Introduction to Philosophy & Logic (1999)
Lecture 41: The Second Figure of the Syllogism and Conversion

Transcript
================================================================================

Let's see if there's only one to work this time, right? If I take dog up here, right, an animal, some animal is a dog, right? That's true. And then I take a C such that every C is an animal and every C is a dog. Let's take a Cocker Spaniel or something, right, some kind of dog, right? So every Cocker Spaniel is an animal and every Cocker Spaniel is a, what, dog. And then let's take the cat and the horse, and every cat is an animal, but no cat is a dog, right? Are I satisfied now with three conditions? And the premise is true when I use those terms. Some B is A, some animal is a dog, true. Every C is B, every Cocker Spaniel is an animal, yep. Every cat is an animal, yep. I satisfy condition number one. The examples are, when they're substituted in, the premises are true, right? And second now, do I have one example where every C is A? Yep, I'll circle that one. Every Cocker Spaniel is a dog, right? I have one example where no C is A, yep. I'll underline that to make sure that are both conditions, right? Now I've shown that there's nothing affirmative or negative that's always so when those premises are true, right? There's no affirmative statement that is always so, because sometimes no C is A left, no cat is a dog, right? There's no negative statement that is always true, because sometimes every C is A, right? Now, some B is A, no C is B, right? You can't try it if you said it at all, because you have a negative statement there. But you have the set of none. Well, the set of none requires a universal negative statement, and something coming under the subject of the universal negative statement. Well, no C is B, it's universal negative, but is anything said to be a C? No. So I suspect that this is not a, what? Syllogism, right? I've got to find examples for A, B, and C to satisfy the two conditions, right? You need to go back and take that B that is always an A, right? And a dog. Yeah. Every dog is an animal. Some dog is an animal, too, right? You can tell what the students have passed, but some students have not passed. Very students have passed. It's true, but some students have passed, too, right? Very students have passed, but some students do. It didn't pass. That's a good thing to do. And I have to find two Cs. Such as no C is a dog, one of them, but every C is an animal. Think of something like that. Cat, yeah. And then a C that's never a dog, but never an animal, either. Stone, yeah. And I always pause, you see, and I say, am I satisfied in both conditions? Well, condition number one is, are the premises true of those examples? Some dog is an animal? Yeah. And no cat is a dog? Yep. And no stone is a dog? Yep. I satisfied condition number one. I satisfied condition number two. You have one example where every C is A? Yeah, I'll start with that. Every cat is an animal. That's true. You have one example where no C is A? Yep. Nope. Underline that. No stone is an animal, right? So I satisfied all the conditions, so I've shown it not to be with syllogism, right? That's a separate thought to say, why do those conditions, why does satisfying those two conditions suffice to show it's not syllogism, right? If you stop and think about it, those examples satisfying those conditions show that when the premises are true, right, there's nothing affirmative that is always so, and there's nothing negative that is always so. Because universal affirmative being true once shows two negatives that are always the case. And universal negative being true once. And universal negative being true once, and there's nothing negative that is always so, and there's nothing negative that is always so, and there's nothing negative that is always so, and there's nothing negative that is always so, and there's nothing negative that is always so, and there's nothing negative that is always so, and there's nothing negative that is always so, and there's nothing negative that is always so, and there's nothing negative that is always so, and there's nothing negative that is, and there's nothing negative that is, and there's nothing negative that is, and there's nothing negative that is, and there's nothing negative that is. Shows that two affirmatives are not always the case, right? So there's nothing affirmative or negative that is always so, when the premises are true. And if nothing is always so, nothing is necessarily so. And if nothing is necessarily so, you don't have a syllogism, right? Okay, let's do that. I don't want to do more precisely, say, if it's a syllogism, then something follows necessarily. If something follows necessarily, it's always so. So, if nothing is always so, if there's nothing necessarily so, then you're not a syllogism, okay? So rather than repeat that, I'll ask the students, you know, a separate question, right? I'll give them examples like this, and you have to find examples, and I'll check and see if the examples sounds like the conditions, right? But then I'll ask them elsewhere, why do those two conditions suffice to show it in the syllogism? Then you have to bring out the fact that the syllogism is necessarily so, or