Introduction to Philosophy & Logic (1999)
Lecture 28: Division, Distinction, and the Rule of Two or Three

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and he says it's in reference to the what eucharist right because that's the the ultimate thing right you define something by the greatest thing it can do right okay and then he reasons that because the priest can what not say mass consecrate and so on and therefore distribute you know the eucharist then he has to have the ability to forgive sins because without that someone would not be suitable to receive the eucharist right okay but the ultimate thing is the eucharist right that's the greatest of the sacraments it's the end of all the other sacraments all the other sacraments contain the power of christ but this alone contains christ himself so it's the end of the combination so the ultimate thing for reason right is to give the reason why it must be so right the ultimate thing is to look before and after in that sense it's the cause before the effect right that's why i call that fifth sense that aristotle gives but back to the fourth you know the crowning sense right now combination of reason but also you know there's the word reason there wouldn't say two reason for the name of the act right now let's look a little bit at division here right and get to the rule of two or three right who knows okay now um using now division here to refer to the not the definition itself but to uh divisions that are not a definition right okay and division in the strict sense is the distinction of the parts of some whole right okay distinction of the parts of some hole so in logic you want to distinguish at least um two kinds of division according to there being two kinds of holes now one kind of hole which is the original meaning of the original meaning of the word hole they sometimes call it the integral hole i like to call it the composed hole right which you'll see both they have the integral or composed hole and then a second sense of all the universal now in distinguishing and in a way defining these two kinds of whole i use an affirmative and a negative part of my definition and the two definitions of just reverse so the integral or composed hole is as the word composed means is put together from its parts but it's not said of so the chair you're sitting on right huh it's put together from the the seat and the legs and the back and so on but chair is not set of the seat the leg is not a chair right the seat is not a chair right the back is not a chair but put together these make a chair right okay okay put together from the letter c a and t right but the letter c is not the word cap the letter a is not the word cap right and so on but the death book is together a definition is a what interval composed hole right the genus is put together the definition is put together from the genus and the differences right but the genus is not a definition the difference is not a definition right but put together in a the way they are in the way they are in the way they are and form a definition right now the universal hole is just reverse huh it's said of its parts meaning instead of each one individually right but not composed of not put together from them so quadrilateral is the universal hole it's said of what square outlong rhombus rhomboid trapezium but it's not composed of them right the animals universal universal whole instead of man and dog cat and horse but it's not put together from them right otherwise when you say man is a animal you say man is something composed of cat dog for itself and so on and so on right it's absurd right okay when you say square is a quadrilateral it's not something composed of square oblong thrombus rhomboid and abenzium okay so if you understand one of these definitions the other one is just the what reverse right okay but the integral composed whole is the first meaning of the word whole and the greek word for for genus or for the general comes the greek word for a hole right and what comes under the whole we sometimes call the reticular but it's stuck apart right so you have these two kinds of holes in every what reasoned out knowledge right what i do when i explain this to students sometimes in college there i'll go through all the sciences inductively right and you see that every one of them divides in both of these ways take for example grammar right which is the first science you learn in the way they are grammar is about the sentence on english grammar is about making sentences in english and greek grammar is about making sentences in greek sentence is the subject of grammar reading now what does he divide sentence into sometimes what they call the parts of speech noun verb adjective adverb right that's a division of the integral or composed all into its parts right so is the noun a sentence or is the verb a sentence but man walks right composed of a noun and verb is a sentence right but then he divides sentence into a firm or declarative sentence baby right question command prayer right and each one of those is a sentence by itself right but it's not composed of these is it sense is not composed of command prayer question declaration is it but it's said that each one of these right okay so the grammarian divides in both ways right does the geometry divide in both ways when euclid divides the circle into two semi-circles the semi-circle a circle he divides a parallelogram into two triangles right the triangle parallelogram that's a division of a whole into its parts right when you divide a parallelogram into let's say square and album and rompus and rhomboid then you divide it into the universal whole so he divides in both ways and sometimes the same thing in both ways right if you go to chemistry and they talk about the atom they might divide the atom into a proton