Introduction to Philosophy & Logic (1999)
34. Second Figure Syllogisms, the Power of the Figures, and Why Universal Negatives Convert
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[0:00] Now, what about no A is B and every C is B, right? Now, notice the set of all instead of none. Either one for the phi that was at stake, right? You can't get the set of all because you have a negative statement there, right? But nothing comes under the subject negative statement. Nothing is said to be an A, right? But can you turn this around and say there's no B is A?

[0:31] Yeah, the universal negative turns around. So no B is A. I'm showing my little arrow here, my conversion, right? And keep this other one the same. Every C is B And lo and behold, I imagine I'm back in the first figure, right? And it seems 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?

[0:59] 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? I go back to the first figure. That's the way he calls that the first figure. Okay. And the sense conversion is a way of making clear what is not clear, right? Okay. Now, over here, every A is B and every C is B.

[1:35] Well, now you'd have to try for the set of all, right? Okay. But the subject is universal affirmative. Nothing comes under it today. 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 me anywhere.

[2:01] So I suspect that those case that form is not a syllogism, right? Okay. But want to confirm my suspicion by finding examples for A, B, and C.

[2:19] So the premises are true. We just substitute examples in. In one case, you have what? Universal affirmative. In that case, you don't. Well, take something like animal for B.

[2:35] 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.

[2:46] 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 cat.

[3:04] Now, if I satisfy 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 cocker spaniel is an animal. That. Every cat is an animal. That. I satisfy condition number one. I have condition number two: If you have one example where every C is A.

[3:31] Yeah, I'll circle that example, right? That knocks out the two negatives as being always so. One where no C is A. Yep, I'll underline that. No cat is a dog. That knocks out two affirmatives, right? So there's nothing affirmative or negative that is always so. When those statements are true, nothing is always so, and nothing is necessarily so. If nothing is necessarily so, that's it. Kind of getting to see

[4:07] the music here. Why we're not going to get the affirmative conclusion, right? You can't convert the universal affirmative, as you can the universal negative, right? Now, this third case is more involved. It's going to have to convert a couple of times, right? We're asking whether there's any conclusion with C as the subject and A as predicate is necessarily so. Well, now since you have universal negative, you're going to have to try it with a set of none, right?

[4:40] And no C is B is universal negative, but nothing comes under C. But you could convert that to what? No B is C, right? And then bring every A is B underneath that, right? Every A is B. Now you have the range of the first figure again, right? Except C and A are reversed. So no B is C. You know that first conversion, right? We're given that every A is B.

[5:14] With all the A's under the Bs and none the Bs are C's, then no A is what C. But we were asking, did anything follow with C as the subject and A as predicate, right? Not the reverse. But I can convert, right? So no C is A. So this requires what? Two conversions. This other one, that is syllogism, requires what? One conversion, right? Okay. So what I'm saying here is that with no A is B and every C is B, necessarily no C is A.

[5:53] 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 that is 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 it again and get no C is A, right?

[6:20] Okay. Now these syllogisms are very common. You find something that is said of one and denied of the other, right? Either way, you're going to do what? You have the syllogize, right?

[6:38] But changes is composed. God is not composed, therefore God is unchanging Very simple. But you have it again, and again, on the negative. Socrates is reasoning that way, right? And Phaedo, you read the Phaedo, there, right?

[7:06] Yeah, we did. He's arguing that soul is not the harmony of the body, right? 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 the harmony of the body.

[7:24] The argument is so arranged that way. There's a harmony of the soul, but there's no harmony of the body. Therefore, the soul is not the harmony of the body.

[7:38] Or you 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, 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 of A and denying universally C, or vice versa, right?

[8:06] The two possibilities. So actually, there's more ways to conclude universal negative than universal affirmative. There's only that one way in the first figure to reason to universal affirmative. There's 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, the way to conclude the universal affirmative, and the three ways to reason to the universal negative.

[8:41] 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 syllogism at all. But if you have a particular and universal, you have a universal

[8:56] conclusion. Now let's jump the other end and look at the two particular ones. Some A is B. Some C is B. Some A is B.

[9:34] Some A is let me say some C is not B. Some A is not B. Some C is B. Some A is not B. Some C is not B.

