Introduction to Philosophy & Logic (1999)
27. Compound Statements and the Square of Opposition
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[0:00] The hypothetical conditional, they call it sometimes. You have two simple statements put together with if and what, then, right? So you say if this number is two, then this number is even, okay? Or this number is after four, something, right? So this number is two, this number is even. Taken by themselves, would be two simple statements, right? But they're combined by means of if and then, right?

[0:51] So the simple statement in the if part is called the antecedent. Ante meaning what before, right? And in the then part, it's called the what consequent, right? You see, antecedent and consequent.

[1:20] Now the either or statement, again, like the if then statement, is put together from at least two, but it can be more than two, right? Simple statements joined by either or, right? You might say a number is either odd or even, right? Okay.

[1:50] And there you try to exhaust the possibilities, right? Okay. Say a triangle, though, is either equilateral, or isosceles, or what? That's the way you probably usually write these things. But you could expand in this. You could say either a number is odd or a number is even, right? Either a triangle is equilateral or a triangle is isosceles or a triangle is what? Yeah. So here we're kind of contracting it, so you don't make explicit, right?

[2:48] You don't spell out the simple statements the way we did in the if then, right? We could have wanted to, right? But usually we what, contract it right? Okay.

[3:01] Going back to a simple statement, you know, either the affirmative statement is true, or the what, negative statement is true, right? Okay. Either or, right? Okay.

[3:23] So as I say, the way this is stated here, you don't seem to have two simple statements or three simple statements combined, right? But it's kind of a contraction of that, right? Okay.

[3:38] I notice in the sixth theorem there in point of Euclid, that's the theorem where he says that if the angles are equal, then the what, sides will be equal, right? Okay. And so, essentially, it's either or statement there, right? Either these two sides are equal, or they are what, unequal, right? Okay. And eventually he's going to show that if they're unequal, he's going to end up with part equal to the whole, and therefore he's going to what, conclude they must be what, equal, right?

[4:18] Okay. Notice again, you might probably say either these sides are equal or they're unequal, right? But you're really putting together essentially two statements: either these sides are equal or these sides are unequal. But that's

[4:37] a little bit more wordy views, right? But it's better to kind of spell out sometimes to see what you're actually doing, right? You're combining two or more what, simple statements, right? Okay. But by means of or, or for Shakespeare, or either or, or, that's right. Okay.

[5:00] That's pretty simple, right? Now there are other kinds of compound statements. If I say that John and Paul are students, right? In a way, I'm saying John is a student and what? Paul is a student. Yeah. Rather than saying John is a student and Paul is a student, I'd say John and Paul are students, right? In a way, I've got two different statements there, right? Okay. But that kind of a you know conjunctive one I call statement is not important for reasoning, huh?

[5:31] Okay. Okay. But the either or and the if then statement are very important for reasoning. We'll see them when you get to take up the syllogism and so on. Okay. So it's because we have in mind reasoning eventually that we emphasize those two among the compound statements, huh?

[6:02] What if you say that man is the only animal that learns logic? Is that a compound statement or a simple statement? It seems to be a compound. statement. Yeah, Some might say it implies that man learns logic and no other animal learns logic, right?

[6:34] Now, let's go to true and false. Now, what does true and false mean in the if-then statement, right? Well, you know, if you're baking or cooking or something, you mix two sweet things together. You expect the resulting thing to be what?

[7:05] So, if an if-then statement you put together two true simple statements, you might expect the compound to be true, right? But actually, it's possible to make a what true if-then statement out of two what false simple statements. Take the statement, for example, that Socrates

[7:30] is a mother. A simple statement and it's false, right? And the simple statement, Socrates is a woman. That's false too, right? Okay. And now, if I combine these and I say, "If Socrates is a mother, then Socrates is a woman," that seems to be true, right? Okay.

[8:04] But now, let's if I say, "If Mary is a mother," let's take Mary, mother God, mother. Is it Mary? Mary is a mother, right? Excuse me. Mary is a woman.

