because you have a negative statement there, right? But do you have the set of none? Do you have a universal negative statement and something comes under the subject of that universal negative statement? You've got a universal negative. No C is B, but does anything come under C? So, you don't have the set of none, do you? So, I suspect that this is not a socialism, right? But now, I'm going to try to show that by example. Now, I have to find examples for A, B, and C. And if you want, you could have, you know, two sets of three examples, right? But that's unnecessarily, you know. You can usually just, you know, have two turns of C and keep the same for A and B, huh? Okay? So, if you think of a B and A such that every B is an A, huh? Well, let's say plant, right, huh? Let's say three, right? Okay? Every tree is a plant. That's true, right? Okay? Now, you've got to find two examples for C. And you've got to satisfy two conditions, right? None of them can be trees. Right? But one of them has to always be a plant and the other never a plant, right? So, you're taking one by one. Can you think of something that is never a tree but still always a plant? Okay? Can you think of something that is never a tree but never a plant? Oh, stone. So, let's say stone, right? Okay? Now, if they tell the students about nausea, you know, take your time. Now, pause wisely and slow, as Shakespeare says. They stumble to run fast. Have I satisfied the two conditions? Every B is A. Every tree is a plant? Yeah, that's true. No C is B. No bush is a tree? Yeah. No stone is a tree? Yeah. Okay? Have I satisfied the second condition? Do I have one example where every C is A? Yeah. Right here, right? Do I have one example where no C is A? Yeah, right there. Okay? That's enough to show it's not a syllogism, but I might ask the student in a separate question. Why is that enough to show, right? What that shows in those examples is that when these statements are true, there's no statement we see as a subject and as a predicate that is always so. Sometimes the affirmative, the universal affirmative is so, which means that the two negatives are not always so, right? And the universal negative is sometimes so, so the two affirmatives are not always so, right? So there's nothing affirmative. Negative or negative is always so, and therefore nothing is necessarily so, right? And if nothing is necessarily so, it means that it's illegitimate, right? Okay? Do you see that? Now, you know, if you put this in a matter where the premises are true, you know, sometimes I'll say to them, let's give you an example here, every mother is a woman, right? Okay? Okay, I'll give you a student like that. Every mother is a woman, and no man is a mother. Students will think it follows that no man is a what? Woman. Yeah. Every mother is a woman, right? And no man is a mother, therefore no man is a woman. Is that foul? But they're deceived, in a sense, because all the statements are true, right? It is true that every mother is a woman, and no man is a mother, and no man is a woman, right? And there seems to be some connection between the fact that no man is a woman, and no man is a mother, right? But it doesn't really follow, does it? We've shown that this form is like a syllogism, huh? But you give a person a matter, wouldn't that matter, okay? Now, if I had given them a matter like saying, every dog is an animal, no cat is a what? Dog, they wouldn't think that no cat is a animal, right? But because it's true that no man is a woman, then they're deceived, huh? The matter seems perfectly good, but it isn't so. I know as a crafty magician, right? If I want the students to answer incorrectly, or show them that they don't really, you know, more logic, I'll give them this form of statements where they're all true, right? You can just follow the number. It's a very common statement. Now, if you have two negatives, huh? We said before that even the set of none requires an affirmative statement, putting something under the subject of universal negatives. So, two negatives, you can get the set of all the other set of none you have, huh? So, right away, I suspect this is not a, like, syllogism, huh? So, now I can find examples for A, B, and C. Can you think of a B and A such that no B is A? Plants and animals. Yeah. In case you've got an animal here, you've got a plant. And no plant is an animal. Now I've got to find two Cs. Can you think of a C that is never a plant, but always an animal? That's why I enjoy it, right? Yeah, right? Can you find something that is never a planet, but never an animal? Yeah, stone, right? Now we pause, huh? We say, can we satisfy the two conditions? What are the two conditions? That when you substitute in your examples for A, B, and C, the premises will be true. And secondly, you have one example where every C is A, excluding the two negatives, from always being so, right? And one where no C is A, excluding the two affirmatives, from always being so. And therefore, together, excluding every possibility. It's nothing affirmative or a negative. In a verse, or a particular, you can say about C as a subject and me as a friend. Well, no B is A, no plant is an animal, yeah, that seems to be true. No C is B, no cat is a plant, no, that seems to be true. No stone is a plant, that's true. I have satisfied condition number one. Now condition number two, huh? Even Professor Burkowitz has to do this, I tell him, right? Sure, right? Here, okay, I have one example where every C is A, yeah, every cat is animal, we're all separate on that, yeah? Here's a one example where no C is A, yeah? No stone is animal. I have satisfied with the two conditions, that's not A, what? Syllogism, right? So the four cases with two universal statements in the first figure, two of them are syllogisms. And two of them are not syllogisms, right? Sometimes people call this an invalid syllogism, but it's not really a syllogism at all. It's better because they call it not a syllogism. We'll see eventually that this case here that concludes every C as A is the only one of all the 48 that you can conclude every C as A. There's going to be more than one to conclude with no C as