[10:09] Notice here you're still in the second figure. Your middle term is the predicate in both cases. That's the arrangements of the figure, there, right? 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 into a syllogism, right? Conversion never gives you a syllogism. You can't convert the particular negative at all.

[10:37] In particular affirmative, you can only convert to itself. 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?

[11:01] So we could take animal, let's say for A, dog for C, and what stone for what C, right? And I satisfy one condition: every dog is an animal, and no stone is an animal. Now I have to find a B such that some animals are and some are not B, and some dogs are B and some dogs are not, and some stones are and some are not.

[11:30] Now I go back to my dear friend Porphyry right? And maybe he's not my dear friend because he's not a Christian. But anyway, my teacher Porphyry. And I say, okay, some animals are white, and some animals are not white. Take a predicable accident, right? Or it could be present or absent. That's the definition of predicable accident, right? So animal may or may not be white. So some animals are white, some are not.

[12:07] Some dogs are white, some are not. Some stones are white, some are not. Right? So it works for any possibility, right? Okay. And I satisfy the conditions here, right? Every dog is an animal. No stone is an animal.

[12:34] Okay, sorry. Let's say example. 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 particular, right? So let's take the four in which the major premise is universal and the minor is particular. The converse form. So you've got every A is B or no A is B

[13:17] as your major premise. 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.

[13:43] Same over here with no A is B. Some C is B and some C is not B. Those are the four mixed cases in which the universal premise is the major premise, right? And the minor premise is particular, right? Okay. Now, which one are you sure right away is not going to be a syllogism? Double the double negative, yeah. So, write down not a syllogism, but you want to show it now by examples for A, B, and C that satisfy the two conditions, right?

[14:32] Well, maybe you could take an example where some no C is B, right, if you wanted to, right? So you can take, say, for A, animal, for C, dog, and then let's take what? Stone, right? Okay. Now we 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?

[15:03] What could that be? Tree? Tree. Okay. Now it's good to always pause and say, have I satisfied the two conditions? What are the conditions? Are the premises true with those examples? No A is B, no animal is a tree. And sometimes what I would do, you know, in class or if I'm a student, I might, you know, write these in like this so it's easy to follow, right?

[15:31] You know? No animal is a tree, and what? Some dog is not a tree, and some stone is not a tree, right? That doesn't mean that some are and some are not, just some.

[15:47] And I have one example where every C is A, which means the two negatives are not always the case, and one example where no C is A, no stone is an animal, which means the two affirmative possibilities are not always the two B's. So nothing is always so. Therefore, nothing is necessarily so. Therefore, you don't have a syllogism, right? Now, no A is B, that's universal negative, and you know the strength of that, right?

[16:17] That's where most likely you'd be able to get something from. Now, if you convert, as it stands, now you don't have anything coming under the subject universal negative, but you can convert no A is B to what? No B is A.

[16:38] Keep some C is B, and lo and behold, by magic, you're back in the first what? Figure. Figure, yeah. And that's obviously Ferio, right? But notice it's a negative conclusion again, right? All we've had so far in the second figure is negative conclusions. Sum C is not B.

[17:02] Give myself my finger. Okay? Sum C is not B, right? Excuse me, and not A, yeah. Okay. You see that? Erased it to make it more clear. Okay. 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 the universal negative, namely some C, right?

[17:32] So if the some C's are B's and none of the B's are A's, then that some C cannot be. Now, what about every A is B and some C is B? Now, if you could 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?

[17:59] 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 I'll have to satisfy them, okay? Now, can you think of A, B, and C, examples to satisfy those conditions, that satisfy the conditions.

[18:35] Dog for A. Okay, animal would be. Now, what could I have for C? Um, a cocker spaniel. Okay, since every cocker spaniel is a animal, it's also true that some cocker spaniel is an animal, right? And then you could have a cat or something, right? Getting tired of cats. You must get as much mileage as you can on the same examples, huh? So now I stop and I say, have I satisfied the two conditions?

[19:10] Well, what's the condition? Are the premises true with those examples? Every A is B. Every dog is an animal. Seems to be true. Some C is B. Is some spaniel 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?

