Introduction to Philosophy & Logic (1999)
42. The Dialectician's Tools: Probable Premises, Senses of Words, and Differences
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[0:12] Let's look a little bit more at dialectic. Aristotle, in dialectic, there, in the first book, he talks about the two arguments that the dialectician uses on. Because the dialectician uses not only the

[0:38] dialectical syllogism, but also uses induction. Now induction is considered more an argument of the dialectician than of the demonstrator, because induction by itself doesn't tell you that something must be so.

[1:00] You know, if you cut open a thousand frogs and each frog has a three-chambered heart, you would guess that all frogs have three-chambered hearts, right? But does it follow necessarily?

[1:15] That because I've cut open a thousand frogs and everyone had a three-chambered heart, becomes very probable, you might say, right? That all frogs have a three-chambered heart. but do you see it as being necessarily so?

[1:28] And I could produce, and I could produce a thousand numbers that are odd, right? And that doesn't make it what necessarily so that all numbers are odd, right? In fact, there's an infinity here of numbers, right, that are odd. This doesn't prove that they're necessarily odd, right?

[1:46] I think we mentioned before when we talked about induction that we always come to universal statements through induction, right? But in some, but in some cases, it's only through induction that we come to accept the universal statement. In that case, we have a kind of probability about it, right? But in other cases, it's not through induction alone that we come to see it, but through understanding better the what terms which is composed.

[2:22] So, in the case of the accidents, for example, we come to understand that a whole is larger than its part, right? Through seeing holes and parts and experiencing them, right? But we understand enough about what a whole is and what a part is to see that a whole must be what more than one of its parts, right? Because a whole as such has parts, right? So to say that it's no more than one of its parts is to say that it has parts.

[2:53] That's a real contradiction, right? Okay. And likewise, we might, you know, originally come to understand what an odd number is and what even number is, and that an odd number is not even through induction, right? But we understand enough about what an odd number is and what even number is to see that no odd number is what even, right? Okay. You know, by definition, right? This is impossible.

[3:27] So the dialectician has two arguments: the dialectical syllogism and induction, right? And you could contrast that with what you have in rhetoric, right? Because in rhetoric you have two arguments too, but they're the enthymeme and the example, right? Okay. Now dialectic and rhetoric have a certain likeness, the two of them, and in both of them, their argumentative part of rhetoric,

[3:59] enable you to make a what reasonable guess, right? Okay. But dialectic is an art for guessing about the universal, and rhetoric, in its argumentative part, and is an art for guessing about the singular in human affairs. We're not so much interested in the singular elsewhere, right? But we are interested in knowing whether this man is guilty or not, right? Whether we should go to war with this man or this country, right?

[4:30] Okay. And so enthymeme and example are arguments for guessing about the what singular contingent, right? But dialectical syllogism, and induction, are, arguments for guessing about something universal, like whether virtue can be taught, and so on. Okay.

[4:55] Now, in the first book of the book about places there, so-called topics, Aristotle distinguishes the four tools of the dialectician, okay. And as Albert the Great says in his commentary, and that this, in a way, is an art by itself, right? The four tools, huh? But then in the later books he gives places for using the where you can use the four tools, right? Okay. But in accordance with the idea of not trying to teach you the whole logic, but some basic things that you can use, right?

[5:30] Well, the four tools of dialectic are very basic here. Now the fundamental tool of the dialectician is the ability, the ability to select probable statements, right? Statements, right? The ability to select. Probable premises, huh?

[6:20] Of course, that involves recognizing what we mean by a probable statement, right, or a probable opinion, if you want to be more precise. You have the opinions of all men or most men, right? For all most men to give an art and the most famous, huh? Now that ability, as far as all most men are concerned, you're going to run into these things, simply from what living with men, right?

[6:47] But the opinions of all most men in an art or science, the most famous one, to some extent, you'll be exposed to these without searching these, without searching them out, right? You're Einstein or somebody quoted, right? But you might have to go what read some of these, men, right? Okay. I mean, some of the famous things like no, smoking is dangerous to your health. You're going to have a thing, or most doctors may be saying that, right?

[7:13] Huh? Okay. So you recognize this as a probable opinion, right? Smoking is dangerous to your lung. But? Aristotle speaks of dividing these probable opinions, right, according to the matter, right, huh? Okay. And in ways like dividing almost, let's say, the sciences, right? Some are opinions in natural philosophy, some in logic, and some in ethics or political philosophy, right? Okay. And that sounds kind of obvious, but you'd be surprised.

[7:52] how it's ignored, right? Huh? I remember after the close of the Second Vatican Council and so on, there are people trying to cash in now on the Second Vatican Council, so they're writing books that are in accord with the Second Vatican Council, so to speak. And this extended even down to philosophy, right? And so I was getting, you know, as you often get as professor, free copies of books I hope you'll adopt for your courses, right?

