Introduction to Philosophy & Logic (1999)
Lecture 40: The Second Figure: Conversion and Validity

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

Now, it seems kind of unsymmetrical, doesn't it, right? Because universal negative is the most useful for conversion. Because you convert and keep universality. A particular negative is useless. You can't convert it at all, right? So one negative is the best for conversion, and the other one is useless, right? Two affirmatives are kind of in between, right? And then, they will give you no universality when you convert. But it's a particular, right? Now, in the second and third figure, the arrangement of terms is never such that you can see the set of all and the set of none as it stands. So what you have to do is to see, can I get the set of all and the set of none by conversion, right? And you have to kind of fiddle around, maybe not to a depth of doing this at first, right? And then you might try to find examples. And if you have not seen how you could convert, you won't be able to find examples. But if you do find examples, it says by the conditions, you know that conversion never gets anywhere. You've got to kind of figure it around. Aristotle calls these imperfect syllogisms, right? Because they're not clear as they stand, and you have to do some manipulating or converging of them. Now, let's look at that in the universal cases here of the second figure. Now, the second figure, your middle term is predicated in both cases. It's in the second spot, as I let you say, right? So, let's put the four cases with two universal statements. Every A is B, every C is B. Two universal affirmatives. And then, every A is B, and no B. Universal negative is a major premise. No B, excuse me, no A is B. Every C is B. No A is B. So, these are the four cases in the second figure, where the middle term is the, what, predicated cases that have two universal statements, right? Two universal affirmatives or two universal negatives, or one affirmative and one negative, and two possibilities there, right? Okay? Now, how many of these are syllogisms? Has something to follow necessarily, and some not. Which one would you know right away is not a syllogism? Two negatives. Yeah. No way you get the set of, even the set of none, not the set of all negatives, right? So, right away, I suspect this is invalid, right? So, I want to find examples for A, B, and C that satisfy the two conditions. So, can you think of A and C such that every C is an A? Animal. A dog. Yeah. And then, C is never an animal. Stone, let's say, right? Okay. I have to find a middle term, B, that an animal, a dog, and a stone is never. Everyone, right? Now, again, I always pause and say, if I satisfy the two conditions. But what is condition number one? Well, the premises are true when you put in the examples, right? So, no A is B. No animal is a tree. I guess that's right, yeah? No C is B. No dog is a tree. I guess that's correct. And no stone is a tree. I have satisfied condition number one. The premises are true when I substitute in the... examples. Now we have one example where every C is A. Every dog is an animal, and that knocks out the two negatives, which shows they're not always true, right? And we have one where no C is A, yeah, no stone is an animal, that knocks out the two affirmatives as being always the case, right? So there's nothing affirmative or negative that is always so, and therefore there's nothing that is necessarily so. If there's nothing that's necessarily so, there's no syllogism. That's in the very definition of syllogism, okay? This is not a syllogism, huh? Now, what about no A is B and every C is B, right? Now, notice the set of all and the set of none, either one could apply for as it stands, right? You can't get the set of all because you have a negative statement there, right? But nothing comes under the subordinate negative statement. Nothing is said to be an A, right? But can you turn this around and say that no B is A? Yeah, the universe of negative turns around. So, no B is A. I'm showing my little arrow here, my conversion, right? And keep this other one saying that every C is B, and lo and behold, I imagine, I'm back in the first figure, right? And I see this is valid in the first figure, right? If no B is an A, and all the C's come under the B's, then no C is A, right? Put that one here and say, no C is A, right? So, notice what I have to do here. By conversion, I can get the set of none to apply here, right? And I go back to the first figure. It's the way to call it that the first figure, right? And in a sense, conversion is a way of making clear what is not clear, right? Now, over here, every A is B and every C is B. Well, now I'm going to have to try for the set of all, right? But the subject of these universal fermetures, nothing comes under to that. Now, if I could turn around, every A is B, and say every B is A, then I'd be back to the first figure, right? But we saw that