[8:37] Then Mary is a mother. Okay. Let's take Mary, as the mother of God. Okay. If Mary is a woman, then Mary is a mother, right? Now, Mary is a woman. That simple statement by itself is true or false. True. And Mary is a mother, true or false. True. But is the statement true if Mary is a woman? Then Mary is a mother. Well, often. So here you have a false if-then statement made out of two true simple statements, and up here you have a what true if-then statement made out of two false ones, right?

[9:23] This is not always the case. Obviously, I've taken these extreme cases to show that truth and falsity must mean something different than truth and falsity in a simple statement. Mean something else. Then you can't judge it by the truth or falsity of the what simple statement. Making it up, right? You must look for something else that tells you whether it's true or false, right? Okay. And you'll see sometimes there's an argument of Aristotle in the seventh book of Natural Hearing and some people criticize because he's using a simple statement and a compound. statement The simple statement is false.

[9:59] and Thomas says, so what? That's not what, truth or falsity means, right? In the if then statement, right? Okay. If you look at that argument in the sixth theorem there, in the first book there, he's going to reason that if these sides are unequal, then this will be so, right? And that's true. But what follows seems to be false, right? But the if then statement is true.

[10:34] Otherwise, you couldn't use it further. Okay. Now, what does truth and falsity mean then here in the if then statement? Can you figure out from these examples. The consequent follows necessarily from the antecedent. Yeah. In other words, if Socrates is a mother, not saying that he is in fact a mother, saying if he is a mother, then he will be a woman, right? Okay. Not saying that in fact he is a mother or in fact he is a woman.

[11:10] You're saying if this is so, then that will be so, right? Okay. Down here, it's like false, right? Because doesn't follow that if she's a woman, that therefore she's a mother, right? That follows that any woman would have to be a what? A mother, right? Okay.

[11:37] So notice, the, what Shakespeare said. the. Reason is the ability to look before and after, right? Here, the before and after is very important, right? If I reverse this and I said if Mary is a mother, then Mary is a woman, that would be true. But sometimes when you reverse it, it's not, right? Okay.

[12:03] Now it could be that going both ways sometimes it's true. Take the example this here. If this number is two, then this number is half of four.

[12:29] Is that true? Okay. And now there the reason goes back to that half of four is what Porphyry would call a property of two, right? A property in the strict sense, right? Now if you reverse that and you say if this number is half of four, then this number is two, would that also be true? No. It's because half of four is a property of two in the strictest sense, right?

[12:59] It belongs to only two, to every two, and always, right? But now, if I took a property in a looser sense, if I said two is if the number is two, then this number is let's say less than ten, that would be true, right? But now, if you reverse it and you say if this number is less than ten, then this number is two, now it's false, right?

[13:24] If less than ten is not convertible with this, huh? Okay. If you had a definition in one property, you can turn it around, right? If this is a square and this is an equilateral right angle quadrilateral, and, vice versa, right? Okay.

[13:54] But if you had something more universal there, like like woman is more universal than mother, right? It doesn't happen to work for both ways, is it? If I say if this number is two, then this number is let's say even, right? It's true. If this number is even, then it's two. No, it's more universal, right? Okay. So if you stop and consider these examples, you can see that truth here means that the consequent follows necessarily from the antecedent, and falsehood means it doesn't follow necessarily.

[14:35] So even if both simple statements are true, but one doesn't follow necessarily from the other, if this number, yeah, well, we already exemplified that. What about Shakespeare now? He says, "What is a man if his chief good and market of his time be but to sleep and feed? A beast no more." He's saying if if man's chief good is but to sleep and feed, then man is no more than a beast.

[15:17] Is that true or false? True. Yeah. Even though it's false that man is no more than a beast, it's false that man's chief good is but to sleep and feed, right? But yeah, right. Or like in my comparison, I say, "What is a three if it be half of four?"

[15:42] Yeah. So if three is half of four, then three is two. Too false. It's true, even though it's false that three is half of four, and it's false that three is two. But if it were half of four, it would be two, right? If man's chief good was to sleep and feed, then man would be a beast, right?

[16:10] No, it's just that you know, is arguing from the proposition that if the sides are equal, right, then you can cut off from the longer side one equal to the shorter one. True or false?