[19:40] Yeah, I'll circle that. Every spaniel is a dog. That knocks out the two negatives, right? That is being always the case Do I have one universal affirmative where no C is A? Yep. Undermine that cat, right? No cat is a dog. And that knocks out the two affirmatives as being always the case. So there's nothing affirmative or negative that is always the case. Nothing is always so.

[20:05] Nothing is necessarily so. Not 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. Is it? Okay. Now, what about this fourth case here?

[20:26] Well, here you can see that you can't get the set of all or the set of none. Obviously, can't get the set of none. Set of all because you have negative statement, right? You can't get the set of none either because what? You can't convert the particular negative at all, right? You don't have a universal negative statement to get it, right? Okay. So according to the rules I've given you up to this point, right?

[20:52] Which are sufficient for the most part. You might guess that this is not a what? Syllogism, right? But do you think you could find examples to satisfy the two conditions in this case?

[21:11] No. 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. Try as you will. Now, maybe you know, you just kind of think about it and say, look, if every A is a B, every one of the A's is a B. But some C is not a B.

[21:38] Could that some C be an A? Yeah. Is every A is a B, right? Yeah. I kind of 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?

[22:01] How can I show a set of all and some set of none? One of them, at least, that some C is not A. I can see it. It won't go away, right?

[22:22] Well, I could take the what? Contradictory of some C is not A, which is what? Also every C. Every C is A, right? Okay. And if every C is A were true, then since every A is B,

[22:45] then every C would be B, which contradicts this, right? You see what I'm doing? 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 premise. So, in other words, if you take the contradictory of this, which is every C is A,

[23:16] 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? 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. Cannot be so with those two, and therefore you must accept that.

[23:55] Yeah, that's the contradictory, right? Contradictory statements are opposed such that both cannot be true, both cannot be false. One must be true and the other false, regardless of whether you know which is the true or the false one, right? You see how you do that? That's very interesting, right? And that's kind of the exception to the rules I gave, right?

[24:22] For the most part, let's say in the second and third figure, if something is a syllogism, you can get it clearly by converting, right? When the premise, is. in this case, you can't convert and get syllogism, right? But if you went on to think, therefore, it's probably not a syllogism, and you try to find examples, you would never find examples that satisfy the conditions right? I know when I correct, you know, the logic.

[24:52] And they, you know, are kind of disprove and show that some form that is a syllogism is not a syllogism, right? Their examples are never satisfying the conditions. They never do. If they satisfied the condition, they would have made history, at least. But the point is that since it is a syllogism, they must have made a mistake in their example, and they didn't, as I told them to do.

[25:19] even Professor Ferguson had to do it to, you know, check over his calculation, right? Just like Einstein, you know, Einstein, you know, made some false calculations, you know, and some, some nobody corrected him, you know. Einstein, you know, finally realized he had his calculated right? huh? Heisenberg tells the time. He got a great idea, right? He was very excited. He came back to the institute there and he started, you know, calculating.

[25:51] He made all kinds of stupid mistakes, you know, and finally he calmed down and he started calculating slowly and came out just perfect, you know, And it's on the way, to the Nobel Prize, right?

[26:03] 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 is not a syllogism, it is, right? And therefore, look for examples. You will not have found examples that satisfy the two conditions. If you had done the simple check to take your time and some kind of, you know, damned it, satisfy condition number one, yes, I was like, you know, but you have to do that, right?

[26:28] And then you will see that the examples don't, and you might rethink it, right? So this is a very interesting thing here, right? It does follow necessarily that some C is not A, right? But the set of R and set of none is not in it as it stands, right? And you can't get the set of R and 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, the one premise to contradiction of the other premise.

[27:03] So that must be true then of all the other valid syllogisms that you can do that. Yeah, yeah, same exact idea. Yeah, yeah. This one here is just a proof that. More difficult, right? And notice the way we did that, in the 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.

[27:28] That's obvious, right? But if B is not so, then A is not so, right? And we said, well, if A were so, that by the first case, the obvious case, B would have been so, right? But you're given 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 with the if-then statement would necessarily include B being so, which contradicts the other premise, right?

