Introduction to Philosophy & Logic (1999)
Lecture 31: Truth and Falsity in Compound Statements

Transcript
================================================================================

You know, in various ways, huh? Let's give you an example of what I mean here. Three is an odd number. Three is an even number. You've contradicted yourself. But now, strictly speaking, three is an odd number and three is an even number. Are they contradictory statements as we define contradictory statements? Again, they're both, what, affirmative statements. But contradictories are one affirmative, one negative. But also, they don't have the same, what, credit, right? And contradictory statements have the same subject and the same credit card. So these are not contradictory statements, strictly speaking, right? Now, you might say, you know, speaking more loosely, right, that when I say in one place that three is an odd number and in another place that three is an even number, I've contradicted myself because, in a sense, to say that three is an even number is to deny it, but it's to say what? Yeah, in a sense, it follows from this that three is not an odd number, right? You see that? But that's really contradictory, strictly speaking, of three as an odd number. So you want to learn to speak very precisely, right? Better on when we talk about the word some, you know. Now, if I say, some women are beautiful. Now, when I say some women are beautiful, have I said that some women are not beautiful? Someone might infer from my saying this that I think some women are not beautiful. And maybe I do. You bet, right? But have I said that? If I say, some twos are half of four. Have I said that some twos are not half of four? Well, even if every two is half of four, aren't some twos half of four, too? Example, I get in class sometimes, you know, where I can sing a lot better grade, you know. Okay? So suppose I give an exam one day, right? Okay? And it's a final exam or something. And I see the next day or so, and I say, Oh, how'd we do? And I say, some of you have passed. Have I said anything about the rest? I might only have corrected, you know, half the papers or something, right? Right. So what am I saying? I'm saying that some of you have passed. But that guy went off and told the other guys, right? He's more so-funky, you know. Let me get out of the degree here. See? Well, I didn't say that. I didn't say that. There might be a reason why. So, the statement that some women are beautiful, is that contradicts every woman is beautiful? And the statement that some students have passed, is that every student is bad? No. So you've got to realize, this here is kind of inference, right? Now here, there really is a contradiction, but it's because this is really a consequence of that, right? Okay. Now, I mean, we may speak, and I might, myself, some may say, you know, you say this, and you say this, and you should contradict yourself. But, strictly speaking, these are contradictory, and so the conviction is a consequence of saying this, right? Now we're going to turn to the compound statements, and first, again, there are composing parts, right? And then we'll talk about truth and falsity in them. And, again, you have to understand the composing parts before you can understand what truth and falsity means. The pedicure, conditional, they call it sometimes. You have two simple statements put together with if and what? Then, right, then? So, you say, if this number is two, then this number is even. Okay? And this number is half a four or something, right? So, this number is two, this number is even. And then, taken by themselves, are two simple statements, right? But they're combined by means of if and then, right? So, the simple statement in the if part is called the antecedent. Ante, you mean what? Before, right? And in the then part, it's called the what? Consequent, right? The antecedent and the consequent. 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 I or even, right? And there, you try to exhaust the possibilities, right? Okay? Say, a triangle, though, is either equilateral, absosceles, or what? That's why you probably usually write these things. But, you could expand on this, you could say, either a number is odd, or a number is, what? Even, right? Either a triangle is equilateral, or a triangle is a sosceles, or a triangle is, what? Yeah. So, here we're kind of contracting it, so you don't make explicit, right? You don't spill out these simple statements the way we did with E.F. Benby, right? We could have wanted to, right? But usually, we, what? Contracting, right? Okay, then, okay? 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? 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? And notice, in the sixth theorem there, in book one of Euclidean, that's the theorem where he says that if the angles are equal, then the, what? Sides will be equal, right? Okay? And so, in a sense, he has an either-or statement there, right? Either these two sides are equal, or they are, what? Unequal, right? And then she's going to show that if they're unequal, you're going to end up with part being equal to the whole. And therefore, he's going to, what? Include they must be, what? Equal, right? So, again, you might probably say, either these sides are equal, or they're unequal, right? But you're really putting together this into some statements. Either these sides are equal, or these sides are unequal, but that's the need to use, 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? But that means, or, or, if you're Shakespeare, or either or, or. That's right. Okay. 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? 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, when I call it statement, is not important for reasoning, huh? 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 soldiers and so on. Okay? So it's because we have in mind, reasoning eventually, that we emphasize those two among the compound statements, huh? 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? Okay. 