Introduction to Philosophy & Logic (1999)
Lecture 17: The Five Predicables: Genus, Species, Difference, Property, Accident

Transcript
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Okay, for the question again, name said with one meaning, many things, signifying something, we divided into two, inside of nature, and outside there. We subdivided inside of nature into three, right? The genus, the species, and the difference, right? Okay? Now, the outside of nature, they divide into two. And again, by a complete division, either something that is connected to nature and follows upon nature, like an effect that follows upon the thoughts, right? Or something that is not connected to nature and doesn't follow upon it then. So, we could say, either it follows upon the nature, it does not follow. And the first is called, in magic, a property. In Greek, the Greek word is idiosum. And it does not fall upon the nature, they call that an accident, in Greeks, in bebekos. That's, both in Greek and Latin, it has the word for happening, right, then, by an accident. Now, when Porphyry defines property, he'll define property in the strictest and fullest sense, right? And that's the property that belongs to only one species, right? To every member of that species, and always. So, it belongs to the one species, to every member, and always. But you can use the word property in a less full sense, providing still its connection with the nature. Now, most properties, it's hard to find one name for them. Usually, you have to use a speech, you know? The standard textbook example of one name is that man is visible. And that means that man is, what, subject to laughter, right, then. And this is not the nature of man, but follows upon his nature. Because he's an animal, he can make the sounds of laughter. But because he has reason, he can see the absurdity of something that makes him laugh. But usually, we lack a name for most properties, and we have to fall back upon any, what, a speech, right? Okay? It's still a function, I guess. So, it's a property, for example, of the number two to be, what, half of four, right? Now, half of four is expressing the relation of two to four, right? It's not what two is, but something followed upon two, right? In other words, because it's two, it's half of, what, four. In the third of six, in the fourth of an A, you can be. But all these things are followed upon as being two, right? Now, to be half of four is a property of two. In the strict sense, it belongs to only the number two. The other number is half of, what, four. And every two whatsoever is half of four. And a two is always half of four. Now, if you said that two is, let's say, less than five, and that would be connected with the nature also of two, right? It belongs to every two to be less than five. And two is always less than five. But not only is two less than five. Four are two, right? But still, we classify as a property because it seems to be, what, connected with the nature, right? The most common example in geometry of a property is that the triangle has its interior angles equal to what? Yeah. And that belongs to every triangle, and always, right? But as you learn geometry more, you realize it belongs only to the triangle. If you had another side, you're going to have more than that. Because we have a, let's say, quadrilateral, you can always, what, divide that into two triangles and we end up with more right angles than just two. So, to have the interior angles, you get two right angles, belongs to only the triangle, to every triangle, and always. Now, if you say it's a property of man to be a magician, well, man is the only animal that is a magician. But it doesn't belong to, what, every man to be a magician. And even those strange ones who are, they were not always magicians, right? Okay. Yet we classify that as a property, right? Because to be a magician is tied to the nature of man as an animal with reason, right? It involves both his animal nature, that's why we have these sensible sounds, we're talking about words, right? Vocal sounds. But obviously it's tied to the fact that he has reason on it. But now, if you say of the triangle, or the square, or the pentagon, that it's green, right? Can you see any connection between the nature of the triangle and being green? Or is that something that actually happens? And likewise then, for a man to be white, is there a connection between the nature of man as an animal with reason, that he should be white? I don't think so. Okay. This seems to be something accidental, right? So this is the magician's contribution to the good racial harmony, right? Did we say before that rational is a property? Well, yeah. Now this is the question that I want to ask. Why do we go into a subdivision of names signifying something outside the nature? We're interested in definition, right? Well, the text explains for two reasons, right? Because the property, right, has a connection with the nature, then it has a two-fold relevance definition. Sometimes, if you don't know the difference, right, but you know the property, you can reason from the property back to the difference, reasoning, as it were, from