Logic (2016)
35. If-Then Arguments, Conversion of Statements, and the First Figure of the Syllogism
Class of 2017-03-16
https://berquistcourse.com/course/philosophy/2-logic-2016/035-if-then-arguments-conversion-of-statements-and-the-first/

[0:01] 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, order and illumine our images and arouse us to consider more correctly. Saint Thomas Aquinas, angelic doctor. Pray for us. Help us to understand all that you've written.

[0:28] Okay. Just 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, huh? 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, huh? So you have the either-or argument and the if-then argument, but fits in with what we were saying when we talked about looking before and after, right?

[1:10] 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 they are the most important ones for for reasoning, huh? Why the conjunctive statement like Berquist is a philosopher and Berquist is a grandfather, or Berquist is a philosopher and a grandfather, right?

[1:51] Those don't have much importance, you know, for reasoning, right? So when Thomas 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?

[2:22] 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?

[2:52] Yeah, that's kind of common sense, no problem, right? Huh? Okay. 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 to talk about the form rather than the matter, if you say if A is so, then B is so, right?

[3:21] 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?

[3:47] Was it three and four? No, two or three. Yeah, yeah, yeah. Now, so as you sit on the board for a second, if A is so, then B is so. And A here and B are standing for the simple statements, right? They're combined. Okay. Now, if you find out that A is so, right, that's kind of obvious that B must be so, right?

[4:27] 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 asserted that it's in fact so, but now that it is so, since you also admit this, you must admit that B is so.

[4:55] That's kind of the obvious form, right? Okay. Now, if you say though A is so, then B is so, and A is not so, by reason of the form of the argument, does it follow necessarily that what? B is not so.

[5:26] If you go to the matter, right, if A were convertible, if it's the number two, then it's half of four. It's not what the number two, therefore it's not half of 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.

[5:59] Still be less than ten, right? Okay. So as far as 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?

[6:35] So if you have one example where these premises are in fact true, right? But B is what could be so or 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 necessary notice when I argue that way, I'm kind of making use of what

[7:18] if A is so, then B is so. So if it's necessarily so, then it must always be so, right? But it's not always so, right? Okay, but I'm using another form here, aren't I? I'm saying if A is so,

[7:34] 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. You're not saying that A is so. You're not saying 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?

[8:15] A is so. 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?

[8:28] Because if A were so, then B would have been so, right? That contradicts this, right? So this is a syllogism in if-then syllogism, if you want to call it that, right? Aristotle always uses the word syllogism for an argument with just two simple statements, right? Simple statement, 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 isn't necessarily so, right?

[9:01] So I call it if then syllogism, right? You have an 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 followed, right?

[9:19] But up here now, you see, if something followed necessarily the fact 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 once? Can these both be true and this be false once or true once? You know, either way you want to do it, right?

[9:54] You have ones where this is true, this is false ones. If it wasn't where this is true, this is false ones, then Nothing about B is always so, and nothing about B is necessarily so. Therefore, you have no what. Yeah, yeah. See, if I'm using the form over here, right? If it follows necessarily, it is. It's always so. It's not always so. Therefore, it's not necessarily so.

[10:20] 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 in that matter, and all the problems in the form. I'm talking about the form now, right? Okay.

[10:40] So, if I am a dog, then I'm 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.

[11:03] Okay. Now, if I am a dog, then I'm a four-footed animal. I'm not a dog, but I am. No, I'm not. Am I a four-footed animal? No. So can you can you have one where this is is true? We had this example here with the example of dog and animal, right? If I am a dog, then I'm an animal. I'm not a dog, but in this particular case, I am an animal, right?

[11:42] If I am a dog, then I'm a four-footed animal. I'm not a dog. I'm 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?

[12:15] If something follows necessarily, then it follows always. But it doesn't follow always, as examples show. Therefore, right? Okay. Now, can you show that some form like this here is a syllogism by examples?

[12:36] I can show by examples that this is not a syllogism, right? But would examples show that something is necessarily so? It's kind of the definition of syllogism. Yeah, I mean,

[12:54] does a one or two example 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, the large, yes, a large induction, right?

