Introduction to Philosophy & Logic (1999)
40. Analyzing Arguments in Euclid and Aquinas: Main Syllogisms, Continuous Syllogisms, and the Matter of Syllogism
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[0:00] Now, proposition six here is perhaps the first one in Euclid that I remember anyway, and that uses all three kinds of syllogisms. It uses the what Aristotle calls the syllogism period, what some people call the simple categorical syllogism, and it uses the if-then syllogism. Maybe a couple of them if you want to make it spelled out any more, and it uses the what? Either-or syllogism, right? Okay.

[0:35] And notice he wants to prove that in the triangle ABC that when the angles at B and at C the angles ABC and ACB are equal, that the two sides AB and AC will be equal, right? Okay. Now how does he go about showing that, right? See. What is the main syllogism? See. What kind of syllogism is the main syllogism?

[1:06] No. When I say main, the one that has the main conclusion, right? See. Oh. Yeah, yeah. In other words, he he proves that AB and AC

[1:25] must be equal because they can't be unequal, right? Okay. Now you say, how's that proving it? Well, you have to realize that they're either equal or unequal. That two straight lines either equal or unequal. There is no other possibility, right? So if you exclude. Unequal, then you must conclude equal, right? So he doesn't prove we say directly that they're equal, right? But he shows that the other possibility is impossible, right?

[2:01] Okay. So notice here you have an either-or syllogism, where you know that the two lines, right, must be one or the other, right? So it's the form of the either-or syllogism where you eliminate all but one possibility, right? Because you know that one of them must be the case. The two lines are either equal or unequal. Two straight lines, you can see that, right? There's no other possibility.

[2:28] So if you eliminate, you know, them being unequal, then they must be equal. Now you don't always spell this out because it's so kind of obvious this form, as they say, right? You say these two straight lines are either equal or unequal, but they cannot be unequal, therefore they must be equal, right? It sounds almost childish, but in a sense that's what he's arguing, right? Okay. So the main syllogism then is an either-or syllogism.

[2:56] Okay. Now, how does he eliminate the alternative, the possibility that they are unequal? Well, basically it's by the if-then argument. Yeah. He's saying if they are unequal, right, then the lesser will be equal to the greater, but you go through several steps to see this. Right. You might have, you know, okay. What he's saying is that if they are unequal, it makes no difference which one is longer, right?

[3:33] Let's say AC is longer, right? If they are unequal, then you can cut off the lesser one or the greater one, a line equal to the what? Lesser one, right? Okay. So he says let AB be longer than AC, right? Okay. If one of them was they're unequal, right? One would be longer. Then could cut off on AB starting from B a line BD equal to AC, the shorter one, right?

[4:01] Okay. That's a consequence of the fact that they're of unequal, right? And as you know it's an earlier theorem, where you cut off a greater line, a line equal to a given line, right? Okay. And if you did that, then you could draw a line from what? D to C, right? And then you'd have two triangles DBC, ACB, one of which is clearly a part of the other and therefore lesser than the other, right?

[4:28] Okay. Now where's the difficulty, right? Well, because of proposition number four, you could say, or you'd have to say, that DBC and ACB have to be equal, though. Why? Because DBC and ACB, the two triangles, have equal angles. DBC we said was equal to ACB, right? They have equal angles, contain equal sides. BC is common to both, right? Common to both, right? And DB was constructed equal to AC.

[5:07] So though they're turned around, they're two triangles with equal angles, contain equal sides. So by Proposition Four, it follows necessarily that they're equal. But that's absurd, because the lesser can't be equal to the greater, right? Okay.

[5:28] Therefore, they you've overthrown the idea that they're unequal, right? Okay. So now you could make that an if then argument ifs if you want to, but basically you're saying if the lines are unequal, then the part will be equal to the whole, the less to the greater, right? But that's impossible. Therefore, they can't be unequal, right? Okay. So you have an if-then argument, at least at least one, if not more.