[28:00] We're doing something similar here, right? And taking the contradictory of what I say is necessarily so, and showing that that is incompatible to two premises, because with one of them, it by the set of all, syllogizes to the contradictory of the other, right? So in a sense, you show it through the first figure, but not as easy as you do when you convert, right? You show it by taking the contradictory conclusion and adding it to one premise and adding it to another premise, And therefore you see that the contradictory conclusion is not compatible with those two premises, because added to one of them, it contradicts the other.

[28:46] If every C is A and every A is a B, then every C is a B, and that contradicts some C is not B. So you can't take the contradictory of the conclusion with those two premises. So you must reject every C is A with a hold on to these two, and therefore. you must. This is not true. Neither an affirmative statement nor a negative statement is always the case, right?

[29:17] 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 them in the form of a syllogism, right? If something is necessarily so, it's always so. But nothing is always so, therefore nothing is necessarily so.

[29:46] Another if something is necessarily so, it's a syllogism, but nothing is necessarily so, So you haven't got a syllogism. I think that's a syllogism,

[29:55] okay? So right away, now does the set of all apply 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 like some C is A, right? So this is a syllogism by the set of all.

[30:37] 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 here.

[30:58] You have the universal negative statement, no B is A. That's the first thing required in that you have set 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. Now,

[31:27] how about this over here? Does the 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 said some C is not B, right? We don't say the same thing So you suspect that you can't say necessarily that C is a subject and B is a predicate.

[31:55] So you look for examples then for A, B, and C to show that nothing is always so, and they have to be 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. So if you think of an A and a B such that every B is an A, A more dog.

[32:20] Yeah. So every dog is an animal. That's true. Now I've got to find a C such that some C is not a dog, and it can be even the no C is a dog. Right? We get every C is an animal. Take what? Cat.

[32:43] And then something that is not a dog. dog, that's not an animal either. A Stone, right? Oh, gee. So two. Where's the one stone, right?

[32:57] So in the mixed forms, right, where the major premise is the universal one, right, and the minor is particular, there's two cases that are syllogisms and two that are not, right?

[33:16] Just like there are two among the universal one there, but none among the two particular one there. Now we've got to look at the four remaining cases where the particular is on top as the major premise and universal below, right?

[33:48] So the particular is 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 every c is b. Every c is b.

[34:16] Or you have universal negative. Some b is a. No c is b. Some b is a. And no c. Excuse me. Some b 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?

[35:06] Now right away you know if it's two negatives down here that that's going to be not a syllogism, right? There's no way to get the set of none either with two negatives. So I look for examples for a, b, and c

[35:23] such that the premises are true when I put them in, and yet in one case every c is a, and every other case no c is a, right?

[35:35] Again, 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 with some b is not a?

[35:52] Just some? Either way. They say universal negative is true. The particular negative is true as well. Why? Okay. You know, or just some, right? You could say some animals are not what? Dogs, right? Okay. Now that works for that thing. But I find examples for c that satisfy the conditions. And I find something that is never an animal but always a dog. So the initial examples aren't going to work, right?

[36:41] I've got to think of 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.

[37:22] You tell the students. You say what? Some students have failed the course. Doesn't mean some have passed either. If I say some have passed, it means some have not passed, right? If you press the first exam. I say, some have passed.

[37:41] 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 are not, or some stones are not animals.

[38:07] 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?

[38:22] Plant. Plant. Yeah. So 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? No. It's not really a subset of all, do you?

[38:53] So I suspect that it's like invalid, right? Okay. So I look for examples for A, B, and C, and. Let's see if there's anyone to work this time, right? If I take dog up here, right, an animal, some animal is a dog, right? That's true. And then I take a c such that every c is an animal, and every c is a dog. Let's take a cocker spaniel or something, right?

[39:26] Some kind of dog, right? So every cocker spaniel is an animal, every cocker spaniel is a dog. And then let's take the cat and the horse. And every cat is an animal, but no cat is a dog, right? Are we satisfied now with the two conditions.

[39:49] Are the premises true when I use those terms. Some b is a. Some animal is a dog. True. Every c is b. Every cocker spaniel is an animal. Yep. Every cat is an animal. Yep. I satisfied condition number one. The examples are when they're substituted in, the premises are true, right? And second, now do I have one example where every c is a? Yep. I'll circle that one.

