Introduction to Philosophy & Logic (1999)
31. Three Kinds of Syllogism: The Four Forms of If-Then Speech, the Meno and Euclid
https://berquistcourse.com/course/philosophy/1-philosophy-1999/031-three-kinds-of-syllogism-the-four-forms-of-if-then-speech/

[0:00] Um, we've got some things here. now. I taught logic. a long time now. so some of these things I copied and some I don't.

[0:14] Is it nobody wants to learn logic anymore? Well, no, no. there are people teaching logic. but first I was chairman, and that wasn't even on my schedule for logic. And then now, let me fill that too. Okay.

[0:31] I'm going to have copies here. How many copies you need? Two, three, four, five, six, seven, eight, eight, two extra, eight, oh, six, seven, eight.

[0:50] Stop. This is the first thing. And it's going to distinguish the three kinds of syllogism, right? One that Aristotle calls a syllogism, period, and then the hypothetical syllogism and the either-or syllogism, okay. And I talk about the forms of the either-or syllogism and the forms of the if-then syllogism, okay? Because they're fairly brief, right? Okay. But first, we'll be finishing the form and matter of the syllogism, right?

[1:25] And going into those forms, okay. Now we may have that extra we may not, but okay, Okay. Now this next one here you want to copy it again. This is the form of the simple syllogism, the regular Aristotle just calls it syllogism, period. That's going to be involved now. Okay, So four, five, six, seven, eight. I'll give you now but maybe we shouldn't read this the next time Let's read the first one I passed out.

[1:57] Okay, but okay, and okay. I mean, eventually we'll we'll look at a few simple theorems in Euclid, and you'll see, like for example in the sixth theorem of Euclid, the first

[2:30] book, that uses the the syllogism and uses the hypothetical syllogism and uses the either-or syllogism, right? Sometimes they call it categorical syllogism, like that categorical syllogism, the hypothetical

[2:51] condition, the if-then syllogism, and the disjunctive with the either-or syllogism, right? But even in a simple theorem like theorem six, the first one, you start to see it using all three, right?

[3:06] In the first theorem, say in Euclid, he uses just one kind of syllogism, right? The first theorem, in Euclid, is, is the one where on a straight line you're going to construct, a, what, equilateral triangle. And so he takes this end as the center of a circle, and this is the radius, and draws a circle like that, right? Then he takes this as the center of a circle, and this is the radius, and draws another circle, right?

[3:36] Then, when they cut, he draws lines from there to there, right? And he's got to prove that the triangle here is what? Equilateral, right? Okay. Well, he's going to prove that these two lines are equal because they are what? Let's give them numbers, A, B, and C. He's going to prove that A B and A C are equal. And what's the statement which he's going to reason to that?

[4:12] AB and AC are radii of the same circle, and all radii of the same circle are equal, right? Okay. So he has two universality there, right? All radii of the same circle are equal, right? A B and B C are radii of the same circle, therefore they're equal, right? Very simple, right? Okay. Then he wants to prove that A C and B C are equal, right? Okay.

[4:48] And it's exactly the same syllogism, right? Because they AC and BC are radii of the same circle, and radii of the same circle, are equal, right? That's that's what Aristotle calls a syllogism period. Okay.

[5:03] Now what remains to be proved? AB and AC are. Yeah, you have to really declare everything, right. Now how's he gonna prove that AB is equal to AC? You can't prove that in the same way. And then the second statement is saying that A is so, right? Well, then necessarily you must say what B is so, right?

[5:33] So this form here is a syllogism, and it's obvious, huh? Right, which I'm just going to say, right? Okay. You have to go back to something obvious because you can't prove everything, right? If you had to prove everything, you could prove nothing. Okay.

[5:51] Now, what about if A is so, then B is so. A is not so. Does it follow necessarily that B is not so? See, now a lot of times students will think that follows that B is not so, right? And as far as your imagination, you know, is something which we call in philosophy false imagination, right?

[6:27] False imagination is the main cause of deception on the side of the knowing powers. Now notice, huh? If A is so, then B is so. A is so, B is so. If A is so, then B is so. A is not so, then B is not so. It's imagination. It seems to what exactly the same, right? Doesn't it? Yeah. That's false imagination, right? Now, how do we show that this form here is not a syllogism?

[6:59] Just find the examples that that are contradictory. Okay. You got to find examples where, when you for A and B, right? Such that when you substitute them in, right? You get a false. Well, first of all, the two statements will be true, right? Okay. And you need one example where B is in fact so, and one example where B is what? Not so, right? Okay. Okay. Now, why do you need that?

[7:41] See. Well, if something followed necessarily about B, that's what syllogism requires, right? It's in the very definition of syllogism, that the conclusion follows necessarily. If something follows necessarily about B, it would have to be always so, right? Okay. In a sense, you see the connection there between necessary and always, right? And you can see that in simpler things than syllogism. If I say that.