[16:24] Yeah. But in fact, they have to be equal when the angles are made to be equal, and you can't cut off. Therefore, you know, a line equal to the other one. There's only part of the person, right? But if one was longer, then you could do that, right? So he's reasoning from a true if then statement. The parts of which are what? False. Yeah. But one follows from the other, right?

[16:50] Shakespeare doing the same thing, right? But you can see why the truth of the if then statement is not the basic truth or act, right? Because you know you go on all dizzying. If this is so, then that would be so. If this is done, that's all right. If I won the lottery, you know, if I want to, have I won the lottery, right? I know is Socrates in fact a mother or not, right?

[17:19] Is it true or not, right? Is he in fact a mother? And this doesn't tell me, right? So later on, you study reasoning, and you study what we call the hypothetical syllogism, or the if then syllogism, as I call. You have an if then statement, but you'll add to that a what? A simple statement, and your conclusion will be a simple statement, right? Okay. That's what Shakespeare is reasoning, right?

[17:50] If man's chief good is to sleep and feed, then man is a beast, right? No more, just a beast. But man is more than a beast, right? Therefore, it can't be like his chief good is about to sleep and feed, right? So your conclusion is a simple statement, and your second premise is a simple statement, but your first statement was a what? If then, right? And Euclid will do the same thing.

[18:17] He'll say, if these sides are equal, then you could cut off from the greater one, right, one equal to the lesser one, and then you could draw a line from that over to the side. You know, he does that. This his angles equal, and if one of these is longer, say this is longer than that, this one here, I could cut off this longer side, one equal to this, right?

[18:42] And then I could draw one like this, and then I have two triangles with an equal angle containing equal sides, and therefore this part would be equal to the whole. to observe. So you conclude that these sides cannot be what unequal, right? They must be equal, right? And you see if part cannot be equal. But notice you had an if-then statement, right? If these are unequal, you could cut off the greater line.

[19:14] Well, that's not. And then you could draw that line, right? And follow something you know is not so, right? But you have to know that something is or is not so, so you can draw a conclusion that something is or is not so in reality. Things. So we don't have, we don't put together the if-then syllogism to two of the if-then statements and draw an if-then conclusion, Doesn't get us anywhere as far as knowing the way things really are.

[19:45] But you have an if-then statement, and then you have a simple statement, and you can draw a simple statement conclusion. We'll see how you do that. Take that up. Okay So notice now when you say that a statement is speech signifying the true or the false, if you say statement of the if-then statement as well as the simple statement, it's not really the same definition, is it? It's not purely equivocal of the word true, right?

[20:18] There's a connection between the two, right? And here you can see very clearly the connection because every mother is a woman, It's based on that simple statement. And the falsity of this is because not every woman is a what mother, right? Goes back to the simple statement. You can reason to a simple statement from one of these if-then statements, but not a if-then statement alone. You've got to know something about the way things are.

[20:47] Can you say in some way if you have a true if-then statement, you're knowing something about the way things are. You could say, well, that's the way things are. If Socrates is a mother, then Socrates is a woman. Yeah, it's not really saying this is the way things are or are not. You want to know the way things are or are not. Should I say more like it's based on the way things are or something?

[21:11] Yeah, yeah, that's the connection. Not purely equivocal. These are true here, right? But it doesn't mean the same thing. It's a word equivocal by reason true instead of the simple statement instead of the if-then statement. Now the either-or statement likewise will have

[21:32] a third meaning here of true, right? That's where I used to do it between. If I say a triangle is either equilateral or isosceles,

[22:00] okay, is that true or false that statement? False. I say a number is either odd or even. True or false? True. Why is this false and why is this

[22:27] true? Yeah. So in a way, the either-or statement depends upon the logic of division, right? And the idea that some people, you know, see the word divide

[22:45] for the word empty, right? But whether that's good etymology or not, I don't know. But the idea that a division should empty out. Everything that's in the thing, right? Okay. So it's like I got a basket here with triangles in it, right? And I pour out and the equilateral triangles, and the isosceles triangles. That's that's all there is, right? And something still in the basket, right? I haven't emptied the whole thing out, right?