electron neutron neutron and that's what kind of division yeah the proton is not an atom electron is not an atom right maybe an atom is composed of these right when i divide atom like you do the periodic table into the hydrogen atom and the nitrogen atom and oxygen atom and so on then you have a universal whole right so the chemist divides in both ways right does the biologist divide in both ways huh how does he divide tree what kind of division is that of tree i think divides tree into the broad leaf tree and the what kind of division is that of tree right that's the same thing right what kind of division is that of tree right that's the same thing right what kind of division is that of tree right that's the same thing right what kind of division is that of tree right that's the same thing right right that's the same thing right right that's the same thing right right A noun is not a statement, right? But an affirmation is a statement and a negation is a statement, right? But a statement is not composed of affirmation and negation. And since, you know, Locke is mixing up these two kinds of division when he's saying the triangle in general is what, all or none of these, right? It's species, right? Like it's kind of, you run them all together, right? It's like, you know, I get the idea of animal by taking a dog and a cat and a horse and so on and run them all together and they're all mixed up and that's animal. No, no, it's not that, right? You separate out what they have in common and you left aside their differences. Now, let me clarify one more thing here about the three words here we've used, the letter D here. You take the word distinction, which I use in the definition of division, division, and division, and then the definition. Since a definition has parts in a way to break up something into parts, right? You could say the definition in a way is a division, right? Okay? Okay? But not every division is a, what? Definition. So division is more universal than what? Definition, right? Okay? So in the first chapter of the physics, Sarah Stout speaks of the definition as diary, that divides into parts, right? Now, sometimes division and distinction, you know, are used loosely to mean the same thing, right? But in the strict sense, only some divisions are, the strict use of the word division, are distinctions. I mean, only some distinctions are divisions, but not vice versa. The division, in the strict sense, is like I define it there, a distinction of the parts of some whole. So you need a whole, really, right? In order to have division, right? Okay? Right, distinction is the most universal, right? So every division is a distinction, but not every distinction is a, what? Division, right? And sometimes I have talked to him, in the first act, about those distinctions are not division. But usually I, you know, or temperate, I leave it to the consideration of the famous fallacies, right? Because they're based upon the most fundamental ones, upon how we're looking to the distinction of the Son. And you'll see Thomas Aquinas say, sometimes he uses division, you know, and distinction kind of interchangeably, but sometimes he's very careful about it, right? One place he's very careful about it is when he talks about the Trinity, right? You distinguish between the Father and the Son and the Holy Spirit. You distinguish between the three of them, right? But is that a division? Well, he'll deny that this is a division, right? God is not divided into the Father, the Son, and the Holy Spirit, right? But nevertheless, you know, that would imply that the Father and the Son and the Holy Spirit are somehow like a, what? Heart, right? In which case, you'd be thinking of God as a composed whole, composed of the Father, the Son, and the Holy Spirit. And that would be a very serious mistake about the time. But there is a real distinction between the Father and the Son and the Holy Spirit, right? There's a real distinction between the Father and the Son, and then a real distinction between the Father and the Son and the Holy Spirit that proceeds from that, right? So, distinction is really broader in meaning than division, right? Every division is a distinction, but not every distinction is a division. And so, when you distinguish the senses of a word, like we did the word before a long ago, or as we did distinguish the sense of the word whole, is that really a division? Or should it be called a distinction, but not a division? Distinction of a whole. Or when you distinguish between the Parasean and the Karachi dance, right? The such and the accidental and so on. When you distinguish between the simply and some respect, right? There are certain distinctions that are very important in thinking, but there are not really divisions in the strict sense. There are only a part that are in the strict sense, I mean, a whole part of the strict sense, the Y or something. As they say, a lot of times we use the word division and distinction kind of interchangeably, you know, somewhat the same thing, you know? Is there a distinction? Between them? No, a distinction. What is it? Define a distinction? A division is a distinction process. Yeah, well basically a distinction, the basis of distinction means opposition, right? You have to have some kind of opposition, so you have to go into a study of the four kinds of opposites, right? To explain distinction. Yeah, yeah. But in the case of division, you add something to that, right? The idea that where being distinguished are really parts is some whole, right? So, these are ascended from more universal to the last year and so on. You'll notice that how we mentioned before how a distinction comes before order, right? When Aristotle in the categories and in the fifth book of wisdom, before he takes up before, he talks about