A, but only one to conclude every C as A. This is very important for demonstration because you prove a property of its subject through the definition of the subject, right? You tend to use this form as Aristotle says in the question of the difference. Did you see that? When we get through all 16, we'll put the ones that were syllogisms back on the board, you know, all together, right? We'll see some interesting things about them. That'll be the result. Now, let's go to the opposite extreme and take the four cases where you have two particular statements. And as you might suspect, if you have no universal statement, you can have neither the set of all nor the set of none, right? We require a true universal theory for the set of all and the set of none. So let's put down those four cases. Now, you're in the first figure. In the first figure, your terms are in this arrangement now. Okay, so you have sum B is A, sum C is B. You have sum B is A, and sum B, excuse me, sum C is not B. You have sum B is not A, sum C is B. Now, if you like to do a lot of work, we could look for examples for each of the four, right? Okay, and there are four sets of examples. But if you're lazy, then maybe you can find examples for A, B, and C that will satisfy, what, all four examples, right? Now, how can I do that? Well, I studied a book called the Isogobia of Porphyry, right? Okay, and I've seen the difference between genus and difference and species and property and accident, right? And I know that accidents can, what, be present or be absent, right? Now it's something that will belong both to some and not to some, right? Okay, so let's take for A, let's take animal. And for C, let's take something that's always an animal, let's say a dog, right? Okay? And then something that is never an animal, stone, right? Now, I satisfied one mission, right? Every dog is an animal, no stone is an animal, right? But now I need a middle term, D, such that sum B is A and sum B is not A. And that will work for both of these, right? And where sum C is B and sum C is not B, and that will work for, what, both of these, right? You see that? Well, now I go back to corporate and say, let's take something accidental, right? Like white, okay? A white thing, if you want to. Yeah, okay? Well, some white things are animals, some white things are not animals. So this will work whether your major premise is sum B is A or sum B is not A, right? Okay? And sum dogs are white and sum dogs are not white. So that will work for sum C is B and sum C is not B, right? And sum stones are white and sum stones are not white. Ah! See, I saved myself later, right? So that will work for you, right? So that will work for you, right? So that will work for you, right? So that will work for you, right? So that will work for you, right? So that will work for you, right? So that will work for you, right? Instead of four sets of examples, just one set of examples, right? That's all these things are work, right? Okay. Now, again, huh? I pause and say, have I satisfied both conditions? Well, the major premise is either some B is A or some B is not A, right? And this is true, right? Some white things are animals, some are not. And the minor premise is some C is B or some C is not B. And some dogs are white, some are not. Some stones are white, some are not. I satisfied condition number one. Not condition number two. If I have an example where every C is A, yeah. I'll circle that, right? Every dog is an animal. If you have an example where no C is A, yeah. Underline that. No stone is an animal. So, the fact that the universal affirmative is sometimes the case excludes both negatives from always being so. And if the universal negative is true just once, that's all I need. It means that no affirmative statement is always the case. So, there is nothing affirmative or negative that is always so. And those statements are true, right? And if nothing is always so, nothing is necessarily so. And if nothing is necessarily so, you ain't got a solution, right? You can see that. Now, we'll go through the eight forms that are mixed from the universal and the particular statement, right? And we'll consider those eight at the time. Now, we'll begin with the four that have the major premise, right? So, you have every B is A, the second one is particular, some B is A, or you have some C is not B. Every B is A, and every B is A, and some C is not B, okay? And which of those four do you know right away is not going to have the set of all or the set of none? It's not going to have the set of all or the set of none. Two negatives. Yeah. You can never have the set of all or the set of none with two, what, negatives, right? Because even the set of none requires an affirmative statement, placing something under the subject of universal negatives. And therefore, it's your set of all. You can never find it. Now, in finding examples to disprove this, examples for A, B, and C, remember that even if no C is B is true, it's still true that some C is B, right? Remember what he said? When you say some C is B, you're not saying some are and some are not. You're saying some are not, right? So, this is true whether the rest are or are not, right? So, when you think of a B and A such that no B is A, now, think of two Cs. A C that is not a plant, but it's always an animal. Yeah. Okay. And now think of something that is never a plant, never an animal either. So, it's true that no B is A, no plant is an animal. It's true that some cat is not a plant, right? Because no cat is a plant. If you took some of them, they would not be. And it's true that no, that some stone is not a plant. And satisfied condition number one. Something else you could know from this, too, if no B is A is false instead of true, right? What about no A is D? Would that be true? Because then you just go backwards. Yeah, yeah. If universal negative is true, it's converse is always true, right? So if universal negative is false, it's converse will always be right. Now there's an infinitive statement where no B is A is the form of it, and the statement is false, right? In every one of those, the converse is necessarily what? False. If the