[13:24] But does that prove that it's always odd? No, no. Or I say every man is white. How many black men do you need to disprove that? Okay. How many white men do you need to prove that man is always white? No matter how many white men you have, you still haven't shown this necessarily so, right? So you have to. You can't realize that this form here is valid by examples, can you?

[13:49] 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, but you're 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?

[14:14] And then we use this to show this form here, 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 contradiction is not so. Right? See, by not not proving it by examples, isn't it? No, I'm showing this true universally, right? Okay. Whenever A is so, B is so. So this is impossible when B is not so, right?

[14:42] Okay. 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. It doesn't follow what? What's the last case here now?

[15:07] You could find that B is so, right? A is so, then B is so. B is so, right? Now, I'm teaching about the students there at the Hutter monastery. There, you start at first and start A is so, right? Okay. A lot of people will say that at first, you know, look, right? Okay. But is that necessarily so? Take more simple examples, huh? You find a dog, then I am an animal.

[15:49] If I am an animal? If three is an even number, then three is four. No, okay. So it could be so that A is so, right?

[16:13] If I am a man, then I am an animal. I am an animal. I mean, it happens, isn't it? Yeah, yeah. Because of the example, but not exactly. I am a man, right? But if I am a dog, then 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.

[16:40] 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 a syllogism either, okay?

[17:10] So, not too hard, huh? To see these four cases, right? But two of them are syllogisms. Something followed necessarily, and something that follows does 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 of four. But it's not half of four, right? Or something, right?

[17:49] Therefore, it's not two, right? Okay. And that's just the one over here. Yeah, yeah, yeah. But now someone says here, if this number is two, then it's half of four. Let's wait here. I was talking here. If this number is two, this number is half of four. This number is half of four. Therefore, it's what? Two. Yeah. But you're doing that by the matter, right? Because you realize that two and half of four can be turned around, right?

[18:29] So if if two is half of if the number is two, then it's half of four, and if the number is half of four, it's two, because that's a proper sense, right? But you know that by the matter, right? Huh? Okay. And sometimes a person will argue, you know, thinking of the matter, and

[18:50] justified by the matter, right? The argument, right? But the form is not necessary, right? Okay. By something other than the form, right? You knowing that two and half of four are convertible, right? Okay. If you have uh instead of an affirmative and a negative. we have. If not A, if A is not so, then B is not so. That's the statement. Yeah. Yeah. We're complicating this. Oh, I.

[19:25] That's okay. Okay. Just to really just consider these four, right? Okay. So the question about this, huh? Why, with an either-or statement, you're going to reason a different way, right? Either you eliminate all possibilities for a negative, or you eliminate all but one for an affirmative, right? That's simple enough. Right? It's Kind of common sense, huh? Now Aristotle uses the word syllogism usually just for a argument that has two simple statements, right?

[20:15] No, you get it that all last time I remember. You didn't get that part. Okay. Aristotle distinguishes three figures of syllogism with simple statements. Three is all there is. But stop and think, right? If you had a statement, you know that now A and B these letters are going to stand for the what

[20:53] 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?

[21:15] 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 got to have some term, either a subject or a predicate, right? That is the same in both, right? So

[21:41] 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? Every C is an A Yeah, every C is an A, right?

[22:08] Now this term that appears in both, this common term, because of the first figure, the first figure, they call this a what middle term, right? Okay. I used to compare it to the middle man in economics. Okay. So he knows the producer, let's say, and he has contact with the customer, right? But the producer and the customer don't come into contact, right? Okay. So I give that kind of yeah for the monetarily business business finance.

[22:45] Yeah, 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 predicate in both cases, huh? Like if you had this here. Every

[23:06] A is B, and every C is B, right? Okay. And this would be called the second figure, okay? Or it could be the subject in both cases, right? Okay.

[23:27] Every B is A, and every B is C, right? So Aristotle calls this position of the middle term the first figure, right? Which arrangement, right? And this here the second figure, and this here the third figure. Okay. You can read my writing, of course. Second figure, third figure, right?

