Logic (2016)
Lecture 38: Conversion of Propositions and the First Figure of the Syllogism

Transcript
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You know the difficulty, losing power, right? And so the particular has less power than the universal, right? Now, let's look at the particular statements now, right? If sum B is A, can you always turn it around and say then sum A is B? Can you turn that around? Necessarily? Shaking your head there, Father. Press impressions are not. Notice, if sum A is B is not necessarily so, when sum B is A, right? If sum A is B could be false, then no A is B could be what? Would be true, right? If this were false, then the propositions, no A is B. If no A is B, then no B is A. By this thing we showed here. Can no B is A, B would sum B is A? So sum A is B must be so, right? So the particular affirmative converts, right? And stays particular, right? And now the particular negative, right? If sum B is not A, then what about reverse? Can you say that sum A is not B? Does that follow? Necessarily? If sum B is not A, then sum A is not... You could say, you know, if sum B is not a dog, then sum B is not an animal. It isn't always the case. So this doesn't convert, right? So the universe of negative stands out, right? Because it converts and stays universal, right? This converts, but it's particular and still it's particular. This converts partially, right? And this doesn't convert at all, right? So the particular affirmative is better for conversion than the particular negative. Because the particular negative doesn't convert necessarily at all, right? The particular affirmative always converts, but it stays particular, right? But the universal negative is better than the what? Universal affirmative, as far as conversion is concerned, because it necessarily converts in stage universal. And this loses power, right? And everyone's down, right? You see the point? So this is, this is, might be the reason why in the second figure, you can have universal negative conclusions. Universal affirmative, right? If you turn this around simply, that's a conclusion, right? So isn't that fun enough? Thomas is commenting on one of Boethius' works, right? I guess the De Trinitatia or something like that. And Boethius says, he's what? He's drawn from the inward parts of philosophy, right? Thomas says, what does the great Boethius mean by the inward, you know? He's speaking of inward philosophy. Thomas understands that logic and wisdom. See? Because in mathematics, right? At least you've got the imagination, right? So it's not hidden, right? And in natural philosophy, you've got the senses, right? But in wisdom, you're talking about immaterial things, right? And therefore, you've got to go into your mind. And logic is doing that same thing, right? Okay? So in one way, logic is more like what? Wisdom. In one way, natural philosophy is more like wisdom, right? Because wisdom is a knowledge of the first causes. And the natural philosophy is going further than the causes, right? In terms of universality, logic is more like wisdom. And in terms of being, what, immaterial, right? So Thomas calls it inward, I mean, the great Boethius, he's a great man, calls it inward philosophy, right? The intimate disciplines of philosophy, right? Which is referring to logic and to wisdom, right? Inward philosophy. So I'm an inward philosopher. I'm also a natural philosopher and a geometer right now. I got my two students there from, another student there from TEC. And I'm going to talk about, you know, Euclid has a thing about inscribing an oblong in a circle. But I think you can do that, right? You can circumscribe a circle around it. But you can't circumscribe a circle in an oblong. And you can't circumscribe. But let's see if these guys agree with me. You can have a little theorem in there, you know, like a circle and square. So you can inscribe a circle in a square. You can circumscribe a circle around a square. You can inscribe a square in a circle. And you can circumscribe a square around a circle. So you have four theorems, right? You can take each one of these figures, circle and square, or you can circumscribe a circle around the other one. But these other ones, you can do it as a bottle. You can put it as a triangle. You can do something like that too. So this is conversion now, right? In all these days you didn't know it. But it's turning something around, right? And you can say now, what? Universal affirmative, universal negative is the winner of it. It's the prize. It converts necessarily, right? And stays universal. The universal affirmative converts necessarily, but drops in power to the particular. Particular affirmative converts and stays particular, right? The particular, what? Negative doesn't convert at all. Also, right? If you look at Aristotle's prior analytics, right? Where he takes up the simple soldiers under the three figures, right? To begin with, you're going to have 16 cases in each figure, right? Because each of the two premises can be either affirmative or negative and universal or singular. So there's four possibilities for the first premise and four for the second. And four times four is what? 16, right? There's 16 possibilities. Because each of the four