you can't necessarily do that. The syllogism requires necessity. So, I don't think conversion is going to get anywhere. So, I suspect that those, that case, that form, is not a syllogism, right? Okay? So, I want to confirm my suspicion by finding examples for A, B, and C. So, the premises are true, and you substitute the examples in them. In one case, you have, what? Where's the fermeture? In other case, you don't, huh? Well, take something like animal for B. And then, for A and C, you want to take one set of examples where both of them are animals and every C is an A. So, I can take a dog and a cat or a spaniel, right? Every dog is an animal, and every cat or a spaniel is an animal. Every cat or a spaniel is a dog, right? And I can take something else, like a cat. Okay? Have I satisfied the two conditions? Well, condition number one is that when the premises are true, excuse me, when the examples are substituted in, the premises are true, right? Every A is B. Every dog is an animal. That's true. Every C is B. Every pack or a spaniel is an animal, and that. Every cat is an animal, and that. I satisfy condition number one. I have condition number two. If you have one example where every C is A, yeah, I'll circle that example, right? That knocks out the two negatives as being always so. One where no C is A, yeah, underline that. No cat is a dog. That knocks out the two affirmatives, right? So there's nothing affirmative or negative that is always so, when those statements are true. Nothing is always so, when nothing is necessarily so, we ain't got a subject. Okay? I'm kind of getting to see the reason here, why we're not going to get the affirmative conclusions, right? You can't convert the universal affirmative as you can the universal negative, right? Now, this third case is no more involved, because you're going to have to convert a couple of times, right? We're asking whether there's any conclusion with C as a subject and as a predicate that is necessarily so. Well, now, since you have universal negative, you're going to have to try it with a set of none, right? And no C as B as universal negative, but nothing comes under C. But if you convert that to what? No B is C, right? And then bring every A as B underneath that, right? Every A is B. And now you have the range in the first figure again, right? Except C and A are reversed, right? So no B is the C. You know that by conversion, right? We're given every A as a B. We'll follow the A's from under the B's, and then the B's are C's, then no A is what? C. But we were asking, did anything follow with C as a subject and A as a predicate, right? Reverse. But I can convert, right? So no C is A. So this requires what? Two conversions. This other one, that is socialism, requires what? One conversion, right? Okay. So what I'm saying here is that when no A is B, and every C is B, necessarily no C is A. In order to see that that is so, not just looking at it as it stands, you have to see that one can be what? Converted, right? And then you can see that it's necessarily so, right? You're back in the first figure, right? In this other case, you convert, and you get back to the first figure too, but your conclusion is the reverse form. But since it's universal negative, you convert again, and get no C as A, right? Okay. Now these syllogisms are very common now. You find something that is said of one, and denied of the other, right? Either way, they're going to do what? The other syllogize, right? But changes is composed. God is not composed. Therefore God will change it. Very simple. But you have it again and again, and I'm going to get to you now. Yeah. Socrates is reaching that way, right? And the Phaedo, you have the Phaedo down there, haven't you? Yeah, I'm reading it. But he's arguing that the soul is not to harm the body, right? And so he syllogizes on the soul resists the body like the man fasting and so on. But the harmony of the body doesn't resist the body. Therefore the soul is not to harm the body. You're already still trying it that way. The harmony of the soul is not to harm the body. Therefore the soul is not to harm the body. Or we already used to know from the previous argument that the soul existed before the body. The harmony of the body does not exist before the body. Therefore, it's very common to use these two. So notice you have two ways of reasoning that no C is A in the second figure here, right? Either by finding something that is said universally or A and denied universally is C or vice versa, right? There are two possibilities, right? So actually there's more ways to conclude universal negative of C than universal affirmative. There's only that one way in the first figure reason to universal affirmative as a way to reason to universal negative. But there's two ways here in the second figure, right? Those four are the most important, right? Remember, right? The way to conclude the universal affirmative and the three ways to reason to universal