[5:54] And then you say, but how do you show that that you would follow from these admissions that they're equal? Well, you'd say that the triangle DBC and the triangle ACB are triangles having equal angles, contain equal sides, and triangles having equal angles contain equal sides are equal. That's the regular syllogism, right? That's the one he used back in number one, right? So all three kinds of syllogisms are used in what is a very simple and very elementary geometrical demonstration, but shows you you can't really analyze it without knowing all three kinds of what syllogisms, right?

[6:35] Do you see that? Where's the simple one at the same time So? The simple one is you're you're giving a syllogism that triangle DBC, which is a part of ABC, right, is nevertheless equal to it, because DBC has an angle DBC which is equal to the angle ACB in the greater triangle, right? And that angle contains equal sides because DB is equal to AC, and BC, of course, is common to both, right?

[7:08] So sides DB, BC surrounding the angle at B, are equal to AC, CB surrounding the angle at C, ACB. So you have two triangles then, right? Having what equal angles contain equal sides, and all such triangles are equal by Theorem Four, right? Right. Therefore. Is the middle term Proposition Four? Well, Proposition Four is the middle term, yeah, yeah. Proposition Four is the premise. Well, the premise is really right.

[7:40] It says triangles having equal angles contain equal sides are equal. And then we can show from the constructions a man has made that these triangles fulfill those two conditions, right? And therefore they must be equal. But they can't be equal because one is clearly a part of the other, right? Euclid says, you know, lesser to the greater. I mean, it goes back to really the axiom that the whole is greater than the part, right?

[8:04] So he's resolving all the way back to the axioms, which are the statements known to themselves by all men. You see that? Okay. I mean, the triangles are turned around. You know, I mean, you know, you can turn around and see easily they're the same. Okay.

[8:32] Now, this is not a particularly interesting or profound proposition, number twenty-nine. But again, just to show you how a very elementary theorem can sometimes involve. All three kinds of syllogisms,

[8:52] so the second page is different from the first page. They're all just one kind of syllogism, right? But here you have all three kinds. Okay. Now he says, "Any prime number is prime to any number which it does not measure. " It's almost obvious, right? Okay. So he says, "Let a be a prime number. " You know, you know what a prime number is, right? A prime number is a number which is measured only by one and not by any other number.

[9:18] So one and two and not one, but two and three and five and seven and eleven, so on are prime numbers. But four is composite, right? Measured by two. Six is composite. Eight is composite. Nine is composite. Ten is composite. Twelve is. You know the difference, right? Okay. Okay. Now a number is said to be prime to another number when there's no number that what?

[9:47] Measure them, right, huh? Okay. Now sometimes Euclid will, you know, if you if you have three and nine, are they prime to one another? Well, Euclid will say no because three and nine are both measured by what? Three. Okay. You got to understand that way of speaking, huh? Okay. So he says, "Let a be a prime number and let it not measure b. " I say that b and a are prime to one another, right?

[10:15] They're measured by no number, right? Now, for if b, a are not prime to one another, what kind of syllogism is going to be using here as his main one? If then. Well. Either or. It's going to be either or because yeah, you see. Now when he says the main syllogism, we don't mean it's the one that is the most interesting or the one that really is the great discovery, right?

[10:39] But we mean the one whose conclusion is the main conclusion, right? Okay. Okay. And either these two numbers are prime to one another or they're not, right? He wants to prove they're prime to one another, but he shows that if they're not prime to one another, some difficulty follows, right? Okay. So he's going to be using the either or syllogism, eliminating the one of possibilities and concluding to the rest, right?

[11:13] Okay. So he says, "For if b, a are not prime to one another, some number will measure them. " Okay. That's what. That's obvious, right? Okay. Let c then measure them, right?

[11:31] Now, what kind of a syllogism is this? Since c measures b, right? According to his hypothesis, right? And a we're given does not measure b. That's one of the things we're given at the beginning, right? therefore c is not the same with a. If then. Since C measures B, and A does not measure B, therefore C is not the same with A.