an effect back to its cause. So the property can be a middle term, as we see in logic, a beginning or reasoning towards the difference, right? But secondly, if you don't know the difference, you can use the property in place of the difference. This gives you an imperfect, right, definition, but something that might be, what, separate the thing you're trying to define. And suppose someone said to me, define the dog and cat, let's say, right? And I say, well, dog is an animal. So is a cat. Well, dog is a four-footed animal. Well, so is a cat. Separates it from the chicken maybe or the man, right? Now, at this point, my ignorance is going to show up, right? So finally I say, a dog is a four-footed animal that barks. And a cat is a four-footed animal that meows, huh? Okay? Now, I'm not sure, maybe, that the dog is the only animal that barks. Even if the dog were the only animal that barks, or the cat was the only animal that meows, huh? Would barking and meowing be the species making difference of these two animals? In other words, is he a dog because he barks? Or does he bark because he's a dog? Or does he bark because he's a dog? Or does he bark because he's a dog? Or does he bark because he's a dog? Or does he bark because he barks? Or does he bark because he barks? Or does he barks? Or does he barks? Or does he barks? Or does he barks? Well, if he barks because he's a dog, then barks is really like a, what, properly following upon his nature as a dog. But because I'm too ignorant to see the essential difference between the dog and the cat, right, by the barking and the meowing, you know, strike my senses, right, rather than be completely ignorant of the difference, you know, between the dog and the cat, I say that one is a four-foot animal that barks, and the other a four-foot animal that, what, meows, right, huh? You see that? So there's two reasons, then, why even with a view to definition, we want to see the distinction between these two names. You see? The accident is useless, right, both for defining even imperfectly something and for investigating the essential differences, huh? But the property can be useful as a starting point, right, to look for the difference, right? And if you don't have the difference, or before you have the difference, it can be used in place of difference to give you a speech that's imperfect, but something like definition. Sometimes Aristotle will call that a perigraphe, or an insertion, right, then? It's like if I didn't know, you know, it made two to be two, and I say, well, it's a number, a half, a four. Well, a number, a half, a four, and two are convertible, right? That's what it separates two from everything else, huh? Now, if you look at the Nicomarckian Ethics, then, when Aristotle sees the need to take up human virtue, when he first speaks of human virtue as a praiseworthy quality, right, okay, and human vice as a blameworthy, right, quality, well, there he's given more of the property than the difference, right? Because praiseworthy is something connected with what virtue is. Well, it's not what really makes it to be virtue, right? It's praiseworthy because it is a virtue. And vice is blameworthy because it is a vice, right? But what is the difference, right? And it's not until the second book of Nicomarckian Ethics that he starts to bring out the real differences. But the fact that he knows that fruit is a praiseworthy quality, right, is a starting point for investigating the essential difference, right? What is it that is praise, let's say, in eating, let's say, in drinking, right? Let's say, well, the man who eats and drinks the right amount for himself in the circumstances, right? That's praiseworthy, right, huh? But he eats too much or brings too much for himself in the circumstances, right? Or not enough, maybe, right? He's classing the job and so on, right? Then this is, what, blameworthy, right? Well, then you're starting to reason out that virtue is a praiseworthy quality. What is praised is the middle, right? Okay, the reasonable middle. And you're starting to see more essentially what virtue is. So you start to reason your way in to the property, right? So, but notice, huh? Now, the little child who doesn't yet see the intrinsic goodness or badness of things, right? But they know certain things that they are frowned upon for doing, right? And other things that they are applauded for doing, right? So, when we have the definition, man is a rational animal, do we use the property because we don't know the specific things? Yeah, yeah. In a sense, you know, in a way we're trying to get what's in the nature of man where a body he has the ability to call it reason, right? So, that's why we tend to have, like, just one difference there, right? That it's really a property, a property district sense. Make it just probably not a property like that. Yeah, yeah, yeah. But it just, even in the physics here at Stavro often refers to man as just a two-footed animal, right? Now, if you stop and think about these five names, right? In reference to definition, you can see there's something very complete about these five names. Species is the name of what is being