negative. But we haven't seen that those are the only ways that you might suspect at this point, right? if you have two particular statements you don't have any solutions at all but if you have a particular university you don't have an universal conclusion. Now, let's jump the other end and look at the two particular ones. Sum A is B, sum C is B. Sum A is B, sum A is C. It's not B, B, sum C. Notice here we are still in the second figure. Your middle term is the predicate in both cases. That's the arrangement of the second figure there, right? The second slot. And these are the four ones that have, what? Particular premises, right? Particular statements. Now, you know that by conversion, you're never going to get any universality, right? There's no universality. Your conversion will never be given any universality. You can't convert the particular negative at all. In particular affirmative, you can only convert to sum. So you know conversion is not going to give you a syllogism. So I suspect that all four of them are not syllogisms. Now, if I can be clever like I was before, maybe you can find one set of examples to eliminate all of them, right? Then, so we could take animal, let's say, for A, and dog for C, and what stone for C, right? And I satisfied one condition. Every dog is an animal, and no stone is an animal. Now, I have to find a bee such that some animals are, and some are not bee. And some dogs are bee, and some dogs are not, and some stones are, and some are not. Now I go back to my dear friend, Porphy, right? And I say, okay, some animals are white, and some animals are not white. I take a pretty cool waxing, right? Or it could be present or absent. That's the definition of a pretty cool waxing, right? So an animal may or may not be white. So some animals are white, and some are not. Some dogs are white, and some are not. Some stones are white. So, it works for any possibility, right? Mm-hmm. Okay. And it satisfies the conditions here, right? Every dog is an animal, and it's your own person. Now, let's start to go and look at the mixed forms. The mixed forms are the ones where one premise is universal, and the other is what? Particular, right? So take the four in which the major premise is universal, and the model is particular, so you've got every A is B, or no A is B. The second one now, in the mixed case, is particular, right? There's two possibilities. Either particular affirmative, some C is B, or some C is not B. The same over here with no A is B. Some C is B, and some C is not B. And those are the four mixed cases in which the universal premise is the major premise, right? And the minor premise is particular, right? Okay. Okay. Okay. Okay. Okay. Okay. Okay. Now, which one are you sure right away is not going to be a syllogism? It's a negative, yeah. So, not a syllogism, but you want to show it now by examples where A, B, and C that satisfy the two conditions, right? Well, maybe you could take an example where no C is B, right? You want it to, right? So, you could take, let's say, for A, animal, for C, dog, and then let's take what? Stone, right? Now, we've got to find a B, since there's no animal is a B, and some dog and some stone is not it. And that'll work even if no dog and no stone is it, right? What could that be? Three? Three. Okay. Now, it's good to always pause and say, have I satisfied the two conditions? What are the conditions? Are the premises true of those examples? No A is B. No animal is a tree. And sometimes what I would do in a class, or a student, I might, you know, write the symbol like this, so it's easy to follow, right? You know? No animal is a tree. And some dog is not a tree, and some stone is not a tree, right? Because that doesn't mean that some are and some are not. It doesn't mean that some are a tree. And I have one example where every C is A, which means the two negatives are not always a case. And one example where no C is A, no stone is an animal, which means the two affirmative possibilities are not always a piece. So nothing is always so. Therefore, nothing is necessarily so. Therefore, you're not a syllogist, right? Now, no A is B, okay, universal negative, and you know the strength of that, right? That's what they're most likely to be able to get some of that. Now, if you convert, as a standard, you don't have anything coming under the subject of universal negative. But you can convert no A is B to what? No B is A. Keep some C as B. And lo and behold, by magic, you're back in the first, what? Figuring out. And that's obviously a talent, right? But notice it's a negative conclusion again, right? All you've got so far in the second figure is negative conclusion. Some C is not B. Tell my finger. Okay? Some C is not B, right? Excuse me. Not A. Okay? Erase that and make it more clear. So, notice, you have to convert, in this case, you convert the major premise, no A is B to no B is A, right? And then you have something coming under the subject of