[12:02] Even the syllogisms we're using and talking about the syllogisms. Yeah, yeah. And what figure is it in? And what figure is it in First, second or third? See a second. Yeah, see. Now there you have a syllogism, the second figure, right? Yeah. You're affirming and denying the same thing, right? You're affirming measures B of C, and denying measures B of A, right? Okay. You were given above that, you know, any prime number is prime to any number, which does not measure, right?

[12:38] So you're given that A does not measure B, right? So if A and B were not prime to one another, something would measure both of them, namely C. And C couldn't be A, because C measures B in this hypothesis, and A doesn't measure B, right? Okay. Therefore, C is not the same with A. Now, since C measures B and A, then it follows that it measures A, which is prime, right?

[13:09] But no prime number can be measured by any other number, only by itself, right? Which is impossible. Okay. Therefore, no number measures B and A. Therefore, the other alternative must be true that they're prime to another. Okay. So you have actually in that you have all what three kinds of syllogisms, huh? The major, I mean, the what do you call it? The chief syllogism, the main syllogism, is that A and B are either prime to one another or not prime to another.

[13:49] But they can't not be. They cannot be what? Not be prime to another. Cannot have something measuring them, right? Therefore, they must be prime to one another, right? Okay. Now the either or statement is obvious, right? Either prime or you might say either prime or composite to one another, right? Okay. And therefore they can't be composite. They must be prime to one another. How do you know they can't be composite?

[14:18] What? Well, if you say that they're composite, right, then would follow that some number C measures both of them, right? Okay. And this number C that measures both of them couldn't be A, because we're given that A doesn't measure B, right? Therefore, there must be some number C that measures both A and B. But A is a prime number, and no number can measure a prime number, except itself, perhaps, right?

[14:48] You can speak that way, right? Three measures three, right? So therefore. That's impossible, right? Therefore, the antecedent that they're composite to another, right? That they're measured by some number, is impossible. Therefore, the other alternative must be the true one, that they're prime to another, right? Where is he at? He's saying, if if they are what, composite, right? Then there's some number that measures them, right?

[15:21] And if there's some number that measures it, then that number measures a, right? But no prime number can be measured by any number except perhaps by itself. But he excludes that possibility, right? Because a doesn't measure b, right? And c is the number that measures both a and b. So a can't be c, or c can't be a, because a doesn't measure b, c does measure b.

[15:50] So you have a a a syllogism in the second figure, right? Here we got the second figure syllogism, unlike the first page where they're all in the first figure, the affirmative. And you got the basically either or, and you got the if then. Now I was mentioning that was the last time I was here. I happened to be reading the my favorite book there, the Summa Contra Gentiles, and Thomas was arguing that the relation of God to creature is not really in God, okay?

[16:28] And he says that a relation, if it was really in God, either has to be the substance of God, right? Or an act, or an accident of God, right? Okay. But in a previous chapter, he's shown there's no accidents in God. Therefore, eliminates that possibility, right? Okay. And then he eliminates the possibility of being the substance of God, because if it's the substance of God, then God would be what?

[17:02] His very substance towards creatures, and therefore depend upon creatures, huh? But God in no way depends upon creatures. Therefore, he can't be that, huh? So he he he's got what? All the kinds of syllogisms there, right?

[17:32] So these things always drive students crazy, but that's the that's the reason. But no, the reason why I give it here is not so much to drive them crazy, although that's that's laudable too. no, I mean it's to show that even in a very elementary theorem like these two. It might require as many as all three kinds of syllogisms, right?

[17:56] Now, when the conclusion of one syllogism is a premise, another syllogism I call those what continuous syllogisms. See, just like the mathematician, you know, sometimes you know he calls Um a proportion like this: four is to six, and six is to nine. You call that proportional because you put the end of one as the beginning of the next one, right? But if you have two is to three, and four is to six, that's not proportional because it's not the same thing that the end of one is the beginning of the other.