defined, right? Genus is the name that begins the definition, right? Difference is the name that is used to complete the definition, or a perfect definition. Property is the name that's useful for completing an imperfect definition, or for investigating, right? An accident is the name that is useless both for naming and defining. See, how complete that is, right? We want to separate and distinguish the name of the thing being defined from the names used in the definition, right? And they are mainly the genus and the difference, right? But then the property has some usefulness for investigating the differences and for being used in place of them for hanging and defining definitions. But we also separate the name that is useless, right? Green is useless for what? Naming triangle and useless for what? Defining it and investigating the definition triangle. Nowhere is in green. Nowhere in geometry would it reappear, right? Now, as Porphy says, these names are useful also for dividing, right? Especially for dividing a genus into a species by its, what, differences, right? But if you didn't know its, what, differences, right? You might use divided by its properties, right? It's also useful, he says, for demonstration. A demonstration is, we'll find out when we get into the logic of the third act. In a demonstration, you show that a property belongs to a species through an understanding of its nature, which genes are differences, right? So we demonstrate, for example, in geometry, that a triangle has its interior angles equal to right angles, right? We demonstrate a property of a species through an understanding of the nature of that species, that involves a genus in different systems. Which means help and calling. Okay. Good. Thank you. Thank you. So the Book of the Five Names, as they called it, the Five Vocal Sounds, the Book of the Five Vocal Sounds, the Five Names, Pente Fona here, it's useful for definition, for division, for demonstration, and also, we'll find out for that Book of Arts Talents called the Categories. Okay, let's see if that's about later on. So it's quite a thing he did, right? And I noticed Thomas Aquinas there in the Summa Contra Gentiles, in the first volume there, where he has a chapter showing that no name is said univocally of God and creatures. There are some names that are said of God and creatures, but, equivocally by reason, yeah. But no name is said of God and creatures univocally. And Thomas gives many arguments to show this. Well, one argument is an either-or argument, then. He says, every name said univocally of two things, it's either their genus, or their species, or their dirhams, or their property, or their accent. And then he eliminates each one of those individually. And then he concludes that no name is said univocally of God and creatures. So he's going all the way back to, what, the first book of logic, right? The eisogogy of Corpher, right? And, you know, in terms of faith and reason, too, this book is very interesting, because I can see Thomas here, in the highest part of our knowledge there, he's about knowledge in theology. He's using, right? It is, right? Albert the Great, his teacher, writes a paraphrase of this. Kajetan, the cardinal that the church sent up to talk to Martin Luther, he has a commentary on the eisogogy, right? The artist thing, huh? And the Mohammedans, of course, use this on the native Greek philosophers. But Porphyry was violently anti-Christian. And Augustine, you know, talks about this, right? Now, the Christian emperor is burnt with anti-Christian books, and we don't have them, right? But Augustine, I think, in the City of God, if I remember right, there he speculates as to why Porphyry was, whether it was pride or something, right? But despite the disagreement between, you know, Porphyry about religion and the Christians, obviously, right? Even the Mohammedans, right? They all took this book, the eisogogy, right? It's common, but teaching, right? Okay. And of course, it really grows out of what you can see people doing naturally. When I go into class, and I say to students, what is a dog? They'll say, an animal. They'll naturally give it to genius, right? I go into love and friendship, and I say, what is love? And they say, well, it's emotion. Well, one kind of love. Yeah. Is it emotion? But I say, so is anger, so is fear, so. They naturally give it to genius, and they've got to add, what? Differences, right? So this is something really common to us and it's natural with Europe. A pagan, like maybe Porcri was, right? Or a Christian, like Thomas Aquinas, or Albert the Great, right? Or Kajetan. Or even a Mohammedan, or Islamic, you know? Or even, you know, one of the native Greek philosophers, who went up to a Christian, right? But they all take this as the fundamental word. And it's called, as they say in Greek, you know, the Isogogae Kherostylus categories, but for short, it's called Isogogae. The introduction, and eventually it started to be used by Antonio Masia, is the introduction, not into logic, but the whole philosophy, in a way. Because you're defining, and dividing, and demonstrating. That's a good philosophy, define, divide, and demonstrate, but