Logic (2016)
Lecture 37: If-Then Syllogisms and the Three Figures of Categorical Syllogisms

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You could say, though, nevertheless, if I predict something very accurate, right, like 1006, it'll start, right? There's more probability to my hypothesis, right? That's why it would work better in physics than in sociology, right? The prediction would be very sloppy and not be very precise, right? How do you get, you know, even predicting it would be a 1006 and exactly what it's supposed to be? That's really amazing, you know? So, the more things that fall from my hypothesis, right, that turn out to be so, the more, what, probability to my hypothesis, right? The form of the argument never had necessity, right? But it had a great probability, right? It's a little bit like an induction, right? You never see all the symptoms, right? But the more you see that it's so, makes it more, what, probable, right? You never saw it when the induction would be more italical argument, right, than the demonstrative argument, right? So, in terms of being a syllogism, a argument which some statements laid down, another falls necessarily, it doesn't fall necessarily, but A is so, right? So, these two are not a syllogism, right? This one is, this one is then a syllogism, right? How if A is so, then B is so? And you find out that, what, B is not so. Does anything follow about A necessarily? You're shaking your head, right? Now, before you thought something followed and it didn't follow. Now you're thinking something doesn't follow and it does follow. But this case is not as obvious as the first case, right, huh? If B is not so, then how could A be so? Because if A was so, then B would have been so. So B is not so. So A must not be so, right? So it follows necessarily that A is not so, right? And so, notice, I say, sit kind of a fair effect from a scientist, right, huh? He says, if my hypothesis is correct, then it will be a, what, eclipse at 1006 phenomenon. And the eclipse at 1006, then it will be very positive. But there's no eclipse at 1006. So there's more rigor in the objective hypothesis, right? And its consequence, then your hypothesis, your conclusion, or your prediction, becomes true, right? Seems unfair, right? So you have these two ways of reasoning, right? Well, sometimes you can bring something else in, right, huh? Yes, in the convertible, right, huh? If you say, if this number is, what, two, then this number is half a four. But this number, what, two is half a four. Therefore, it's true, right? Here, I think, because the things are convertible, right? But it's not from the form here, as it's stated here, right? It can be so. So Shakespeare is arguing, in a way, with necessity, right? He's arguing, it explains this to us, you know, He's saying that, what, three is half a four. What's false holds that three is half a four, and it's no more than two. But if it were half a four, it would be no more than two. So if man is chief good, it's the chief good of the beast, then man is more than the beast. Man is more than the beast, right? If it's chief good, it's not the chief good of the beast, then. But notice in these forms, not recently, the two betters are for the technique. Some man I read, you know, he said in the recent books, before and after, right? Now, in these two syllogisms here, right? How many befores do you see? In these two syllogisms, in each of these two syllogisms, how many befores do you see? How many befores and afters are there in this syllogism? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? 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How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? How many befores do you see? It's kind of obvious right now. It's based upon this principle, a set of all, which I sometimes state in that order. If every B is an A, whatever is a B is going to have to be an A, right? It's kind of obvious, right? So the conclusion is every C is A right now. But they call this one the major premise and this is the, what, minor premise, right? So this premise comes naturally first, right? So the major premise is before the minor premise, right? That's one before and after, right? But the main before and after is that the premises are before the conclusion, right? That's where we draw a line that they help us distinguish, right? What's before and what's after, right? But in the if-in syllogism, you've got three, you know? So we people who love to look before and after, we really like that. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. God, our enlightenment. Move us, God, to know and love and praise you. Help us, God, to know and love and praise you. Guardian angels, strengthen the lights of our mind. Odor and luminar images, and arouse us to consider more correctly. St. Thomas Aquinas, Angelic Doctor. Help us to understand how it's written. Okay, just to review a little bit here, a few things. I was thinking about the simple statements, and then about the compounded statements, right? They're made up of two or more simple statements. And, of course, it always strikes me that the ones that are most important for reasoning are the either-or statement, right, and the if-then statement. So you have the either-or argument and the if-then argument. But it fits in with what we were saying when we talked about looking before and after, right? Looking before and after presupposes looking for distinctions, huh? So the either-or statement involves what? Seeing a distinction, right? And then the if-then statement, it says involves a before and after, right? This