[8:22] And they have one on the philosophy of man, whatever that's supposed to be, you know, and one in ethics, and one something else, right, and so on. And you know, in this sort of ecumenical spirit, you know, you have selections from a bunch of different thinkers, obviously, right, who are thinking different things about the same thing, presumably. But anyway, in the one here, you've got a selection from Aristotle's three books on the soul, right?

[8:51] And then you have a selection from Karl Marx, who's talking about man in society, right? Well, that's kind of a false comparison, isn't it? You see, Aristotle's somewhere up there talking about the soul, and Marx is talking about man as he meets him in society. Well, no, you should be comparing Aristotle's politics to that, right? You know, Aristotle says man is by nature a political animal and talks about man in society, right?

[9:19] You know, in other words, you're making kind of a false dichotomy, right? You know, as if Marx thinks that man is a social animal or a political animal, right? And Aristotle thinks he something that has a soul. I think it's kind of a false dichotomy, right? If you just, you know, so Aristotle in this first tool, he's talking about selecting statements, you know, that are held by all or most men or all or most men in a given field are the most famous, right?

[9:51] But dividing them according to the what? Subject, yeah, the matter, yeah, okay. And there he distinguishes between the natural and those things that contribute to choice and avoidance or choice, right? Which would pertain to ethics. And then the ones that are pertaining to logic, right? Okay. And then the third thing he speaks of is what, and ordering them from the general to the what? Particular, yeah, yeah. And see, all of that enables you to select quickly, right, the opinions you want in a given matter, right?

[10:36] And the most basic ones, which influence all the other thinking of the person, because you've what, ordered them from the general to the particular. And especially if the man has a definition of something, right? As you know, that's very basic, right? In all his thinking, right? Okay.

[10:58] You think philosophy is in the search? Okay, that's his opinion, right? You see, but if you think philosophy is in the search, that's going to influence all the rest of your thinking, right? And I can be looking for demonstrations for one thing. You see, so at

[11:19] first, tool is the ability to select probable premises, right? But that involves recognizing what a probable premise is, and also dividing them and ordering them, right, according to the subject matter. Okay.

[11:39] Could you say a little more about that aspect of ordering them according to general and particular? Well, you know what general and particular means, right? Okay. So let's see, you're in ethics now, right?

[11:53] Well, what's most basic in ethics is good and better. What do you mean when you say something is good? What do you mean when you say something is better, right?

[12:10] That's more general than what do you mean by justice or what do you mean by courage or what do you mean by temperance, right? Okay. That's more general than what do you think of premarital sex or what do you think of, you know, something more particular, right?

[12:28] So what does a man's opinion about good, huh? Okay. Now, when we first define good, right, the first definition really of good is the good is what all desire, what all want. Okay.

[12:55] But Socrates, you know, in the Euthyphro, he teaches us how to ask a very fundamental question about such things, huh? The question Socrates would ask, you know, after we define the good as what is desired, what is wanted, right? Do You want it because it's good, or is it good because you want it?

[13:24] And of course, the answer you eventually see, if you proceed correctly, is that we want it because it is good. Okay. But there's all kinds of difficulties before you can see that's right. But now, if a man thinks that something is called good because we want it, that's going to influence all his thinking about good, right?

[13:48] Nothing in itself is good, or for that matter, bad, right? But we call it good because somebody happens to want it. There's all kinds of people running around with that idea about the good, right? But that opinion about the good is going to influence all thinking about these things, huh? Okay.

[14:11] And notice if something is good because you want it, then it's better or good because you want it more. Okay. Now, when I take that up in in in ethics, the first thing I show is that something is not good because we want it, right? Right. But that we want it because it is good, but it takes some time to show that. And then after you show that, right?

[14:44] Then you can syllogize that something is not better because you wanted more. Because it's not good because you want it. See? Okay. You always talk about something very simple. You say something is good because you want it, and everything you want was good for you, wasn't it? And I take always the example you're at the party, and the host says, "You want another drink?" And you say, "Yes."

[15:11] And after you drink this last thing, you go, "Ooh." I was the one too many, right? You've got to go. You're embarrassed, and so on. And you recognize now that what you wanted was not, in fact, good for you. It's not other people saying it's not good, but you yourself recognize that. Or the kid that wanted to drive his car 100 miles an hour down the curvy road and ended up, you know, in a tree, right?

[15:36] And he's still alive, you know. He's a wreck, you know. His car is ruined, you know. Even he might think it wasn't good for him to drive his car 100 miles an hour, right? But he wanted to do it, right? Okay. So there are very very obvious examples, you know, to show that something is not good for us because we want it, right? Okay. But then you have to, you know, do other things to show the opposite, right?

