Introduction to Philosophy & Logic (1999)
33. The Syllogism: Set of All, Set of None, and the Sixteen Moods of the First Figure
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[0:00] of that, you know, to that reading, reading, become second nature to me now, you know. I thought it was your schedule or something. Yeah, just to read your own script.

[0:12] Okay. Okay, let's do the day of love. In the name of the Father, and the Son, the Holy Spirit, amen. God our enlightenment, guardian angels strengthen the lights of our minds, order and illumine our images, and arouse us to consider more quickly. Saint Thomas Aquinas, angelic doctor, pray for us and help us to understand all we should. In the name of the Father, the Son, and the Holy Spirit, amen.

[0:42] So today we're going to look at the form of the syllogism, which Aristotle simply calls the syllogism, right? And in a way, if you look at the definition of syllogism, um, it's speech in which some statements laid down, another follows necessarily because of those laid down. In the regular syllogism here, the conclusion is in no way really in the premises actually, right? It's, uh, in the power of the premises, right?

[1:16] Mm-hmm. In the case of the either or, it's already there, but not yet asserted, right? Oh, yeah. In the case of the if then statement, it's already there, but not yet said in fact to be so. Right, right. But here, when you have this simple syllogism, like an example there, every animal is alive, every man is an animal, but the conclusion every man is alive is not actually found in the premises.

[1:40] So the idea that the conclusion is another statement, right, from what's on the premises is more true in the syllogism than in the if-then syllogism or the either-or syllogism, right?

[1:56] Just seeing a nice either/or syllogism this morning in Thomas Aquinas. He's, he's reasoning that the relation of God to creatures is not a real relation, but a relation of reason. And he says, well, if it's a real relation in God, it either have to be an accident in God, or it have to be the very substance of God, right? Well, in the first book, the previous part, he showed there are no accidents in God, right?

[2:23] So if it isn't an accident, it'd have to be the very substance of God, and then his very substance would be towards the creature, and there'd be some dependence of God upon the creature there to be, and we'd shown in the first book that he's what's necessary to be through himself, right? So it's a beautiful way, you know. There are two possibilities, right? Either it's something existing in God, but not what he is, or is what he is, and it excludes both possibilities, therefore it can't really be in God.

[2:53] So... Okay. Now, um, we recall the definition of state of syllogism. Now, the regular syllogism, we take it apart. And Aristotle wrote two books, which in English are called the Prior and the Posterior Analytics, two books about demonstration, the kind of syllogism you have in geometry and elsewhere to some extent too. And, in the Prior Analytics, which means, you know, before, analytic means take apart in a sense, he Takes apart these arguments to see if the conclusion follows necessarily from premises, and then the posterior analytics he takes them apart again to see if the premises are necessarily what true, okay?

[3:44] And that's, as we said, is proportional to a man checking his calculation to see if he added or subtracted correctly or multiplied correctly, and then checking to see that he had in fact the correct numbers, okay? To add or subtract.

[4:03] So, when you take apart the syllogism, we call this the syllogism period. analysis, you always find that it has two what premises, huh? Okay. And if you take the premises apart into their subject and predicate, you find not four different terms, but what three, right? There's one term that is found in both premises, and then two that are found just in one, right? Okay. And that term that is found in both is going to what is if it's really syllogism, it's going to enable you to put together the other two, right, in the affirmative statement, or separate them in a what negative statement, huh?

[4:56] And so we sometimes compare the middle term to the middle man economics, right? That's inconvenient for, let's say, the consumer to get together in many cases with the producer, right? So there's a man then who has a shop, and he has contact with the producer, and he has contact with the what consumer comes into his shop. So he kind of what brings them together, right? Okay. Or like a matchmaker, right?

[5:25] huh? Maybe, someone who thought this man woman were just right for each other, and so he arranged to have a party at which everybody there was a married couple except for these two. So, for necessity, there they started talking, and they were very suited for each other, and in a week or so they knew they were going to marry, you know. They did eventually get married, and they married and had children.

[5:49] So, um, so he's what he knew the guy from his work, and he knew the woman, and he just said, "Let's just, you know." Okay. So that's what the middle term is a little bit like that, okay? But then sometimes too you have two guys fighting, and somebody comes in and what separates, you know, and separates them, right? And that's like the negative statement, right? You know, separate.

[6:11] This is not that, right? Okay. Now, that middle man or that middle term, as they call him in logic, he can have three relations to the terms, huh? He can be, as it were, in between the two, right? The subject of one of them, and the predicate set of the other, or he can be what the predicate in both cases set or denied of the other two, or he can be what subject, right?