is so, then that is so, right? So it's not surprising that they are the most important ones for reasoning, huh? Why the conjunctive statement, like, Brookhurst is a philosopher and Brookhurst is a grandfather, or Brookhurst is a philosopher and a grandfather, right? Those aren't much importance, you know, for reasoning, right? So when Tamarist was reasoning, for example, that the father is not before the son, right, he used a what? Either-or statement, right? Either he would be before him in time or duration, right? Or before him in being, or before him in cause and effect, or in knowledge, right? Or in goodness, right? And then you can either, what reason? By eliminating all possibilities, right? In which he cases, he confirmed, concluded that God, the father, is not before God, the son, right? But you might have an argument, say, you know, where a guy would argue, this straight line must be greater than, less than, or equal to this straight line, right? And then he would try to reason that it's not greater than for this reason, some other reason, and then it's not less than, and therefore it must be what? Yeah, that's kind of common sense, no problem, right, huh? Now, as far as either-or argument, right, huh? We put that on the board, I think, before, right? Let's just recall it for a moment, huh? If you say an either-or argument, the first premise is an if-then statement, right? So using letters, huh, to talk about the form rather than the matter, if you say if A is so, then B is so, right, huh? Then your second premise can look at either A or B, right? And your second premise might be that A is, or it might be that A is not so, right? And likewise, there's two possibilities. If you look at B, it might be so, it might not be so, right? Now, of those four combinations, in how many does something follow necessarily by the form of the argument? Yeah, yeah, yeah, see? Now, let me board for a second. If A is so, then B is so. And A here and B are standing for the simple statements, right there, combined, huh? Okay? Now, if you find out that A is so, right, it's kind of obvious that B must be so, right? Because here, when you say A is so, then B is so, you're not saying that, in fact, it's true that A is so, or it's true that B is in fact so, but if A is so, then B is so. You're admitting that, right? And then since you admit that A is in fact so, that's the difference between this and that, right here, it's not being a third, that it's in fact so. But now that it is so, since you also did this, you must admit that B is so. That's kind of the obvious form, right? If you say, though, if A is so, then B is so, A is not so. By reason of the form of the argument, does it follow necessarily that, what, B is not so? If you go to the matter, right, if A were convertible, if it's a number two, then it's half a four. It's not, what, a number two, therefore it's not half a four. But that's because of the matter, right? Because it's convertible, right? But this can be true, even if B is something more, what, universal, right? So if it is two, then it is less than ten. But it's not two, it'll still be less than ten, right? Okay. So, as far as the form is concerned, it's no, what, syllogism, right? Now, how do you prove that this is not a syllogism? In that form, right? Well, go back to another if-then statement, right? If a statement or a conclusion follows necessarily, is going to follow just some of the time and sometimes not, see? If something is necessarily so, it's always so, right? So, if you have one example where these premises are, in fact, true, right? But B is what? It could be so or it might not be so, right? All you need is one example where these are true and B is what? So. Another example where B is not so, right? And then you realize there's nothing about B that is always so and therefore there's nothing, right? Yeah, yeah. And notice when I argue that way, I'm kind of making use of what? If A is so, then B is so. So if it's necessarily so, then it must always be so, right? It's not always so, right? Okay, but now I'm missing another formula, aren't I? I'm saying if A is so, then B is so and if B is not so. From that form of an argument, does anything follow about A? Necessarily. If you admit that if A is so, then B will be so. I'm not saying that A is so, I'm not saying that B is so. But you are admitting that if A is so, then B will be so, right? And if B is not so, does it follow necessarily about A something? Yeah. But it's not as obvious as this first case up here, is it? So we use this first case to back up this case here, right? Because if A were so, then B would have been so, right? That contradicts this, right? So this is a sojism in... It's demisojism, if you want to call it that, right? Aristotle usually uses the word sojism for an argument with just two simple statements, right? And a simple statement is a conclusion, right? But sometimes we call this the... They call it the hypothetical syllogism. I don't like that word because it suggests there's something hypothetical about it and it isn't necessarily so, right? So I call it the if-then syllogism, right? You have the if-then syllogism, I mean the statement and then something either affirming that A is in fact so or affirming that B is in fact not so, right? And you have something following, right? But up here, you see, if something followed necessarily to write to B, it would be either that B is, what? Not so, which you don't think, right? Or B is so, which you probably wouldn't think, right? But those are the two possibilities. But can these both be true and this