opposition, right? That's the basis of the distinction. But there are four kinds of opposites and I'd be able to study it in itself. Would they be used in that sense or in a different sense? For the Trinity, you mean, or what? More afraid of the main. Yeah, that's a little different sense there, you know. I mean, involves making distinctions, right? You see? But it's like the word, you know, speech and rhetoric, It's a little different sense than what you've got in mind. They would involve it, you know, with distinctions in there, yeah. He's thinking more that you're investigating the question and coming to a more distinct knowledge of whatever the question is about, right? Now, one more thing that we point out there towards the end of that part on division there is that a division should exhaust, right? All the parts, right? And some people want to say that the word division comes from the word or the word divide and divide comes from empty, right? So in a way, when you divide something, you're emptying it out, right? Nothing left there, right? There should be nothing left in there, right? And if I divide triangles into equilateral say and isosceles, there's all something in the basket still, right? Scalene, right? You know, you didn't have a lot, right? Now we've come to my invention here, the rule of two or three, right? Okay. And I begin with the question, you know, whether there's any number into which you should divide, right? And can you give any rule about this, right? I don't think you can give a rule that is true universally for any number to divide into, right? I think you can give a rule that is true for the, what, most part, right? Okay. But the way you approach this is kind of dialectically. And if you go down to the history of philosophy, you see that there are two famous empires who are known for always dividing into a particular number, right? In the dialogues of Plato, when you get into division there, in the dialogue called the Sophist and so on, Socrates is always dividing into, what, two, right? Okay, Socrates or Plato, right? They don't know who are original there. Okay? But Socrates or Plato in the dialogues is always dividing into two. In Aristotle, the pupil of Plato, in the biological works there, he criticizes the Platonists for always dividing into two, right? because Plato is one of the chief philosophers. Socrates is one of the chief Socrates is one of the chief of Plato, right? Socrates is one of the chiefs Socrates is one of the chiefs Socrates is one of the chiefs And you can give a reason why you should divide into two. And that is that the basis of distinction is what? Yeah. And opposites are two, right? So you can syllogize there, right? We should divide by opposites, but opposites are two. Therefore, you should divide into what? Two, right? So it makes a lot of sense what he's doing there, right? And so you have all these divisions, like, you know, the arithmetician. He divides numbers into what? Odd and even, right? Well, those are opposites, right? Divisible into two equal parts? Not divisible. You divide the numbers into prime and composite. Measured by another number? Not measured by another number. I pin the biologists down. They have to admit that the division really makes sense. It's an animal into vertebrate and invertebrate. Into two, right? Okay. Now, if you don't divide by opposites, it doesn't seem to be a good division, right? We fight the vitamin beings into male and good. Both the male and good, right? So it makes some sense to divide human beings into male and female, right? Because their opposites are good and bad, right? But male and white. Female and white, right? You've got to have opposites, right? You can't belong to the same thing. And opposites are two. So there's a lot of truth in saying you should divide into two, right? But sometimes it seems more natural to divide into three, like when you divide triangle, right? And two equilateral isoses and scaling, right? It's a little force to divide into two and then you're good, I mean, you know? It seems more natural to divide it into three, right? Or the Greeks divide government into three, you know? Monarchy, oligarchy, democracy. Rule by one, by the few, or by the many, right? Seems to make sense, right? Nevertheless, as I say, there's a very good reason to divide into two, right? There's a lot of truth in what's going to say, huh? And there are all kinds of divisions into two, right? As I say, Aristotle criticizes them for always trying to divide. Now, the only other philosopher who's known for always dividing into certain numbers is Hegel, right? In modern times, 1770, 1831, right? In Hegel's system, he's always dividing into what? Three, right? Now, it's more difficult to give a reason for always dividing into three, right? But there's a sign which Aristotle himself gives, right? In the Book of the Universe, that three is the first number about which we say, what? All, right? As we were saying before, when you divide, you want to have, what? All the parts, right? And if three is the first number about which we say all, with two we just say both, right? You've got to have all the parts, well, then three would seem to be, like, that's a sign, right? You divide into three, huh? And some people see it, too, in the connection of beginning, middle, and end. That seems to be all there is, right? A reflection of the Trinity, in a sense. And Karl Marx says that as a young man under the influence of Hegel down in the general universities, you're always dividing into three, right? And then after a while, we begin to suspect that there's something kind of artificial in force. It's kind of forcing even in the three other time. Maybe he wasn't always adapting to himself to that, right? Now, to my knowledge, there's no famous thinker other than these two who's