[23:16] But here I haven't emptied them all, right? All the numbers, right? As I put all the triangles in the basket and I pour out and I get just these two. I haven't emptied the basket out, right? So obviously, what truth means here is that you've exhausted the possibilities, right? And maybe two may be enough, but two may not be enough, right? As you know from the rule of two or three, usually two or three are enough, but it's not what universally true, right?

[23:48] Okay. Um. And I was giving an example there from Thomas there, where following Porphyry says that every name said univocally of many things is either a genus or species or difference or property or what accident. And then he eliminates each one of those five and concludes that no name is said what univocally of God and creatures, right? But the goodness of the argument depends upon genus, difference, species, property, accident being an exhaustive right distinction of names said univocally of many things.

[24:37] Question? Okay. Um. it's harder to see then this is exhaustive right? Or if you said equilateral or isosceles or scalene, right? It's easier to see in mathematics that something is what Exhaustive, right?

[24:59] But sometimes even there you have to what think about, right? You could divide these triangles into equilateral or isosceles and scalene, and they divide them into right angle and obtuse and acute angle. They might say, could we crisscross these divisions and get nine different kinds of triangles, right? Right? No, I don't think so. Like good men, bad, men, good human beings, bad ones, male, female. You can crisscross them in four real things.

[25:41] But is there such a thing as a right angle equilateral triangle? No. But you have to know that a triangle always has its interior angles equal to two right angles. And then you have to know that an equilateral triangle, the three angles must be equal, right? And so if one of them is right angle, the other two couldn't be equal to it, right? You have three right angles, right?

[26:07] But you have to do some reasoning to realize that um, I haven't left out a possibility: of the right angle you know, equilateral triangle. There's no such animal. Now, if you get to an isosceles triangle, where you could have a right angle isosceles triangle, like half of a square, but you could have a sort of an obtuse angle one, right? So you have to

[26:39] do some thinking sometimes, even in math. Generally speaking, it's easy to see that division is complete, right? Every statement is either what simple statement is either affirmative or what negative. That's exhaustive, right?

[26:59] Two straight lines are either equal or unequal, right? So when Euclid in that sixth theorem shows the impossibility of there being unequal, you can conclude they must be what equal, right?

[27:18] Well, Thomas is arguing, you know, as to how the members of the Trinity are distinguished, right? Well, he says two kinds of distinction: formal distinction and what material distinction. Material distinction is based upon the division of the continuals. Well, there's nothing continuous in God, right? It's not a body, so eliminate that. So it must be formal distinction. Now, formal distinction is based upon opposition. Thomas says there's four kinds of opposition.

[27:48] Going back to the categories of Aristotle, he distinguishes the four, and then he eliminates three of them as impossible in God. Therefore, it must be a relative distinction.

[28:02] But I say now, are there only four kinds of opposition contradictory, privation and lack, or you know, lack and having, contraries and relatives? That's harder to see, right?

[28:17] I can't think of any other kind of opposition. You know, You have to kind of stop and, you know, think about this, right? Or we say God is a cause. There's only four kinds of cause: matter, form, mover, and end. right?

[28:36] The only kinds of cause are... It's harder to see, right? You know. And some cases, you know, I mean, we may use an either-or statement that's only what probable, right? huh? We don't see that these are the only, right?

[28:54] Now a man is either white or black or brown or red or yellow, right? I think it's all possibilities, right? Yeah. If they find you know a green man on Mars or something, I'll be surprised. I don't say,

[29:15] why can't there be purple people, or you know? Yeah, yeah. But there, I don't see that these are the only possibilities, right? So sometimes an either or statement there is only probable. Even a simple statement can be only probable.

[29:41] So we're talking here about on page six now. We get to it what true and false mean in either or statements. That's pretty easy to see. But at the end, of reason is to know the truth of a simple statement. That's what I'm saying, right? Because then you're saying that something is or is not, and things, right? And you can see when I swear to. Tell the truth, the whole truth, and nothing but the truth, right?

[30:02] The kind of truth interested in the courtroom is the truth of the simple statement, right? He's guilty or not guilty, right? Everybody knows that if he murders so and so, then he should go to jail or something, right? The question is not so much that, but did he or did he not, right? Should he or should he not go to jail, right? You want to get back to the to the truth of the simple statement, right?