be false ones? And these both be true and this be false ones or true ones? You know, either way you want to do it, right? You have one where this is true, this is false ones. If it wasn't where this is true, this is false ones. And... If nothing about B is always so, then nothing about B is necessarily so, therefore you have no what? Yeah, yeah. See, if I'm using these forms over here, right? If it follows necessarily, it's always so. It's not always so, therefore it's not necessarily so. I use this to kind of disprove this here from being a syllogism, right? I have to find examples where both of these are true, so there's no problem when they matter, and all the problem is in the form. I'm talking about the form now, right? Okay. So, if I am a dog, then I am an animal. True or false? True, right? But I'm not a dog, therefore I'm not an animal. So this is sometimes not the case, right? I'm still an animal. Okay. Now, if I am a dog, then I am a four-footed animal. I'm not a dog, but I am. Am I a four-footed animal? So, can you have one where this is true? We'll have this example here with the example of dog and animal, right? If I am a dog, then I am an animal. I'm not a dog, but in this particular case, I am an animal, right? If I am a dog, then I am a four-footed animal. I'm not a dog, then I am not a four-footed animal. So, sometimes this is true, sometimes that is true. And all I need is one example for each, right? And then what have I shown? There's nothing you can say is always the case with B. It's not always so, it's not always not so, right? So, if nothing is what? Always so, then nothing is necessarily so, right? If something follows necessarily, then it follows always. But it doesn't follow always, as the examples show. Now, can you show that some form like this here is a syllogism by examples? I can show by examples that this is not a syllogism, right? But would examples show that something is necessarily so? Yeah, I mean, does one or two examples show that something is always so? No. Does a million or infinity of examples show that it's always so? I mean, I say, I maintain that a number is always odd, right? Three, five, seven, nine, eleven. I can go on for infinite examples. I can go on all day here, you know, backing up this, you know, with a large, yes, a large induction, right? Does that prove that it's always odd? Or I say every man is white. How many black men do you need to disprove that? How many white men do you need to prove that a man is always white? No matter how many white men you have, you've still been shown that it's necessarily so, right? So you have to, you can't realize that this form here is valid by examples, can you? You've got to see what it means to say, if a is so, then b is so. If you admit that, you lay that down, you're not saying that a is so or that b is so, which is saying that if the first is so, the second will be so, right? And then in the second statement, you're admitting that, in fact, the first is so. Well, then it's obvious, you must admit that b is so, right? And then we use this to show this formula, which is not quite as obvious, right? It's impossible that a be so and b is not so. Because then by this, b would have been so, and that positive x b is not so! Ta-da! Right? See? I'm not proving it by examples, am I? I'm showing this true university, right? Whenever a is so, b is so. So this is impossible when b is not so, right? But here I can disprove it by what? Examples, right? If it's so necessarily, then it's always so. It's not always so, it's not so necessarily. If it doesn't follow, it's the last case here now. Finally, b is so, right? A is so, then b is so. B is so, right? Now, I'm teaching my other students there from the other monastery there. At first they thought that a is so, right? A lot of people say that at first, you know, look, right? But is that necessarily so? Take more simple examples, huh? You find a dog, then I am an animal. If I am an animal? If three is an even number, then three is what? So, it could be so that a is so, right? If I am a man, then I am an animal. I am an animal? I am an animal. I am an animal. Therefore, I am a dog. No. So, my examples show that it's not always the case that a is so. It's not always the case that a is not so. So, no conclusion about a, there's only two possibilities, right? Either a is so or a is not so. No one of them is always the case, is it? If nothing is always the case, then nothing is always the case, then nothing is always the what? Necessarily so. If nothing is necessarily so, you ain't got a syllogism, right? And you haven't got any syllogism either. Okay? So, not too hard on, you see these four cases, right? But two of them are syllogisms, something called necessarily, and something else is not, right? Now, sometimes a person gets mixed up when they think of the matter, right? If this number is, what, two, then it's half a four. If this number is two, then it's half a four. If this number is two, then it's half a four. This number is half a four. Therefore, it's what? Two, yeah. But you're doing that by the matter, right? Because you realize that two and half a four can be turned around, right? So, if two is half, if the number is two, then it's half a four. And if the number is half a four, it's two. Because that's a probabilistic sense, right? But you know that by the matter, right? And sometimes a person will argue, you know, thinking of the matter, and it's justified by the matter, right? The argument, right? But the form is not necessary, right? By something other than the form, right? You're knowing that two