ever tried to always divide into some definite member. Okay? But there's something probable about both, right? So if you combine the two and say, don't always divide into two, don't say that, like, don't always divide into three, right? But divide into two or three, right? That has more probability than either, what, what Plato's saying or Hegel's saying, right? Still, I wouldn't say one should always divide into two or three, huh? But it seems to me that for the most part, one should divide into either two or three, huh? And that if you divide into more than three, right? You are probably, right, either crisscrossing two divisions, right? Or you're dividing and then subdividing and kind of other things, right? And so inductively, I see, you know, that most divisions that are into more than three are a result, right, of one of those two things, right? And so when you analyze it, you have to divide, see how it's based on divisions into two or three, huh? And sometimes they'll say Aristotle divides government into six kinds of government, monarchy, tyranny, oligarchy, aristocracy, democracy, republic, right? That's six kinds of government, right? But really, he's combining two divisions, right? Division of government into good and bad, that's the name of the good of the whole or the good of the part. That's the one division. And then rule by the one, the few, or the many. And he crisscrosses those two divisions and you end up with what? Six, yeah. We saw Thomas dividing order into four, remember that? Okay. Which he examined more closely, and he distinguishes the order not made by reason and then three orders made by reason. That's really a division into two, isn't it? Order not made by reason, made by reason. Perhaps, that's right. And then he divides the second of those, not by the maker, but by that in which they make. In the act of reason itself, which logic will consider, right? In the act of the will, we're next here, right? So he really has a division into two and one of them subdivided into three and that's four, right? Now, take a theological example here, right? You might say that there are seven sacraments. You might divide sacraments into seven, right? But that's not the way Thomas would divide it in theology. How would you do, how many divisions or how many distinctions does Thomas use to arrive at your big seven sacraments? I've got the seven sacraments and see that there's a distinction one from the other. How many divisions do you use? You use five distinctions, right? You divide, what? You know, the sacraments are something sensible, right? And they have a certain likeness to the natural life, right? And the first distinction Thomas points out is that there are those who have to be fed and directed, right? In a particular life, like my grandchildren, they have to be fed and directed and so on, right? Then there are those who, what? Propagate and direct these little ones, right? Okay? That's the fundamental distinction, right? And so he divides five of them against two of them, right? And the sacrament of matrimony and the sacrament of orders are analogous to what? Those who propagate and direct others, right? And then the other five to those in themselves, right? So that's a division into two, right? You distinguish matrimony and orders against that five. The other five of them ordered to the good of the individual, I might say, in himself, and the other two ordered to what? Directing others, right? So that's a division into two, right? And then you can distinguish between matrimony and orders, right? That's the second distinction, right? In that group, right? Then in the first group of five, he divides baptism, confirmation, Eucharist against what? Confession or what you want to call it? And the last one, even when you're loyal, right? Yeah, okay? That's the third distinction, right? And then he divides his first three according to an analogy of bodily life. You have to be generated, you grow, you nourish, right? And he compares it to the three parts of the soul, the plant, the soul, right? Reproductive, power, and then the power to grow, the power to feed itself. And then the ones where we went into trouble, right? You know? You become sick or something, right? And then he distinguishes between the way penicillor to that and the way we went into this, huh? So, your first divide, the first division is along where you divide matrimony and orders. you know, you know, you know, you know, you know, it's the other what five right that's one distinction right okay and since matrimony and order is a little bit like distinction in philosophy between the family and the city right then he subdivides those right and then these five he divides three things necessary for the you know spiritual life according to analogy of what's inside my life and then when the defect takes place right so he divides the three against the two right and then he subdivides um penance against uh the last morning oil and then he divides these three right okay so you've got one division two divisions three divisions four divisions five divisions right so in order to you know see distinctly the distinction of seven sacraments you need five divisions right but notice every division is into either two four three if you want to i mean you can you know you can bypass that and then say we've got seven sacraments right but you don't understand the seven sacraments and their distinction and relations and so on uh like you do when you go through these divisions right so how do you get from sacrament to these seven to make five separate divisions of distinction and then you see exactly which one of the words possesses just like when we went to you know from names said you typically have many things right you can go to the five they're five and really try to understand why