[6:50] Okay. Now, when Aristotle does this, he actually uses the Greek. If you look at it, uses different letters for these three arrangements, and to example, that in English, you know, he said, "Well,

[7:10] in this first figure, as he calls it, the middle term is in the middle position, right? And then in the second figure, that middle term is the what predicate in both cases, huh? The predicate seems to be more universal than the subject usually. It's more apt to say man is an animal, than an animal is man, right? In the third figure, the middle term is what subject, Subject, right?

[7:41] Okay. But now we're sometimes accustomed, and we do it like in here to use A, B, and C for all three arrangements, right? But then we have to realize that the order of the letters doesn't necessarily correspond to the order of the what terms, right? It doesn't in the first, right, but not in the second and the third.

[8:07] So using the same letters, then to say you have this arrangement, and he calls that arrangement the first figure, and then you have this arrangement where

[8:24] the middle term. In each of these arrangements, we're going to be asking: Are these statements such that you can say something about C and A, with C as a subject and A as a what? Predicate, right? Is any conclusion? C is a subject, and A is a predicate. Can you see necessarily that every C is A, or can you see necessarily that no C is A, or can you see necessarily that some C is A, or see necessarily that some C is not A, right?

[9:11] Is there some statement, in other words, universal or particular, affirmative or negative, right? Being those four possibilities we have in square opposition, is anything that is necessarily so, because of these statements? Okay. Now,

[9:30] the complication arises because you've got those same four possibilities for each of these statements in this arrangement. You have every B is A, no B is A, some B is A, some B is not A, right? And that gives you four. What? They're down here. Every C is B, no C is B, some C is B, some C is not B. You've got four possibilities here and four there.

[9:56] How many possible combinations do you have in that arrangement? Sixteen. Yeah, not four plus four, it's, four times four, right? Okay. You've got four men and four women. How many possible marriages? Sixteen possible different ones.

[10:16] Each one of the four men could possibly be married to one of these four women, right? So sixteen possible possibilities, right? It's complicated, right?

[10:25] Okay. So in each figure, there are sixteen cases or sixteen moods, they call them sometimes, in the language of modes, to be considered and ask the question: Does anything with C as a subject and A as a predicate follow necessarily, right? Okay. And you'll find out in all three figures that in most cases, nothing does follow. Okay. Bad for most part. The good is rarely huh. Okay.

[11:06] And sometimes when they speak of these sixteen, they'll speak of four cases where you have two universal statements, right? And four cases where you have two particular statements, and then eight mixed cases where you have the first premise be. Universal second particular or vice versa, right? Okay. So we're going to be considering those. And of course, the most important ones to think about are the ones that have two universal statements.

[11:34] and they're not the only ones, we have a syllogism. Now, you might suspect when you reason out things, you have to start with something that is obvious,

[11:55] and there is a couple of obvious things that will be found in the arrangements that are in fact a syllogism. And sometimes we call these two obvious things the set of all and the set of none, which fits more Aristotle's grammatical way of saying these things. So let's take these two principles

[12:26] and ask yourself: Is this something needs to be proven, or something that's kind of obvious if you understand them? This is the set of all.

[12:41] If A is said of all B, then A will be said of whatever B is said of. If animal, for example, is said of all dogs, right? Then whatever dog is said of, animal will also be said of, right? Is that statement here obvious, or does it need a proof?

[13:39] Seems to be obvious. Yeah. Notice, huh? If B is said of something else, right, and that is not an A, right, then that B, because B is said of it, it is B, will not be an A, right? And that kind of gets what you're given here, right? So if A is said of all B, then A will be said of whatever B is said. If B is said of all C, then A will be said of all C.

[14:13] If B is said of some C, then A will be said of some C. Now you can state this another way. dramatically, it's more the way we're accustomed to do it now. But you could say, if

[14:29] every B is A, then whatever is a B is also an A. Now, if you understand what it means to say every B is an A, right, that's in the form of a universal affirmative statement, right? There's no exception, right? Every B is an A. If every B is an A, right, you have to just say yes. If every B is an A, then whatever is a B must also be an A, right?

[15:09] Again, that's just stating the same principle, just a little differently, formally. Again, is that obvious? Yeah. Yeah. You can see how how to deny that would be to argue this, right?

[15:27] Because if something is a B and it's not an A, then not every B is an A, right? Okay. So this is one of the two beginnings principles upon which the syllogism is based, right? If the set of all is found in the arrangement of the terms, right, then something follows necessarily, right? Okay.