that's the three Ds, right? In English, define, divide, and demonstrate. And these are important for that, understand definition. So think about those five names, then, in comparison to the definition, how complete that is, right? The separating, and distinguishing, and defining the name of the thing being defined, as opposed to the names used in the definition. And you're distinguishing between the name that begins the definition, and the names that complete the definition, and between the names that complete it perfectly, and imperfectly. And all these are being distinguished from the names that are useless for naming a thing, or for what? Defining it, or for investigating a definition, right? So, you'd be surprised that many philosophers don't know the Isagoge, right? Some, you know, some of you haven't even heard the name of the Isagoge. Others have heard the name, but they've never read the book. You know the Isagoge out of TC, didn't you? So, Porphy would define all these, and then he'll compare them, you know, and what they have in common, and so on. So, any questions up to where we are now? Notice that the definition of genus has a genus, right? What's the genus of genus? Name, yeah. Oh, okay. Name. And you have these differences, right? Said with one meaning, right? Of many things, other in kind, signifying what it is. Those are all differences, right? The genus of all five of these is name, right? Name said with one meaning. It's common to them. We need break. We need... Say the genus is that which is said of many other species, signifying, you know, what it is. And so, it kind of left up in the air whether you're talking about the name, genus, right? Or the general kind of thing signified by it, right? And I think that the word genus and, say, species are equivocal by reason, right? Because sometimes you use the word genus to mean the name of a general kind of thing and species the name of a particular kind of thing. And sometimes you use the word genus to mean the general kind of thing and species to mean the particular kind of thing. So when you're dividing a genus into species, I think you're dividing a general kind of thing into particular kinds of that thing. You're not dividing a name into names, right? Okay. But because the definition is composed of names, and when we ask for a definition, we use a name. That's why we distinguish the names here. And sometimes you use the word genus and difference even when you have a name that's equivocal by reason, right? But that's a little bit different, right? A little different meaning of the word, a looser use of the word genus, huh? So when you first distinguish these five names, we're taking them in the strict sense, huh? Whether each of them is a name said with one meaning of many things. Okay? Now I might make a little footnote here about the difference between logic and what? Grammar, huh? Take this example. Green triangle and equilateral triangle. Now the grammatical analysis of the phrase green triangle and equilateral triangle would be exactly the same. The grammarian would say, triangle here is a noun, right? And green or equilateral is an adjective, modifying triangle, as far as it goes. There's no difference between those two from a grammatical point of view, is there? Now the magician comes on the scene, right? He says, there is an enormous difference, right? Triangle here can be taken as a genus or a species, right? Green is a what? Accident. But equilateral is a species making what? Difference. Difference. Oh, yeah, yes. Okay? Now, the magician would ask, you know, what is a triangle? A triangle is a plain figure contained by three straight lines. Now is green one way of being three-sided? No. Nothing to do with three-sided at all, right? But equilateral is one way the three sides can be, right? Isosceles is another way, right? Or just two of them are equal. And scalene, or none of them are equal, right? So scalene, isosceles, and equilateral, they determine three different ways of being, what? Three-sided, you know? Green or red, white, and blue might sound patriotic, but it's nothing to do with being three-sided. Genus, or species for that matter, more by a noun, right? And the accident, or even the difference, by an adjective, right? Because the difference does signify how it is, what it is, right? Okay? We're more apt to signify how by an adjective than by a what? Noun, right? Okay. Now, another thing that we've touched upon to come back to here. In a definition, can there be more than one genus? If there was more than one genus in a definition, you wouldn't be talking about one thing, right? Okay? But there is usually, if not always, more than one what? Difference, right? Now, if you read the dialogue called the Mino, right? When Mino tries to eventually define virtue, and then Socrates examines the definition, right? And Socrates makes, you know, one critique of the definition. But when I first hear the definition, he defines virtue something like this. The desire for good things and the ability to achieve them. Sounds pretty impressive, right? But when I examine that, it seems to me that he has two, what? Genus, right? That's