guys can get married and there's four girls, among which he could be a husband, right? And so four times four is 16 possibilities, right? Of marriage, right? That kind of gets you interested. But nowadays, if they think that guys can marry guys and girls can marry girls, it's a lot of, right? I understand. They'll destroy my home. My home. It's very example. Yeah. But I think, you know, they're not too confused. You just go through the four cases that have two universals, right? The universal moves, right? So the first figure, you have a little term, subject. The first one. And the predicate in the second one, right? Okay. It's in a slant position, right? B is A. B is B. No B is A. Regrees C is B. And then, no B is A. And no C is B. So, these are the first four cases in the first figure, right? The case is where you have both premises being universal, right? So, either you have two affirmatives or two negatives, right, universal. Or else you have a mixed one, universal affirmative and one universal negative, right? In one case, the universal negative is the major premise, right? And in the other case, it's... Oh, what did you do, right? Premise A. Oh, my goodness, huh? How could I have been so, so stupid? I know it before fall, right? So, either both premises are universal affirmative, like every B is A and every C is B, right? Or both are universal negative, like no B is A and no C is B. Or one is what? Affirmative and the other is negative. In one case, it's the major premise, right? It's negative and the other one is the major premise is what? Affirmative and the other one is the other one. That's clear enough, right? And Aristotle is going to ask, does anything follow what? Necessarily, right? With C as a subject, right? And A as a, what? Predicate, huh? Anything you can say necessarily, right? With C as a subject and A as a predicate. Or does nothing follow necessarily, right? Now, he's basing himself upon the said of all and said of none for a valid syllogism, right? Most of the ways to take it grammatically on Aristotle says if A is said of all B, right? But I say if every B is A, right? Then whatever is a B is a what? A. And that's really obvious, isn't it? If you understand what it means to say every B is an A, if you can discourse about the universal, right? If you understand what it means to say every B is an A, that means there's no exception, right? Every B is an A, right? Well, then whatever is a B obviously must be a what? An A. And then the said of none, huh? If no B is an A, you understand what that means, right? No means what? None, right? So if no B is an A, then whatever is a B must not be a what? A. So every B is A, and here you're told that every C is a B, well then what follows? Yeah. That's kind of obvious, right, huh? So Aristotle calls these syllogisms, incidentally in the first figure, he calls them perfect because you can see in the way it stands that the set of all or the set of none applies and does not apply, right? In the second and the third figure, you've got to kind of try to convert and, you know, clear perfect, you know, in some way, right? You have to rearrange before you really see it. Over here, you can see the set of none, clearly, right? If no B is an A, then whatever is a B is not an A, right? And so, since you're told that every C is a B, then none of those Cs can be a what? A. So you can see in the first figure here, in these two cases we've considered so far, that there's one case to conclude the universal affirmative and one to conclude the universal what? Negative, right, huh? Now, what about this one here? No B is A and no C is B. Right? So you've got two negative parents and not going to have any children, right? Okay. But does anything follow necessarily about C and A, right? Now, we're going to use examples to show that nothing follows necessarily, right? And we want to find examples for A, B, and C that satisfy two conditions, right? One is that when you substitute them in to this form, these statements will be what? True. And you have one set of examples where every C is A, in the case, and another set of examples where no C is A. And that will show, right, that when these are true, nothing about C and A is always true. Because if the universal affirmative is once true, then by the square of opposition both negatives are what? False once. And if the universal negative is true once, then both affirmatives, every C is A and some C is A, are false at least what? Once, right? And therefore, nothing affirmative or negative is always the case with C and A in this arrangement of terms. And therefore, nothing follows necessarily about C and A, right? Now, if you like to do unnecessary work, you can find two sets of examples for A, B, and C, right? But I'm an easy son of a gun. And so, I like to find the same example for A and B and just two examples of C. Like that, right? So, let's take animal for A. And let's take, let's say, stone. Simple example. No B is A. No stone is an animal. Makes sense to me, right? I'm going to take two examples for C, right? One where no C is a B, but every C is an A. Well, that's simple. Dog. No C is B. It satisfies that, right? No dog is a stone, but every C is an A, right? And now, let's take my friend