[15:59] And you've got to recognize it right away. I mean, if someone has, for example, every B is A and every C is B, it should be clear right away that every what

[16:16] C is an A, right? It's obviously involves this principle here, right? If every B is an A, you told that in the first sentence, lay it down, right? Then whatever is a b will have to be an a. Since you're told all the c's, every c is a b, right? And every one of those c's will have to be an a, right? Now, if you were told only that some

[16:47] c is b, then you would conclude necessarily only what? The some c is a, right? That's rather kind of obvious, right? Okay. Now, in the first figure when we study that, you'll find out that if it is a syllogism, the set of all or its twin principle we're going to talk about here, the set of none, will be really obvious the way it stands. Okay. In the second and the third figure, it will not be obvious the way it stands.

[17:24] And therefore, Aristotle calls those syllogisms imperfect in sense; they're not clear. And you have to, even to get them to see what follows, you have to turn statements around. That's why we had to study conversion before we can do the second and the third figure. But in the first figure, if it's valid, you see the set of all or the twin principle, set of none, in it as it stands.

[17:46] And if either set of all or set of none applies as it stands, you can guess it's not going to be a syllogism. But then we're going to prove that's not a syllogism by examples, right? That will show that when the premises are true, nothing is always so, and therefore nothing is necessarily so. Let's take the set of none, which is the other principle.

[18:12] If a is said of none of b, then a is denied. Of whatever B is said, of. Okay? So if cat is said of none of the dogs, right, then cat will be denied of whatever dog is said, of, right. If dog is said of all cocker spaniels, then cat will be denied of all of them. Now it might be better to state the other way around because that's the form we're using more.

[19:03] So if no B is A, then whatever is a B is not an A, right? Okay. If no B is A, then whatever is a B is not an A. Again, is that obvious? Yeah. So in the first figure, if it says none applies, you'll see it right away. You have a statement like no B is A,

[19:43] and you will have either every C is B, in which case none of the C's are A's, right? Or you have some C is B, which case some C is not A. Okay.

[20:02] Now notice, huh? In all of these forms here, where you have the set of all instead of none, you have a universal statement, right? And then you have an affirmative statement, right? Placing something under the subject of that universal statement. Now, if you don't have a universal statement, you can't have the set of all instead of none, right? So in all those arrangements in the first figure, where the two premises are particular, you would guess right off the bat, right, that they're going to not be syllogisms if the syllogism in fact depends upon the set of all instead of none, because there's no universality, right?

[20:48] Okay. Likewise, if you have two negative statements, can you have the set of all instead of none? No, because even the set of none requires an affirmative statement placing something under the subject of the universal negative statement. So right away, you might guess, huh? That if there are two negative statements or there are two particular statements, it will not be a syllogism, right? But we won't just guess that.

[21:19] We'll use examples to show that it is not so, right? Okay. We'll talk about what you need in the way of examples. Now, how do you need the way of examples?

[21:43] You need two. you need three terms that will make Yeah the premises true. Well, take an example of this figure, okay? Okay. let's take this example here. Every B is A, and no C is D. Okay. Now, can you draw any conclusion with C as a subject and A as a predicate? That's the question they're asking, right? Okay. They customarily call the predicate conclusion the major term

[22:28] because the first figure tends to be more universal, and the conclusion, the minor term, and the term that's found twice but not in the conclusion is called the middle term. Okay. This is where the middle term is in the first figure. Now,

[22:53] I'll call this first statement the the major premise because it has the major term in it, and I'll call this the minor premise. Okay.

[23:09] Now, what you want to show is that there's nothing you can say necessarily about C as a subject and as a predicate when these statements are so right. In other words, you can't say necessarily that every C is A. You can't say necessarily that some c is a. You can't say necessarily that no c is a. And you can't say necessarily that some c is not a, right?

[23:41] So there's nothing affirmative or negative, right? Universal or particular, right? That you can say, right? So you've got to find examples that show that none of these is always true when these two are true, right? Okay. So you've got to find examples for a, b, and c that one satisfies the condition when you place them in this form, right? The statements are true, right? And one example where this is false.

[24:17] One example where this is false. One example where this is false. One example that is false. But now we simplify life, right? Instead of finding four sets of examples, right? If I find one example where no c is a is true, right? Then these two would both be what? False. Yeah. If I find one example where every c is a, these two would be false then, right?

[24:52] Okay. So what I have to do then is find examples for a, b, and c such that one, when I place them in this arrangement, the statements are true, right? And one of examples for every c is a, which makes both negatives false ones, right? And one where no c is a is true, which makes both what? Affirmatives false ones. Do you see that? Okay. So you have to kind of guard with that anytime it comes to second nature, okay?

[25:29] So notice, huh? When I look at this form, every b is a, no c is b, the first figure I say to myself Do you have the set of all or the set of none? Well, obviously you don't have the set of all because you have one A distinctly there, right? And the set of all involves one. Do you not have the set of none? Because the set of none says you have a universal negative statement and something comes under the subject of that universal negative statement.