converse could be true, then by the conversion we already showed here, the original one would have to be true, right? Isn't that interesting, right? The universal negative statement is true, its converse is always necessarily true. If it's false, its converse is always necessarily, again, infinitive, or possibly true universal negative statements, and infinity is false. In yourself, if the infinity is incredible, right? Now, let's take the universal affirmative. Every B is A. Now, if every B is A is true, if you're given that every B is A, whatever B or A might be, it's true that every B is A. Anyway, is it necessarily true that every A is B? And one example where it isn't so, right, is enough to show it isn't always so, and therefore it isn't necessarily so, right? So I take the example, every dog is an animal, right? Animal is a dog. Every woman is a human being. And therefore, a human being is a woman. So, the universal affirmative does not convert necessarily into a universal way, right? But, if every B is an A, can you say necessarily that at least some A is B? Notice, in some cases, both are true. Like, every two is half a four, and everything half a four is a two, right? You have a thing in its definition, right? Every square is a two lateral, right angle, quadrilateral, and every three lateral, right angle, quadrilateral is a square, right? If you have a definition, it should be converted, well, it's not a definition. That's why Socrates will turn around when somebody thinks they have a definition, right? If you ask them, what is a dog? They say, well, my definition of a dog is a four-footed animal. And Socrates would say, well, every dog is a four-footed animal, but it's every four-footed animal dog. If you can't turn around, it's not a definition yet, right? So, if you have a definition, you should be able to turn around. And if you have a property, in the strictest sense, right, that belongs to every member, always, and necessarily, like half a four to two, then you can turn around, right? Okay? But, it ain't always so, because B might be something less universal than A, like every dog is an animal. Every square is a five-footed animal, but not every five-footed animal is a square. So, just one example is enough to show you can't necessarily encourage an universal. But can you say that necessarily some A is B? Well, again, so you can't look at all the cases, right? And see that? But, you can reason to it, right? I say that whenever B is an A, some A must be a B. Now, if you say that that is not true, necessarily, then by the square of opposition, that could be false, right? If some A is B, then by the square of opposition, what would have to be true? No. Yeah, then no A is B is true. If this were false, right? Which I say it can't be. If that were false, even once, then no A is B is true, and then by the square of opposition, What we saw before, no B would be A. So you're saying when every B is A, it's possible that no B is A. Come on now. Is it possible, right? When every B is an A, no B is an A? But that follows if you're not admitting that some A is B is necessarily true, right? If that could be false, then no A is B could be true by the square of opposition, right? But could that possibly be true, that no A is B when every B is A? Well, by the first case, then no B is A would be true. I'm interested in that, right? That's no way possible when every B is A, right? The two universal statements cannot both be true. They can both be false, right? They can't both be true. But if some A is B is false, then no A is B must be true. And if no A is B is true, we solve universal negative. Some no B is A must be true. It can't be true when every B is A. So we say that universal affirmative, right? It converts partially, right? Okay? So it drops from what? Universality to particular, right? That's all you can say necessarily so. So that's going to be part of the reason why you're going to get only negative conclusions in the second figure, we'll see. Because if you convert the universal negative, it remains universal negative. It seems like there's a set of none in some cases, right? But universal affirmative, you convert that, it loses its universality, and therefore you can't have the set of all, right? That's all right. Isn't it? Now, how about some B is A? If that's true, is it necessarily true that some A is B? Yeah. Yeah. We actually showed that in the first one. Yeah. If that's unnecessarily true that some A is B, if some A is B could be false even once, right? Then no A is B would be true by the square of opposition. And then by the conversion universal negative, no B is A would be true when some B is A, right? Was it possible when some B is A is true that none of the B's are A's? No. So something impossible follows, right? So the particular affirmative is convert C, right? Isn't that when you showed the universal negative converts, didn't you use that conversion in that? No, it was the example, right? You said that, you know, the usual example is something that is both A and B, right? It gave a name to the A that was A, B, right? And it's something like that, right? You can probably show it in this way if you like it, right? I could say, you know, that if some B is an A, some A must be B, but what he shows you is usually the universal negative, right? Because that's very clear what he's saying. Okay, but kind of implicitly you showed that because you said we had an X, therefore we have an A that's a B. Yeah, yeah. I was just wondering what's the first one. Bairstock shows the universal negative first, that's the most important conversion, and then to that you will show what you're used to, right? Now, what about the particular negative? Suppose it's true that some B is not A. Can you turn around and say that some A is not B? One example is enough to show you can't necessarily say that. Some animal is not a dog, therefore some dog is not an animal. Some number is not an odd number, therefore some odd number is not a... One example suffices to show that it isn't necessarily so. So the particular negative does not convert, right? 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