[8:21] Two is necessarily half of four, right? Then two is going to be always what Half of four, right? But if I say that man is necessarily, let's say, white, if man were necessarily white, man would always be white, huh? So if you produce one man who is what not quite a black man, you've shown that man is not always right necessarily what right, yeah, because he's not always so, right?

[8:52] He's not necessarily not quite either, but so you got white man, right? So you need that, huh? So you think of examples for both, huh?

[9:11] If Socrates is a dog, then Socrates is an animal. Yeah, that's true, right? If Socrates is a dog, then Socrates is an animal, and it's also true that Socrates is what not a dog. In this case, Socrates is a what? Yeah. But if Socrates is a what mother, let's say, then Socrates is a woman. True. Socrates is not a mother, but he's also not any what woman, right?

[9:57] So it might be that B is so when B is true, like Socrates is an animal. It might be that B is not so, like Socrates is not a woman, right? Now that ain't enough

[10:12] for a what? Necessary for a syllogism, yeah. Because a syllogism means that something's necessarily so and therefore always so, right? So if it's sometimes so, sometimes not so, that isn't enough for necessity, right? If I say that man is necessarily white, some men are white and some are not. Forget about saying that man is necessarily white. You see that? Okay. Now, you can't show that it form a syllogism by examples.

[10:51] Because examples wouldn't show even that it's always so, would it? But you can't draw examples that alone is necessarily so, right? The same way with simpler things. If I say that numbers are necessarily even, see two, four, six, eight, ten, twelve, fourteen, sixteen. Infinite examples, all the time, where numbers even, right? But that doesn't show that they're always even, no matter how many I give, right? Let alone that numbers are necessarily even, right?

[11:23] So you cannot show that it forms if then speeches of syllogism by examples, but you can show by what? Examples that it's not. See the reason for that? But I can't show by examples

[11:45] that every two is necessarily half of four, always half of four, right? That counted every two, right? If I find one two that isn't, that would be enough to just

[12:09] Okay, so you see, that, this then is what? Not syllogism, right? And often people are taken in by that. I'll say to the students, bring the material. Say, if Socrates is a mother and Socrates is a woman, but Socrates is not a mother, therefore he's not a woman. That seems to them make sense, right? Because they know that Socrates is not a mother is true. They know it's true that Socrates is not a woman, right?

[12:43] And is not being a mother and is not being a woman have something to do with each other, right? But does it follow, right? Because he's not a mother, then he's not a woman. Well, then the the woman who in my class of students, you know, who are well, you know, not mothers either, it would not be a woman, right? Very strange conclusion, right?

[13:10] But if you give this, you know, with statements they know are true. Socrates is a mother; then he's a woman. Socrates is not a mother; Socrates not a woman. So sometimes think it follows, right?

[13:25] Aristotle compares that. well, he put that down. So you see the point? Okay. Now, this form up here: if A is so, then B is so. B is so. This is the one that deceives people the most. Most people on first sight will think that what? It follows that what? Yeah. And in the book called the Poetics, Aristotle says this is the way that Homer taught the other poets how to tell a what good lie, right?

[14:04] Okay. I mean, take that expected consequence being true as a sign, right? Of the truth of what went out before, right? If he's losing the argument, then you'd get angry. Well, that's sounds highly probable, right? He got angry; you must therefore what? Be losing the argument, right? We could get angry for more than one reason, right? Not because you're losing the argument, but because the other guy is just, you know, not seeing the obvious or denying the obvious or, you know, confusing the issue or something, right?

[14:40] Many reasons to get angry at somebody besides the the fact that you might be losing the argument, right? But often we'll we'll do that, right? Yeah.

[14:51] Okay. Um. Now, notice you have to disprove this to show that's invalid; it's not a syllogism. You have to do that again through examples, right?

[15:16] I know it, son. If this number is two, then this number is half of four. This number is half of four, therefore this number is 2. Ain't that good, arm done there? Now, in terms of the matter, there you could say,

[15:36] well, half of four and two are convertible. Half of four is a property in strict sense; it belongs only to two, to every two, and always. So, if this number is two, this number is half of four. This number is half of four; it's two. And you can back this form here, right? Well, if you just consider the form itself, this could follow from that, and yet that not be so.

[16:04] And there's an infinity of examples, But any kind of example B is more general, and A is like what species. Let's say B is a genus. If this is a dog, this is an animal. This is an animal. Therefore, this is a dog. We got this pet at home, Tabitha. She's a cat, or she's a cat. If Tabitha is a dog, then Tabitha is an animal. True.

[16:28] Tabitha is an animal. True. Therefore, Tabitha is a dog. No. So it might be that A is so. If Tabitha is a cat, Tabitha is an animal. Tabitha is an animal. Tabitha is a cat. Okay. If Tabitha is a dog, Tabitha is an animal. True. Tabitha is an animal. Tabitha is not a dog. So A might be so; it might not be so. When these are what true, right?