a plural of genus, huh? The desire for good things and the ability to, what? Achieve them, right? Is the desire for good things and the ability to achieve them, are those the same? No. And couldn't a man have a desire for good things without the ability to achieve them? Right. Or vice versa, he could have maybe the ability to achieve good things, but not, what? Desire them, right? So you're really talking about two different things, right? The desire, right? And the ability, right? If I desire money, I may not have the ability to get money, right? Or someone else might have the ability to get money, but not desire it, huh? The famous anecdote about babies, right? When you decide to show you can make money. This is so that a philosopher could make money if you wanted to, right? Makes you have an interest in that, right? So if I've got a gun, I've got the ability to get money, right? But maybe not the desire to get money, huh? Someone else doesn't have, has the desire to get money, doesn't have the gun, so they don't have the ability to get their money, right? And two different things, eh? So really, that's kind of the first criticism I might make of Eno's definition, right? So there should be only one genus and, what? Many differences, right? That's an interesting difference there, right? Okay? When Aristotle defines a tragedy, right, the genus is, what, imitation or likeness, right? It's a likeness of an action that is serious, etc., right? And then it only says act it out, right? So there are many differences, right? They say, like, this is something serious, as in common with epic, that it's acted out, as in common with comedy, right? But the two differences there are separated from both epic and comedy. But there's just one genus on it. Now, a second difference here you might point out, between logic here and grammar, right? In English, we tend to put the, what, adjective before the noun, right? Okay? So if the genus is signified by a noun, and the difference by an adjective, the difference will be given before the genus, right? So when we say that the genus is the first word in the definition, what does that mean? Should that be understood grammatically? First in our understanding of the thing, right? And unless you understood the genus first, you wouldn't really see why these are differences, right? Unless they understood, let's say, what a quadrilateral is, right? I wouldn't see how equilateral or right-angled are differences of quadrilateral. Unless I understood what a triangle is, I wouldn't see how equilateral and isosceles and scalene are differences of triangle, by red, white, and blue are not. Red, white, and blue are not three ways of being three-sided, but equilateral, isosceles, and scalene are, right? So unless you really understand the genus, right, you don't understand the differences of them. Aristotle talks about tragedy and common ethic. He points out that each of these are a likeness, right? An imitation. And therefore they can differ by that in which they imitate, or what they imitate, and so on, right? But you wouldn't see those as being the sources of differences unless you saw in general that it is a, what, a likeness or imitation. So, the genus, let me say, it's the first word in the definition. We don't mean that it's the first word in necessarily, you know, in a language like English, for example. When you write it out grammatically, right, then? So if I say in English that a triangle, excuse me, a square, is an equilateral and right-angled quadrilateral, right? Quadrilateral is the last word in the sentence, right? In the speech, right? I say the definition of square is equilateral and right-angled quadrilateral. But nevertheless, quadrilateral is the first in our, what, understandings that logic is concerned with, right? Right, okay. But maybe in Latin or in Greek, you could put the, what, adjective after the noun, right? Play around with this in English, you could, you know, do with a phrase, right? I say man is an animal, what, with reason. There I did the genus first, right? But if I used the adjective and said with reason, I said man is a rational animal, I'd have to put, as English grammar, rational before animal. It sounds right English to say man is an animal rational. You know, I seem to have to say man is a rational animal, right? Even in my opinion, I could say anima or ratzkanalus or something like that, right? So the grammatical order is not the logical order, right? And it doesn't mean you have to, you know, go against the grammatical order by using phrases, you know, you understand it, you have to, right? It's like when I was explaining the definition of name there, I began with it's a sound, right? And then I add it's a vocal sound. It's a vocal sound that signifies, a custom, part of which signifies. Right self, right? But I start with the word that logically speaking is before the rest, right? As I say, when you ask the child there, you know, when you ask a, you know, classic student, what is it? We call it a common student out of the blue. Say, what's a dog? They'll begin by saying animal. They won't begin with four-footed. You see? They'll say animal