[30:32] Now, the next part here, starting in the bottom of page six and going to the end there, is a extension on the idea of contradictory statements because there's a little problem, right? A contradictory statements as you recall, regular statements, simple statements, one affirmative and one negative, right? But the problem arises

[31:07] when you have a universal subject rather than a singular subject, right? Now, if I say Socrates is wise, the contradictory to that is Socrates is not wise, huh? Okay.

[31:32] No problem with your contradictory that. But now suppose I say man is wise. What's the contradictory of that? Man is not wise, right? Okay.

[31:49] But now, if you think a little more carefully, you say, well, man is not a singular, right? Man is said of all men; it's a universal. Okay. And so, when I'm saying man is wise, what do I mean?

[32:07] Do I mean that every man is wise? If I say man is not wise, do I mean no man is wise, or do I mean some men are not? Right? If I say women are beautiful, what do you mean, Mr. Burquist? Well, women are beautiful. Oh, it's a couple. See, you mean everyone is beautiful, Mr. Burquist? Well, you see. Okay. Now, usually to talk about this, they take a universal subject and they make a little square.

[32:49] Okay. We call this the square of opposition. We allow squares in logic, but no circles in logic. Okay. I always tell the students tell the students.

[33:04] In one corner we put the universal affirmative statement, statement form of let's say every B is A. Every woman is beautiful, for example. And then in the other corner over here, the universal negative

[33:39] form: no B is A, like no woman is beautiful, right? Then down here we put down the particular affirmative, like some B is A,

[34:04] some woman is beautiful, okay? And the fourth corner, the particular negative: some B is not A, like some woman is not beautiful, okay? But mix, you know, take any subject predicate you want, right? Every man is wise, some man is wise, no man is wise, some man is not wise, right?

[34:46] So notice if you go back to the definition of contradictory statements, they're statements with the same subject and predicate, right? One affirmative and one what? Negative, right? And they're opposed such that both cannot be true, both cannot be false, right? But one must be true and the other must be false. Now to see which affirmative and negative are such with a singular subject is no problem, right?

[35:23] President Bush is sitting, President Bush is not sitting, obviously just a contradiction, right? But now what is the contradictory to the statement every B is A?

[35:36] Is it no B is A, or is it some B is not A? Well, if you think an example here: every woman is beautiful and no woman is beautiful, they might both be what? False, right? Every man is wise. No man is wise. They could both be what false, yeah. Now it may be true in some cases that the universal affirmative, universal negative, is what true, right?

[36:14] Like every two is half of four, right? And no two is half of four It's false, right? But sometimes both universals are what false. Every man is sitting.

[36:31] Which different on room here? It's false, right? No man is sitting. False, right? Okay. So these two here can both be false. Okay. So if you know that you have two statements in the form every B is A, no B is A, same subject B, same predicate A, right? One true and, one negative. They are not necessarily one true and then a false, right? Regardless of which you know is the true, which is false, then for all you know, they might both be what false, right?

[37:13] Okay. Now, how about some B is A and some B is not A? Maybe some woman is beautiful, and some woman is what not beautiful, right? Maybe some man is wise, and some man is not wise. Maybe some man is sitting, and some man is what not sitting to take it. This controversial example, right? So, is it true that some man is sitting or some man is not sitting?

[37:50] Both true. And both true. Some man is wise, and some man is not wise. Some man is good, some man is not good. Okay. So the two particulars could both be what true. Both can be true.

[38:14] So as far as the form of the statement is concerned, every B is A and no B is A are not opposed as contradictories.

[38:27] You don't see that one of them must be true and the other must be false, regardless of which way you know is true or false, right? It's possible for both to be false.

[38:41] Every man is white. No man is white. Both are false. Every man is black. No man is black. Every human being is male. No human being is male. Right? They're both false, right? Some man, some human being is male. Some human being is not male. Both are true. Some man is white. Some man is white. So what? are, What is the contradictory then of every b is a?