[17:02] I've given example of each. So there's nothing that is always true when those premises are true. You can't say that B is always so. You can't say that B is. Excuse me. A is always so. You can't always say that A is not so. right? Tabitha is a cat. Tabitha is not a dog, right? And if nothing is always so, then nothing is what necessarily so, right?

[17:32] And therefore, you don't have a syllogism because syllogism means something necessarily so as a term. Now, what about this fourth form here? If A is so, then B is so. B is not so, right? Does anything follow necessarily about A?

[17:57] That's not so clear as up here. So we show that A is not so necessarily through the more obvious case. You show what is not obvious to what is obvious, right? And we say that either A is so or A is not so, right? Now, if A were so by the first case, then B would have to be so, right? Necessarily. What you're given is B is not so, right?

[18:30] So the statement A is so, B is not so, and A is so. Those three statements are they compatible? No, because by the first case, statement one and statement three

[18:53] contradict statement what? Two. So you can't hold on to those three statements, can you? It's just like in calculating. You know, if I say that the the the length, let's say, is ten, and the width is two, and the area is let's say forty, you say I don't know which of those numbers is wrong, but they can't all be right, right? You know, because two of them would contradict the what?

[19:22] Third. Two times ten would not be forty, or forty divided by two would not be ten, right? Okay. So statement one, if A is so, then B is so, and statement A is so, they necessarily by the first case lead to conclusion B is so, or suppose against B is not so. So if you're given that B is not so to start with, that's laid out. Then B is so must be eliminated, right?

[19:46] You're also given that if A is so, then B is so, so you can't hold on to what? A is so. You must reject that, and therefore you must say that A is not so. Either A is or not so. You see that? Yeah. And often we reason this way, right? We'll we'll take a a statement we know is so and a statement that is false, at least it's something we know is false, and then we know that that the one statement that's unknown must be what?

[20:15] False, right? So if given as true if A is so then B is so, you're given that B is not so, then A is so could not also be true, because A is so and the if then statement would have led to B is so, but that's false. So the cause of the falsehood has to be that A is so. Okay, you see that then? Okay. So there's two forms of if then speech that are syllogisms, and two that are what, not.

[20:45] Okay. And so you have to become very familiar with those four forms, and you're going to find them repeated, right, infinity of times. If you read the first book of Natural Hearing, what they call the Physics of Aristotle, and he examines the argument of what, Melissus, right? And he argues thus: if being had a beginning, then being had an end. But being doesn't have a beginning, therefore it doesn't have an end.

[21:24] This is the way Melissus argues, right? If being had a beginning, then it has an end. There's some probability of that. But it doesn't have a beginning, therefore it has no end. He's using this form, which is not a what, syllogism. His conclusion doesn't follow, right? Because matter is bad too, Aristotle himself. But either defects enough to you know to overcome somebody's argument, right? His conclusion doesn't follow, or the premises and uses aren't true or even probable.

[21:57] And Melissus eventually is shown to have both defects, right? Poor Melissus But it shows the need for logic, right? The philosophers need logic, right? It's the last part of philosophy to be discovered, right? But they begin to see the need for it just now. Okay. Now, when I first explained this, and most times I go just using the letters A and B there, right? Now it might be if you wanted to follow the order of letters and the order of what, the antecedent and consequent, right?

[22:42] Then you could state this a little differently, right? You could say reason looks before and after, right? Now, if you have a simple statement

[23:00] which we'll call B, if you have a simple statement called B, if you want to prove that B is so by an if then syllogism, right? Would you look before B or after B? You look for some statement that comes before B that B follows upon, right? Like A, or would you look after B for a statement that follows upon it, C.

[23:30] Well, if you want to establish that B is so, you look before, right? You have something from which B follows, right? Now, if you knew or could prove that A is in fact so, then you could what? Syllogize that what? B is so, right? Okay.

[23:58] If you want to overthrow B, right? We're going to try to overthrow B. Then you want to look for something that follows upon B, right? Okay. So you look after B and say, if B is so, then C is so, right? Now, if you know that C is not so, or you can prove that C is not so, right? Then you can syllogize that B is not so, right?

[24:30] Okay. You can see how well the phrase of Shakespeare fits this, right? Looking before and after, right? Notice, you know, the kind of obvious before and after in syllogism. The premises are before the conclusion, right? That's one before and after, obviously. Okay. The word premise indicates that pre, before. But then, in the if-then statement, you have before and after because the antecedent is before the what? Consequent, right?

[25:02] Okay. But then you see that you look before if you want to establish the statement. You look after if you want to what? overthrow, it right? And so it represents the fact that I'm looking before and after B. I use A for what is before B and C for what is after, right? Okay. But that's to just tie it up with where you look. Okay. So these are the two ways you can reason, right?