universal negative, namely some C, right? So the some C's are B's, and none of the B's are A's, and that some C cannot be. Now, what about every A is B and some C is B? Now, if you convert the universal affirmative, you could go back to the first figure, right? And say every B is A, and that some C, being a B, would have to be an A, right? But you can't necessarily convert the universal affirmative, right? If you convert the universal affirmative, it becomes particular. So, I suspect this is not a syllogism, right? So, I'm going to put down my suspicion, not a syllogism. The professor requires examples, so it's satisfying. Now, when you think of A, B, and C, it's satisfied with conditions, it's satisfied with conditions. Dog, right? Okay. Now, what could I have for C? A carcass spaniel. Okay. Since every carcass spaniel is an animal, it's also true that some carcass spaniel is an animal. Once again, as much as you can on the same examples, huh? So, now, I stop and I say if I satisfy the conditions. What's the condition? Are the premises true with those examples? Every A is B. Every dog is an animal, that seems to be true. Some C is B. Is some carcass pangelo an animal? Yeah. Is some cat an animal? Yeah. So I satisfy condition number one. Premises are true when I substitute those examples. Now condition number two. Do I have one example for a universal affirmative where every C is A? Yeah, I'll circle that. Every partner's spangelo is a dog. That knocks out the two negatives, right? That is being always a case. Do I have one universal affirmative where no C is A? Yep. Undermine that cat, right? That cat is a dog. And that knocks out the two affirmatives as being always a case. So there's nothing affirmative or negative that is always a case. Nothing is always so, nothing is necessarily so. Nothing is necessarily so. You ain't got it. You haven't got it. It's syllogism, right? That's the very definition of syllogism. You see? Now, what about this fourth case here? Well, here you can see that you can't get the set of all or the set of none. Obviously, you can't get the set of all because you have negative statement, right? But you can't get the set of none either because you can't convert the particular negative at all, right? You don't have any universal negative statement together, right? So according to the rules I've given you up to this point, which are sufficient for the most part, you might guess that this is not a syllogism, right? But you think you could find examples to satisfy the two conditions in this case. So, on the one hand, you don't have the set of all or the set of none, even by conversion. On the other hand, you can't find examples to satisfy the conditions as prior as you will. So, now, maybe, you know, if you just kind of think about it and say, well, look, if every A is a B, every one of the A's is a B, but some C is not a B, put that some C, B, and A. Because every A is a B, right? I've got to figure out that some C is not an A, right? But how can I, you know, show this by the set of all or the set of none, right? How can I show, instead of all or the set of none, or one of them at least, that some C is not A, I can see it, I know it will go away, right? Well, I could take the, what, contradictory of some C is not A, which is what? Almost every C is A. Every C is A, right? And if every C is A were true, then, since every A is B, then every C would be B, which contradicts this, right? You see what I'm doing? Okay. I'm taking the contradictory of the, what? What I say is necessarily so. I'm taking the contradictory of that and joining that to one of the premises and a contradictory of the premise. So, in other words, if you take the contradictory of this, which is every C is A, every C is A and every A is B contradict, by the first figure, some C is not B, right? Okay? So, these three statements, you could say, are, what, cannot be held together, right? They're incompatible, right? So, if every A is B is laid down, and some C is not B is laid down, you must reject every C is A. It cannot be so with those two. And therefore, you must, but. Accept that. Yeah, that's the contradictory, right? And contradictory statements are opposed, like, say, both cannot be true. Both cannot be false. One must be true and the other false, regardless of whether you know it's a true or false one, right? You see how you do that? That's very interesting, right? And that's kind of the, what, exception to the rules I gave you, right? For the most part, let's say, in the second period figure, if something is a solism, you can get it clearly by converting, right? When the prince, in this case, you can't convert and get a solism, right? But if you went on to think, therefore, it's probably not a solism, and you try to find examples, you'll never find examples that satisfy the conditions, right? And I know I correct, you know, since they're out of logic, and they, you know, are trying to disprove and show that some