[24:04] It's called the first figure. Okay. 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, the one the second, and one the third, right? And we will see the reason for that in a moment, huh? Okay. But

[24:35] 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 can have both universal affirmative conclusions and universal negative conclusions, as well as particular affirmative, particular negative, right? Okay. 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.

[25:09] Okay. So it's not by chance that he calls them the first, the second, the third, right? But that there being just three figures, right? He says well the middle term can be either what in between the major and the minor term, right? Or it can be above them, as a predicate is said, to be above them. Or can be below them, right? Any other possibility? Just one or the other.

[25:41] Yeah. So three is the first number about which they all agree. So don't make you know, I just get crazy guys that have four figures, but it's a figure, okay?

[25:54] You're playing checkers, excuse me, I would say. Okay. So as we go through these three figures, we'll see that here you can get both right. Now, what is the syllogism based on, right?

[26:17] Well, in Latin they'll say the dictum de omni and the dictum de what nullo, right? The set of all, I call it for brief, and the set of what of none, right? Aristotle will state the principle of the said of all and the said of none like this. He'll say if A is said of all B, then A will be said of whatever B is said of.

[26:57] And we saw that in categories, didn't we? If substance is set of animal, right? An animal set of dog, then substance is set of what dog, right? Okay.

[27:12] And then the other principle is the set of none. If A is said of none of the Bs, right? Then A is set of none of the things that B is set of, right? Okay. Now grammatically, I sometimes turn around a little 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?

[27:41] And I'll state it. Well, 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? Okay.

[28:05] Some statements, in fact, most statements that are proven are proven by other statements, right? But that can't be going on forever, can it? Every statement doesn't

[28:18] need a proof. Could any statement be proven? It had to be immediate right? Immediate Okay. Now, what we're going to see is that the set of all. What's going on with the set of all and the set of none here? First of all, I have to state that kind of fun of that if then statement. I'll state the set of all in this way: If every B is an A.

[29:06] Then whatever is a b is also a what? An a. So if every b is an a, then whatever is a b is also an a. Kind of obvious, right? This is the set of all.

[29:36] And if set of none, if no b is an a, then whatever is a b is not what? An a. An A. But notice in the first figure here, right away you can see the set of all is in there, right? And if you had in the first figure no b is a,

[30:17] and every c is B, what would follow? No c is a, right? Makes sense, okay? Because the first sentence is 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?

[30:53] Okay. Just think 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. Those some c's are b's must be a's, right? Follow? Okay. Okay. So it's right away you can see by the set of all, the set of none, that you have that in the first figure, right?

[31:21] Over here. Yeah, yeah. Now, if you could turn around every a is b and see every b is a, 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. No, it doesn't necessarily turn around. Yeah. Yeah. So you have to talk about conversion, right?

[31:58] Because in the second figure and the third figure, right, it's not in the form that the set of all, 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? Okay.

[32:21] So now we've got to go into conversion, right? That's good. Yeah. Yeah. Yeah. That's a good topic for models. Conversion is a word that's equivocal by reason, right?

[32:47] So if every a is b, in the form of the statement, 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, okay?

[33:19] But could you turn this around at least partially and say, then some A is B? That might be possible. Or is it? No, not just possible. It is necessarily so.

[33:49] What do you think about that? Well, if every B is an A, yeah, A may be larger than what B is, but you could say at least some of the A's B. I think it's true necessarily. Yeah. 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 a universal, right? Necessarily, right, huh? Okay.

[34:16] So not necessarily, right? And again, how do I show that it's not necessary? One example. If it's necessary, it'd 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. It's not. But just one example is all I need, right? Okay.

[34:47] Every woman is a human being. Therefore, every human being is a woman. No. All you need is one example, right? So this ain't necessarily so when that's so, right? Because it's not always so, right? Okay. And now you get the thing. What if no B is an A? Then must no A be a B? No, not necessarily. Aha! Got you now. Now I'm going to show. He's going to

[35:37] have to accept this here, right now. 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? If this is false, right?