they're five you have to divide the three against the two right proper and hexagonal distinguish right and then you have three divisions there we get to five right and even the categories you know thomas when he divides it it's always into two or three right that's how he gets the name something said of individual substance a reason of what they are a reason of something existing in them or something outside of them right he said divides and the second and the third it's always into either two or what three right so i think there's a lot of truth um there's some lot of truth in in both what hegel does and what plato does and in fact no other famous philosophers try the other number right but if you combine the two right for the most part again i don't say always i mean we know solid dividing quality aristotle divides it into four right and as i mentioned there you know i worked for years on dividing shakespeare's plays you know and setting aside the ten history plays which has some special problems of classification and divide the 27 plays there into four kinds so i don't always divide it two or three i'm not a fanatic right i think there's an awful lot of truth in there right and for the most part i think it fits you know what you're dividing but also i think it fits our mind for the most part too our mind is a very hard time understanding division to more than three it ends up being what we call an enumeration and not a division right you see i can enumerate more than three but do we not have a sense of table's reasoning for his well it's um there's a perverse imitation of the trinity there's a perverse imitation of the trinity oh really yeah yeah yeah and then there's later on a perverse imitation of the incarnation in the uh that in hegel's system yeah yeah yeah yeah you see the trinity the trinity the son proceeds from the father right and then when the father son proceeds the holy spirit well hegel has a perverse imitation of that right where he has the you know the thesis maybe the antithesis yeah and then you have something there negation negation right and you have this thing and you have the same thing you get it right so it's kind of a bizarre imitation of the trinity yeah very genius but very imaginative right but i mean it doesn't really and then he you know he does it for a while and this is all mental and then he suddenly jumps to matter this is kind of a perverse imitation of the word was made flesh right but understood in in the sense of which it's heretics you don't say the word was made flesh that the word itself became flesh right yeah yeah you know you know thought becomes matter then you have the union of thought and matter rising up in the spirit you know you get the ego system which is a combination of all history right so i mean there's more theology there than philosophy right but perverse imitation yeah you see then right after him comes comes uh forebock right and forebock there's a bridge between hegel and marx and forebock's famous work is the essence of christianity right where he says the real meaning of christianity is that uh god that uh that man is god you know so i mean you know it's really a perverse that goes on and you know a lot of the modern philosophers you know they could go on but hey it was really spent theological right you've been circles and so on but i think part of the problem with the modern philosophers is is is that they've given up the faith and they've given up uh therefore theology right and therefore they can't distinguish between theology and philosophy because i think i mentioned this principle before i think it's a universal principle but i i show it usually inductively first that it always belongs to the higher knowledge right and the knowledge that is more the character of wisdom to distinguish between itself and the lower knowledge and to show the order between them and i can you know shows inductively now the first distinction we make it between the reason and the senses right and it's belong to reason of the senses to see that distinction but the senses can't distinguish between the reason and the senses but the reason is the is the higher knowledge and it's more it has a character of wisdom right okay and uh does belong to uh even the inward senses like the imagination because the distinction between imagination and reason belong to the magnification of reason to make that distinction yeah and therefore the order of the senses to reason or the imagination of reason belongs to only what reason okay then um uh the distinction say between mathematics and natural philosophy what is that made is it reason or what science yeah it's made in mathematics or natural philosophy natural philosophy yeah yeah and that is more the character of wisdom in fact it was thought to be wisdom right but the distinction between wisdom and natural philosophy mathematics is made in wisdom right so um if you compare theology and philosophy yeah theology is wisdom in a higher sense than philosophy as aristotle himself would agree if you knew about right because the the great great greek philosophers all say that god either alone is wise or god most of all right so theology um it's based on revelation right based on the word of god it's a sharing of god's wisdom has more the character of wisdom so thomas shows in the first question summa right the theology the real theology has more the character of wisdom than philosophical wisdom so therefore the distinction between philosophy and theology doesn't belong to philosophy but to theology and to show therefore the order of the one to the other so now the greeks aren't in the need to distinguish between philosophy and reveal theology because they don't have revealed theology right yeah they don't mix them up right but the modern philosophers in a civilization that