[40:20] Every cocker spaniel is a dog, right? I have one example where no c is a. Yep. Underline that. to make sure I have both conditions, right? Now I've shown that there's nothing affirmative or negative. It's always so when those premises are true, right?

[40:40] There's no affirmative statement that is always so, because sometimes no c is a, like no cat is a dog, right? There's no negative statement that is always true, because sometimes every c is a, right?

[41:02] Now some b. Let's put that somewhere. Some b is a. No c is b, right? You can't try the set of all, because you have a negative statement there. But you have the set of none. The set of none requires a universal negative statement, and something coming under the subject of the universal negative. Well, no c is b is universal negative, but is anything said to be at c?

[41:29] No. So I suspect that this is not a what? Syllogism, right? I got to find examples for a, b, and c to satisfy the two conditions.

[41:49] It might be easy to go back and take that b that is always an a, right? A dog. Yeah. So every dog is an animal. Some dog is an animal too, right? Take the students. Every student passed, but some students did not pass.

[42:15] Every student passed. It's true that some students passed too, right? Every student did pass, but some students you did pass. See, that's not good thinking there. And I have to find two C's. Such that no C is a dog. One of them. but every C is, a animal. Think of something like that. Cat. Yeah. And then a C that's never a dog, but never an animal either.

[42:47] Stone. I always pause. See, and I say, "have I satisfied both conditions." Well, condition number one is: are the premises true of those examples? Some dog is an animal. Yeah. And no cat is a dog. Yep. And no stone is a dog. Yep. I satisfy condition number one. I satisfied condition number two. You have one example where every C is A. Yeah, I'll circle that. Every cat is an animal.

[43:19] That's true. I have one example where no C is A. Yep. Nope. Underline that. No stone is an animal, right? So I satisfy all the conditions. So I show it not to be a syllogism, right? That's a separate thought to say why do those conditions? Why does satisfying those two conditions suffice to show it's not a syllogism, right? If you stop and think about it, those examples satisfying those conditions show that when the premises are true, right, there's nothing affirmative that is always so, and there's nothing negative that is always so.

[43:57] Because universal affirmative being true once shows that the two negatives are not always the case, and the universal negative being true once. Shows that two affirmatives are not always the case, right? So there's nothing affirmative or negative that is always so when the premises are true. If nothing is always so, nothing is necessarily so. If nothing is necessarily so, you don't have a syllogism, right? Okay, let's do that?

[44:28] I don't want to do more precisely, say, if it's a syllogism, then something follows necessarily. If Something follows necessarily. It's always so. But nothing is always so. Therefore, nothing necessarily so. That's a bad syllogism. Okay.

[44:44] So rather than repeat that, I'll ask the students you know the separate question, right? I'll ask, give them examples like this, and they have to find examples, and I'll check to see if the examples satisfy the conditions, right? But then I ask them elsewhere, why do those two conditions suffice to show that it's a syllogism? Then you have to bring out the fact that a syllogism means necessarily so, necessarily so requires something be always so, and the examples show that nothing is always so, right?

[45:12] You have to go back and wait to the square of opposition. You're seeing that if universal affirmative is true, any negative statement is not always true, right? And if universal negative is true once, no affirmative is always true, right? To exhaust the possibility, right? Nothing affirmative, so the universal particular is always the case. Nothing negative, right? Is always the case. Okay. Rather than find one example for each four where it's false, I find two where universal affirmative and universal negative.

[45:47] I can kill two birds with one stone. Suppose we kept the same order of the particular affirmative as the one we just did, and the universal negative. Yet, instead of some B is A, we had some A is C, no C is B. Would you then have a valid set of none? Well, if you could turn around, no C is B, and say no B is C, right?

[46:22] Okay. Then you turn this around. You could say that some A is not C, but you can't turn around particular negative. So there's no conclusion with C as the subject and A as the predicate. When you get to the second and third figure, you'll see that the set of all and the set of none never apply to as it stands. You have to convert them, right? And so there's an extra step there in the second and third figure.