[35:59] You don't have to be every A is B. You got to be some A, right? So if this is not necessary, then this is what, yeah, okay, this is possible, right? Now 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 they just be one of them. It is a what, b.

[36:31] 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, which contradicts this. So if you say that no b is an a, then you must say that no a is a b.

[37:06] 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, or the square of opposition, some a must be a b. And let's give a name to the some a that is a b. It's just one little a, right?

[37:31] We'll call it x, huh? Whatever you want to call it. Call it a drafy one, whatever you call it. But x is both an a and a b. Therefore, there is some b, namely x, that is an a, which contradicts that, right? So you've got to admit that if you admit that, right? Right?

[37:53] So the universal negative converts simply, they say, right? Not only is it converted negative, but universally so, again, right? But the universal affirmative, what, does not convert completely. Converts at least in part, right? Because if you said that it doesn't

[38:22] convert, which would cancel convert necessarily to every, right? But then at least some a must be a b, right? Because if some a is not a b, then what? No a is a b. If no a is a b, you can turn it around, right? Yeah, you contradict that, right?

[38:46] So universal negative, huh? Ah, much, much. All right. It converts simply, right? It stays universal. And from that you can show that the universal affirmative at least converts partially, right? Because if some A is, if you don't admit that some 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.

[39:14] So whatever you B is an A, no B is an A. Ridiculous, right? Okay. So you must admit, right? Do you see it? So we see the universal affirmative converts partially, right? But it drops two. 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 universal negative, but not the universal affirmative. You can turn around universal negative, and you end up, like magic, in the first figure again.

[39:48] But you can't turn around the universal affirmative. And so, it's simply right. And if you lose power, right, you know, you're having for power questions over here, right? You know? Losing power, right? And so the particular has less power than universal, right? Particular statement. Okay.

[40:12] Now let's look at the particular statement. Now, right? If some B is A, you always turn it around and say then some A is B. You turn it around necessarily?

[40:42] Shaking your head. My first impression. First impression, no. Notice if some A is B is not necessarily so when some B is A, right? If some 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.

[41:24] Why? Can no B is A be when some B is A? So some A is B must be so, right? Okay. So the particular affirmative converts, right, and stays particular, right? Okay. And now the particular negative, right?

[41:47] If some B is not A, then what about the reverse? Can you say some A is not B? Does that follow necessarily? If some B is not A, then some A is not B. You can

[42:30] say, you know, you say if if some animal is not a what? Dog. Then some dog is not an animal. It isn't always the case.

[42:52] So this doesn't convert, right? So the universal negative stands out, right? Because it converts and stays universal, right? This converts, but it's particular and still is 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 stays particular, right?

[43:26] But the universal negative is better than the what? Universal affirmative, as far as conversion is concerned, because it necessarily converts and stays universal. And this loses power, right? Okay, the battery runs down.

[43:46] You see the point? So this is this is going to be the reason why in the second figure you can have universal negative conclusions, but not universal affirmatives, right? You can turn this around simply. You can have universal conclusions, right? Okay.

[44:09] Isn't that fun now? Thomas is commenting on one of Boethius's works, right? I guess the De Trinitate 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. You know what Thomas understands that is? Logic and wisdom. See, because in mathematics, right, at least you've got the imagination, right, so it's not hidden, right?

[44:47] 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 one way logic is more like what? Wisdom. In one way, natural philosophy is more like wisdom, right? Because wisdom is the knowledge of the first causes, and the natural philosophy is going further than the second causes, right?

[45:17] But in terms of universality, logic is more like wisdom. And in terms of being what? Immaterial, right? So Thomas calls that inward. I mean, oh, I mean the great Boethius. He's a great man. Calls it inward philosophy, right? The intima disciplines of philosophy, right? So he's referring to logic and to what? To wisdom, right? You know, inward philosophy. So I'm an inward philosopher. I'm also a natural philosopher and a geometry, right now.

[45:47] I get my two students there from. Another student now from TEC. And. Going to talk about you know Euclid don't think has the thing about inscribing a a oblong in a circle, but I think you can do that right, and you can circumscribe a circle around it. But you can't circumscribe a circle in a oblong, and you can't circumscribe a pentagon in a oblong around a circle.