[25:58] You've got a universal negative statement here, but nothing is said to be a c. So neither the set of all nor the set of none applies to this, huh? So now I suspect that it's not a what? It's syllogism, right? So now I'm going to try to find examples for a, b, and c

[26:22] such that this will be what? True. These two statements when I substitute my examples in. And I want one example where every C is A, and one where no C is A. I think of something obvious where every B is A. I'm just going to kind of second nature right here. Every dog is an animal. No problem, right? Animal for A, and dog for B. Every dog is an animal.

[26:54] That's true, right? Okay. Now I've got to think of a C such that every C is an animal, but no C is a dog. Can. You think of something that is always an animal but never a dog? Man, Man yeah. Or cat, or horse, right? Take cat, right?

[27:22] Now, can you think of an example of something that is also never a dog but never an animal either? Stone, right? Yeah. Now, I tell the students again and again. I say, you think you found an example that satisfies two conditions, right? Now, check it out. Yeah, right. So you say every B is A. Every dog is an animal. Yep, that's true. No C is B. No cat is a dog.

[27:56] Yep, that's true. No stone is a dog. Yep, that's true, right? I have satisfied what condition number one, right? With my examples, these premises are both true. When both of these are true, sometimes every C is A, right? Which means that sometimes both negatives are what? Not so, right? And sometimes no C is A. Which means both affirmatives are what false ones, right? So every possible thing, right, is not so always, right?

[28:38] Now, if nothing is so always, then nothing is so what necessarily. And if nothing is so necessarily, you don't have any syllogism, right? Notice the way I'm reasoning, it, I'm saying, if it's a syllogism, something's necessarily so. If something's necessarily so, it is always so. My examples show that nothing is always so. Therefore, nothing is necessarily so. Therefore, there's no syllogism, right? Okay. So this is what we're going to do in the 16 cases of the first figure, right?

[29:15] We're going to look at all 16 of them, and we'll ask ourselves, does the set of all or the set of none apply to them as they stand, right? If the set of all or set of none applies, it should be obvious right away. What follows necessarily? and if something does follow necessarily. If the set of all or set of none does not apply to them, then you suspect they're invalid, right?

[29:40] And then, like in this example here, you look for examples for A, B, and C that fulfill these two conditions. You substitute them in. The statements are true. In one case, every C is A. In the other case, no C is A. right? Now, when I, you know, give exams to students on these things, you know, I'll say to them, I'm going to check your

[30:12] examples to see if they fulfill those two conditions, right? Okay. But in a separate question, I'll ask you, why do those two conditions, right? Why do they suffice to show it's not a syllogism? And then you have to bring other things in, right? You have to point out that that the universal affirmative being so excludes the two negatives, right? And the universal negative being so at some time means universal the two affirmatives are never always so, right?

[30:43] Then you have to take a connection between necessary and always, right? Between necessary and syllogism, right? Okay. So I ask them then a separate question to see if they understand why examples fulfilling those conditions are sufficient to show this is not a syllogism. And then you have to go back and say, hey, if it is a syllogism, something has to be necessarily so. Something necessarily so would have to be always so, right?

[31:12] And then these conditions that I say the examples have to have show that nothing is always so, right? The fact that universal affirmative is sometimes So shows that both negatives are not always so, right? And the universal negative is sometimes true, right? Shows that two affirmatives are not always so, right? So there's nothing with "c" as the subject and "a" as the predicate. You can say it's always so; when these are true, right?

[31:41] There's nothing, therefore, that is necessarily so when they are so. See that? Okay. Now let's look at the 16 cases in the first figure.

[31:57] Now the most important ones are the universal cases. So let's put those in order. Every "b" is "a". Every "c" is "b". Notice this is the arrangement in the first figure under terms of a schematic form this way.

[32:27] No "b" is "a". Every "c" is "b". Every "b" is "a" and no "c" is "b". And no "b" is "a" and no "c" is "b".

[33:04] I ask myself in each case: Does the set of all or the set of none apply as it stands? And notice for the set of all or the set of none to apply, you have to have one a universal statement, and then secondly, an affirmative statement putting something under the subject of the universal statement. So let's look at this first one. Every B is an A, and every C is a B.

[33:31] You have a universal affirmative statement. Every B is A. And is anything said to come under the Bs? Yeah, every C is B, right? So obviously the set of all applies here, right? Every B is an A, so whatever is a B must be an A. We're told that every C is a B. So necessarily then, every C is A. So this is by the set of all, right?

[33:59] Recognize the set of all involved in it, right? Okay. Now over here, no B is A. Every C is B. Well, now you're going to have to try the set of none. Do you have a universal negative statement? Yes, no B is A. Does something come under the subject of the universal negative statement? Yeah, we're told that every C comes under the Bs. Well, now it follows necessarily that none of the B is.