[18:29] Just by kind of a metaphorical likeness there to to put continuous syllogisms. In the continuous, the end of one is beginning of the next, right? The end of America is the beginning of Canada, or the end of America is the beginning of Mexico, vice versa. So, when syllogisms, when the conclusion of one syllogism is a premise, the next one I call those two syllogisms what continuous. Or when what is defined, you know. One definition is in another definition that I call those definitions what continuous right like the definition of quadrilateral square continuous right okay the square is defined by quadrilateral which is defined by its own definition right okay so you have continuous even simple geometrical theorems you have continuous what syllogisms right even that first

[19:21] one in Euclid you had two syllogisms whose conclusions were necessary to see the minor premise in the third syllogism in Thomas here you had what a syllogism and then a syllogism backing up the major premise and a syllogism backing up the minor premise right okay and these other two examples from Euclid you have a either or syllogism which is backed up maybe by an if then syllogism in one of its premises and backed up if then syllogism in one of its premises by a simple syllogism right okay

[19:58] so those eight forms are going to be used again and again sometimes just one sometimes two or three right go back to to Socrates there in the the Meno right you take the first argument there the argument for virtue being able to be taught right okay the main syllogism that he gives there is an if then syllogism if virtue is knowledge then virtue can be taught right but virtue is knowledge therefore it can be taught right okay now the if then statement is kind of obvious if virtue is knowledge then it can be taught it's knowledge it's taught the simple statement the second statement there in the if then one if virtue is knowledge he shows by a regular syllogism virtue directs us to the

[20:50] good what directs us to the good is knowledge therefore virtue is knowledge right so simple syllogism backs up the second premise of an if then syllogism huh you see that huh so you have the different kinds of syllogism sometimes as well as the same kind of syllogism with one term backing up another one right so as you go through Book One of Euclid I mean you know there's one continuous syllogism after another right A is used to prove B and B C and then on down the line right and it becomes a long chain there at the end huh to go through all the steps okay

[21:37] the thing to do is is to is to know this and and to use it right it's better as Aristotle says in the logic there to know a few basic things and to use them and to use them often and use them well because to use them for so many things right and You don't always stop and analyze the argument, but you suddenly just stop. Oh, I see, and it's all put together, you know.

[22:01] And then, then, I should type it down. There was an argument in the the Summa the other day. I think it's using all three kinds, you know, and Summa Contra Gentiles. But those will be used again and again.

[22:17] But everywhere from from Euclid, the first science to theology, the queen of the sciences, the last one, excuse all the way to the forms. Now, of course, you know, in the theater you also have, a, inductions, right? So Socrates induces the changes between what, contraries, right? And then he syllogizes from that to the immortality of the soul, the first argument.

[22:53] And with with the induction example too, sometimes we find them used in a second sense. The first meaning of induction is an argument from many singulars to what one universal, right? But sometimes you have something like an induction where you proceed from many what

[23:16] less universal things to one more what universal thing, right? Okay. And suddenly you have a kind of philosophical example too. Instead of going from one singer to another singer of the same kind, you go from one particular kind of thing to another particular kind. Okay.

[23:37] So those other forms of argument will be used, right? And as much as Dionne would often point out, Aristotle sometimes he'll give the syllogism, then he'll give a kind of a sign, John, from the weaker arguments. Why does he do that, right? To take us back to the senses, huh?

[23:59] The mind wants to return always to the senses. And the alpha and the omega, right? You want to go back to your beginning, and the senses are the beginning of our knowledge. Do we

[24:15] have more time, or you want to stop, or what? I know what your schedule is. Yeah. Okay. Now, next time, maybe we should talk a little bit about the matter of syllogism. This is very sketchy, but just a little bit here, huh? By this time, the course is over. Logic, so that's time for too much, right? Okay. But that's how many copies you need.

[24:50] Eight, five, six, eight copies here. Okay, let me talk logic for a while. So these are run down these supplies of these things here.

[25:13] So I'm touching here upon some of the more important aspects of what they call the matter of syllogism, right? Because you can have a syllogism, you know, from statements that are known to be true, right, and known to be necessarily true. You can syllogize from statements that are only what probable, right? Okay. People forget that they forget that the rigor of the syllogism doesn't mean that the conclusion is necessarily true.