was mentioned a christian right and where there's still believers around and where they're exposed to theology and so on but in having given up revealed theology with the faith for theology they've given up their very knowledge whereby they could distinguish between the two and so they start to mix up the two right then and they start to use philosophy also as a substitute for theology and they seek in philosophy things that they've given up in theology and they can't get them you see and there's a tendency you know sometimes um you know sometimes people you know they get uh that they have a uh they're a philomuthas as aristotle said right they have a love or the myth of the stories and so on and there's something like a philosopher right but then philosophy is too high for them or too abstract or whatever right and they kind of seek in literature a substitute for a philosophy right And that's kind of understandable, right? So the modern philosophers seek, in a sense, a substitute for theology in philosophy, right? And that's why you find the demands of modern philosophers, they're always criticizing the senses, because the senses are in a perfect way of knowing, right? And the Greeks knew that senses were in a perfect way of knowing. But it becomes a chronic complaint of the modern philosophers, right? And they're demanding a certitude that doesn't in any way depend upon the senses. And as a human being, naturally, you can't get in like that. And so they end up all in despair, or skepticism like that. They're demanding a certitude that's more than human. They're demanding the certitude of theology. In philosophy, they can't get it there, right? Theology is more certain than being based on the Word of God than philosophy. And so, in a sense, they're seeking a substitute, right? In philosophy, for the certitude, they gave up, and they gave up the faith and theology. They can't get it there, right? So, I mean, I hope they're staged. They're staged. Plato, in the dialogues, he's always distinguishing between rhetoric and what? And political philosophy, right? And to whom does it belong to the distinguish between those two? Or political philosophy. Yeah, it has more the character of wisdom, right? But the rhetoricians can't make the distinction, right? And Aristotle says in the book on rhetoric that some rhetoricians, you know, who are good at persuading people, they think they're politically wise, right? Partly because of boastfulness, partly from ignorance of how these kinds of knowledge proceed, right? Not the human weakness, as he says. But the lower can't do this, right? You know, when he was distinguishing before between speaking appropriately and speaking metaphorically, right? Or figuratively, right? To whom does it belong to make that distinction? Yeah. Yeah. In other words, you can't define what a metaphor is using metaphors. You can exemplify, you know, metaphors, but you can't define what it is to speak metaphorically using metaphors. It doesn't know itself, right? So it doesn't have the character, it must have the character of wisdom, huh? Right. But speaking properly, I can say what it is to speak properly. See? I can define definition and so on, huh? Or distinguish between knowledge in words and statements and knowledge in mathematical symbols and equations. Now, to whom does it belong to distinguish between those two knowledges? Yes. Yes. I can say in words what it is, what a word is, right? Like we've been saying what a name is here and so on, huh? But you can't write an equation that says what an equation is. So the knowledge in mathematical symbols and equations doesn't know itself, right? That's the first thing for a wise man to know himself, right? So the knowledge in words and statements is more the character of wisdom than the knowledge and mathematical symbols and equations, huh? Of course, Heisenberg recognizes that, huh? He's talking about trying to put into words what the scientist is. That is the symbols and equations, right? He often speaks of how the mathematics is wiser than we are. You don't know what it means. You know? And they try to state it in words, you know? And he says, well, we have to go out and talk to other people. But even for the scientists, he says that's a criteria on how much he understands. His ability to express it in words. So, I think that's a kind of a universal principle, huh? That it always belongs to the higher knowledge, and the knowledge has more the character of wisdom, to distinguish between itself and the, what? Lower knowledge, right? And their consequences, since distinction before order, right? It always belongs to the higher knowledge to say what the order is, right? So, it belongs to the natural philosopher, the natural scientist, to say to what extent mathematics is useful in natural science, right? And how it can be used, right? And the limitations of math was seen by the modern scientists, huh? There's a table, too. So, that kind of curves this kind of, in a way, huh? Now, let me give you the, I'm going to be away for a little while, you know. I've got a, I guess, grandfatherly concerns, right? Now, I'm going away this Thursday, I mean, Thursday this week, and then I'm coming back on the 31st, right? So, yeah, I don't, I'm probably getting late that day, but yeah. So, you have to be the following week, okay? So, that'd be the first, yeah, yeah, okay. Let me give you the text here on statement here, which is the easiest part of logic here, about statements, huh? So, we'll start with the statement, and I'll come back, then. Okay? Okay. God, our enlightenment, guardian angels, bring from the lights of our minds, order and illumine our images, and arouse us to consider more correctly. St. Thomas Aquinas, angelic doctor. Pray for us. 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