first, right? Okay? So even though later on they might say four-footed animal, they didn't, what? Think first of four-footed, right? They thought of it being an animal. Okay? Or if they were, you know, thinking of tragedy and comedy, we'd say it's a play, right? You know, well, they wouldn't use four-footed or two-footed to separate one play from another, as I thought it would say. Okay? Um, okay. Now, we've come to a nice subtlety here now, at the bottom of page 12, huh? Can the same name be both a genius and a species? And the answer to that question is going to be yes. Species as we define it. But in order to understand that answer, I distinguish between two kinds of distinction that you find in sciences. and one we could call an absolute distinction and the other a, what? Relative distinction, right? Because absolute means in itself or by itself, right? And relative means, what? Towards another, right? Okay? So I give a number of examples of this distinction, right? Between two distinctions. Um. The distinction in a human being between a man and a woman is an absolute distinction. A man cannot be a woman, a woman cannot be a man. But what about the distinction between a father and a son? Well, the same man can be a father and a son, but in relation to different people, right? But you can't be a man who refers to one and a woman who refers to somebody else. You're either a man or one. Or take numbers, right? The distinction between an odd number and an even number is what? Yeah. You're either an odd number or an even number, huh? A number. But double and half is what? Yeah. It's possible for a number like four to be both what? Double and half, but in comparison to different things, right? So is the distinction between genus and species as we define them, right? Is that an absolute distinction of names? Or is it a relative distinction of names? It's relative, huh? So that the same name could be a genus in comparison to these, and a species in comparison to those. Take our stock example, quadrilateral, right? Well, if you compare quadrilateral to rectilineal, plane figure, plane figure contained by straight lines, quadrilateral is a what? Species, right? Along with trilateral or triangle, right? Okay. Pentagon and so on, right? But quadrilateral, in comparison to square and rhombus and oblong and so on, is a what? Genus, right? Okay. Do you see that? I know it's relative here. It doesn't mean that this thing is subjective or not precise or not certain, right? That two is half of four is very certain, right? And the quadrilateral is a genus of square is very certain, very precise, right? It's a species of rectilineal figure. But it's a different distinction, right? Than an absolute distinction. I am really the son of Reno Victor Berkowitz, right? But I'm really the father of Paul Berkowitz. So I'm both the father and the son, but not towards the same man. Now, this is a similar thing. Once you see this, then arises kind of the ultimate questions. And this happens in other areas too, right? If you realize that the same man could be both a father and a son, you might go on and ask now, is every son also a father? And is every father a son, right? And there are some sons who are not fathers, as we know, right? Is there a father who is not a son? Well, we're told about Adam, right? Right? And so Adam is a father, but not a what? Son. But he doesn't have a father like you and I have a father, right? But notice, that may be hard to know. Is there a father, right, who is not a son, right? Okay? We might know that there's a son who's not a father, right? We're not always sure. But you have two questions, right? Looking at either end, right? Okay? I'll take another example of this sort of thing. Can something be both a cause and an effect? Of course, there's an absolute distinction between cause and effect. The same thing can be a cause and effect. Yeah. Okay. Now you ask the question, though, but is every cause a what? An effect, right? And the answer to that's eventually going to be no. There's a cause that is not an effect, right? Maybe there's an effect that's not a cause to the end. Okay? Now, take whole and part, right? Is there an absolute distinction, whole and part? Or relative? Yeah. So, Massachusetts can be what? A part of New England, right? Okay? So, New England is a whole, which Massachusetts is a part. But New England is a part of the United States, right? But again, Massachusetts has parts. You know, I live in Worcester County, right? So, Worcester County is a part of Massachusetts, which is a part of New England, right? So, now you ask the ultimate question, right? Is there a part that's not a whole? And vice versa, is there a whole that's not a part? Now, in modern science, in the 20th century, the physics of the 20th century is distinguished from that of the 17th, 18th, the 19th. In the first part of the century, there are two great things, right? One was the discovery of the quantum, right? And Max Planck in December of 1900, right? The dawn of the 20th century. He proposed a quantum hypothesis, right? It took about 30 years of quantum theory to be, which is full development, right? And then 1905, Einstein