[16:00] Okay. But once you've shown that. Then you can show that something is not better because you want it more. So when he takes up Socrates' disagreement with Athenians in the in the Apology, right? You know, the Athenians want the goods of the body and exterior goods more than the goods of the soul, and Socrates wants the goods of the soul more than the goods of the body, exterior goods.

[16:29] Socrates thinks they're better, and these are better, and the Athenians think the goods of the body and exterior goods are better. I say a false solution would be to say, you know, well, Socrates, if you want those more, go for those. We want these more, so then go for these. But if you've already shown that something is not good because we want it, then you can't say it's better for us because we wanted more.

[16:57] The ball game is over. You've got to give a reason for it. Once you have to give a reason for saying the goods of the body and exterior goods are better, you'll find out how to give a reason for that. There's one fallacious reason you can give, which is not too hard to solve. But there's all kinds of reasons you can give for saying that the goods of the soul are better than the goods of the body, exterior goods.

[17:17] You see? So anyway, you know, you can find little thinkers who will say, you know, that the only reason we can call something good is that people actually want it. See? So that's that's an example of most general opinion about

[17:41] good, right? Okay. Or take the better now. When Aristotle takes up the better, there he says that the end is always better than what is for the sake of the end,

[17:57] and the whole is always better than the part. But those are very general statements about the better, right? Okay. So if you arrange Aristotle's opinions, you know, you start with the opinion about the good and about the better, and then you go down to, you know, his opinions about the chief good of man, right? And eventually his opinions about courage and temperance and justice and all the way down, right?

[18:27] Okay. Now in the same way in political philosophy, right? I mean, Aristotle begins by saying man is by nature a political animal, right? You know, he needs to live in society, right? He's not self-sufficient, and so on, right? He doesn't need society of other men. He's either a beast or a god, but not human being, right? And then you start to go down to more particular things, you know, the need for government, but then the various forms of government, right, so on.

[18:57] So the most common opinions are going to come up again and again and again, right? So when they have those. Down and you you know select those when you talk about these things, huh? Okay.

[19:15] So that's the first tool of dialectic, right? The ability to select probable opinions, right? And it requires to some extent the experience of of men in general, but also some experience of men in these particular things, right? Okay. So I'm always quoting Einstein when I talk about science, or quoting Mozart when I talk about music, right? But I have you know that might require some reading on my part, right?

[19:43] There's not everything maybe that Einstein said or Mozart thought and read it actually, right? Okay. But you tend to arrange these things, right? So I always keep in mind what Einstein says about the hypothesis, huh? That's freely imagined, right? Because that's a very general opinion. And Einstein has opinions about quantum theory in particular, right? You know, and so on, right? And it's good to know those too, but this first one is more useful to know because it's going to be brought up again and again in any hypothesis, right? Einstein says, creating that in theories like writing a novel, it says it's something you know much more magnitude than philosophy, right?

[20:37] It's closer to literature, really, than philosophy. It's not a means of knowledge. Now, the second, the third, and the fourth tools are presuppose this first tool, but they're very interesting, huh? Very interesting. The second tool is the ability to

[20:57] distinguish the senses of a word. Okay. Now, when a man makes a a a statement, it's always in words, right? If you don't have the ability to distinguish the senses of a word, you could what misunderstand the statement, right? Or it would not be so what clear, right? Okay. But likewise, you could easily become what fall into the fallacy of equivocation, the most common mistake in thinking according to the father of logic, right?

[21:43] Okay. Become a sophist in a way, right? Even knowingly or unknowingly, right? Because you'd be reasoning from words in different senses, right? To the current refutation of somebody, right? Okay.

[22:09] But at the same time, right, distinguishing the sense of a word might give you a way of multiplying probable opinion, right? So the good is desirable, right? But good can mean what the useful or the pleasant or the reasonable. Now you got three probable statements, right? The reasonable is desirable. the pleasant is desirable. the. Useful is desirable, right? Okay. Um.

[22:51] So there's many reasons why it's useful to be able to distinguish the senses of a word. Okay. For clarity of what you're actually saying, right? For what avoiding the equivocation, right? Okay. But it's also a way, in a sense, of multiplying sometimes probable statements, right?

[23:16] Now, when Aristotle takes up the second tool, he distinguishes a number of ways in which one can what distinguish the senses of words, right? Okay. Without going through all of them, the most basic ones are to look at the what opposite, right? Okay.

[23:38] Or to look at the word in combination, right? So you take the word like dry, for example, just let's do this. And the opposite of dry in cloths is wet, huh? Okay. But the opposite of dry in wines is what sweet, yeah? Okay.

[24:05] So when I say the cloth is dry, dry is the opposite of sweet, of wet, rather, huh? When I say the wine is dry, dry is the opposite of what sweet, right? So sometimes there's even different words for the opposite, right? Well, then clearly the word dry has more than one sense, right? But if you recognize, if you had one word over here for both of these meanings, right?