and half of four are convertible, right? If not A, if A is not so, then B is not so. That's the statement. Yeah. The statement. Yeah. We're complicating that. Oh, okay. Okay. Just consider these four here, right? So, there's a question about this, huh? Well, with an either-or statement, you're going to read it in a different way, right? Either you eliminate all possibilities for a negative, or you eliminate all but one for an affirmative, right? Okay. That's simple enough, right? It's kind of common sense, huh? It's kind of common sense, huh? Okay. Okay. Okay. Okay. Aristotle uses the word sojism usually just for an argument that has two simple statements. Did you get it that far? Aristotle distinguishes three figures of a sojism with simple statements. Three is all there is. But stop and think, right? If you had a statement, you know, that a and b is going to stand for the simple terms rather than for statements, right? If you had, for example, every b is a and every x is y, is there any way you can reason from those two to any conclusion? Can you make any connection, affirmative or negative, of x and a in this one here? There's no connection between the two, right? You've got to have some term, either a subject or a predicate, right, that is the same in both, right? So, if I had something like this here. Every c is b, right? Well, then I can make a connection maybe between c and a, right? Every b is an a and every c is a b. What would you think follows necessarily? Yeah, every c is an a, right? This term that appears in both, this common term, because of the first figures, the first figure, they call this a, what, middle term, right? Okay? We used to compare it to the middle man in economics. Okay? So, he knows the producer, let's say, and he has contacted the customer, right? But the producer and the customer don't come into contact. So, I give that kind of, you know, for the monetarily. Business. Business. Finance. Yeah. Yeah. But now, you could have a middle term which would be, instead of the subject here and the predicate here, would be the, what, predicate in both cases, huh? Like if you have this here. Every a is b and every c is b, right? Okay? And this would be called the second figure, huh? Or it could be the subject in both cases, right? Every b is a, is c, right, huh? So, our style calls this position of the middle term, the first figure, right? First arrangement, right? And this here, the second figure. And this here, the third figure. Second figure, third figure, right? First figure. This the second and this the third, right? You might suspect that Aristotle is looking before and after, right? And there's a reason why he calls one the first, and one the second, and one the third, right? And we will see the reason for that in a moment, huh? But, you'll find out that the first figure is more powerful than the second. And the second is more powerful than the third. You'll find out that in the first figure, you'll find out that in the first figure, you can have both universal affirmative conclusions and universal negative conclusions. As well as particular affirmative, particular negative, right? In the second figure, you can have universal negatives, right? But no universal affirmatives. And the third figure, you can't have any universal conclusions. So, it's not by chance that he calls them the first, the second, and the third, right? But that there would be just three figures, right? He says, well, the middle term can be either, what? In between the major and the minor term, right? Or can be above them, as the predicate said to be above them. Or can be below them, right? Any other possibility? So, three is the first number above if you say all, right? So, don't make, you know, as long as you get crazy guys say, I don't have a fourth figure. Okay. You're playing checkers as much as you say. Okay. So, as you go through these three figures, we'll see that here you can get both, right? Now, what is the syllogism based on, right? Well, in Latin, they'll say the Dice De Omni, huh? And the Dice De, what? Nulo, right, huh? The set of all, I'll call it for brief. And the set of what? Of none, right? Aristotle will state the principle of the set of all and the set of none, like this. You'll say, if A is set of all B, then A will be set of whatever B is set of. And we saw there being the categories, didn't we, huh? If substance is set of animal, right? And animal is set of dog, then substance is set of what? Dog, right? Okay. And then the other principle is the set of none. If A is set of none of the B's, right? Then A is set of none of the things that B is set of, right? Now, grammatically, I sometimes turn around a bit, huh? And state it in the form of a what? If then statement, right? Instead of saying that A is set of all B, I say every B is an A, right? And I'll state it where I state these two here. If every B is an A, then whatever is a B is a what? Yeah. Now, is that obvious? Yeah. And all reasoning goes back to statements that are what? Obvious, right? Some statements that are proven are proven for other statements, right? But that can't be going on fabricated. If every statement, does it need a proof? Could any statement be proven? It had to be an add in the need, right? Add it in the need, right? Now, what we're going to see is that the set of all, that's plenty for the set of all instead of none here first of all. I like to state that kind of fun, but I just didn't state it, you know. I'll state the set of all in this way. If every B is an A, then is a B is also an A. So, if every B is an A, then whatever is a B is also an