[25:39] Now, let's take an example of this. Aristotle in the Metaphysics there he says that Socrates was trying to syllogize, and he gives us a sign that Socrates is trying to syllogize. That Socrates is trying to define, and definition is very much the beginning of syllogism, especially of what we call the syllogism period, right? The simple categorical syllogism. Now, if you look at the dialogue called the Meno, in the Meno, Socrates is asked by Meno in the beginning of the dialogue, "Can virtue be taught?"

[26:23] Right? And Socrates says, "I don't know." Furthermore, I don't even know what virtue is, right? Furthermore, I've never met anybody who does know what virtue is. And Meno says, "Well, I know what virtue is, right?" Socrates says, "Well, come tell me what it is, then, right?" And so Meno tries to say what virtue is. At first, he gives examples, and Socrates says, "Boy, every time what virtue is, right?"

[26:51] I ask you what water is, and he says, "Well, there's the rain, and there's the lake, and there's a faucet, and so on." You know, you give me examples, but not tell me what water really is. So then, finally, Meno tries to say what virtue is, and Socrates points out that his definition of virtue doesn't even separate the good man from the bad man. So by the end of that first conversation, which is an examination conversation, it's clear that Meno doesn't know what virtue is.

[27:21] Then begins the second part of the dialogue, where Socrates says, "Now, let's put our two heads together and find out if we can what virtue is." Right. At that point, Meno gives the sophistical objection: you can't look for what you don't know. Then Socrates tries to show how you can by trying to show that learning is what, in a way, recalling what you do already know. Okay.

[27:55] He has some difficulties there too, because on close examination it may appear that the slave boy is not really recalling how to double square, but coming to know it through things he already knows. But coming to know it for the first time, right? In fact, he's so far from recalling it in the beginning that he's actually mistaken as to how it's done, right?

[28:20] But then, in the third part of the dialogue, Meno still wants to know whether virtue can be taught, right? Even though they don't know what virtue is, and he's not willing to discuss what virtue is, right? My brother Mark says, "What do you do with a guy like that? You're going to find people in daily life, in the academic world, who just like Meno, right? They want to know this, but you have to know this before you can know that.

[28:48] And you explain that, and then you show them that they don't know this that you have to know beforehand. Okay, okay, okay. I still want to know this. Right? " So, a student came to me and he said, "I heard this Pythagorean theorem, you know. " I said, "I know the Pythagorean theorem. " I said, "All I know is Euclid's proof in Proposition 47. " Now, we got to do 46, argument 46.

[29:12] "Oh, I don't do that. 46, right? " You know, I'm sorry. You know, I see. It's illogical, right? But Socrates says, "Well, if you, you know, twist my arm, right? I'll, at least I'll discuss it, right? " But since we already know very well what virtue is, let's look on both sides and see if there's any reason to think that virtue can be taught, right? Any reason to think that virtue cannot be taught, right?

[29:43] So you're going to proceed dialectically, right? Reasoning from probable opinions, right? Even to contradictory conclusions. Now, the demonstrator never does that because he knows what things are, but the dialectician is on the way to find what things are. So, the dialectician very often and naturally adopts one of these two forms, right? It's sort of appropriate to his matter, right? Okay. So, Socrates' first reasons that virtue can be taught. And he uses what this form here, and then he reasons that virtue cannot be taught, and he uses this form here.

[30:28] The other two forms that we saw were not syllogisms. He doesn't quite use right? Okay. So he says, without knowing, he says what virtue is. What sort of a thing would virtue have to be in order to be taught? Well, it seems that we teach what we know, right? Okay. So if virtue is knowledge, then it seems that it can be taught, right? Okay. And that's not saying that in fact virtue is knowledge.

[30:58] It's not saying that in fact virtue can be taught, but it's saying if it is knowledge, then it seems it could be taught. as other forms of knowledge, right? Okay. So if virtue is knowledge,

[31:14] then it can be taught. Now that statement there is a statement that seems to be kind of obvious. on the very surface, right. Okay, it's knowledge that's taught, right? Okay. so if virtue is knowledge, then it can be taught. So the question is, is virtue knowledge, right? Okay. Now, Socrates is going to eventually say well, there's some reason to think that virtue is knowledge, right? He's going to have to show that or back that statement up by an argument, right?

[31:56] The first statement kind of stands by itself, right? Okay. Now, eventually you're going to give a reason for thinking that virtue is knowledge. If you can give a reason for thinking that virtue is knowledge, statement of fact, then you can syllogize that virtue what can be taught. Yeah. Yeah. I think that virtue can be taught. Therefore,

[32:32] virtue can be taught. Now, a rule you have to do when you're examining somebody's reasoning. Very often we start from their main conclusion, right? And then we see the statements that are next to the main conclusion. Huh? That these two statements here lead up to the main conclusion. Now, sometimes one, sometimes both of those statements are needed for what some proof or manifestation, right? In this case, the if-then is pretty clear because people know that virtue, that knowledge, is something that you can teach, right?