[46:53] But after you fiddle around, converting, if you don't seem to be able to turn it into it, then you suspect it's not a syllogism, and then you look for an example, right?

[47:05] And maybe you know one one case is a little tricky because that's a little roundabout we can't show it. We'll see that, right? I mean, for the most part, that's all you have to do, right? The first figure is a set of all or a set of none apply to it as it stands. If it does, it should be clear right away that it's valid. If it doesn't apply to as it stands, then it's going to not be a syllogism.

[47:27] But you can always find examples to show that it's not a syllogism. Okay. In the second and third figure, the set of all and the set of none will never apply to it as it stands. But sometimes it can be made to apply by conversion. But if doesn't apply by conversion, then you suspect it's invalid, and you find examples. Okay. Now, I've done this for years, you know.

[47:54] I used to say to students, for those four forms that we showed were syllogisms, right? You can never find examples for those forms that satisfy the two conditions. Because they stay up all night trying to find examples, and you know, I you know used to be very flamboyant I take out my billfold, you know, and I wave, you know bills, you know, and say, "to the first man who can do this, right?

[48:19] " I put a lien on my salary you know, and he'll collect my salary for the rest of my life. See, and so I said, "If you're down in South Africa, you know, 40 years from now, you think of examples, you know, telegram me, and I'll send you the money, right? " You know, I used to say, you know, this is a crude age. They don't believe in truth, but they believe in money.

[48:40] So you know, I'm not sure about these things. see? And uh, it's impossible to do so. But we got one more case to look at. Sum B is not A, every C is B. Well, obviously, you can't get either the set of all or the set of none. can. you? You have a universal affirmative, but the other statement is negative. So you can't have the set of all, which requires. And obviously, don't have universal negative, so you suspect that this is not a syllogism, right?

[49:14] So you find examples for a, b, and c that satisfy the two conditions. So can you think of a, B, and A such that some b is not a?

[49:34] Hmm? Plant animal, okay. Since no plant is an animal, some plant is not an animal, right? And every c is b. Can you think of a c such that every c is a plant and every c is a plant?

[49:57] There's a micro. What is that micro organism? As for chlorophyll. Does that work? I think you got to stop and think of examples. You have to double the a.

[50:20] Maybe that's one way to do it, right? Yeah. Go. Okay, something. Oh, I said you mean yeah. Okay, that's one way to do it, yeah. Let's double it. Instead of taking two c's. Okay, which example for a? Animate, animate. Right. Animate and inanimate. Animate and inanimate. Yeah. Okay. Inanimate.

[50:46] And this is plant. No, that's white. White. And this is what. Snow. Snow. Yeah. So let's check now. Some B is not a. Some white things are not what animate, and some are not inanimate. Then, so that's okay. Every c is b. All snow is white. Okay. And snow. Animate universal what? Negative, right? Snow animate universal negative. Okay. Can you also find two examples for c? Are we

[51:30] doing that? Or you might you might take our friend there. Animal. And dog, right? Some animal is not a dog, right? That's true, right? And every every spaniel is an animal, right? And every cat is an animal. I satisfy condition number one, right? The premises are true. Some animal is not a dog. And every every spaniel is an animal, and every cat is an animal. That's condition number one.

[52:07] Do we have an example where every C is A? Yep. I'll circle that. Every cat is a dog. Do we have an example where no C is A? Yep. Underline that. No cat is a dog, right? So none of these ones are syllogisms, right? Okay.

[52:29] So notice that in the four cases with two universal statements, there were two syllogisms, right? And in the four where you had two particulars, there's no syllogism, right? In the eight where you have a mixed one, well, in the four where you have the minor, the major premise being particular, you have no syllogism. But in the four where you have the major premise being a universal, you have two, right?

[53:00] There are two that are not a syllogism. So out of the sixteen cases, then, we have four that have the form of a syllogism, right? And twelve that do not have the form of a syllogism, right? Okay. Now, let's put the four forms that are

[53:21] syllogisms. Four forms in the particular. You had every B is A, and every C is B. And you had no B is A, and every C is B. And then in the mixed forms, you had every B is A, and some C is B, and no B is A. And sum C is B. Okay. Now, if you look at these four,

[54:18] you notice that all four possible statements, that C is a subject and a predicate predicate, are found here, right? From every B as an A and every C coming under the Bs, every C must be an A, right?