[39:16] Yeah, yeah. These, what they call the diagonals, right? Every b is a, and some b is not a, are opposed, regardless what b and a are, without knowing what b or a is, right? I mean the same thing by b in both of these, and by a the same thing, right? Those two statements cannot both be true. That's kind of obvious stuff, but think about it. If it's true that every b is an a, that means without an exception, right?

[39:46] Every b is an a. It must be false that some b is not a, right? But if even one b is not an a, right? It can't be true that every b is an a, right? Okay um

[40:06] So those are contradictories, those diagonals. Likewise, the other diagonals: no B is A and some B is A. Can they both be true? Well, if you stop and know what it means to say no B is A, that's universal negative, right? A is denied of all the Bs, right? So if it's true then no B is A, it would have to be false that even some B is A, right?

[40:34] And vice versa, if some B is A is true, if even one B is an A, in other words, then it must be false that no B is A, right?

[40:48] Now, because in reasoning we're concerned with subjects like the universal more than the singulars, right? That's why we have to consider the square of opposition, right?

[41:03] What statements, affirmative and negative, with the same subject and predicate, where the subject is universal, are really opposed in the way we call a contradictory. They cannot both be true, cannot both be false. So one must be true, the other must be false. Well it's the diagonals, right?

[41:21] Now, as I was pointing out earlier today, you've got to realize that when I say some B is A, it's true. I'm not denying that every B is A. I'm not asserting that some B is not A either.

[41:45] If I'm in my room at home, all by myself, sitting down, some man is sitting. I don't see anybody else that I'm wondering about. I can guess that some, you know,

[42:01] but for all I know, everybody else might be sitting right now. So some B is A is true, but there some is not, or every is right. Now sometimes it's an exercise, you know, we'll take each of the four corners, right? And we'll say if, yeah, if then, if this is true, what about the other four corners, right?

[42:38] And maybe if you know that something is true or false because of that, or maybe unknown, right? So just do that simply there. If it's true that every B is A, then no B is A must be what? False. Yeah. And some B is not A must be what?

[43:00] But some B is A. It's true, right? If every student is sitting, then some student is like sitting too, right? Okay. If it's true that no B is A, then some B is A must be what? If every B is A, obviously must be false, right? But some B is not A would be what? True. Now, if it's true that some B is A, what can you say?

[43:39] Anything else necessarily true or false? No B is A. False. Yeah. Some B is A is true, then it's obviously false that no B is A, right? But if some B is A, is it necessary that every B is A? So we say unknown, right? And some B is not A is like.

[43:59] No, it's both things would have to be unknown if one is unknown. It's a one or no, the other be no. By this pair opposition, right? Okay. And if some B is not A is true, what do you know is false?

[44:15] But no B is A is that false. And some B is A. in two diagonals, One is unknown, and the other is unknown. If one is unknown, the other is unknown.

[44:27] Then you go around with the opposite and say false. If it's false that every B is A, what about no B is A? Unknown. It's unknown. And it's unknown that some B is A is going to be what? No. The fact is false that every B is A. It could be false because some are and some are not. Or it might be false because none are. I don't know.

[44:54] So if this is false, you know that some B is not A is like true. But the other two are unknown. The same way here, it's true that no. Excuse me. If it's false that no B is A, you know the diagonal is like true. But the other two are unknown. It's even nowhere. If This is false. No B is A is true. And some B is not A must be true.

[45:26] And every B is A must be false. Yeah. That's one way of kind of you know exercising your mind. You know, go around and do with each corner, like an exam I did with the students out. Okay, this is true. What about that corner? And I say, right, true, false, or unknown, right? If this is known to be true, this is known to be true, right? The universal is known to be true.

[45:55] The particular, right? But the particular is known to be true. The universal is not known to be true, right? But if the particular is false, the universal is false. If the universal is false, the particular is not necessarily false. I mean, the most important thing to see about this is that the what diagonals are what posed as contradictories. That is to say, no matter what B and A are, they can't both be true.

[46:27] They can't both be false. One must be true, and the other must be false, right? And when I use the letters here, I don't even know what what B and A are, right? But I know if you had the same B in in both of those, the same A, that one of those is true, and the other is what false, huh? And so when they get into reasoning later on, Aristotle is contrasting, let's say, demonstration and dialectic.