necessarily so, requires something we always so, right? And the examples show that nothing is always so, right? So you have to go back and away to the square of opposition. You're seeing that if universal affirmative is true, any negative statement is not always true, right? And if universal negative is true once, no affirmative is always true, right? So you exhaust the possibility, right? Nothing affirmative to the universal particular is always the case. Nothing negative, right? It's always the case. But rather than find one example of which four, where it's false, I find two where universal affirmative and universal negative, because I can kill two birds with one stone. But if you turn around, no C is B and say no B is C, right? Okay, okay. Then you turn this around, you can say that some A is not C, but you can't turn around a particular negative. So there's no conclusion that C is the subject and A is the right thing, right? When you get to the second and third figure, you'll see that the set of all and the set of none never apply to it as it stands. You have to convert C, okay? And so there's an extra step there in the second and third figure. But after you flip around and bring it, you don't seem to be able to turn it into it, then you suspect it's not a syllogism. And then you look for the examples, right? And maybe, you know, one case here is a little tricky, because that's a little random all. We can't show it. We'll see that, right? You know? I mean, for the most part, that's all you have to do, right? The first figure is a set of all or a set of none apply to it as it stands. If it does, it should be carried away with all of us. If it doesn't apply to it as it stands, then it's going to not be a syllogism, but you can always find examples to show this high syllogism. Okay? In the second and third figure, the set of all and the set of none will never apply to it as it stands. But sometimes it can be made to apply by conversion. But if it doesn't apply by conversion, then you suspect it's invalid. You know, find examples, right? Now, you know, I've done this for years, you know, but I used to say to students, for those four forms that we've showed were so just right, you can never find examples for those forms that would satisfy the two conditions. Because they'd stay up all night trying to find examples. And, you know, I used to be very flamboyant here. I'd go forward, you know, and I'd wave, you know, bills, you know, and say to the first man you can do this, right? And I'd put a lien in my salary, you know, and she'll collect my salary for the rest of my life, see? And so I said, if you're down in South Africa, you know, put a yourself down and take an example, you know, telegram me and I'll send you the money, right? I used to say, you know, this is a crude age, you know, they don't believe in truth, but they believe in money. So, you know, I'm not sure about these things. Some B is not A, or C is B. Well, obviously, you can't get either the set of all or the set of none, you can't. You have a universal primitive, but the other statement is negative. So you can't have the set of all, which requires... And obviously you don't have universal negatives, so you're suspecting that this is not a syllogism, right? So you find examples for A, B, and C that satisfy the two conditions. So, can you think of a B and A such that some B is not A? Like that. Like that. Like that. Like that. Like that. Since no plant is an animal, some plant is not an animal, right? And there you see it as B. Can you think of a C such that every C is a plant and every C is a what? There's a micro... What is that micro-rugging? That's what the horror film is. I think you've got to stop and think of it. You have to double the A. Maybe that's one way of doing it, right? Which I have something... Oh, I said you need it. Oh, that's one way of doing it, yeah. What's the double-word? Instead of taking two C's. What's the example for A? Animate and inanimate. What? Animate and inanimate. Animate and inanimate? Yeah. Okay. And this is plant? No, that's white. White. And this is what? Snow. Snow? Yeah. So, let's check now. Some B is not A. Some white things are not what? Animate and some are not inanimate. So that's okay. Every C is B. All snow is white. And snow, animate, universal what? Negative, right? And snow, inanimate, universal primitive. You could also find two examples for C. Or you might take our friend there. Animal and a dog, right? Some animal is not a dog, right? That's true, right? And every Parker Spaniel is an animal, right? And every cat is an animal. And it satisfies condition number one, right? The premises are true. A man was not a dog. And every Parker Spaniel is an animal. And every cat an animal. That's condition number one. Do we have an example where every C is A? Yep. I'll circle that. Every Parker Spaniel is a dog. Do we have an example where no C is A? Yep. Underline that. No cat is a what? A dog, right? So none of these ones are syllogisms, right? Okay. So notice that now. In the four cases with two universal statements, there were two syllogisms, right? And in the four where you have two particulars, there's no syllogism, right? In the eight where you have a mixed one, well, in the four where you have the minor and