[24:32] And you recognize one of these opposites as having two meanings, right? Then the other one would have what two meanings, right? Okay. Um. What's the definition of genus, huh? What's the definition of difference? What's a name said with one meaning, right? Of many things, right? Other in kind, signifying what it is. That's what genus is, right? A difference is a name said of many things, said with one meaning of many things other in kind, signifying how they are what they are, right?

[25:13] Okay. I know, so that definition you got the word many, right? Okay. I suppose we're dealing with probable opinions here. Well, we got the opinion of Porphyry, right? Or the genus, right? Okay. But in that definition, he uses the word many, right? Now, does the word many have more than one meaning?

[25:45] If someone says he has many children, what would you think he means? Number. Yeah. What would you think of? Just two children? I think more than two, right? Is I have many children, right? Okay. Okay. Be a many chairs. Be a many chairs in my house. We have many chairs here, right? Maybe more than two, isn't it? Well, there are many as opposed to what few, right?

[26:19] Okay. You have a few days off. I got many days off, right? Okay. In the other sense, many is opposed to what one, one, one, yeah. See, see.

[26:41] Okay. Two different senses of many, right? So when genus or difference is defined by many, right? Which sense of many is it? Second, yeah, yeah. So sometimes trying to avoid equivocation, there I'll say more than one, right? Okay. Okay. So a genus always has many species, meaning it always has more than what one, okay? But it may have only two, like numbers divided into odd and even, maybe, or habit into virtue and vice, something, right?

[27:23] But all you need is more than what one, right? Okay. Or take another one from logic. We're in logic now, right? We distinguish sometimes between a nivocal word and an equivocal word, right? A nivocal word has one meaning, right? And an equivocal word has what many. Yeah. What does many mean here? More than one. Yeah. A word is equivocal if it has more than one meaning, right?

[27:54] Doesn't have to have five, six, seven, right? Okay. Some of them do, but doesn't have to have that, huh? So even when somebody, you know, we quote somebody's definition or we quote, you know,

[28:13] his opinion, it's in words, right? We've heard these words, right? But now we have to what, you know, sometimes clarify exactly what those words mean, right? And the words he uses are very often critical words, right? So what sense is he what using that, right? Okay.

[28:38] So you look at the opposite, and if that is more than one meaning, right? Or you look at it in combination, like you say, dry wine and dry cloth, right? Now, if you were to define what you mean by dry wine, right, and then drop off the wine from it, what's left would not be the same as if you define what you mean by dry what cloth, right?

[29:03] Okay. So these are places where that's why the book is called about places, right? Not topics in Greek, right? Even here in the second tool, there are places. Where you might look to see if a word has more than one meaning, you look at the what? Opposite, right? Let me take another example of that. A little more difficult,

[29:30] right? Take this word "liberal," right? Okay. Now you find the word "liberal" in ethics, right? In fact, as a virtue, we take up in ethics called "liberality, yeah. and we speak of liberal. Also, we speak of liberal education, right? Liberal arts, right? And then we have liberal in the sense of in politics, huh? In the sense of politics, politics is liberal, right? Okay. Now,

[30:07] is the word "liberal" equivocal in these three uses, huh? If I say Senate Kennedy is liberal, right? You know, and we have. uh. a Liberal arts college, whatever that means, you know. And then we speak of someone being liberal, ethics, right? What is the opposite of liberal, right? Well in ethics, the opposite of liberal is what? Miser. Huh? Miser.

[30:42] Yeah. Stingy, right? Okay. Stingy. Okay. What's the opposite of liberal in education? You speak of liberal education originally, liberal arts. What's the opposite of liberal? Something like servile. Yeah, yeah, yeah. Aristotle speaks of the liberal education opposed to a servile, against slavish. And so, you know, even as late as Shakespeare, I see, you know, in in in the editions they have of, Shakespeare, you know, they're referring to works of the time, right?

[31:22] And you know, he surveys the liberal and servile arts, common distinction, right? Okay. Now, in politics, we contrast the liberal with what? Conservative. What? Conservative. Conservativeness. Yeah. Yeah.

[31:46] Well, these three obviously are different things, right? So these have three different what meanings, right? The word liberal, right? But the liberal might believe that the conservative is both servile and stingy. Yeah, but there might be some, you know. Aristotle, in the second book of wisdom, there, he says some people don't like, you know, precision or certitude and argument. You know, being very careful about everything, they think it's kind of what?

[32:22] You know, the liberal arts, stingy or something, right? That's kind of, you know, almost metaphorical likeness there between the two, right? Shakespeare in the sonnets, you know, he speaks of niggard truth, and niggard there is another word for stingy, right? Okay. You know, you go to somebody's house for dinner, you know, and you praise the meal, even though it may not be too good, right? You see, or sometimes professors praise the students' paper, even though it's not very good, right?