[34:31] None of the Cs are A. See that? Instead of none saying, "Hi kitty, you want some logic want some logic? " So no B is A, right? Whatever comes under the Bs cannot be an A, right? You're told that every C comes under the Bs. So what follows necessarily is that no C is

[35:01] A, right? And this is by the set of none. Now every B is A and no C is B is a form that is apt to deceive people, right? But do you have the set of all or the set of none as it stands?

[35:28] Well, obviously you don't have the set of all because you have a negative statement there, right? But do you have the set of none? Do you have a universal negative statement? And something comes under the subject of that universal negative statement? You've got a universal negative, no C is B, but does anything come under the C? No. So you don't have the set of none, do you?

[35:49] So I suspect that this is not a syllogism, right? But now I'm going to try to show that by examples. Now I have to find examples for A, B, and C. And if you want, you could have, you know, two sets of three examples, right? But that's unnecessarily, you know. You can usually just, you know, have two terms for C and keep the same for A. and B Okay.

[36:17] So if you think of a B and A, such as every B is an A, huh? Let's say plant, right? And let's say three, right? Okay.

[36:32] Every tree is a plant. That's true, right? Okay. Now I've got to find two examples for C. And I've got to satisfy two conditions, right? None of them can be trees, right? But one of them has to always be a plant, and the other never a plant, right? So take them one by one. Can you think of something that is never a tree but still always a plant?

[37:00] Bush. Okay. Can you think of something that is never a tree but never a plant? Dog. stone. Okay. Let's say stone, right? Okay. Now, if I tell the students, "about nausea," you know, take your time. Now pause. Wisely and slow, as Shakespeare says, they stumble to run fast. Have I satisfied the two conditions? Every B is A. Every tree is a plant. Yeah, that's true. No C is B.

[37:35] No bush is a tree. Yeah. No stone is a tree. Yeah. Okay. Have I satisfied the second condition? Do I have one example where every C is A? Yeah. Right here, right? Do I have one example where no C is A? Yeah. Right there. Okay. That's enough to show it's not a syllogism. But I might ask the student a separate question: Why is that enough to show, right?

[38:00] What that shows in those examples is that when these statements are true, there's no statement with C as a subject and A as a predicate that is always so. Sometimes affirmative universal affirmatives are so, which means that the two negatives are not always so, right? And universal negatives sometimes so, so the two affirmatives are not always so, right? So there's nothing affirmative. Either negative, that is always so, and therefore nothing is necessarily so.

[38:33] Nothing is necessarily so. Syllogism, right? Okay. You see that? Now, you know, if you put this in matter where the premises are true, you know, sometimes I'll say to them this example here: Every mother is a woman, right? Okay. If you do something like that, let's write it down here.

[39:10] Every mother is a woman, and no man is a mother. The students will think it follows that no man is a what? Woman. Yeah. Every mother is a woman, right? And no man is a mother. Therefore, no man is a woman. Does that follow? No. But they're deceived in the sense because all the statements are true, right? It is true that every mother is a woman, and no man is a mother, and no man is a woman, right?

[39:44] And there seems to be some connection between the fact that no man is a woman and no man is a mother, right? But it doesn't really follow, does it? We've shown it this form this way, syllogism. But to give a person one of that matter, now if I had given them matter like saying every dog is an animal, no cat is a what? Dog. They wouldn't think that no cat is a what?

[40:09] Animal, right? But because it's true that no man is a woman, then they're deceived. The matter seems perfectly good, but it isn't so. I know a crafty logician. If I want the students to answer incorrectly, or show them that they don't really you know more logic, I'll give them this form where the statements were they're all true, right? It just seems to follow, doesn't it? Very common thing.

[40:42] Now we have two negatives. We said before that even the set of none requires an affirmative statement, putting something under the subject of the universal negative. So two negatives, you can get either the set of all or the set of none. So right away, I suspect this is not a what? Syllogism.

[41:07] So now I can find examples for A, B, and C. When you think of a B and A, such as no B is A.

[41:21] No plant is an animal. Yeah, this is an animal here. Is that plant? No plant is an animal. Now I got to find two Cs. Can you think of a C that is never a plant, but always an animal? No, it's a door.

[41:45] Can you find something that is never a plant but never an animal? Stone. Now we pause. We say if we satisfy the two conditions, what are the two conditions? That when you substitute in your examples for A, B, and C, the premises will be true. And secondly, you have one example where every C is A, excluding the two negatives from always being so, right? And one where no C is A, excluding the two affirmatives from always being so.