major premise being particular, you have no syllogism. But in the four where you have the major premise being a universal, you have two, right? There are two that are not a syllogism, right? So out of the 16 cases, then, we have four that have the form of syllogism, right? Twelve that do not have the form of syllogism, right? Okay. Now, let's put the four forms that are syllogisms. Four forms in the perfect picture. You have every B is A, every C is B, and no B is A, every C is B, and then in the mixed forms, you have every B is A, and some C is B, and no B is A. All right. All right. All right. And the sum C is B, okay? Now, if you look at these four, you'll notice that all four possible statements that C is a subject and has a predicate are found here, right? From every B is an A, and every C coming under the Bs, every C must be an A, right? No B is an A, but all the Cs coming under the Bs, none of those Cs can be an A. So no C is an A. Now, the other two are like that, except you're told only that some C comes under the Bs, right? So if every B is an A, you know some C is a B, you know that some C at least is an A, right? In the last case, you're told that some C is a B, but none of the Bs are A's, so you know at least that some C is not A-what? An A, yeah. So in the first figure, you can draw all four possible conclusions, huh? Universal affirmative, particular affirmative, universal negative, and particular negative, right? And none of the other figures will seek to draw these four conclusions. In the second figure, you'll find out that all you can draw is a negative conclusion, no affirmative. Just there's two here, no C is A, and some C is not A, we'll find out in the second figure. In the third figure, you can't draw any universal statements, just particular, some C is A, and some C is not A, right? So there's a falling off of power in the second and third figure. A syllogism that has a universal conclusion is more powerful than one you can only conclude in particular, right? So in the third figure, you have no universality in your conclusion at all. So that's the reason why you call it the third figure, or one reason why you call it the third figure. It's the weakest, right? In the second figure, you do have universality, but only universal, what? Negative, right? But in the first figure, you have both universal negative and universal fermeture. So in terms of power, the first figure, we'll find out, is more powerful than the second, and the second is more powerful than the third, right? And that's why it's more important to remember the first and second figure and the powerful cases, right? Because those are the ones that are used over and over again in reasoned out knowledge, in geometry, in natural philosophy, in theology, and so on, right? Okay. Now, what about the premises when you have your soldiers in the first figure? Let's make an induction here now, right? Okay. What can you say about the major premise, the premise that contains the predicate and the conclusion? It's always what? In the first figure. You know, so as you do so. And the minor premise is always what? Permanent, yeah. Okay. Now, I'll write that down. The major premise is always universal, because of the syllogism, and the minor premise is always affirmative. affirmative. Now, is there any other case that we considered, any of the other 12 that have those same two conditions? No. Because if the major premise is universal, there's only two possibilities, universal affirmative or universal negative, right? And if each of those two possibilities, if the minor premise is always affirmative, there's only two possibilities, right? It'd be the universal affirmative or particular affirmative, right? So two times two is four, right? So in four cases, right, the other 12 are not what? The syllogism, right? Although I know and you know now the one that were deceived as students, right? So Elijah teaches you that in the same way. But the two most important ones are these two universal ones. Now, not only is there a falling off of power in the second and even more so in the third figure, but the cases that are syllogism in the second and third figure, it's not clear right away that the set of all is not the same. So there's none involved, right? What you have to do is, what, turn them around, right? Put the subject where the credit it is, or vice versa. So now we have to study the topic of conversion, right? Because conversion is a way of making an imperfect syllogism, in the second or third figure, clear, right? And when you convert, you get back in something you don't have the first figure, okay? Now let me just exemplify, before I go into conversion here, this thing here. Take the second figure here. Suppose you had, on the second figure, this is the rank of terms. The middle term, B, is the credit it in both cases, right? And the third figure would be the reverse. And the middle term would be the subject in both cases, huh? Okay. When I was first learning this, back in high school, my harmonic device was that in the second figure, the middle term is in the second spot. There's no third spot. Once I do that, then the third verse is to reverse, right? Okay. That's why I'm going with the first figure. That's why I kept the second and third figures. That's