[32:52] So we often tend to flatter people a bit, right? You know, where the truth, you know, would be like might hurt, you know. The truth seems to be kind of what, you know, stingy, right? Stingy in your praises, you know, niggard in your praise, no more than you deserve, right? Huh? So niggard

[33:16] truth is what for Shakespeare? How does he use that Yeah, yeah, As opposed to flattering, right, the sweet, you know, flattery. He speaks of as, you know. Preserving sometimes peace among people, right? Oh, okay, okay, so, so I'm praising you beyond, but you you really have, right? And he's saying, "Nigger truth, truth doesn't do that. Truth wouldn't flatter. " No, no, no. A flattery means you're praising them beyond, but it is, right?

[33:44] Yeah. Who's that famous professor now at Harvard? There, you know, he gives two grades. Oh. He's opposed to all this grade inflation, right? Oh yeah. That's rampant at Harvard, right? You know, being maybe below B or something, right? So he gives them the inflated grade, right? So that they can go on the record with the inflated grade, right? And then he gives them the real grade, so they know.

[34:05] that they really deserve. Ah, ah, ah, ah, ah. He can do that, right? He's very controversial, right? You know, he's. You see. If he gave his students

[34:19] what they deserve, you know, most of most monks, then you'd be out of business as a college. Retention rates would be no good, right? You see, I'm very just this this this grade

[34:35] inflation right now. When I first started out teaching in college, just at St. Mary's College in California, one of my colleagues was was very tough marker, right? You know, and I get called into the dean's office. You know, you having trouble with your teaching? You know,

[34:52] like you know, he's not getting the material across. The students are just giving them what they they deserve, right? But you know, you know, dean is protesting this, right? You know, so I mean, you always you got to to some extent compromise there, see? You know, so people, you know, so make that a job, right? You know. If it's proven inferior to your

[35:17] teaching that your some of your students have flunked in your course, we had a professor there when I was at the College of Saint Thomas in the old days. His name was Ross, right? And he taught constitutional law. Did I tell you what his grading policy was? He announced at the beginning of the course. In this course, he said A is for God. B is for Ross.

[35:40] That was his own. C is for geniuses. And D's and that's the rest of you. Remember, remember, you know, you know, these colleges, they they some professors, you know, kind of, you know, where you are compared to other professors in your grades, right? So you kind of, you know, see your outline like that. My teacher Casuric, right? He he gives his stuff, and he says, "He says I'm the next to the lowest teacher," he says.

[36:10] So he looked at Ross. He said, "I wonder who the other guy is." Yeah. Okay. So no, these have quite different meanings. So I think you know conservatives are very much interested in what liberal education, right? I think more so than liberals in politics, or I mean from my experience of them, right? And so I mean they're quite different meanings, right? And of course there's a joke about the liberal in politics, right?

[36:40] That he's he's liberal with other people's money. That's the joke. They say by the the true liberal in the virtues, obviously a man who's generous with his own money, right? Things, right? It's kind of funny to see that professors and good professors who are politically very liberal, right? And they want to you know obviously spend other people's money and you know tax industry more and give it away and wherever they want to give it away to.

[37:09] But when it comes to you know let's say faculty raises and there's all these different ways of calculating how the raises will be distributed, right? Then they vote exactly their their own pocketbook, right?

[37:23] I remember one one year, you know, one year where I mean they can do you know cross the board thing or they can you know try to even ranks and so on. And I happened to be in in two different things. One I was with you know advanced professors, right? And they might want like maybe not a certain financial increment, but a percentage increment, which would be more for them than for the guys in lower ranks because the percentage would be but same percentage would be more money for them, right?

[37:53] And then you go to the other one where you have professors, you know, more of lower ranks, right? And they were voting to to have it more at the you know certain amount of money, right? And so they're all voting their you know putting it in their pocketbook, right? And so not really being liberal with their own money, right? But

[38:16] so that's kind of a pun on the two different meanings of liberal, right? You see. And the liberal politics is when it's liberal other people's money. But three different meanings of the word.

[38:32] And so you got to be aware of that, right? You know. And sometimes we kind of conflate these these meanings together, right? Sometimes we say liberal education in the traditional sense, and then liberal education the sense of what befits a free man or something like that, right? And combine the two.

[39:00] That's the second tool of the dialectician. It shows you how close the budget is tied here to words, right? The opinions of all men or most men or all or most men in a given field and most famous, all these opinions are expressed in words, right? And words, for better or for worse, are for the most part equivocal, and so there's a lack of clarity there. If you can't distinguish the senses of these words, right, and there's a possibility of falling into equivocation.

[39:41] We'll get some more examples of that when we get down. We should stop and look at the exercise. Did you have a chance to look at the exercise? Let's just go over it and see.

[39:53] Dry set of wine and cloth is that unequivocal or equivocal? Yeah. Okay, it's obvious. Number set of two and three. Number set of two and one.