[42:26] Together, excluding every possibility, is that a affirmative or a negative, universal or particular? You can say about C as a subject and B as predicate. Well, no B is A. No plant is an animal. Yeah, that seems to be true. No C is B. No cat is a plant. No, that seems to be true. No stone is a plant. That's true. I have satisfied condition number one.

[42:51] Now condition number two. Professor Burks has to do this. I tell him, right? I mean, sure. Okay, have one example where every C is A. Yeah, every cat is an animal. Well, so far, that is a one example where no C is A. Yeah, no stone is animal. I have satisfied the two conditions. That's not a syllogism. So the four cases with two universal statements in the first figure, two of them are syllogisms. And two of them are not syllogisms, right?

[43:30] Okay. Sometimes people call this an invalid syllogism, but not really a syllogism at all. It's better just to call. it not a syllogism. Okay

[43:47] And we'll see eventually that this case here that concludes every C is A is the only one of all the 48 that you can conclude every C is A. It's going to be more than one to conclude that no C is A, but only one to conclude every C is A. This is very important for demonstration because you prove a property of its subject through the definition of the subject.

[44:13] We tend to use this form as our style of syllogism. Watch your habits. Aristotle says in the Posterior Analytics Okay you see that? Okay.

[44:24] When you get through all 16, we'll put the ones that are syllogisms back on the board, you know, all together, right? And we'll see some interesting things about them. That will be resolved. then.

[44:37] Now let's go to the opposite extreme and take the four cases where you have two particular statements. And as you might suspect, if you have no universal statement, you can have neither the set of all nor the set of none, right? Require true universality for the set of all and the set of none. So let's put down those four cases. Now you're in the first figure, and the first figure, your terms are in this arrangement, okay?

[45:07] So you have some B is A, some C is B. You have some B is A, and some B, excuse me, some C is not B.

[45:32] You have some B is not A, some C is B, and you have some B is not A, some C is not B. Now, if you like to do a lot of work, we could look for examples for each of the four, right? Okay. And four sets of examples. But if you're lazy,

[46:14] then maybe you can find examples for A, B, and C that will satisfy what all four examples, right? Now, how can I do that? Well, I studied a book called The Isagoge of Porphyry, right? Okay, and I've seen the difference between genus and difference and species and property and accident, right? And I know that accidents can what be present or be absent, right? Now, something will belong both to some and not to some, right?

[46:49] Okay. So let's take for A, let's take animal, okay, and for C, let's take something that's always an animal. Let's say a dog, right? Okay.

[47:13] And then something that is never an animal, but stone, okay. Now I satisfy one condition, right? Every dog is an animal, no stone is an animal, right? But now I need a middle term D such that some B is A and some B is not A, and that will work for both of these, right? And where some C is B and some C is not B, and that will work for what both of these, huh?

[47:42] You see that? Well, now I go back to Porphyry and say, let's take something accidental, right? Like white, okay, a white thing, right? Well, some white things are animals, some white things are not animals. This will work whether your major premise is some B is A or some B is not A, right? Okay. And some dogs are white, and some dogs are not white, so that will work for some C is B and some C is not B, right?

[48:18] And some stones are white, and some stones are not white. Ah, see, I saved myself labor, right? huh? Instead of four sets of example, just one set of example. That's all these things are worth, right? Okay. Now, again, huh? I pause and say, have I satisfied both conditions. Well, the major premise is either some b is a or some b is not a, right? And this is true, right?

[48:48] Some white things, are animals, some are not. And the minor premise is some c is b or some c is not b. Some dogs are white, some are not. Some stones are white, some are not. I satisfy condition number one. Not condition number two. If I have an example where every c is a, yeah, I'll circle that, right? Every dog is an animal. If you have an example where no c is a, yeah, underline that.

[49:17] No stone is an animal. So the fact that universal affirmative is sometimes the case excludes both negatives from always being so. And if the universal negative is true just once. That's all I need. It means that no affirmative statement is always the case. So there's nothing affirmative or negative that is always so. Those statements are true, right? If nothing is always so, nothing is necessarily so. If nothing is necessarily so, you ain't got a syllogism, right?

[49:54] You see that? Now we'll go through the eight forms that are mixed from the universal and the particular statement, right? And we'll consider those eight. at the time. Now we'll begin with the four that have the major premise, right? Universal and the minor premise, particular, right? Okay. So you have every b is a,

[50:39] and the second one is particular: some b is a, or you have some c is not b. Every b is a. Some c is not b.

[50:58] You have the universal negative: no b is a. Some c is b, and no b is a, and some c is not b. Okay. Now, which of those four do you know right away is not going to have the set of all or the set of none? It's not going to have the set of all or the set of none. Two negatives. Yeah, it can never have the set of all and the set of none with two what?