[33:13] So it seems right away acceptable to say that if virtue is knowledge, then it can be taught. But Socrates has to give some kind of reason to think that virtue is knowledge, right? Okay. And he uses, it seems, you know, what we'll call a syllogism period, right? And say that something like this to back it up. We'll say,

[33:41] ah, we take as kind of at least a probable statement that virtue directs us to good things, right? Okay. Right. Vice leads us to what bad things. Now that's something you might accept on the surface, right? Virtue directs us to good things.

[34:07] It would also seem to be probable that it's some kind of knowledge that directs us to good things, right? It's knowledge that directs you, right, to good things. So what directs us to good things is knowledge, right?

[34:33] So the medical art is a form of knowledge that directs us to good things like health, right? Economics directs us to wealth, and military art to victory, and so on, right? Okay. So at least this probability here, right, that what directs us to good things is knowledge, and it also seems probable that virtue directs us to good things, right? We all know that. Therefore, virtue is knowledge, right?

[34:58] Okay. That's another kind of syllogism backing up the what second premise of the main or chief syllogism, right? The main or chief syllogism is the one whose conclusion is the main conclusion, right? And then the backup syllogism, they call it the pro syllogism sometimes, but it's backing up the second premise, right? And sometimes we have to back up one of the premises in the second syllogism, right? Okay.

[35:26] But here both statements seem to have some probability, right? So notice Socrates is doing. He's looking before, the statement virtue can be taught, right? And what sort of thing about virtue from which it would follow that it can be taught, right? Well, it seems obviously the connection between knowledge and what teaching, right?

[35:55] We teach what we know, right? If we teach something to someone, we know something about what we're teaching, right? Then Socrates syllogizes on the opposite side. Okay. Now, here he has an if-then statement, which is, I think, somewhat weaker than the if-then statement in the first argument. But nevertheless, one that, if you stop and think about it, has some

[36:30] probability, right? He's saying, if virtue can be taught, right, then there are teachers of it. Okay. Okay. Now, why does that seem reasonable, right? Well, we modern dummies we have, you know, kind of a very narrow understanding of virtue, right? But the Greeks would say, well, look, the most obvious virtue is what courage, right? Okay. In fact, the word virtue is, you know, sometimes identified almost with courage.

[37:21] Virtue is a Latin word meaning what manhood, right? Okay. And the Greek word arete is like that. But this first off is a sense of courage, huh? Okay. Now, notice, huh? If the citizens of the city are not courageous, they're going to what? Be unable to defend the city. The city will be what? Taken by enemies, right? What do they do? They'll kill the men, and if they don't kill the wife and women and children, they'll enslave them, right?

[37:55] This is a horrible thing, right? To have your city what? Yeah. So courage is extremely necessary for the good of the city, right? So if virtue can be taught, you can be sure that there will be people what teaching it, right? That makes sense, right? Okay. And temperance and justice, huh? How can men live together? Without being temperate, right? If a man was around raping his neighbor's wife or his neighbor's daughter, right, then have turmoil around the city, right?

[38:29] If men go around robbing from each other and so on, and stealing from each other, then they have chaos in the city, right? Okay. So it seems that you know things like justice and so on can be taught. They would surely be taught, right? Okay.

[38:48] Now, the second premise would be, but there are no teachers of virtue. Now, how does Socrates, in the conversation with Meno and so on, and Anytus gets in there, gets very angry with Socrates. He's one of the guys who brings the cartoon appearance against Socrates, huh?

[39:27] Well, when they discuss are there teachers of virtue, there are two groups of men that are possible candidates to teach the virtue, right? And one is those men called sophists, right? Who are going around Greece, right? But they already have a somewhat unsavory reputation, right? Especially to the nobility like Meno and Anytus and so on, right? So in the conversation with Meno, Meno's would deny that the sophists really teach virtue, right?

[40:02] Socrates maybe too, but for other reasons, huh? Okay. But what about the great men of Athens, right? Well, now the great men of Athens were known for their virtue, for being courageous men and being just men and so on. And it seems that if these things could be taught, they should surely have taught their sons, right? But here is great men of Athens who were courageous, and their sons were cowards, right?

[40:33] Here is a great man of Athens who was just, and his son is a what thief, right? So if I am a sober man, I don't want my son to be a drunkard, right? If I am a just man, I don't want my son to be a thief. If I am a courageous man, I don't want my son to be a coward, right? So it seems that if these could be taught, these men would have had their sons taught these, right?

[41:00] But here are these great men that taught their sons how to ride a horse, right? They didn't teach them how to be just. They hired somebody to teach them how to ride a horse, but they didn't hire anybody to teach them to be just. Is it because it's more important to be able to ride a horse than to be just? It's more important to ride a horse than to be courageous?