[46:27] I don't see these guys agree with me, you know. But Euclid will have four things in there, you know, like the 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 in a circle. So you got four things, right? You can take each one of these figures, circle and square, you can inscribe and circumscribe around the other.

[46:51] But those other ones, he doesn't bother those. But the triangle can do something like that too. Amazing, amazing things. Okay. So this is conversion now, right? We're talking about conversion all these days, and you didn't know what

[47:11] it meant. Okay. But it's turning something around, right? And you can say now what universal affirmative, universal negative is is the winner of this prize, you know. It converts necessarily, right? And stays universal. The universal affirmative converts necessarily, but drops in power to particular.

[47:36] The particular affirmative converts and stays particular, right? But the particular what negative doesn't convert at all. It's not repenting. Yeah, yeah. He's a lost soul, right? Okay.

[48:03] If you look at Aristotle's Prior Analytics, right, he takes up the simple syllogism and the three figures, right. To begin with, you're going to have sixteen 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 what second, and four times four is what? Sixteen, right?

[48:32] Okay. So I say the guys tell the kids they got four guys and four girls. There's sixteen possibilities. Because each of the four guys can get married, and there's four girls, he can among which he could be a husband, right? And so four times four is sixteen possibilities, right, of marriage, right?

[48:57] But nowadays, if they can, guys marry guys, girls marry girls. It's a lot more guys. You're gonna destroy my whole, my whole, my whole experience here. Yeah. It shows the whole illogic. But I think right to you know, not too confused. Let's go through the four cases that have two universals, right? The universal moods, right? So the first figure,

[49:25] you have the middle term is the subject in the first one, and the predicate in the second one, right? Okay, it's in this slant position, right? Okay.

[49:48] No B is A, and every C is B, and No B is A, and every C is B, and then no B is A, and no C is B. These are the first four cases in the first figure, right? Cases where you have both premises be 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?

[50:44] In one case, the universal negative is the major premise, right? And the other case, it's. Oh my goodness! How can I be so so stupid? I thought

[51:09] before fall, right? Okay. So, 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, and one case it's the major premise, right? It's negative, and the other one it's the major premise is what affirmative, and the other one is

[51:43] the other one. Okay, that's clear enough, right? Aristotle is going to ask, does anything follow what necessarily, right? With C as a subject, right? And A as a what predicate? Anything you can say necessarily, right? With C as a subject and A as a predicate, or does nothing follow necessarily, right?

[52:09] Now, he's basing himself upon the said of all, and said of none of none, for a what valid syllogism, right? Now, 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?

[52:43] 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?

[53:09] A. So every B is A, and here you're told that every C is a B. Well, then what follows? Every C is A. Yeah, that's kind of obvious, right? huh? So Aristotle calls these syllogisms, incidentally, in the first figure, calls them perfect, because you can see in the way it stands that the set of all or the set of none applies; it does not apply, right? In the second and the third figure, you've got to kind of try to convert, and you know, so they're imperfect, you know, in some way, right?

[53:43] You have to rearrange before you really see it. Now 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?

[54:19] Negative, right? Huh? Okay. Now, what about this one here? No B is A, and no C is B. I used to say to students, You've got two negative parents; I can't have any children, right?"

[54:40] 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

[55:02] that satisfy two conditions, right? One is that when you substitute them into this form, these statements will be what? True. And you have one set of examples where every C is A, happens to be the case, and another set of examples where no C is A. Okay. 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?

[55:52] 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? Okay. 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? Okay. Now, if you like to do unnecessary work, you can find two simple examples for A, B, and C, right?

[56:33] But I'm an easy son of a gun, and so I like to find what same example for A and B, and just two examples for C, and then it satisfies my my exageration, okay?

[56:49] So let's take animal for A, okay? And let's take, let's say stone, simple example, okay? No B is A, no stone is an animal. Makes sense to me, right? Okay. 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 or cat or giraffe, or whatever. No C is B. It satisfies that, right?