the tree. No tree is a stone, but no tree is an animal. So, these examples satisfy but two conditions. But when you substitute them in there, in that form, no B is A, no C is B, those premises are both true. There's no defect to the matter. Nothing false about saying no stone is an animal, is there? Nothing false about saying that no dog or no tree is a stone, right? So, when these premises are true, it might be that every C is A. And I'll circle that example, right? And when these premises are true, it might be that no C is an A, right? I'll underline that. I think I satisfy the two conditions, right? No B is A and no C is a B are true with those examples, right? And I have one, in the second case, I have one example where every C is A, which means no negative is true always. Therefore, no negative is necessarily follows, right? And then I have another example, no tree is an animal. No C sometimes is an A. So, no affirmative statement is always true, is it? Well then, no affirmative statement is always necessarily so, right? So, nothing affirmative, nothing negative. I have now shown that this is not a what? Syllogism, right? But notice, you don't see the set of none in there, do you? You have an universal negative, no B is A, but nothing is said to be a B, right? The set of none is that if no B is an A, whatever is a B is an A, right? Or not an A, rather. But there's no, nothing is said to be a B, is it? I have some affirmative to get any conclusion, right? Okay, you see how I did that, right? Thank you. Thank you. Thank you. And you'll find out in any socialism, or any figure rather, that two negatives give you nothing, right? If you have two negative parents, you have no kids, right? At least if one is affirmative, there's a possibility you may get children, right? Sorry, I spiced up my logic. Now here, I'm a little bit more hesitant now, you see? You certainly can't get the set of all, because you need two affirmatives for that, right? Every B is an A, but nothing is said to be a B, so you can't get the set of all. You can get the set of none as it stands, huh? When no C is B, you've got a universal negative. Is anything said to be a C? Now, I could turn around, no C is B, to what? I could turn around, and I just, you know, hope it's going to be true. No B is C. What about this up here? You could say sum A is B. I could say that sum A is not C, right? You can do the reverse, right? But I wasn't asking whether you could make one A as a subject and C as a predicate, right? I was asking whether you could say anything from C as a subject and A as a predicate, right? Okay? And here you're reading back in the form of the first figure, right? But you've got a, you know, A, where C is usually in C where A is. You're kind of just doubling your effort there, right? So we're saying, is there any necessity that of a statement that C is a subject or A as a predicate? Here, you're just playing checkers, right? Continue to be honest, right? Black and white and blue brown, huh? Well, as a standard, I don't see the set of all or the set of none, right? So I try to find examples for A, B, and C that satisfy what two conditions. That the premises will be true when I substitute them in for A, B, and C. But I have one example where every C is A and one example where, what, no C is A, right? That'll knock out everything, right? Because if every C is A is true once, no negative is true always. And if no C is A is true once, then no affirmative is true always. And if no affirmative and no negative are true always, then nothing is true always, it's true necessarily, right? Okay? So, let's say animal. Let's say dog. I'm going to study like simple examples. Every dog is an animal. That's true, right? Okay? Now, I've got to find two examples for C, right? Can you think of a C such that no C is a dog but every C is an A? Yeah, very good, very good. Can you think of an example where no C is a dog and no C is an animal? Okay, yeah. See how simple that is after a while, right? Now, I'm kind of stupid in the sense that makes two mistakes, so I'm going to check to see if I'm satisfied in both conditions, right? Are the premises every B is A and no C is a B true with those examples? That's the first condition, right? Every B is an A. Every dog is an animal. No C is a B. No cat is a dog. No tree is a dog. Well, I have satisfied the first condition, right? When you substitute this into this figure, you have premises that are true, right? There's no difficulty in the matter, right? Considered about the form, right? Now, do I have an example where, one example where every C is an A? I'll circle that universal affirmative example, right? Now, I've knocked out any negative statements being true always, right? If it's just one example where every C is an A, then cat an animal is one example, right? Then no negative is true always, right? And therefore, no negative is true necessarily at this point. And I also have an example where no C is A, right? Now, underline that, huh? That's nice to assume. Okay. No tree is an animal. That's true, right? So, I have at least one example, that's all I need, where no C is A, right? And that means that any affirmative statement is false once. There's no affirmative statement that's