[46:55] In demonstration, you know which of the contradictory is true. You know that it must be so, and therefore you know the other one must be what false. But in dialectic, you don't have the necessity. You have only probability, and so you're inclined to to one of the two, right? With some fear that the opposite might might be true. Yeah, yeah. And in dialectic, you sometimes even reason both to the affirmative and to the negative, right?

[47:23] There's something to be said on both sides, right? And maybe one side outweighs the other, right? In the same way in rhetoric, right, which is even more problematic, right? Huh?

[47:36] And so in the Inchon Landing they came up there, right? My favorite example there, the head of the Naval operations and the chief of staff were both opposed to the Inchon Landing MacArthur was for it, right? MacArthur went out, right? It was a great a great thing he did, but I mean, it was a very dangerous thing he was doing, right? The second thing, right? Huh? McCarthy said, "The Admiral, I have more confidence in Navy than you have.

[48:08] They could do this, right? They could win." Probably, you know, basically failure, backup plan. He says, "We're it'll be lost, except for my reputation."

[48:22] I mean, that's nature of rhetoric, so you can argue on both sides, right? Simply on both sides. So that's kind of contrast there between demonstration and dialectic, right? In demonstration, you argue to only one half of a contradiction, and seeing that that must be so, of course, naturally, the contradictory must not be so, right? But in dialectica and in rhetoric, you can reason from probable opinions or from likely opinions, right, to opposite conclusions.

[49:00] You might be able to reason more strongly to one than the other, but you don't see one as necessarily so, and therefore, the other is necessarily false, right? You see, one is maybe more probable or more likely, So there's some probability left to the other. If there's no probability at all, the other would be manifestly false, and the first one would be more than just probable; it would be necessarily true.

[49:24] So, if I see one half of a contradiction, if we speak of the two contradictory statements as halves, right? If one half of a contradiction I see is only probable, then the other half I can't see as necessarily false, can I? Then the first one would be more than just probable; it would be necessarily true, right? So, once you understand the opposition of contradictories, you can see how Aristotle uses that, right?

[49:49] The contrast demonstration and dialectic In the third book of Wisdom, Aristotle reasons for and against things, right, before he determines the truth. But in geometry, where the demonstrations are very clear, you don't reason on both sides; you're just on one side. So you understand the square opposition, you got to think about it a bit, huh? We did some exercise. Yeah. And give you that exercise, thinking about this.

[50:29] These two universals, they say, sometimes call those contraries, right? So you can say the man who thinks that everyone is beautiful is contrary to thinking the man who thinks no one's beautiful. The man who thinks that every man is good, right, is contrary to thinking the man who thinks that no man is good, right? But you know, in a sense they're further apart, right? But still, they're not what contradictories, right?

[51:04] The statement cannot oppose that one must be true and the other false. That's why we thought it should be the trapezium of opposition, because the top has to be longer than the bottom. Well, something you said to that, but I mean, the square's been around for a long time. I thought I'd use those awful

[51:28] circles. It's awful. One thing kids get into in probably the circle, you know, they want to say like, you know, this represents maybe everything, right? And this represents some, you know, right? Okay. The trouble is, this makes some definitely some are and some are not, right? It's misleading, right? Because if every man is an animal, then some men are animals, are still true. If you represent it by coloring the part that is, then you're saying you're imagining that the part is not, and you get to a false understanding.

[52:07] of it. No, no, never, never let me kick with those circles, right? These Venn diagrams and all that sort of stuff. That's that's very bad. Very bad. And you end up dissolving into the imagination like Socrates does in Parmenides, right? And the Parmenides, Socrates is a young man who's learning the Socratic method from Parmenides. Parmenides is a guy who emphasizes contradiction, so on, right? And avoiding it, Socrates is trying to imagine universal to be like one of these diagrams, right?

[52:43] Oh. So it's spread over everybody, right? So man is spread over all of us. You only have part of what man is, and you have another part of what man is. And actually, what man is, the whole that is found in you, but also you. So you can't imagine it like a quantitative whole, like a sail over covering us. But the moderns often get into that problem.