[57:30] No dog is a stone, but every C is an A, right? Okay. And now let's take my friend the tree, okay? No tree is a stone, but no tree is an animal either. So these examples satisfy the 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 in the matter.

[58:04] 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 an A. 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. That's what would happen, okay?

[58:35] So I think I satisfy these two conditions, right? No B is A and no C is B are true with those examples, right? And I have one and second case. I have one example where every C is A, which means no negative is true always. Therefore, no negative necessarily follows, right? And then I have another example. Every no tree is an animal. No C sometimes is A.

[59:07] 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've now shown that this is not a what? Syllogism, right? Okay.

[59:27] But no, you don't see the the set of none in here, do you? You have a 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?

[59:49] You kind of have some affirmative to get any conclusion, right? You get some, my love. Okay. You see how I did that, right? Okay. And you'll find out in any syllogism or any figure rather, that two negatives give you nothing, right? Okay.

[1:00:09] If you have two negative parents, you have no kids, right? Okay. At least if one of them is affirmative, there's a possibility you may get children, right?

[1:00:23] Sorry, I spiced up my logic. Now, here I'm a little bit more hesitant now. you see, I, see, you certainly can't get the the set of all because you need two affirmatives for that, right? Every B is an A, but nothing's said to be a B. 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 got universal negative.

[1:00:52] Is anything said to be a C? Okay. Now I could turn around no C is B to what? No, I can I can try around. I just you know hope spring the truth and do the best. No B is C. Okay. And what about this up here? Well, no A is C. There's no C there, as you would say. You could say some A is B. I could say that some A is not C, right?

[1:01:35] You can do the reverse, right? But I wasn't asking whether you could make one is a subject and C is a predicate, right? I was asking whether you could say anything with C as a subject and A as a predicate, right? Okay.

[1:01:57] And here you're reading back in the form of the first figure, right? But you've got a you know A or C G G and C R A S. You're kind of just doubling your effort there, right? So we're saying is there any necessity that I'm mistaken with C as a subject and A as a predicate? That's the question I'm asking, right?

[1:02:21] Here you're just playing checkers, right? I'm seeing you the other way. Black and white, putting 'em around. Well, as it stands, I don't see the set of all, the set of none, right? So I try to find examples for A, B, and C that satisfy what two conditions? The way you did it over there Yeah? 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?

[1:03:13] No C is A, right? And I'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 is true necessarily, right? Right? Okay.

[1:03:42] So let's say animal, and then what? Let's say dog. I haven't studied simple examples. Okay. Every dog is an animal. That's true, right? Okay. Now I 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.

[1:04:15] And 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? Okay. Now I'm kind of stupid, since I make stupid mistakes, so I'm going to check to see if I've satisfied both conditions, right? Are the premises every B is A and no C is B true with those examples?

[1:04:41] That's the first condition, right? Every B is an A. Every dog is an animal. That's true. No C is a B. No cat is a dog. No tree is a dog. I have satisfied the first condition, right? You substitute this into this figure, you have premises that are true, right? There's no difficulty in the matter, right? But I'm concerned about the form, right? Okay. Now do I have an example where one example where every C is an A?

[1:05:21] Yeah. I'll circle that universal affirmative example, right? Now I've knocked out any negative statement being true always, right? If it's just one example where every C is an A, and cat and animal is one example, right? Then no negative is true always, right? And therefore no negative is true necessarily. with this form. And I also have an example where no C is in, right? Now underlying that.

[1:05:52] huh? That's I taught my students too. Okay. Ah, 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 at least once. There's no affirmative statement that's always so, is it? Right. Because no tree is an animal.

[1:06:21] That makes any affirmative statement from being necessarily so, being always so. And then from nothing being always so, you argue that way. I go back from the if then, right, and say if something follows necessarily, then it follows always, right? Okay. But nothing follows what? Always, right?

[1:06:47] That's that's necessary. That then is not necessary, right? It's necessarily not necessary, right? Fun things, right? Peter Piper picked a pickle pepper. Peter Piper picked a pickle pepper. Where's the pickle peppers? Peter Piper.