A. Have obvious, right? If every B is an A, then whatever is a B is not but an A. If every B is an A. If every B is an A. If every B is an A. Well, notice in the first figure here, right away you can see the set of all's in there, right? And if you had in the first figure, no B is A and every C is B, what would follow? No C is A, right? Make sense? Because in the first sentence you're saying no B whatsoever is an A. Now you're saying every C comes under B, therefore you must admit that every C is an A, right? If you were told just that some of the C's are B, those some C's would have to be what? A's, huh? Because whatever is a B is not an A, right? Just like up here too, if you had instead of every C is a B, just that some C is a B, you'd say what? Well then those some C's have to be A's because every B is an A and those some of those C's are B's. So those C's are B's must be A's, right? Follow? Okay. So right away you can see by the set of all and the set of none that you have that in the first figure, right? Over here. Yeah, yeah. Now if you could turn around every A is B and see every B is A, how can you necessarily do that? See if the matter was convertible you could turn around, right? If you just take anything in that form it might not be convertible. Every dog is an animal and every animal is a dog. It doesn't necessarily turn around, yeah. So you have to talk about conversion, right? Because in the second figure, in the third figure, right, it's not in the form that the set of all or the set of none can be applied to it as it now, what? Stands, right? And so you have to see, can you at all turn these around, right? So now we've got to go into conversion, right? Conversion is a word that's equivocal by reason, right? If every A is B, if every B is A, then can you turn that around? Not necessarily, right? Every dog is an animal, but not every animal is a dog, right? Every saint is a man, but not every man is a saint. But could you turn this around at least partially and say then some A is B? Or is it, not just possible, is it necessarily so? At least some of the A is B. But let's go over to something which is kind of interesting, huh? We saw that you can't turn this around simply and keep it universal, right? Necessarily, right, huh? So, not necessarily, right, huh? And again, how do I show that it's not necessary? One example. If it's necessary, it would always be so, right? But sometimes every B is an A, but not every A is a B. In fact, that's true most of the time. That's not. But just one example is all I need, right? Every woman is a human being. Do they refer to a human being as a woman? No. I need one example, right? So this ain't necessarily so when that's so, right? Because it's not always so, right? And now you get the thing. What if no B is an A? Then must no A be a B? Ah, gotcha! I'm going to show you're going to have to accept this here, right? If you don't accept this as being necessary, right? Then by the square of opposition, if this is not true, at least some A must be a B, right? This is false, right? We don't have to use it every A is a B, but at least some A is, right? So, if this is not necessary, then this is what? Yeah, this is possible, right? Now, if some A is a B, let's give a name to the A that is a B. We'll call it X, right? So, X is the A, and it may just be one of them, that is a what? A B. Well, then X is both a what? A B and an A. And therefore, there is some what? B that is an A, namely X. So, if you say that no B is an A, then you must say that no A is a B. That shows you, right? That the universal negative converts simply, they say, right? If no B is an A is true, always no A is a B will be true. Because if that's not true, then by the square of opposition, remember the square of opposition? Sum A must be a B. And let's give a name to the sum A is a B, even if it just wasn't all A, right? We'll call it X, son, whatever you want to call it. We'll call it a drafty one, whatever you want. But X is both an A and a B. Therefore, there is some B, namely X, that is an A. Which contradicts that, right? You've got to admit that if you admit that, right? The universal negative converts simply, they say, right? Because not only is it converted negative, but universally so, again, right? But the universal affirmative, what? Does not convert completely. Converts at least apart, right? Because if you said that it doesn't convert, we sure it doesn't convert necessarily to every, right? But then, at least sum A must be a B, right? Because if sum A is not a B, then what? No A is a B. If no A is a B, you can turn around, right? You can kind of do that, right? So, universal negative, huh? Much. It converts simply, right, and stays universal. And from that you can show that the universal affirmative at least converts partially, right? Because if sum A is, if you don't admit that sum A is a B, you're going to have to say that no A is a B could be so, right? And if no A is B could be true, then no B is A could be true. So, when every B is an A, no B is an A. Ridiculous, right? Okay? So, you must admit, right? You see that? So, we say that the universal affirmative converts partially, right? That's very important, right? Because that's the reason why in the second figure, eventually, you're going to be able to, what, syllogize the universal negative but not the universal affirmative. You can turn around the universal negative by imagining the first figure again. But you can't turn around the universal affirmative simply, right? And if you lose power, right, you know, you're going to put power questions out here. Right?