[40:13] That's Marcel, right? When when Euclid defines number, he says what? It's a multitude composed Of units, right? Aristotle says it's a multitude measured by the unit, right? Measured by one, right? And one is not a what? Or the unit is not, therefore, a number is not a multitude, right? Okay. Now that's the first meaning of the word number, huh? In daily speech, huh? A multitude, huh? So I would not say that I have a number of heads, right?

[40:52] Or that I have a number of wives, huh? Unless I want to be misunderstood, right? That I'm one of those guys down in Utah, okay? Getting investigated no more. There's still Mormons, right? Who have you know, a plurality of wives, right? Okay. So I would not say that I have a number of wives because I have only one wife. You see? And Shakespeare in the in the love poem there, Phoenix and the Turtle, he speaks of love as what?

[41:27] Eliminating number, right? Because love makes the lover and the loved in some way one, and of course one is not number. Okay. So that's original meaning of number. You still see it in daily life, huh? So if I have one child, I wouldn't say I have a number of children, right?

[41:55] Now even in Euclid, although he doesn't say explicitly, sometimes he uses number to include what? One, right? Okay. And the word number said of two and one is not purely what? Equivocal, because two is composed of one, right? And there's a likeness there because one has to two, a ratio like what? Three has to six, let's say, right?

[42:28] So there'd be reason to call more reason to call one a number than to call the point a line, right? Because a point doesn't have to a line, the ratio of a line to a line, but one can have to a number, the ratio of a number to a number, so it's something like a number, right? But now you're extending the word number, right? But there's some theorems in Euclid, you know.

[42:49] There's a theorem in the books on numbers where he says that numbers prime to one another are the least of all numbers that have the same ratio, and then that they measure all the other numbers that have the same ratio. Say, like for example, two and three are prime to one another. No other. number that measures them, and all numbers that are in the same ratio as two to three, let's say four to six, or ten to fifteen, they're all measured by two and three, the smaller by the smaller, and the greater by the greater, right?

[43:26] But he actually proves that right. Any numbers that are in the same ratios two to three, they're measured by each right. But then you see, what about this here? Two to four and

[43:43] six to twelve and so on. What are the least numbers in that ratio? One to two. Yeah. If you don't consider one to be a number, then that theorem breaks down, right? Because, well, I took a particular example there. Say three to six, right? Well, two doesn't measure three, right? But two measures six, right? But one measures of course every number, and two measures four, six, and so on, right?

[44:14] So the theorem would not be true unless Euclid's understanding by number, right? One to be a number, right? But now he's using the word number equivocally, right? But equivocal by reason. There's a reason why one might be called a number, right? As a certain likeness to a number, right? And also because a number is composed of ones, right? So the connection between one and number. And because one can have to a number the ratio that number has to a number.

[44:45] Okay. So most people don't realize that, right? Because we're so accustomed, the moderns calling one a number, right? But you can still see in daily speech that, you know, number means what more than one, right? Have a number of children. Do you have any children? They say I have a number of children. Everybody's going to understand that to be more than one, right? Okay.

[45:15] Do you have a number of heads? You have a number of arms, right? Number of legs, right? You have a number of heads too. Huh? You wouldn't say that daily speech, would you? I'm not sure. You know, speaking some kind of metaphorical sense, but you wouldn't say a number of heads like you have a number of arms, right? Number of fingers, right?

[45:39] You have a number of fingers, but only one thumb in a hand, right? There's a number of fingers on a hand, but only one thumb. Right. Not a number of thumbs on a hand, there. You see me? So that's the first meaning of number, right? It's always a multitude. You see that? So it's equivocal instead of two and one, but not equivocal by chance, right? There's a reason for it, huh?

[46:05] Number set of odd and even number. Equivocal. Equivocal, yeah. Yeah. Square set of four and nine. We never call. Square set of a number and a quadrilateral. Yeah. Then there's a certain likeness there, but it's still equivocal, right? Doesn't Euclid call, you know, what we call in modern math sometimes the factors of a number, you know? When you call like two and three the sides, it's six.

[46:40] Huh? Can you do that? The way of speaking. Yeah. What? I think so. Yeah, but it's not the same sense of the word side as when I speak of a triangle as being three-sided, right?

[46:58] Lines set a straight line and curved line. Yeah, because they're both length without what? Width. Lines set a straight line and line of argument.

[47:14] Yeah. Yeah. Okay. Limits set a point and line. Never call. Never call. Let's go on, huh? Okay. I have to define it. Extremity. Continuous set of line in the proportion four to six, six is nine. Equivocal. Yeah. You see, when they distinguish between arithmetic and geometry, they do because of the kind of quantity they consider, right? And as we learn in the categories of Aristotle, there's two kinds of quantity: the continuous, right, and the discrete.