[53:06] They want to talk about class rather than universal. And class is what a collection, universal is something what. One said of many, but it's not a collection of the many.

[53:27] The difference between, let's say, you know, man and mankind, right? You know, the biologists, you know, talk about the animal kingdom, right? When they think of the animal kingdom, they're they're thinking of the collection of animals, right? Rather than something one said of many. If you ever, you know, study Porphyry, he talks about genus. You know, he gives two meanings that are before logic, right? The genus is one man from whom a multitude of men have descended, and another meaning of genus is the multitude of men that have descended from one man.

[54:06] Okay, as if we were to call Adam a genus, right? And all the Adamites, right? You know, all descend from him a genus, right? And both of them have something like the meaning of genus, but it's something one like Adam, but it's not individual like Adam, right? And has some reference to the many, because it's something one set of many, right? But it's not a multitude either.

[54:33] you see It's something one said of many. And it's something different from the individual from whom they descend, and the multitude that have descended from him.

[54:50] See all my offspring, all my grandchildren there, right? The multitude who have descended from me, you know. Like this friend I saw in church, there, and he saw all these grandkids. Now. You're responsible for this. See, but I'm an individual and not said of the multitude descended from me, right? By man is said of all individual men, right? But there's a one-to-many relation there. So, Porphyry leads you from those earlier meanings of genus, which Aristotle gives in the fifth book of Wisdom, too, right, to the one that we're interested in, logic?

[55:27] But you have to kind of transcend your imagination, which wants to imagine a collection rather than universal, which cannot be imagined. Understand universal. Now, the logic of the second act is easier than the logic of the first act, much easier than the logic of the third act.

[55:53] Now, when I was going through the things there, I found that I had some of the things copies of, but the first one I don't have a copy. I'm going to give you a copy of it, and you can reproduce it for the gentlemen. Okay. The ones on syllogism in particular, I have those. I think copies. I think. yeah, I haven't taught logic for a while, so these things get reproduced.

[56:16] But the first thing I do is a kind of general thing on the four kinds of arguments: syllogism, induction, enthymeme, example. So this one that I'm going to give to you, you can reproduce. You're the one reproducing. Reasoning and the four kinds of argument, right? So this is what we'll do for for next time. This is altogether about eight, seven pages here, okay? And kind of imitating Monsignor Dionne one time in the course, there, four kinds of argument, but beginning from the ones that are closest to the senses, right, and then moving up to the more profound ones.

[56:56] The main thing we studied in the logic of the third act is the syllogism, but these other forms are important too. Okay. So next Tuesday we'll get done that paper. Okay. I noticed that in the for the particular statements, you would say some man

[57:26] is wise. Yeah, you should really say some man is wise, rather than some men are wise, right? I was wondering, what's wise? Well, because they might both be false, right? Every man is god. False, right? Some men are god. Yeah. Some man is god. Yeah, that's true. Right.

[57:56] Some every man is my father. False. Some men are my father. No. But some man is my father. So if it's false that every B is A, there must be at least one B that is an A, right, or not a not a A. It doesn't have to be a, you know, three or four of them. Okay. I think I think a lot of times, I, even myself, in former years, I might have, you know, said some.

[58:32] Every man is wise. Some men are wise, but that gets a mistake, you know. Say some man is wise, huh? Because those are the what cannot both be what, be true or both be false. Okay.

[58:51] Every man here is the abbot, right? False. Some men here are the abbot. It's just one man, right? False. Oh. So this is they can't both. They can both be false then, but some men are the abbot. Every man is the abbot. Yeah. But some man is the abbot. Must be true if it's false. Every man is the abbot, but not that some men are the abbot.

[59:24] Every bishop is the pope, right? We didn't see that yesterday. We were just arguing. Oh, when you define something, you exhaust it We thought it was exhausting. it was absolutely singular. Every bishop is a pope, right? False. Therefore, some bishop must be the pope. Well, yeah, we need to just one bishop. the bishop of Rome is the pope. right? You see the pope's remarks there on the Transfiguration there.

[1:00:00] All right. All right.