[51:38] negatives, right? Because even the set of none requires an affirmative statement placing something under the subject of that universal negative statement. Therefore, it's already a set of all. We never apply it.

[51:52] Now, in finding examples to disprove this, examples for a, b, and c. Remember that even if no c is b is true, it's still true that some c is b, right? Remember what he said When you say some c is b, we're not saying some are and some are not. We're saying some are not, right? So this is true whether the rest are or are not, right?

[52:23] So you think of a b and a such as no b is a. Animal. And what? Plant. Plant. Okay. Now think of two c's. A c that is not a plant, but is always an animal.

[52:47] Cat. Cat. Okay. Now think of something that is never a plant and never an animal either. Stone. Okay. So it's true that no b is a. No plant is an animal. It's true that some cat is not a plant, right? Because no cat is a plant. We took some of them; they would not be. And it's true that no that some stone is not a plant. I have satisfied condition number one.

[53:21] Okay. Something else you could know from this too, if I just a little induction here. If no b is a is false instead of true, right? What about no a is b? Would that be true? No, because then you just go backwards. Yeah, yeah. If universal negative is true, its converse is always true, right? So if universal negative is false, its converse will always be what? False. And there's an infinity of statements, where no b is a is the form of it, and the statement is false, right?

[54:10] And every one of those, the converse is necessarily what? False. If the converse could be true, then by the conversion we already showed here, the original one would have to be true, right? No. Isn't interesting, right? Universal negative statement is true, its converse is always necessarily true. If it's false, its converse is always necessarily what false, right?

[54:38] We know that for infinity of possibly true universal negative statements, and infinitely possibly false universal negative statements. Amazing, man. You yourself can do infinity, you know? You do that? Incredible.

[55:09] Now let's take the universal affirmative, huh? Every b is a. Now, if every b is a is true, given that every b is a, whatever b or a might be, it's true that every b is a. Is it necessarily true that every a is b? No. And one example where it isn't so, right? Is enough to show it isn't always so, and therefore it isn't necessarily so, right?

[55:41] So I take the example: every dog is an animal. Every animal is a dog. No. Every woman is a human being. Every human being is a woman. No. No. So the universal affirmative does not convert necessarily into the universal, right? But if every b is an a, can you say necessarily that at least some a is b? No.

[56:16] In some cases, both are true, you know, like every two is half of four, and everything half of four is a two, right? Okay. You have a thing in its definition, right? Every square is a quadrilateral, right?

[56:32] And every quadrilateral, quadrilateral is a square, right? If you have a definition, it should be convertible. Why was that definition? That's why Socrates will turn it around when somebody claims that a definition, right?

[56:47] If you ask him what is a dog, he says, "Well, my definition of the dog is a four-footed animal." Socrates would say, "Well, every dog is a four-footed animal, but is every four-footed animal a dog?" If you can't turn it around, it's not a definition, right? So if you have a definition, it should be able to turn it around. And if you have a property in the strictest sense, right, that belongs to every member always necessarily, like half a four to two, then you turn it around, right?

[57:15] But it ain't always so in that form, right? Because B might be something less universal than A, like every dog is an animal, every square is a quadrilateral, but not every quadrilateral is a square. So just one example is enough to show you can't necessarily convert universal. But can you say that necessarily some A is B?

[57:41] Well, again, say you can't look at all the cases, right? And see that, but you can reason to it, right? I say that whenever B is an A, some A must be a B. Now, if you say that that is not true necessarily, then by the square of opposition, that could be false, right? That some A is B. Then by the square of opposition, what would have to be true?

[58:09] No. Yeah, then no A is B is true. If this were false, right, which I say it can't be, if that were false even once, then no A is B is true, and then by What we saw before, no b would be a. So you're saying, when every b is a, it's possible that no b is a. Come on now. Is it possible, right? When every b is an a, no b is an a.

[58:39] But that follows. that you're not admitting that some a is b is necessarily true, right? If that could be false, then no a is b could be true by the square of opposition, right? But could that possibly be true that no a is b when every b is a? Well, by the first case, then no b is a would be true. Converse of that, right? That's no way is possible that every b is a, right?

[59:07] The two universal statements cannot both be true. They can both be false, right? They can't both be true. But if some a is b is false, then no a is b must be true. And if no a is b is true, we saw with the universal negative, some no b is a must be true. That obviously can't be true when every b is a. So we say that the universal affirmative, right?

[59:34] It converts partially, right? Okay. So it drops from what universality to particular, right? That's all you can say necessarily so. So that's going to be part of the reason why you're going to get only negative conclusions in the second figure. We'll see. Because you convert the universal negative, it remains universal negative. You might get the set of none right? But universal affirmative, you convert that, it loses its universality, and therefore you can't have the set of what all, right?