the amount of my device, huh? But there's not close to the second one, because it's less powerful than the first, but more powerful than the third, right? Now notice in the second figure, if you had a form like this, no A is B, and every C is B, right? Okay. Now, does the set of all or the set of none apply to it as a mouse stand? We've got a universal negative state, so no A is B, but nothing is coming under A. You have a universal affirmative, every C is B, but nothing comes under what? C. But now, if I could turn this around, if I could turn this around and say that no B is A, and keep the other one like it is, every C is B, I'd be back in the first figure, right? And I could syllogize that no C is A, right? But now I've got to consider, if it's true that no A is B, is it necessarily true or reverse that no B is A, right? That's what we have to consider, right? And we're going to find out that in fact it is, but the question is, how can we know this for sure? Because there's an infinity, right, of possible statements in the form no A is B that is true, right? And I can't go through all those examples to see that it's always so, right? So how can Aristotle show for infinity, right, of possible statements that are true in the form no A is B that regardless of what A and B are, so long as it's true that no A is B, it will also be true that no B is A, right? Now, what about this now, see? Every A is B, every C is B, right? Now, if I could turn around the statement that every A is B, okay, if I could turn around and say that every B is A, necessarily, keep the other one like it is, I'd be back in the first figure and I could syllogize the universal affirmative statement, right? But is it necessarily true that if every A is B, the reverse is true? Well, just one example is enough to show that it ain't necessarily so, right? So every dog is an animal, but not every animal is a dog, right? Every odd number is an ogre, but not every number is an odd number, right? So you can't convert the universal affirmative and keep it universal, right? But the universal negative you can, we'll see. But again, my monetary challenge, right? I challenge anybody to find a, what, statement in the form no A is B that is true and reverse no B is A that is not true. How can I be sure about that, see? Well, let's see how Aristotle does this. I'm going to look at the four corners there of the square of opposition. I'm going to see to what extent they do or do not convert, right? So let's take the statement. The letter's not important, but let's take B and A. If no B as A is true, I say that the reverse will be true, right? No A is B. Now, I'm just a certain guy. I don't give you any proof, have I? Now, if you don't admit that no A is B is necessarily true, then by the square of opposition, if the universal negative could be false, then what could be true? If this is false, what I'm saying now, you see, if it's true, given it's true that no B is A, no matter what B is, no matter what A is, right? I maintain that it's true that no B is A, it would be true, reversed, if no A is B. I say it could never be false. But if you don't accept that, you've got to say that sometime, somewhere, no B is A is true, but no B is, no A is B is, what, false, right? Now, if no A is B is false once, you know, right? At that one time, then by the square of opposition, some A is, what, B, right? See how the logic of the second act is presupposed to the logic of the third act, right? Okay? So then some A is B is, what, could be true, right? Now, if some A is B could be true, let it be true. You said it could be true, right? And now Aristotle says, let's give a name to the A that is a B. There's just one A that is a B, right? You're going to get told about it. Let's say, let's give a name to the A that is a B, okay? And here's what I call it. Let us call the A that is a B, let's call it X, right? Okay? Now, if you call the A that is a B, X, then X is both an A and a B, isn't it? Okay? You saw that? Then X is both an A and a B. Now, if X is both an A and a B, then there is a B, namely X, right? That is an A. Then there is a B, namely X, that is an A. So, that follows from your saying that no A is B is not necessarily true. I say it is necessarily true. If you say it is necessarily true, then you're saying it could be that some A is B, but it's square of opposition. If X could be, let it happen, right? And let's call the A that is a B, X, right? Then X must be both an A and a B. Therefore, there is some B, namely X, that is an A. So, you're saying that when it's true that no B is A, possibly some B is an A, right? It's not X, right? So, something impossible follows, right? Notice what I'm arguing in a sense, by the argument, right? I'm saying that if no B is A is true, then in reverse, no A is B will be true, right? If you say that's not true, then some A is B could be true, right? And if that was true, there would be an A that is a B, and we'd call it X, right? Then X would be both a B and an A, and therefore there'd be a B and a X that is an A, right? So, something impossible follows, right? If no B is A is true, it's impossible that some B is an A, right? But that would follow your admission, right? So, you're going to be forced to say that if no B is A, no A is