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 resultant thing to be what? 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, huh? Take the statement, for example, that Socrates is a mother. The simple statement and it's false, right? And the simple statement, Socrates is a woman. That's false too, right? Okay. But now, if I combine these and I say, if Socrates is a mother, then Socrates is a woman, that would seem to be true, right? Okay. But now, let's suffice to say, if Mary is a mother. Mary is a mother of God. We think it's Mary. Mary is a mother, right? Excuse me. If Mary is a woman, 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. 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? And 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 or falsity in a simple statement. It means something else that 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 that it's true or false, right? Okay. And you'll see sometimes with an argument of Aristotle's in the seventh book, if not, you'll hear it, you know, that some people criticize because he's using a simple statement in a compost. So here we go. Let's go. The simple statement is false, 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, if you put in the first book there, there's one 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. Otherwise, you couldn't use it for me. 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 create a woman, right? Okay? Not saying that in fact he is a mother, or in fact he is a woman. You're saying if this is so, then that will be so, right? Right, you know? Okay? Down here, it's like false, right? It doesn't follow that if she's a woman, but therefore she's a mother, right? That follows, then you mean, women would have to be a what? A mother, right? 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? Now, it could be that going both ways sometimes, it's true. Take the example of this here. If this number is two, then this number is half of four. Is that true? Okay? And there, the reason goes back to that half of four is what Porphyry would call a property of two, right? And 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? Yeah. It's because half of four is a property of two in the strictest sense, right? It belongs to only two, to every two, and always, right? Okay? But now, if I took a property in a looser sense, if I said, if two is, if this 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? Because less than ten is not convertible, then, 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 a gridado, right angle, quadridado, and vice versa, right? Okay? If you had something more universal there, 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 there's two? No. More universal, right? 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, right? So, even if both simple statements are true, but one doesn't follow necessarily from the other, this number, 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 man's chief good is but to sleep and feed, then man is no more than a beast. Is that true or false? 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'd say, what is a three if it be half of four? Yeah. So, if three is half of four, then three is what? Two. Two of four. It's true. Even though it's false that three is half of four, and it's false that three is two, you know? 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 beast, right? Notice, you could, you know, it's arguing from the proposition that if the sides are equal, right, then you can cut off from the longer side and equal to the shorter side. Two are false. But, in fact, they have to be equal when the name of the base is equal, and you can't cut off, therefore, you know, a line equal to the other one that's only part of the first one, 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? In shapes we do the same thing, right? But you can see why the truth of the if-then statement is not the basic truth reactor, right? Because, you know, you'd go on all this and if this is so, then that would be so, and this is so, and that's all right. If I won the lottery, I don't. If I won the lottery, right, I don't know if Socrates, in fact, the mother or not, right? Is it true or not, right? Is he, in fact, the mother or not? And this doesn't tell me, right? So later on, when you study reasoning and you study what we call the hypothetical syllogism or the if-then syllogism, as I call it, you have a if-then statement but you will add to that a what? A simple statement and your conclusion will be a simple statement, right? Okay? That's what Shakespeare's reasoning, right? If man's chief good is to sleep and feed, then man is a beast, right? Or more. It's a beast. But man is more than a beast, right? And 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 you use the first statement and I say what? If-then, right? And you can do the same thing. You'll say, if these sides are equal, then you could cut off from the greater one, right? One equals the lesser one. And then you could draw a line from that over to the side. Now he does that, but this, his aim is equal. And if one of these is longer, so this is longer than that, this one here, then you could cut off this longer side, and one equal to this, right? And then I could draw them 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 say if the part cannot be to the whole. But then once you had an instant statement, if these are unequal, you could cut off the greater line. So that's the one. 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 to something is or is not so in reality. 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 because anywhere are as far as knowing the way things really are. But you have an if-then statement and then you have a simple statement and then you can draw a simple statement conclusion. And we'll see how you do that when we take that out. 