form that is a solism is not a solism, right? Their examples are never a satisfying condition. They never do. If it's satisfying a condition, they would have made history to the beast. But the point is that since it is a solism, they must have made a mistake in their example. And they didn't, as I told them, to do, maybe Professor Purvis had to do a guess to, you know, check out of his calculation, right? Just like Einstein, you know, that Einstein, you know, made some false calculations, you know, and some nobody corrected him, you know. Einstein's, you know, finally in fact realized he had gets calculated, right? Heisenberg tells the time he got a great idea, right? And he was very excited. He came back to the institute there and he started, you know, calculating and making all kinds of stupid mistakes, you know, and finally calmed down and started calculating slowly and he came out just perfect, you know, and he's on the way because there's no prize, right? So, you know, I tell the students, always go back and check your examples, right? Because if you have made the mistake of thinking something that's not as silly as it is, right? And therefore, look for examples, you will not have found examples of satisfied two conditions. If you had done a simple check and take your time and some kind of, you know, damage to, I satisfied condition number one and I thought, yes, I was like, you know. But you have to do that, right? And then you will see the two examples don't. You might be thinking, right? So this is a very interesting thing here, right? It does follow necessarily that some see it's not it, right? But the set of all or the set of none is not in it as it stands, right? And you can't get the set of all or the set of none by conversion, right? But if you take the contradictory of what follows necessarily, you can syllogize, right? To the contradiction of the other premise, to one premise, to the prediction of the other premise. So that must be true that of all the other valid syllogisms that you can do that. Yeah, yeah. The same, right? Yeah, yeah, yeah. This one here is, you know. Just a proof of that. It's more difficult, right? Yeah. And also, the way we did that in a way with the case of the if-then syllogism that's not so clear, right? We said, if A is so, B is so, A is so, B is so, B is not so, and A is not so, and we said, well, if A were so, then by the first case, the obvious case, B would have been so, But you'd give it that B is not so, right? So the statement if A is so, then B is so, B is not so, and A is so, are not compatible. A is so, B is then statement would necessarily include to B being so, which contradicts the other premise, right? We're doing something similar here, right? I'm taking the contradictory of what I say is necessarily so, and showing that that is compatible with the premises, because with one of them, it, like I said at all, so it ties us to the contradictory of the other, right? So in a sense, you show up to the first figure, but not as easy as you do when you convert, right? You show up by taking the contradictory of the conclusion, adding it to one premise, the contradictory of the premise. And therefore, you see that the contradictory of the conclusion is not compatible with those two premises, because added to one of them, it contradicts the other. If every C is A, and every A is a B, then every C is a B, and that contradicts some C or not B. So you can't take the contradictory of the conclusion with those two premises. So you must be rejected, because C is A and they're allowed to these two. And therefore, it doesn't. going to be going to be going to be going to be going to be Yes, is that true? Neither an affirmative statement nor a negative statement is always the case. Because sometimes universal affirmative is so, which means the two negatives are not always so. And sometimes universal negative is so, so the two affirmatives are not always so. Therefore nothing is always so, nothing affirmative, nothing negative. If nothing is always so, then nothing is necessarily so, right? Put down the form of an if that still is, right? If something is necessarily so, it's always so. But nothing is always so, then nothing is necessarily so. And another if it's a sodism, if it's a sodism, something is necessarily so, but nothing is necessarily so. So you haven't got a sodism. I think that's a sodism, right? So right away. Now, there's a set of all five, a set of none to these first two here. Every B is A, you've got a universal affirmative. Does anything come under the subject of universal affirmative? Yeah. But you're told only that some C comes under B. So all you can conclude necessarily is what? Some C is A, right? So this is a syllogism by the set of all. Set of all means you have what? A universal affirmative statement, and something comes under the subject of this affirmative statement. Okay? It's a predicate of that universal statement. Now, when you say no B is