[41:20] No. So it seems that there are no what teachers of it, right? Okay. And like I might say, you know, I can name the first teacher that I had for my children of the piano, and the second teacher who taught them piano, right? Now, if you ask me, and who did you hire to teach your children courage or temperance? Gee, I didn't hire anybody to do that?

[41:47] I can hire somebody to teach my children to play the piano, but is it more important they play the piano than they be just and temperate and courageous? Why did I pay money for a piano teacher and not a teacher of justice? Well, it seems like there's no what such teachers, right? Okay. So there's a certain probability, at least, to that statement that there are no teachers of virtue.

[42:10] I might mention that in the, I guess, in the Protagoras, right? There you see a little bit the other side, right, of that. And I think Protagoras says, you know, well, it's like who taught you Greek? Who taught you English, right? I can name maybe the first man I had as a teacher in French, but I can't really in English, right? Who taught me English? Well, to some extent was my mother, to some extent my father, to some extent my older brothers, maybe, right?

[42:41] And it was kind of diffused, right? Okay. And to some extent, my mother taught me justice, and my father taught me justice, and so on, right? Okay. But it's not, you know, like you go and hit somebody, right? There are, in a sense, teachers, right? Okay. But Socrates here in this dialogue, he just brings out the way which there seems to be no teachers, right? I had no teacher of justice for my children, only the piano.

[43:10] Is it because I I think that it's more important to play the piano than to be just? No, no, I don't think that. I think it's more important my children be just. But why didn't I hire anybody? Apparently, there isn't anybody. Okay. So now you're saying if A is so, use the second way, say if B. is so then C is so. B is so, then C is so, but C is not so.

[43:34] Therefore, what? B is not so. Okay. So in one case you're affirming the antecedent, and then affirming the consequent. Other case you're denying the consequent, and therefore denying the antecedent. Right. But you can't do the other two ways, right? You can't deny the antecedent, and necessarily deny the consequent, right? I'm sorry. I'm going to say, what about this? You know, if this number is two and this number is half of four, this number is not two.

[44:12] Therefore, it's not half of four. Well, you might say that, right? Because you realize that two and half of four are what convertible, right? But from the form, it isn't necessarily so, right? Um.

[44:27] If I'm a dog, I'm an animal. I'm not a dog, therefore I'm not an animal. No, isn't necessarily so, right? Um. Likewise, you can't reason from the what? Affirmation of the consequent to the affirmation of the what? Antecedent.

[44:54] Now, materially speaking, the two most common examples maybe are where something in the consequent is more general than the antecedent, right? Okay. Or where the consequent is an effect, right? Of a cause, but there are many what? Possible causes, right? The example I was given in class. They say if Burkus dropped dead last night, he would be absent from class today. Absent from class today, therefore he dropped dead.

[45:27] I say that's wishful thinking, but it's not logical thinking, is it? Now, in the experimental sciences, how do they confirm or test, for that matter, a hypothesis? Right? Well, they say if my hypothesis is correct, then such and such will take place. Right? They make some kind of prediction of the sort, right? And, if the prediction comes true, does that prove necessary that the hypothesis is correct?

[46:03] No. He's arguing the form I gave there, right? If Burkus dropped dead last night, he would be absent from class, right? Let's wait and see if he's absent from class. Ah, he is absent from class. But that same conclusion could follow. That same effect could follow from his car breaking down he did the other day. I didn't get up here, right? And so, okay. So you can't know it's necessarily that one because you're about right.

[46:33] Now Einstein, in the he said that Newtonian physics, he said, predicted so many things that came true that the scientists began to think it must what must be true, right? But from the very form of the confirmation, it never follows what necessarily that it is so, right? And Newtonian physics was kind of remarkable. The one that always sticks in my mind was they say that when they were examining more closely the paths of the planets that they knew about and the paths they followed weren't exactly what they should follow from Newtonian physics.

[47:11] Newtonian physics had been so successful, they said there must be what, other planets we don't know about that are interfering with the known planets, huh? And the mathematics is so exact they could predict where those planets would be, right? And sure enough, that's where they were. So by the time or before Einstein, theories of relativity, some of the physicists said this must be true, right? And then Einstein, with the theories of relativity, showed he could predict the same things that Newton predicted with a different theory, and in fact predict some things that Newton could not predict, right?

[47:49] And then he says it became crystal clear that we never know here, right? No in the sense of being certain, right? You know? That a scientific theory is, as he says, a system of guesses, right? But Pierre, I mean Claude Bernard, even in the 19th century, one of the fathers of modern physiology, said that doubt is always intrinsic to experimental method, right? That a hypothesis, no matter how many times it's been tested, can always be what.

[48:17] Tested again, right? Okay. It may always be the most possible thing to do at this point, right? But I mean, it might possibly, but be contradicted, right? Now, notice the more things you predict that come true, the greater probability you might say to hypothesis, just like an induction, right? The more frogs you find with three chambered hearts, the more probable it becomes that all frogs have three chambered hearts, right?