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-them statement as well as the simple statement, it's not really the same definition, is it? It's not purely equivocal to the word truth, right? There's a connection between the two, right? And here you can see a very clear connection because every mother is a woman. It speaks in that simple statement. And the falsity of this is because not every woman is saying, like, mother, right? It goes back to the simple statement. You can reason the simple statement from one of these if-then statements, but not the if-then statements alone. You've got to know something about the way things are. 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 Sarkis is a mother, then Sarkis is a woman. Yeah, it's not really saying definitely this is the way things are or not. And you want to know the way things are. So you'd say, well, like, it's based on the way things are or something. Yeah, yeah, that's the connection. It's not truly equivocal. I mean, it's a bit true here, right? But it doesn't mean the same thing. It's the word equivocal by reason. True instead of the simple statement and so would be if-then statement. Now, the either-or statement, likewise, will have a third meaning here. My phrase here just means, huh? If I say a triangle is either equilateral or isosceles. Is that true or false, I'd say. I'm going to say a number is either, odd, or even. Now, why is this false and why is this true? Sure, 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 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? So it's like I've got a bastard here with triangles in it, right? And I pour out, and the equilateral triangle is going to the sausage triangle. I say, that's all there is, right? And something's still in the basket on that. So I haven't emptied the whole thing out, right? But here I haven't emptied them all out, right? All the numbers, right? And as I put all the triangles in this 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? You know, from the rule of two or three, usually two or three are enough, but it's not, what, universally true, right? And I was giving an example there from Thomas there, where he, following Porphyry, says that every name said univocally of many things is either a genus, or a species, or a difference, or a property, or a 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, and accident, being an exhaustive, right? The distinction of a name said univocally of many things, huh? Question? Okay. And it's harder to see, then. This is exhaustive, right? Or if you said equilato, or isosceles, or scavene, right? It's easier to see in mathematics that something is, what, exhaustive. Exhaustive, right? But sometimes even there, I think it looks great. You can define these triangles into equilato, or isosceles, and scavene, and they divide them into right-angled, and obtuse, and acute-angled. You might say, couldn't we criss-cross these divisions and get nine different kinds of triangles, right? No, I don't think so. I mean, like, good men, bad men, good human beings, bad ones, male, female, just crossing it for real things. But isn't such a thing as a right-angled equilato triangle? No. But you have to know that a triangle always has its interior angles, equal to right angles. And then you have to know that an equilato triangle, the three angles must be equal, right? And so if one of them was right-angled, then the two couldn't be equal to it. Or you'd have three right angles, right? But you have to do some reasoning to realize that heaven left out of possibility, the right-angled equilato triangle is no such animal. Now, if you get to an isosceles triangle, where you could have a right-angled isosceles triangle, like half of a square, which you could have, as used to be, an out-two-sangle one-frencher. And you could have a three-an-one. So you have to do some thinking sometimes, even in math, you know. Generally speaking, it's easy to see that a division is complex, right? Every statement is either, what, simple statement, is either affirmative or what? Negative. That's exhaustive, right? Two straight lines are either equal or unequal, right? So when Euclid, in that sixth theorem, shows the impossibility of they're being unequal, you can conclude they must be what? Well, Thomas is arguing as to how the members of the Trinity are distinguished, right? Well, he says, there are two kinds of distinctions, formal distinction and, what, material distinction. Material distinction is based upon the division of the continuous. Well, there's nothing continuous in God, right? It's not a body. So you eliminate that. So it must be a formal distinction. Now, formal distinction is based upon opposition. Thomas says, no, there's four kinds of opposition. Going back to the categories of Aristotle, the distinctions are four. And then he eliminates three of them as impossible in God. Therefore, it must be a relative distinction. But now you say, now, are there only four kinds of opposition? Contradictory, privation, and lack, or, you know, lack in having, contraries, and relatives. That's harder to see, right? I can't think of any kind of opposition. You know? You have to kind of stop and, you know, think about this, right? Or you say, God is a cause. There's only four kinds of cause. Matter, form, mover, and end. Is that the only kinds of cause there? It's harder to see, right? You know? In some cases, you know, I mean, we may use an either-or statement that's only, what? Probable, right? We don't see that these are the only, right? Now, a man is either white, or black, or brown, or red, or yellow, right? I think it's all possibilities, right? Yeah, yeah, yeah. If they find, you know, a green man on Mars, or something, I'll be surprised. I don't say, yeah. Why can't it 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, right? Being a simple statement can be only probable, right? So we're talking here about, on page six now, we get to it, what true and false mean an either-or statement. 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 would say, right? Because then you're saying that something is or is not to think. That's right. And you can see, when I swear...