proposed a special theory of relativity, right? Which he followed later on with the general theory of relativity in 1915, right? After special relativity and quantum theory were somewhat perfected, these physicists went in opposite directions on it. And the quantum physicists went in the direction of looking to see if there's a part that is not a what? Whole, whole, whole. And so they're going from, what? Atomic physics, which quantum theory in a way completed, right? Down to the study of the elementary particles, right? And they were, what? Accelerating elementary particles and shooting them, right? And trying to split them, right? But they're trying to see if you come to a part that is, what? Yeah. There's no longer a whole. No longer a whole. It's a part that is not a whole, right? Okay. And, um... Which means it? Oh, it's indivisible. Not in one, right? Yeah. Yeah. Okay, yeah. And actually, for a while, they were trying to get the idea that there's a minimum length in the universe. Minimum what? Minimum length in the universe. Oh. Oh, yeah. Yeah. So notice, without, you know, trying to say what the conclusion will be, they're moving in the direction of looking for what? Of the part, right? And that would go on until you came to a part that was not a whole, right? Mm-hmm. Okay? I'm not saying there is such a thing, but... Now, Einstein, with the general theory of relativity, went in the opposite direction. This is a starting point for kind of an explosion in the study of the cosmos as a whole, right? The general theory of relativity led to a great, what? Interest in cosmology, the study of the universe as a whole, right? Okay? And, you know, what came out of the intense understanding of the universe at first from the general theory of relativity, it appeared that the universe was, what? Finite. Contrary to what the moderns have thought since the Renaissance. Aristotle thought the universe was limited, right? Although the first Greek philosophers thought it was infinite. Well, the first modern scientists thought the universe was infinite. And now, with the general theory of relativity, it seemed that the universe was finite, limited. And this was quite a shock to them. And Einstein itself, I guess, didn't fully realize the consequences of its own theory. It was leading this way, huh? Okay? But notice, if the universe is not finite, huh? If it's limited, not infinite, then there is a whole that is not a what? All right. So you notice, the ultimate questions, you might say, right? Are what? Following out, once you recognize that a whole can be a part and part as a whole, the ultimate question is, is there a whole that's not a part? And that takes you in the direction of the study of the cosmos, the universe as a whole. And the other one down into, what? The atom down towards, what? The energy particles, and if there are quarks, you know? Down that direction, right? And you're looking to see, is there a part that is not a whole? Kind of the ultimate question, right? But the most ultimate question is, is there a cause that is not a what? An effect, right? But you have a similar question with the father and the son, right? The same way with the mother, right? The same woman can be a mother and a daughter. Was there a daughter who's not a mother? I think there are daughters, but not mothers. But is there a mother who's not a daughter? What about Eve? But notice now, we're not assuming that any of these series are closed at either end, right? They might be open at both ends, for all we know. To give an example, in geometry, right? Is there a, you could have one straight line be, let's say, half of another straight line, right? Okay. But is there a straight line that is half of some other straight line, but not double of some other line? When you get to study the continuous in Book 6 of Aristotle's book, The Physics, or Natural Hearing, as it's called in Greek, you'll find out that the continuous is divisible forever. So you can always bisect the straight line. So, you never come to a straight line that is double and not what? Reading this half and not double, right? Okay. But again, if you can always geometrically extend the straight line, is there a line that's double of something, but has no line that it's half of? It goes on forever, right? But now with numbers, right? Unless you're a crazy modern who thinks you can divide the one, right? Yeah. But the one is actually simpler than the point. It's indivisible. There's a number that is double, like two, right? Okay. Or there's a number that's half, like three, right? Three is half of six, but it's a double of indivisible. So three is a half that is not what? Double. Right, four is a half that is double, right? But now is there a number that's double but not half? So in the case of number, in the case of double and half, in numbers, right, there is a half that is not double, but there's no double that's not half. It's closed at one end, so to speak, not the other end. But in straight lines, there's no line that is open at either end, right? Okay. Or a father and son without, you know, faith, right? You might say what he was, was there a man who was not, was