A, some C is B, you have the set of none there. You have a universal negative statement, no B is A. That's the first thing required in that you have the set of none. And something has to come under the subject. Something has to be said to be a B. Does something come under B? Yeah. But in this case only some C, right? So you can conclude with necessity that some C is not what? No. Now, how about this over here? Does a set of all apply to that? You've got a universal affirmative statement, that's part of what's required. But does anything come under the subject of that universal affirmative? Is anything said to be a B? No. You say that some C is not B, right? So you suspect that you can't say anything necessarily that C is a subject and he has a predicate of it. So you look for examples then for A, B, and C to show that nothing is always so. And they have to give you true statements when you substitute them in. And there has to be one example where every C is in fact A. Another one where in fact no C is A, right? So can you think of an A and a B such that every B is an A? A more dog. Yeah. Okay. So every dog's A. That's true. Now I've got to find a C such that some C is not a dog. And it can be even if no C is a dog, right? We get every C is a man. Take what? A cat. And then something that is not a dog, that's not an animal either. A stone, right? Oh gee. So two birds with one stone, right? I can't. We'll clung that stone. So in the mixed forms, right? Where the major premise is the universal one, right? And mine is particular. There's two cases that are syllogisms and two that are not, right? Just like there are two among the universal ones, right? But none among the two particular ones, right? Now we've got to look at the four remaining cases where the particular is on top as a major premise and universal below, right? So the particular is the universal premise. So it's in the form either some B is A or some B is not A, right? And then your minor premise is either universal affirmative or universal negative, right? So it's in the form. It's in the form. So it's in the form. So, every c is b, every c is b, or you have the universal negative law, some b is a, no c is b, some b is a, and no c is not a, and no c is b. Those are the four possibilities, right? The major premise is particular, it's either what, particular affirmative or particular negative, right? And the second one is universal, there's two possibilities, it could be c is b and no c is b, universal affirmative, universal negative, right? And right away, you know, if there's two negatives down here, the best one would be not a syllogism, right? There's no way you can get the set of none either, with the negatives, right? So, I look for examples for a, b, and c, such that the premises are true when I put them in, and yet in one case every c is a, and in every other case no c is a, right? Again, it's probably easier to take an example where no b is c, right? No b is a rather than just some b is not a, right? Can you think of an example where some b is not a? Just some? Either way, either way, either way. They said, universal negative is true, the particular negative is true as well, right? If you want to take a, you know, or just some, right, you can say, some animals are not, what, dogs, right? Okay? Okay? Now, that works, for that thing. Now, can I find examples for a seed that are going to set up like different conditions? Can I find something that is never an animal, but always a dog? Okay? No. Okay? So then I say, examples aren't going to work, right? I've got to take another kind of examples, right? Okay? So, what would you take? One stone. Okay? No stone is an animal. No stone is an animal. It's also true that some stone is not an animal. Okay? Some students, if you say, what? Some students have failed the course. It doesn't mean some have passed, either. If I say some have passed, it doesn't mean some have not passed, right? If you press the exam, I say, some have passed. That sounds ominous, right? But if I've only corrected some, and they have passed and they have corrected, right? All I can say for sure is that some have passed, right? Because I'll read into my book what I'm saying, right? So, if I've only corrected some, and they have passed and they have corrected, right? No stone is an animal. Some stones aren't. Some stone is not an animal, right? And then, can you find a C such that no C is a stone, but it's always an animal? Cat? Cat? Dog? And then something else that is never a stone, but never an animal? This is not a syllogism, right? Okay? Now, in these other cases, you do have a universal statement, don't you, right? Like in the first one up here. Some B is A, and every C is B. You have a universal statement, every C is B. But does anything come under the subject of that universal statement? It's not what you have instead of all, do you? So I suspect that it's, like, invalid, right? Okay? So, I look for examples for A, B, and C.