[48:30] Louis de Broglie, the great French physicist, the father of wave mechanics, has a book on the continuous and the discrete in modern science. Very interesting. But the continuous has two definitions. Continuous is that which is divisible forever. And this is a definition Aristotle brings out in the sixth book of Natural hearing, the Physics. And then the continuous is that whose parts meet at a common boundary. So the parts of a line, you might say, the left and the right side are parts.

[49:03] continuous at a point and the parts of a surface meet at a line. Parts of a body at a surface, right? But? We can also show that the continuous is divisible forever.

[49:18] Okay. By numbers, they're not divisible forever. They're divisible down to one, right? And that's the end. And they're not continuous because in the number six, the two and the three, or the two and four, what do you want to say? They don't meet anywhere, do they? Okay. Right, in that see, that line, the point that's the end of the first half is the beginning of the second, right?

[49:46] So it's continuous, right? So there's an likeness to that when you have a proportion like four is to six or six is to nine, as opposed to this kind of proportion where you have, um, like two is to four, three is to six. You don't have any link there, do you? Okay. Two is to three as four is to six. Okay. But as soon as likeness, we speak of a what?

[50:13] Of a continuous proportion, right? Because you have the same number there. That's the end of one, right? Six and the beginning of the next. In that same way, I speak sometimes of continuous syllogisms. Or continuous definitions, right? The continuous syllogisms, are where the conclusion of one syllogism is the premise in the next one, right? Okay. Or you could say the definition of quadrilateral and the definition of square are continuous, right?

[50:46] Okay. Because what is defined by the definition of quadrilateral, namely quadrilateral, is in the definition now of what? Yeah. So reason now of knowledge is really an effect, you know, of continuous, well, is what syllogisms. So Euclid, you know, see, prove one thing and use that to prove something else, and so on, right? You know, define this and what is defined here, you use to define something else, right?

[51:15] So body set of a stone and a geometrical sphere. Equivocal, yeah. Even in different genera. One is in the genus of substance and the other in the genus of of quality. Healthy said of body and soul. Equivocal, yeah. Sick said of body and soul. Yeah.

[51:42] So they shouldn't say that somebody has a psychological problem, right? You know, naming the thing from the science that studies it, right? You see, sick in a soul, right? He's got a sick soul. Okay. Healthy set of body and diet. And diet.

[52:03] Yeah. But it's equivocal. These examples of healthy and so on. Equivocal by reason, right? Because the diet is a connection with the health of the body, right? So it's not purely equivocal. Convertible set of automobile and statement. That's obviously equivocal, right? Good set of pleasing and reasonable. That one's equivocal. Equivocal, yeah, yeah. So premarital sex is good in the sense of pleasing, but not good in the sense of reasonable, right?

[52:43] Yeah. So what is good, simply speaking, for a man? is what is reasonable, right? Okay. Good set of beautiful and useful. Equivocal. Equivocal, yeah. Yeah. See, good said of the end of the means, the same thing, right? Science set of geometry and logic. Equivocal. Science set of geometry and experimental science. Equivocal. No. Equivocal. See, geometry and logic are both a reasoned out knowledge. That's what science means there.

[53:26] They're an effective demonstration, but experimental science is not a reasoned out knowledge. It's not an effective demonstration. Because Einstein says, right, the hypothesis is freely imagined, right? Okay, and then you you test it, you know. But even if it's confirmed, it's not by syllogism. You are saying if my hypothesis is correct, there will be an eclipse of the sun at nine o'clock this morning. There is an eclipse of the sun at nine.

[53:53] Therefore, it's not syllogism, right? If A is so, B is so, B is so, therefore. But the more predictions you make that come true, and the more you know precise those predictions are that come true, the greater probability you might say, right, or possibility, right? But it's not really reason out. Not reason out of knowledge as logic and geometry are. So it's being used equivocally. huh? Seeing instead of the act of the eye and understanding.

[54:23] Yeah, and likewise, seeing instead of the act of the eye and imagining, right? You know how Hamlet says there, I can see my father now, and that'll come off of the ghost, right? And he says in my mind's eye, he means really in his imagination or memory, right? But Gregory the Great, you know, he says what?

[54:49] Anger disturbs the eye of the soul. What does he mean by the eye of the soul? Kind of metaphor for for reason, right? No.

[55:01] Dog said of what? Living dog and a dead dog. Equivocal. Yeah. Dead dog is not really. An animal anymore, right? It's like saying you know.

[55:24] House said of a house that's good condition, right? And a house said of a destroyed house. We call it destroyed house. Isn't it a house? It's a destroyed house. No, it's not really a house anymore. Huh? Any

[55:42] more than melted ice cube is an ice cube anymore, right? It's a melted ice cube. That's the kind of ice cube it is. If you insist upon calling it ice cube, you know you are really being equivocal there, right?

[56:02] So a dead dog is no more a dog than a melted ice cube is an ice cube. Okay. So you can see how this is important to separate different meanings of a word, right? Again, Aristotle has done a lot of that for us.