B, right? I think this is something a monkey can't do. It's kind of amazing to see this, right? It's kind of amazing to see this, right? It's kind of amazing to see this, right? It's kind of amazing to see this, right? It's amazing to see this, right? It's amazing to see this, right? It's amazing to see this, right? It's amazing to see this, right? Aristotle has shown here for an infinity of possibly what? Universal negative statements, there's infinity of negative statements that are true, right? No three is a four, no three is a five, no three is a six, whatever, right? There's an infinity of universal negative statements that are true, right? There are infinity of statements in the form, no B is A, that are true, right? And for every one of them reversed, no A is B is true. You can't possibly go through them all one by one, right? You can't look at them all, right? But this reasoning here shows that if you don't say that, something impossible follows, right? That some B would be A, which, that's correct opposition is, impossible with no B is A. Do you see that? That's kind of marvelous when you stop and think about that, right? You can see it kind of abstract when you say, now, see, no dog is a cat. Okay, they're both true in that case, right? No stone is a, no two is a stone, it's true in that case, you know? No man is a woman, no man is a man, you know? But isn't it always true? And then you can't, you know, begin to go through all of them, right? Again, you can't go through all of them. But here you've shown that no matter what you take, or behave, right? If the universal negative is true, the converse will be right, right? And this is the reason why we'll see that around in the second figure, that you get a negative conclusion, because of the fact you can convert the universal negative and it stays universal, right? You see that? Do you want to go over again, or do you want to go over again? Yeah? Are there examples that people have done? No, no. You can never find, as I already explained to the students, you can never show that something is a syllogism by examples, right? Examples never show that something is necessarily so, whatever. They don't even show that it's always so, right? But one example is enough to show that something is not always so, right? If I say that man is necessarily white, then, no matter how many white men I produce, I haven't shown that man is always white, no one necessarily white. But you produce one black man or one yellow man, you know, goodbye to my claim that it's necessarily true, right? If I say every number is necessarily odd, three, five, seven, nine, nine, one, and infinity examples, I haven't shown that it's always so, right? You produce one number that is not odd, that's not clean, right? So, examples can be used to show that something is not a syllogism, but never to show that it is. That's always a question I ask the students, you know? Why are examples enough, right? Well, one example is enough to show that something is not always so. But one example, or even a hundred or a million, or infinity of them, is not enough to show that something is always and necessarily so. But the same way here. I'm saying, this is not syllogism, but I'm saying that when no B is A is true, necessarily no A is B is true, right? I can't show that that is so by examples, right? But at the same time, if I show it in the way that I showed it, you could not find one example. You know, I leave my money on the table, you know, I'm touching it today, but, you know? Fought my wad of dollars and so on, and sent her my insurance policy in the house. So, it's so weird to send my chat to you, and so. You cannot find one example of B and A, right? Where no B is A is true, and no A is B is not true. I know you can't find one, because I know that necessarily no A is B is true. How do I know that? Well, if that's not true, then some A is B could be so, right? And if that happened, then some A would be a B, and we'd give that A is a B, call it something, an X, right? And therefore, X would be both an A and a B, and therefore, there'd be some B, namely X, that is an A. That's impossible, and no B is A, right? So, something impossible follows if you don't admit, necessarily, that the converse, universally speaking, is also true. two cases that are syllogisms and two that are not, right? But the two that are syllogisms are, of course, particular as we expect, right? But they're both particular, what? Negative, right? On those same four in the first figure, you had one particular affirmative and one particular, what? Negative, right? We could finish this figure. You had one more set of four? Yeah. Yeah, we could do that. Okay. I was just learning this, you know, my thinking was, you know, I want to think about this in a quiet room away from everybody else. I'll figure it out, you know, but I don't want anything else to wrap it. You know, you know, he said, you made it that way by now, let's see. He's going to tell me you can't wait for all of this. It's company and talk and firmly convinced you had to do that, see? Now the four cases were mixed cases, but where the particular one is on top, right? The major premise and the universal on the bottom, right? But you have the second figure now with this arrangement for