[48:47] But it never follows what necessarily, right? Okay. But notice there is more rigor in rejecting an hypothesis, right? Because there you are saying, if my hypothesis is correct, there will be an eclipse of the sun at ten o'clock this morning. There is no eclipse at ten o'clock. There you seem to have the form of the syllogism, right? Okay. So it seems unfair, right? That there is more rigor there in the what rejection of hypothesis than there is in the what confirmation, right?

[49:22] If what you predict by the hypothesis doesn't come true, then you say, well, the hypothesis can't be true, right? But what you predict comes true. Oh, maybe your hypothesis is correct. Maybe, you know.

[49:41] So there is certain weakness there in science, right? And as Einstein says, on the scientific hypothesis, he maintains it's freely imagined, right? It's not by reasoning that the scientist arrives at his hypothesis. He compares it to writing a novel. In fact, one place and

[50:15] read the crazy kinds of fiction that Einstein was reading at the time he invented the theory of relativity. One day. So the fact that it's freely imagined means that it has no justification until it's been tested, right? But when it's tested and confirmed its predictions, right, it's still what not sure, right? But but nevertheless, you can say this is better. But six is the inverse. Six says

[50:47] if the angles here B and C are equal, then the sides AB and AC are equal, right? Okay. Now, how does he prove that that is so, right? Well, he says first of all that if they are unequal, then one of these sides will be longer than the other, right? Now, it makes, as far as I'm concerned, no difference which side is longer. Let's say AB is longer.

[51:16] Well, then on AB, starting at B, we cut a line off equal to the shorter one AC, and starting from B, let's say it's BD, right? Okay. That's the third theorem there, which you cut a line equal to another one. Then you draw from D to C a straight line, right?

[51:41] Now you have two triangles, DBC and ACB, and these two triangles have an equal angle contained by equal what sides? BC is common to both of them, right? And the other side is DB, and let's say this is AC. So you have two triangles having an equal angle contained by equal sides. Well, the fourth theorem, which has been shown already, is that triangles having equal angles contained by equal sides are what equal.

[52:25] So that's a simple syllogism, right? Like the one we had there in the first theorem, right? Okay. So you say triangle DBC and triangle ACB are triangles having equal angles, namely at C and B, contained by equal sides DB, BC, and AC, BC, right? Therefore, they are equal, right? Okay. But it's

[52:54] not, right? Okay. So, but that follows if you say the sides are unequal, right? Okay. So you're going to have an if-then syllogism. If the sides are unequal, you can do all this, right? As a consequence, the part will be equal to the whole, but that's impossible. Therefore, they can't be unequal, right? And if they're not unequal, then they must be what equal. Because you have in a way either or, right?

[53:30] So the basic argument is either they're equal or they're unequal. Can he prove? He's going to prove by separate argument that they can't be unequal. Therefore, they must be equal. How does he prove that's either or syllogism, right? How does he prove that they are in fact unequal? I mean, they can't be unequal. Well, if they are unequal, then the part is equal to the whole. less than the greater.

[53:52] Okay? But then, to show that if they are unequal, the part is equal to the whole, he has to use a regular kind of syllogism. You see? So he's going to use all three kinds of syllogism in one demonstration. right Another example like that in Euclid, but we'll come back to that after we go through the syllogisms I'm just saying anticipation, right? You know, and this is kind of a very early theorem, right?

[54:24] Not a terribly unexpected theorem, right? You expect these sides to be equal at the end of the proof, but in the proof he uses all three. I just

[54:36] had some times Aristotle uses all three too, you know. But here in geometry those are the three kinds of syllogisms you'll be able to find. So that's the three that I teach, you know, and probably the only kind that you hear about.

[54:55] Now, the disjunctive syllogism is the one that is either or one, it's almost common sense, right? But there's a few things to realize about this, right? A couple ways you can reason from an either or statement, right? Because sometimes what you do, you reason from either or statement by eliminating all but one possibility and concluding to be what?

[55:22] A vermillion. Yeah. Other times you eliminate all possibilities, right? And then you deny the thing which is being divided, right? Okay. And the syllogism that Thomas gives there in the Summa Contra Gentiles. He says every name said univocally of two things is either a genus, a difference, a species, a property or an accident. And then he eliminates all five of those, right? Therefore, there is no name said univocally of God and creatures.

[55:55] But here, you eliminate one, right? And include him in one, right? Now, if he had three, it's common sense; he had three, he'd have to eliminate two, right? To get one, right?

[56:20] That's a little different, right? In the one case, you know that something comes under this division, right? But you don't know which member it is. In the other case, you're showing that something doesn't come under something because all the members can be eliminated, right? But in this case, two lines must either be like equal or unequal to straight lines, right? And so, but in the case of is of the name said univocally of God and creatures, we don't know that there is a name said univocally, right?