[56:24] So this is the second tool of the dialectician here: the ability to distinguish the senses of a word. Here's a different example of that. Other way you said about. Seeing the genera of which the word is, we have an example over here. We're talking about body, right? In the genus of quantity, right? We have line, surface, body, and then body in the genus of what substance, right?

[56:53] Okay, That was an example right there. So knowing the categories of Aristotle is a big help, right? Okay. Aristotle is distinguishing the meanings of the word disposition, right? And sometimes disposition is the name of a what quality, right? But sometimes it's almost the name of a genus by itself, disposition or position, right? But then sometimes they speak of of continuous quantities, a quantity whose parts have position,

[57:35] and sometimes they'll speak of quantities a disposition of matter. Okay. So disposition, in one sense, is its own category or position. In other sense, it's a species of quality. Another thing it pertains to the category of quantity, right? So obviously the word disposition is being used in three different words, right? In in in Greek and in Latin, the category for man in particular, being clothed and so on, right?

[58:05] That's called habit hexis in Greek, habit in Latin, habitus, right? So habitus can be its own category, right? If you have a monk's habit, right? That goes back to that sense of habit. But then habitus can be the sense of what a virtue or a vice that you have, a quality that you have, right? So it's a different sense, right?

[58:31] And sometimes Aristotle uses the word habit in the Greek word for anything is had, that you have in positive as opposed to a lack. Okay. So

[58:46] Aristotle is always careful when it's necessary to distinguish the senses of a word, right? And Thomas Aquinas is the same, you know. But the modern philosophers they they never do that. You don't know what they're saying. They're always using words in a new sense without ever bothering to distinguish them. I remember Warren Murray. That was a good example. He was trying to show that value isn't the same as good.

[59:14] So he said, "Well, what's the opposite of good? What's the opposite of yeah? Value is not the same. Okay. Now number three, the third tool Aristotle gives is the ability. To see the difference between things, to

[59:48] see a difference. This is something almost natural, right? So he's not going to give you rules for seeing a difference, right? Okay. But the thing he's going to point out, I think, is kind of interesting in his ability to see a difference, is that when you want to exercise this ability to see a difference, that the mind is exercised more in seeing the difference between things that are close together than seeing the difference between things that are far apart, right?

[1:00:26] Okay. And you might start with things that are further apart because it's easier to see, right? But you're exercising your mind, you want to get things that are very what Close right? Okay. So you take, for example, the Meno. Let's say, where the question comes up in that solution to the arguments to contrary sides, right? Contradictory sides. Well, can't a man be directed to the good by right opinion, right?

[1:01:00] And not just by knowledge, right? Okay. And so, you know, to go back to my simple example there, the fork in the road, right? Would I get to Boston just as well thinking that this is the road to Boston as knowing that this is the road to Boston? Right. But notice you're contrasting now, and then Meno comes in and says, "Well, there's any difference between knowing and having right opinion, right?

[1:01:25] Because they both seem to be, as far as action is concerned, just as good, right? " See? Why there's an enormous difference between false opinion, right, and incorrect opinion, and knowledge, right? See? But right opinion and knowledge, in both cases, you're thinking what is true, right? Okay. So they're much closer. So what is the difference between those two, right?

[1:01:51] The difference between knowing false opinion—that's obvious. One is true, one is false, right? But what's the difference between right opinion and knowledge? Well, knowledge you're certain, as Socrates says, right? Right opinion, you're not certain. What you think in knowledge is tied down, he says, right? And what you think in your right opinion, it's not really firmly tied down in your mind. And he goes on that way to explain what demonstration is, right?

[1:02:20] Because he says it's tied down by knowledge of the what cause, right? And so the kind of reason you give in a demonstration is the reason why something must be so.

[1:02:39] If you see why this must be so, then your mind is what tied down. What your mind thinks is tied down in the mind, right? If you don't see why it must be so, something might change your mind, as Socrates says, right? See? So if I start down that road that in fact is going to Boston because I think it's the road, but I don't know it is, right?

[1:03:01] I don't know it is, right? And some jokester says, "That's not the road to Boston. That's the road to Boston over there. " Oh. I might believe them, right? See? Or someone's put up, you know, a different sign, right? If I'm driving from New York and those buildings are no longer there, right? Well, it can't be New York, right? Something changed my mind, right? I never, even heard about this, you know, World Trade Business, right?

[1:03:29] Okay. And my example in class was you know your mother and father tell you what's right and what's wrong and you go to college and some smart professor convinces you that these bad things that your parents told you are bad that they're okay right and now you change your opinion right so you have to go to college where they'll help you know what you you believe right

[1:03:58] so this is very important obviously when you're asking whether two things are the same or different right they're really different knowledge and right opinion but it's also very important for what defining right okay for those two things