the middle terms, the predicate of both cases in the second slot, so to speak. So you have some A is B, or some A is not B, right? That is your major premise. But for each of those possibilities, you have two possibilities for the second premise. You do universal affirmative, right? Or universal negative. Now you know right away, if you have two negatives, you can't have a syllogism, right? So we'd expect right away that this is not a syllogism. And we have to find examples then for A, B, and C that satisfy the two conditions. So can you think of examples for A, B, and C that satisfy those conditions? Oh, I'm guessing. Yeah, or just take, you know, let's say animal and dog, right? So some animal is not a dog, right? That's true. Now you've got to find two Cs. You find a C that is never a dog, but always an animal? Okay. Yeah. You find something that is never a dog and never an animal either? Oh, okay. Now, as I preach, I say, well, let me pause now, right? Not the pause that refreshes, as they say, but the pause that helps you avoid mistakes, huh? If I satisfy condition number one, the premise is true with those examples. Some A is not B. Some animal is not a dog. That's true, right? No C is B. No cat is a dog. That's true. No stone is a dog. Ah, I satisfy condition number one. The premises are true with those examples, right? Now, if you have one example where every C is A, every cat is animal, I'll circle that. If you have one example where no C is A, I'll undermine that. No stone. So every cat is an animal being true once knocks out the negatives as being true always, right? And no stone is an animal being true once. If there's a negative, it knocks out the two affirmatives, right? Everything's knocked out. Cale, right? Cale, right? Cale, right? Cale, right? Cale, right? Cale, right? Cale, right? Cale, right? There's no affirmative, no negative statement, it's always so. Therefore, there's none that's necessarily so. Therefore, there's not a syllogism, right? Now, let's go up here. The sum A is B, every C is B. Well, now, you can't get the set of none there. You've got no negative, right? Can you get the set of all? Well, you do have a universal affirmative statement, and nothing comes under the C. If you could turn that around and say every B is C, then you'd have sum A is C, and sum C is A. But you can't convert the universal affirmative, right? It isn't necessarily so. So, conversion will not get you anything, right? So, again, I figured this is naughty syllogism, but I've got to prove that with my examples, huh? So, can you find examples that satisfy the conditions? Dog and animal and Cocker Spaniel. So, dog is A, or what? Since every dog is an animal, it's also true that some dog is an animal, right? Cocker Spaniel and cat. Yeah. Now, I'm going to pause and say, have I satisfied both conditions? Condition number one is, are the premises true when I substitute examples? Well, some A is B, some dog is an animal, that's true. Every Carpist Spaniel is an animal, that's true. Every cat is an animal, that's true. I satisfy condition number one. The premises are two of those examples. So, I have one where every C is A? Yeah. Now, if every C is A is true once, every negative is false once, right? No negative is always the case. To have one universal negative, yep. No cat is a dog. If that's true once, there's no affirmative statement, either every C is A or some, it's true always, right? So, there's nothing, affirmative or negative, that is always so. But if something were necessarily so, it had to be always so. So, if nothing is always so, nothing is necessarily so. Maybe the syllogism would have been necessarily so, but nothing is necessarily so. That's not a syllogism, right? Okay. Now, how about this? Some A is not B, every C is B, right? You can't get the set of all, because you've got one negative, and you can't get the set of none, because you don't have any universal negative, right? So, I guess that that's not a syllogism myself, right? I used to, you know, I have students in class, to be one of my general questions, they say, why can you show that something is not a syllogism by examples, but you can't show that it is a syllogism by examples, right? But if you just stop and think about it, it's the fact that you can show by examples that something is not always so, if you have one contrary example, right? But, you know, it's possible sometimes, you know, to produce an infinite example, right? If I produce, you know, I go on producing, you know, examples where numbers are odd forever, right? 5, 7, 9, 11, right? I still haven't shown that they're always odd, so you have to go back to something universal that's obvious, like the set of all and the set of none, to see that the syllogism is valid, or go back to the square of opposition, and so on, right? Okay. If you find examples for A, B, and C, they're going to satisfy the conditions, huh? Dog for A. Dog, white for B. B, white. Animal for C. Why don't you do the reverse there? I have every C is A. Some C is A, right? So you end up with some C is A, right? You do that. And you want to have a reverse one, like this animal. Oh. And dogs, you have an example, every C is A, right? Every... So some animals are not white, and some... Every dog is an animal. Yeah, okay.