[56:54] And Thomas wants to show that there is a possibility, yeah. So, Gotta keep bringing God down. Yeah, yeah. So it's kind of common sense of either or, right? There's no doubt about studying logic, right? And we tend to do that. We don't even sometimes even stop to see what we're doing. But we are, in fact, arguing from either or statement, right?

[57:32] Thomas always does that. Sometimes he does that, Either or, but the if then is a little more interesting, right? And there are four forms of if then speech, right? And two of them are syllogisms, and two are not. And I usually put the forms up on the board when I teach this part of logic. I get four students in the front of the board, right? And ask them, you know, if something follows necessarily, write under the line.

[58:04] If nothing follows necessarily, write invalid, right? And always, some of you get one of them wrong, right? And sometimes you get all wrong You'll think something follows, when it doesn't. It doesn't follow, when it does, right? Now this is a sign that you need. logic, right? Your mind is in serious trouble. It thinks that something follows necessarily when it doesn't, or it doesn't follow when it does, right?

[58:31] You miss out on knowing something you could know, or you think you know something you don't know, right? And I say, all you have to do is consider those four. Cases, right?

[58:46] Which will meet them, you know. If A is so, then B is so. Always have to state it in this form: If A is so, then B is so.

[59:04] If A is so, then B is so. If A is so, then B is so. Now, sometimes the mind looks at the antecedent, the part, the if part, right? And you may find out that A is so, right? You might find out that in fact A is not so, right?

[59:34] Sometimes the mind looks at the negative, and you might find out that B is so. Sometimes you find out that B is not so, right? The question is:

[59:49] Does anything follow necessarily? In these cases, about what B, right? In these cases, does anything follow necessarily about A, right? I used to put those four forms up on the board, and

[1:00:06] what do you think at first sight? You know, just to see what syllogism in the beginning is part of logic, and actually one of these forms is syllogism, and we can say it's obvious, right? It follows. One of them is also syllogism, but not as obvious, but follows, right? We can show it to be obvious, form, it follows. And two of them are not syllogisms at all, right?

[1:00:34] But people are very often deceived, and one of them is not a syllogism, Aristotle. Says, in the Poetics is the way Homer taught the other poets how to tell a good lie. So,

[1:00:49] but you see, we use the either or. over and over again right. It's a very common form of argument. You find it all the way through the dialogues of Plato. you find it in Aristotle. you find it in Euclid, right? And I do these two forms first because they're very simple, right? When you get to the simple syllogisms, that's simple, because then you have to consider 48

[1:01:16] combinations. We'll see why, because the simple statements can be affirmative or negative, universal or particular. We have four possibilities, and then another variations too. We'll see further. So altogether, if you want to do the big number, 48 things to be considered. So I start with the either or. Well, it's in some sense the least important, and because it's common sense, right? You get the if then, people then tend to get kind of mixed up over that, right?

[1:01:48] And then the other one is, you know, the more things to be considered, right? That's what we call the syllogism. Aristotle calls that the syllogism, you know. And uh, we call it syllogism from hypothesis, you know.

[1:02:08] This is used very much in in dialectic right? You go back to the Meno, you see that Socrates uses the if then syllogism a lot. But he uses the either or syllogism, right? In there, he uses the syllogism too. And uh, I use the Meno's introduction to logic. if you have a chance to do that sometime,

[1:02:32] Yeah, so we'll see he uses these different forms in there, you know. But it's especially striking the way he uses the if then, right? If virtue can be taught, then there are teachers of it. But there are no teachers of it, therefore virtue cannot be taught. If virtue is knowledge, then virtue can be taught. But virtue is knowledge, therefore it can be taught. He's using that syllogism.

[1:02:52] right? But then he go back up, you know. If virtue is knowledge, but they, um, they syllogism, simple syllogism, regular syllogism, And uh, so. Yeah, so w- w- so that first reading then that I gave, you, right? Um, it talks about the form of the either or, which is kind of common sense. And then it'll talk about which of these are syllogism, right? So we'll talk about the form of the hypothetical if then syllogism, as I call it.

[1:03:29] And I say this, is something you can use a lot, because this comes up again and again, right? This explains, in a certain way, once you understand what we're teaching about this, why a scientific hypothesis is never known to be true, but remains, as Einstein says always a guess. It explains this

[1:03:52] because once you realize what once there are not a syllogism, you realize that the confirmation of a scientific hypothesis uses, what is not a syllogism. then you realize that it ain't necessarily so. Mm-hmm. Well, I think you get the data to put the hypothesis in right away. I guess. Yeah, yeah, yeah.

[1:04:21] So this. side is deathbed for having understood everything. That's interesting, huh? I like. I told you that story about this trying to understand something in Saint Paul and just kind of dismissed his helpers here, just down the floor. and prayed and wept, you know, and then he woke up. And now I understand it.