always so, because no tree is an animal. That makes any affirmative statement for being always so. And then, from nothing being always so, you argue that what? Yes. I can go back to the if then, right? And say, if something follows necessarily, then it follows always, right? But nothing follows what? Always, right? If necessary, the then is not necessary. It's necessarily not necessary, right? But it is important to realize, huh? That this is a word equivocal by reason. And, you know. That's, reason looks for order, right? So, there's an order among those meanings, right? Well, if order means before and after, it's got to be one of those senses, right? And you've got to stop and think now, right? Great sense, right? Another thing I like to do is like to bring out, you know, that distinction and order, right? And I say, give me the distinction and order, distinction and order. Well, what does distinction mean, right? It means this is not that, right? But order means this is before or after that, right? Okay, so that's the distinction of distinction and order. What's the order, right? Well, you have to see a distinction before you can see an order. And I know that by the axiom of before and after, that nothing is before or after itself, right? And now you know the distinction and order of what? Distinction and order. But you know, this is because the logic is called the philosophy of reason, right? And what's marvelous about reason is that it can know itself, right? And so when reason thinks out, the definition of reason, right? It's coming to know itself much better than it did before, right? So when Shakespeare thinks out that reason is the ability for a larger discourse to be before and after, right? He's kind of obeying the, what? The seven wise men of Greece, right? He meant in the oracle of Delphi, according to the tradition, right? And he put up two things for everybody to see. You came from the oracle of Greece to go to the oracle of Delphi, right? It's like going to Rome or something, right? What's happening? And they put up the thing, you know, nothing too much and know thyself, right? And I used to say to the students now, to whom are the words, know thyself? That's an exhortation. It's not a statement. It's an exhortation. But to whom is that addressed? It's not addressed to God or the angels, because they know themselves first of all, most of all, right? It's not addressed to what? The trees or even to the dog, because the dog can't even know what the dog is. It's addressed to someone that can know himself, but doesn't know himself very well. And that's man, right? But more precisely, you can say it's directed to your reason, right? Because anger doesn't know what anger is. And they just... doesn't know what digestion is. But reason can know what reason is, right? So in a way, it's addressed to reason. Know thyself, right? And this is reflected in the science, in the philosophy of reason, right? There can be what? Not only a definition of, reason can have a definition of reason, reason can know itself, but there can be a definition of definition, right? Kind of an amazing thing, right? So the definition of definition is speech signifying what a thing is, right? And I often like to ask students and say, now, is the definition of definition of definition of definition of definition? Is the definition of definition of definition of definition of definition? What would you say? Yes or no? Is the definition of definition of definition of definition of definition? Yes or no? I'll start to tell you about this, this guy who I met, who's teaching philosophy at the Western University, he's talking about, you know, what the, you see on the faces of the kids who are teaching philosophy, right? It's like saying, why are you doing this to me, you know? That's kind of a joke. You know, it's the definition of definition, right? Is speech signifying what a thing is. And that's true of the definition of square. It's a definition in that sense, right? So it's not a definition of what? The definition of definition. It's a definition of definition in general, right? Okay? So if this is the definition of square, it's a definition of the soul, each of them is speech signifying what a thing is. So the definition of definition is not a definition of definition of definition. It's a definition of definition in general. Is that clear? Did you see it? Did you follow me? I might get fast. I might get fast. But it is interesting to come back upon this, right? So you can have a definition of definition. You can have a statement about statements, right? A statement of speech signifying the true or the false. I've now made a statement about statements, right? But this is reflecting the fact that reason can come back upon itself and know itself, right? And reason is really the only part of me that can know itself, right? And therefore reason is the only thing that can really fully obey what the great seven wise men said. Know thyself, right? It's an easy thing, right? Remember how when Thomas was talking about the distinction of the Trinity, right? And he makes use of a, what? Distinction of distinction, right? The distinction, you've got the formal distinction which is by opposites and then the material