[1:00:13] You see that? Okay. Now, how about some b is a? If that's true, is it necessarily true that some a is b? Yeah. Yeah. We actually showed that in the first one. If that's necessarily true that some a is b, if some a is b could be false even once, right? Then no a is b would be what true by the square of opposition. And then by the conversion, universal negative, no b is a would be true when some b is a, right?

[1:00:56] Is it possible when some b is a is true that none of the b's are a? No. So something impossible follows, right? So the particular affirmative is converts again, right? Okay.

[1:01:17] Isn't that when you show the the universal negative converts? Didn't you didn't you use that conversion in that? No, it was the example, right? He said that

[1:01:35] you know you took the example of something that is both A and B, right? A given name to be A or say B, right? And something like that, right? You could probably show it this way if you wanted to, right? You know, I could say you know that if some B is A, some A

[1:01:55] is not B. But he shows it usually for the universal negative, right? That's very clear. Okay, but kind of implicitly you show that because you said we had an X, therefore we had a name that to be yeah yeah. I was just wondering. First, Aristotle shows the universal negative first. That's the most important conversion, and then to that he will show these two, right?

[1:02:28] Now, what about the particular negative? Suppose it's true that some B is not A. Can you turn around and say that some A is not B?

[1:02:55] Well, one example is enough to show you can't necessarily say that some animal is not a dog. Therefore, some dog is not an animal.

[1:03:08] Some number is not an odd number. Therefore, some odd number is not a number. One example suffices to show that it is necessarily so.

[1:03:20] So the particular negative does not convert, right? Now, this seems kind of unsymmetrical, doesn't it? Right? Because universal negative is the most useful for conversion, because you convert and keep universality. Particular negative is useless. You can't convert it at all, right?

[1:03:46] So one negative is is the best for conversion, and the other one is useless, right? Two affirmatives are kind of like in between, right? Then they will give you no universality when you convert but you can at least keep a particular, right.

[1:04:12] Now, in the second, the third figure, the arrangement of terms is never such that you can see the set of all and the set of none as it stands. So what you have to do is to see can I get the set of all and the set of none by conversion, right? And you have to kind of fiddle around, maybe you're not too adept at doing this at first, right?

[1:04:36] And then you might try to find examples, and if you have not seen how you could convert, you won't be able to find examples. But

[1:04:46] if you do find examples that satisfy the conditions, you know that conversion never gets anywhere. You've got to kind of figure it around Aristotle calls these imperfect syllogisms, right? Because they're not clear as they stand, and you have to do some manipulation and conversion of them.

[1:05:05] Let's look at that in the universal cases here of the second figure. Second figure, your middle term is predicated in both cases. It's in the second spot, as I might have to say, right? So let's put the four cases with two universal statements: every A is B, and every C is B, two universal affirmatives. And then every A is B,

[1:05:44] and no C is B. The universal negative is a major premise. No B, no A is B, and every C is B, and no A is B, and no C is B. So these are the four cases in the second figure where the Middle term is the predicate in both cases. It has two universal statements, right? Two universal affirmatives, or two universal negatives, or one affirmative and one negative, and two possibilities there, right?

[1:06:35] Okay. Now, how many of these are syllogisms? Has something to follow necessarily, and some not. Which one would you know right away is not a syllogism?

[1:06:49] Two negatives. Yeah. No way you get the set of even the set of none, not the set of all negatives, right? So right away I suspect this is invalid, right? So I want to find examples for A, B, and C that satisfy the two conditions. Okay.

[1:07:10] So can you think of A and C such that every C is an A? Animal. Animal dog. Yeah. No big deal. Okay. And then C is never an animal.

[1:07:32] Stone, let's say, right? Okay. Now I have to find a middle term B that's an animal, a dog, and a stone is never. Okay. Okay. Syllogism. Okay. Any children? Everyone, right? Now, again, I always pause and say if I satisfy the two conditions. What is condition number one? Well, the premises are true when you put in the examples, right? So, no A is B. No animal is a tree.

[1:08:09] I guess that's right. No C is B. No dog is a tree. I guess that's correct. And no stone is a tree. I'm satisfied. Condition number one. The premises are true when I substitute in. the... examples. Now, we have one example where every C is A. Every dog is an animal. That knocks out the two negatives. Shows they're not always true, right? And we have one where no C is A.

[1:08:41] Yeah, no stone is an animal. That knocks out the two affirmatives. It's not always the case, right? So there's nothing affirmative or negative that is always so. And therefore, there's nothing that is necessarily so. If there's nothing that's necessarily so, there's no syllogism. That's in the very definition. of syllogism, Okay. This is not a syllogism.