a father but not a son, right? But when you hear about Adam, he must be a father, right? Okay. So, we're going to ask a similar question here now. Logical question, right? Okay. Well, we ask it also when we reason, see? When we reason, we find out that the conclusion of one argument can be a premise in the next argument, right? So to be a conclusion or a premise is a relative distinction, right? So in Euclid, the conclusion of one theorem, right, is a premise for the next one, right? But is every premise a conclusion? No. In that case, you'd never know anything, right? Okay. So, we ask then now, quadrilateral is both a genus and a, what, species, right? Now we go in both directions. Does, is every genus a species? Another way of putting it. Does every genus have another genus above it, right? Well, see, above rectilineal plane figure is like a plane figure, right? Above plane figure is like a figure, right? Does that go on forever, right? See? But you always have a thought about every thought that you have. And then vice versa, you know? Does every, what, species have species below it, right? Or do you come eventually to what we would call a lowest species, huh? A species that has no species below it. In other words, a species that is not a, what, genus, right? And then the other question, the other, other end, right? Do you come to a genus that is not a species? A genus that has no genus above it. Do you come to what we might call a highest genus, right? This is where Aristotle comes on the scene with the categories, right? I said, Bill gave me Aristotle, he's that for lesser minds, right? But he always takes the most difficult thing, right? Just like, you know, he points out about Euclid, right? He takes the most difficult case of these, the easier case for us, secondary minds, okay? So, are there species that are not in general? Yeah. Yeah. Are there different kinds of square? No. But how about oblong? Is oblong a lowest species? Are there different kinds of oblong? Different shapes in the world? Yeah, they are. Yeah, yeah. See? So, this oblong here, and this one here, they have exactly the same shape, do they, right? But if you had an oblong, let's say, where the longest side is to the shorter side is two to one, maybe that's the lowest, what, species, right? How about triangle, right? Is triangle a lowest species? But now, what about scalene, isosceles, and equilateral, any of those lowest? Yeah? What about equilateral triangle? You know, bigger or smaller. Yeah. But that's not really a difference in the figure, right? Okay. So, all equilateral triangles have the same shape, right? Oh, yeah, okay. But not all isosceles triangles, right? Okay. See? Uh-huh. Good. So, that's kind of interesting to see that in the species you get when dividing a genus, one of them might be already a lowest species, like square, but not oblong, right? Yeah. Okay? Um, a circle would seem to be a lowest species, right? You have to have different shapes of circle, right? Uh-huh. Is man a lowest species? See, if you speak of man as an animal with reason, if there are species of man, we have to have different kinds of reason, right? Now, if the white man and the black man and the yellow man and the red man had a different kind of reason, right, then you might argue that man is a genus and not a, what, lowest species, right? But if red and black and so on are not a different kind of reason, but they're really accidental differences, right, then maybe man is a, what, lowest species, right? Okay? Now, it's not as easy to see that as it is that a circle is a lowest species, right? Some men and women don't have the same, don't think the same way. But, you know, I'm not sure that, you know, you can see that they really have a different kind of reason, right? You know, it seems more obscure than it is in math, right? With that, in man with female, it doesn't seem like, it seems more than an accident, doesn't it? Mm-hmm. That, when you feel like a crop. Well, that's a very difficult subject, right? Oh. Well, I'm saying the logician will explain what a lowest species is, right? Okay? It's a species that is, you give a definition of it now. And it's a name, right? You take the lowest species now as a name. It's a name, set of many things, right? Not other in kind, right? It's a set of many individuals, right? Signifying what it is, right? Like circle, for example, is a set of many individual circles, right? So, I get the definition of lowest species here now. Remember, the definition of species we gave before was in comparison to genus, right? Okay. If you look at what's below species, if that species is also genus, well, then it's a genus which affects what's below it. But, if you're coming down to the lowest species, then it has a unique relation of what's below it. So, it's a name, set with one meaning, of many individuals, huh? Not other in kind, huh? Let's understand when you say individuals. Signifying. So, you take a circle here. Set of this circle, and this circle, and this circle, right? It's a name set with one meaning, right? The reason why I call this a circle is the same as using why I call this a circle, and this a circle.