distinction which is the, what? Division of the continuous, right? So when Euclid divides a straight line into two equal parts, right? He's, what? Divided and continuous, right? But when we divide being into substance and what accident and we say, well, accident exists in another there isn't a subject that substance does not, right? There's opposites, right? So there's a distinction of distinction just like there is a, what? Definition of definition. It all goes back to the fact that reason can come back upon itself, right? It goes its own act, right? Does love know what love is? But reason to some extent knows what understanding is and what reasoning is, right? It's kind of a marvelous thing reason in some ways, right? Now the other thing the wise men said was nothing too much, right? So can you love, you know, can you avoid loving God too much? Well, how would you answer that? I mean, if you say the wise men, these are white men, right? They've been profound as know thyself, right? And then they said nothing too much, right? Well, that seems to cover everything, right? Nothing too much, right? So you shouldn't have loved God too much, right? I mean, he's nothing. You know, even Bethlehem says that the trinity is three things. He does, right? Oh, and that can say he's not created. Yeah, yeah. But he's a thing. He's something, right? You have to be careful, you know, because, you know, it's kind of a common thing, you know, we're talking about all the time in the parishes, you know, that you've got to get the guy to not regard the girl as a thing, right? To play around with. But as a person, right? But in some sense, we might divide person against thing. But other times, you would say, what? The person is a thing, right? The person is something, right? They're not lovely, yeah. You know, it's really good in itself as a person is something, right? And so, this is one of the ways that a name becomes equivocal by reason, right? Sometimes a name is said in many things, but one of them keeps it as its own name, right? And a new name is given, right? And the example there that they give in logic is when Aristotle talks about disposition habit, right? Sometimes he says habit is a disposition that is firm and not easily lost, right? And then there's a disposition that is easily lost, right? But sometimes he divides habit against disposition, right? As if the disposition that is easily lost keeps the name disposition as its own sometimes. And then we get a new name to this disposition as firm, which is habit, right? And you find all kinds of examples of that, right? And Aristotle was in the book on places, right? On dialectic, right? And he talks about property and definition, right? And property is what? Convertible with what? Something, right? Like half a four is convertible with two. Every two is half a four and everything half a four is two, right? And then sometimes he divides property into property and definition, you see? And definition, one sense of property, it's a convertible speech, right? Every square is an equilateral and right angle quadrilateral and every equilateral right angle quadrilateral is a, you know? Every man is an animal that has a reason, every animal that has a reason is a man, right? You know? But definition does something more than property, right? It brings out the nature of the important deal, right? So it's got a certain excellence. So it gets a new name, definition, and then the property that is not bringing out the nature that keeps it in property. I told you how my mother didn't like it when I said man is an animal, right? Because sometimes you keep animal for what? The beast, right? But sometimes you say man is an animal and you're not insulting somebody, you know? I always say if your biology teacher says you're an animal and not a plant it's not insulting anything. But if your girlfriend says you're an animal she's probably insulting you, right? Okay, notice here. The two forms of universal forms here, universal premises in the first figure two of them are syllogisms, right? And by one of them you can syllogize a universal affirmative. And this is the only way you can, by the way. Very very syllogism, right? And by one of them you can syllogize a universal negative, right? We'll find out in the second figure you can syllogize sometimes universal negative but in every universal affirmative we'll see in the second figure. And then there are two where nothing follows, right? So our style brings this out, right? But notice how you know that these are valid in a different way that you know these are invalid, right? You can't prove these are valid by examples, right? But you can use examples to show that either of these is a syllogism, right? How can you do that, right? One example doesn't show that something is always so let alone just necessarily so, right? I'm a philosophical grandfather, okay. So every philosopher is a grandfather? But I'm not a philosophical tree or something, right? See what I mean? You can't go over it as many times but you kind of see what we're doing, you know? Now we'll go to the second figure and take the four universal cases there. I'm going to write an example here. I used to get a bunch of kids sometimes an exam, you know?