[25:49] And you can compare that to adding and subtracting, multiplying and dividing, right? Correctly adding two numbers doesn't mean the number you got is correct because the numbers you correctly added might have been the incorrect numbers, right? And so when I define calculating as coming to know or guess a number of other numbers, how could you say that? You know, you you know, it's a dangerous thing adding, subtracting, multiplying, dividing.

[26:17] But the point is that you might not have the correct numbers to add, subtract, multiply, divide. So all the rigor of adding and subtracting doesn't make necessarily that you have the right number at the end, right?

[26:33] And the same way with syllogism, right? You can even syllogize from false statements, right? So if I say every man is a stone and every stone is an animal, it follows necessarily that every man is a what animal? If every man is a stone and every stone is an animal, well, then every man is an animal, right? Okay. But it's not a good argument because the premises are clearly false, right?

[27:03] You know. So you can have premises all the way from what ones that are necessarily true and even seen as necessarily true to ones that are only probable to ones that might even be what simply false, right?

[27:19] Aristotle takes apart Melissus there in the first book of the Physics, right? He says this conclusion doesn't follow and the premises aren't even true or probable, right? Poor guy. Aristotle the great, he's very kind. He says, you know, that's difficult in the first book because he didn't have logic, right?

[27:43] But you can see what Socrates reasons in the in the Meno there. He reasons that the virtue can be taught and which cannot be taught. Well, this can't be necessary reasoning when you're reasoning. But contradictory conclusions. But dialectic enables you to do that, right? You can reason from probable opinions, right, to contradictory conclusions. In demonstration, where you reason from statements necessarily true, you never reason to opposite conclusions.

[28:10] You always reason to one half of a contradiction, right? So we'll talk a little bit next time. You know, just a little induction to the matter of the difference between dialectic and demonstration and and sophistical argument and so on, right? Okay. We'll expand a bit upon it. Very brief. A little

[28:37] exercise on equivocal names. What's that? Where is it? I have it here. Let's see if they're around the office still. Where is it? Exercise.

[28:55] Is the first name being said equivocally or unequivocally of the second name? Okay. Again, a copy's here. Just keep that in mind. Good exercise for next time. You know. Okay. You know, in the case of of the five predicables the Isagoge, you have a name said of many things unequivocally, right, with the same meaning in mind, right? So, but sometimes the name is said of many things, a different meaning in mind.

[29:24] So, see if you can figure all these out here. You also go through the fallacies. Yeah, a little bit. Yeah, but I just do the main ones, you know. The which I say I do

[29:41] here, I just do the the first one in language, the first two outside of language, right? Yeah. Yeah. Okay. The fallacy of the accidental. Yeah. But again, there are thirteen fallacies. Eventually, you have to learn them all. But some of them are much more important, much more common than the others, right? And the three that are most common are the fallacy of equivocation, the fallacy of mixing up different senses of the same word, and the fallacy of the accidental, making up the accidental and the as such, and the fallacy of simply in some respect, right?

[30:21] But they also correspond to three distinctions. They're not really divisions, I think, right? But are made over and over again in philosophy. These three kinds of distinctions, right? You meet them again and again and again, everybody. And if you don't understand these kinds of distinctions, you know, you have a hard time understanding them, right? And once you start to get a hold of each kind of distinction, you'll be meeting it again and again.

[30:45] It's absolutely crucial. At least for all you can look What's what? Your love and friendship course. Yeah. I first invented the course. see. And of course, he was saying, "you have to put sex in there. I said, "No, I'm going to put sex in there. Love and friendship. And so, and you know, getting you know, you know, a couple sections of it, you know, in advance of course.

[31:11] And then one semester, I could only offer one section. What do I do? So I said, "We'll put, you know, eight three in the morning, right? That cuts down the numbers. It was nine three, ten thirty, you know."

[31:28] But a lot of people think, you know, coming to such a course, I think it's going to be fluff, you know, and just you know, talking about you know, social relations, you know, and they realize it's all real loud. So. That was really good. That's good stuff. I got to think about.