[54:36] No B is an A, but all the Cs coming under the Bs, none of those Cs can be an A, so no C is an A. Now the other two are like that, except you're told only that some C comes under the Bs, right? So if every B is an A, you know some C is a B. You know that some C at least is an A, right?

[55:00] In the last case, you're told that some C is a B, but none of the Bs are A's, so you know at least that some C is not an A.

[55:13] So in the first figure, you can draw all four possible conclusions: universal affirmative, particular affirmative, universal negative, and particular negative, right? None of the other figures will see if you draw these four conclusions.

[55:31] In the second figure, you'll find out that all you can draw is a negative conclusion, no affirmative. Just there's two here: no C is A and some C is not A. You'll find out in the second figure. In the third figure, you can't draw any universal statements, just particular: some C is A and some C is not A, right? So there's a falling off of power in the second and the third figure.

[55:59] A syllogism that has a universal conclusion is more powerful than one that can only conclude to particular, right? So in the third figure, you have no universality in your conclusion at all. So that's the reason why it calls it the third figure. one reason why it calls the third figure? It's the weakest, right? In the second figure, you do have universality, but only universal what? Negative. But in the first figure, you have both universal negative and universal affirmative.

[56:28] So in terms of power, the first figure we'll find out is more powerful than the second, and the second is more powerful than the third, right? And that's why it's more important to remember the first and second figure and the and the powerful cases, right? Because those are the ones I use over and over again in in reasoned-out knowledge in geometry and natural philosophy and theology and so on, right?

[56:53] Okay. Now, what about the premises when you have a syllogism in the first figure? Let's Make an induction here now, right? Okay. What can you say about the major premise? The premise that contains the predicate and the conclusion It's always what In the first figure, universal. And the minor premise is always what affirmative. Okay. Now, I'll write that down. The major premise is always universal, syllogism, and the minor premise is always affirmative.

[57:44] Now, is there any other case that we considered any of the other twelve that have those same two conditions? No, no. Because if the major premise is universal, there's only two possibilities: universal affirmative or universal negative, right? And for each of those two possibilities, if the minor premise is always affirmative, there's only two possibilities, right? Either universal affirmative or particular affirmative, right? So two times two is four.

[58:20] Right. Only four cases, right? The other twelve are not what syllogisms, right? Although I know, as we know now, the one that would deceive the students, right?

[58:34] Philosophy teaches you that same thing. But the two most important ones, are these two universal ones. Now, not only is there a falling off of power in the second, and even more so in the third figure, but the cases that are syllogisms in the second and third figure, it's not clear right away that the set of orders. So there's none involved, right? What you have to do is turn them around, right?

[59:12] Put the subject where the predicate is, or vice versa. So now we have to study the topic of conversion, right? Because conversion is a way of making an imperfect syllogism in the second or third figure clear, right? And when you convert, you get back into something using the first figure. Okay.

[59:38] Let me just exemplify before I go into conversion here, this thing here. Take the second figure here. Suppose you had, on the second figure, this is the arrangement of terms. The middle term B is the predicate in both cases, right? In the third figure, it'd

[1:00:04] be reverse. The middle term will be the subject in both cases, huh? Okay. When I was first learning this back in high school, my mnemonic device was that in the second figure, the middle term is in the second spot. There's no third spot. Once I do that, then the third was just the reverse, right? Okay. That's always number the first figure. That's why I kept the second and third figures as my own mnemonic device.

[1:00:34] But there's stuff close to the second one because it's less powerful than the first, but more powerful than the third, right? Okay. Now notice in the second figure, if you had a form like this, no A is B, and every C is B, right? Okay. Now does the set of all or the set of none apply to it as an outstanding? No. You've got a universal negative statement, no A is B, but nothing is coming under A.

[1:01:05] You have a universal affirmative, every C is B, but nothing comes under what C. But now, if I could turn this around, if I could turn this around and say that no B is A,

[1:01:20] and keep the other one, I could say that every C is B. I'd be back in the first figure, right? And I could syllogize that no C is A, right? But now I've got to consider if it's true that no A is B. Is it necessarily true the reverse is no B is A, right? Okay. That's we have to consider, right? And we're going to find out that in fact it is, but question is how can we know this for sure?

[1:01:50] Because there's an infinity, right, of possible statements in the form no A is B that is true. Right. And I can't go through all those examples to see that it's always so, right? So how can Aristotle show for an infinity, right? Of possible statements that are true in the form "no A is a B," that regardless of what A and B are, so long as it's true that no A is B, it will also be true that no B is A.

[1:02:22] Now, what about this? See, every A is B, and every C is B. Now, if I could turn around the statement "every A is B," okay, if I could turn that around and say that every B is A, necessarily keep the other one like it is, I'd be back in the first figure, and I could syllogize the universal affirmative statement, right? But is it necessarily true that if every A is B, the reverse is true?

[1:03:04] Well, just one example is enough to show that it ain't necessarily so, right? So every dog is an animal, but not every animal is a dog, right? Every odd number is a number, but not every number is an odd number, right? So you can't convert universal affirmative and keep it universal, right? Necessity, but the universal negative you can, we'll see. But you know, again, my monetary challenge, right?

[1:03:38] I challenge anybody to find a what statement in the form "no A is B" that is true, and reverse "no B is A" that is not true. How can I be sure about that? See, well, let's see how Aristotle does this, huh?

[1:04:00] I'm going to look at the four corners here of the square of opposition. I'm going to see. To what extent they do or do not convert.

[1:04:13] So let's take the statement. The letter is not important, but just take B and A. If no B is A is true, I say, okay, that the reverse will be true, right? No ASB. I'm just asserting that. I haven't given you any proof have I, right? Okay. Now, if you don't admit that no A is B is necessarily true, right?

[1:04:45] Then, by the square of opposition, the universal negative is could be false, right? Then what could be true? Some A is true. Yeah. Okay. So if this is false, what I'm saying obviously, if it's true, you're given it's true that no B is A. No matter what B is, no matter what A is, right? I maintain that it's true that no B is A. It will be true.

[1:05:26] Reverse, no A is B. I say it can never be false. But if you don't accept that, you've got to say that some time, somewhere, no B is A is true, but no B is no A is B is what false, right? Now, if no A is B is false once, even right? At that one time, then by the square of opposition, some A is what B, right?

[1:05:54] See how the logic of the second act presupposes the logic of the third act, right? Okay. So then, some A is B is what could be true, right? Okay. Now, if some A is B can be true, let it be true. You said it could be true, right? Okay. And now Aristotle says, let's give a name to the A that is a B. There's just one A that is a B, right?

[1:06:22] You can't tell me else. Let's give a name to the A that is a B. okay. And here's what I call. it. Let us call the A that is a B. Let's call it X, right? Okay.

[1:06:42] Now, if you call the A that is a B X, then X is both an A and a B, isn't it? Okay. All that.

[1:06:54] Then X is both an A and a B. Now X is both an A and a B. Then there is a B, namely X, right? That is an A.

[1:07:13] Then there is a B, namely X, that is an A. So that follows from your saying that no A is B is not necessarily true. I say it is necessarily true. If you say it is necessarily true, then you are saying it could be that some A is B by the square of opposition. If that could be, let it happen, right? And let's call the A that is a B X, right?

[1:07:47] The X must be both an A and a B. Therefore, there is some B, namely X, that is an A. So you are saying that when it's true that no B is A, possibly some B is an A, right?

[1:08:04] It's an admission, right? So something impossible follows, right? No, I am not arguing in the sense that if that argument, right, I am saying that if no B is A is true, then the reverse, no A is B, will be true, right? If you say that's not true, then some A is B could be true, right? And if that is true, there would be an A that is a B, and we call it X, right?

[1:08:34] The X would be both a B and an A, and therefore there would be a B in the X that is an A, right? So something impossible follows, right? No B is A is true. It's impossible that some B is an A, right? But that would follow from your admission, right? So you are going to be forced to say that if no B is A, no A is B, right?

[1:08:58] I think this is something a monkey can't do. It's kind of amazing to see this, right?