Logic (2016)
Lecture 39: The Three Figures of the Syllogism and Their Validity

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Every mother is a woman. No man is a mother. Now, what figure is this? What figure is this? Is the middle term in the middle, right? So it's in the first figure, right? It's in the slanted position, you might say, right? Middle term, right? Now, if I say to my students, what follows from this? What do the students have to say, do you think? If every mother is a woman and no man is a mother, then no man is a woman, right? Wouldn't you kind of attempt to say that, right? And that's why I knew Mark was wrong. You think it is no man is a woman. Now, the reason why he's deceived is because of the matter, right? He knows that it's true that no man is a woman. At least he knows that no man is a woman, right? And he knows that there's some connection between the fact that no man is a mother and no man is a woman, right? But does this conclusion in the forum follow necessarily from these two, right? Okay, well, I'll give you another one. Every dog is an animal. That's true, true. And no cat is a dog. And nobody would think it follows that no cat is a, what? An animal, did they say? Because they know it's false, right? They know it's true that every dog is an animal. They know it's true that no cat is a dog. They know it's not true that no cat is an animal, right? Well, why are they deceived by this and not by this? It's exactly the same form, right? If this really followed through these two premises, then would follow here that no cat is a, what? A dog. This doesn't follow from these two, right? Even though it's true that no man is a woman, right? And given there's a reason why a man is neither a woman nor a mother, right? But still doesn't follow by the form, right? See how easy this is to deceive them? You see, sometimes, no man is a woman. If you give a free term right now, say this is A and this is B, right? Sometimes in this form, no man is a woman, the universal negative is true, right? Sometimes in the same form there, every B is A, right? No C is B, same form, right? Every, what? Excuse me. Sometimes every cat is an animal, right? Sometimes every C is A and sometimes no C is A, right? If the universal negative is sometimes true, no affirmative is always true. If the universal affirmative is sometimes true, right? Then what? Yeah, then the two negatives are false ones, right? I put this down here to show what you're saying. This is followed, right? Didn't say the same thing down here, right? That form is invalid, right? It may be that every C is A, it may be that no C is A. There's nothing you can say, affirmative or negative, that is always so. And if nothing is always so, then nothing is necessarily so, right? But then the form of these dead soldiers doesn't be wanted to, right? If something is necessarily so, then something is always so. But if nothing is always so, therefore nothing is necessarily so. He is not so. There he and B are standing for premises. A great pleasure to know. What is deceived there, right? Because he knows that no man is a woman is true. He knows there is a connection between his not being a mother and his not being a woman. I'm going to look at the second figure, right then. On the second figure, the way I used to remember this was that the, what? Little term is in the second place, right? Stared out. So, what are the four universal cases? Every A is B, and every C, two affirmatives, two universal negatives, right? And again, the two mixed ones with, in one case, the universal affirmative on top, right? Every A is B, and every C is B, and every C is B. Oh, okay. See, that's it. I got it. I'm stupid. I make stupid mistakes. You've got to check yourself, right? Now the question is, does anything follow with C as a subject and A as a, what, predicate, right? In any one of these four cases, right? And this so what follows, right? And take that first one. Every A is B, and every C is B. You certainly can't try for the set of none, but that requires a universal negative, right? You could try for the set of all, maybe, right? Because you've got universal affirmatives, right? But does anything come under the subject of this universal affirmative? So the set of all would require every A as a B, and then whatever is an A as a B. But nothing is said to be an A, is it? Anything is said to be a C? So the set of all doesn't apply to this standard now, is it, right? You do have a universal affirmative. In fact, you have two universal affirmatives, right? But the set of all requires not only a universal affirmative, but something else which is said to come under that, what? Subject of the universal. You've got two universal affirmatives here, but nothing is said to be an A or a C. There's no way to get the set of all, right? So you're not going to have the set of none, right? Now, if I could turn this around and say every B is an A, then I'd be back in the first figure and I'd have a universal conclusion that every C is A. But can you turn around necessarily a universal affirmative? It necessarily converts to the particular, but not to the universal. So conversion is not going to get me there either, right? I give up. But I'm a tough guy, you know, and I want to prove that when these are true, it could be that every C is an A. It could be that no C is an A, right? So I'm going to find examples. Give myself a little more room here. So I want to find examples for A, B, and C, right, that satisfy what two conditions? Well, I want to have examples for A, B, and C, such that when you place them into this position, right, the premises will be what? True. So there'll be no defect in that. And I want a second condition that I have one set of examples where every C is an A, and one where no C is an A. And they'll knock out everything. Because there's only two possibilities, affirmative and negative. A universal affirmative knocks out universal negative and particular negative. A universal negative knocks out universal affirmative and particular affirmative. This is a man with a major premise of my practice. I hate the major premise. He's on top. Okay. So I'm going to try to find examples now for A, B, and C to prove my suspicion, right, huh? Now, can you think of an A and a B, such that every A is a B? And can you think of a C, such that every C is also an animal, right? And every C is a dog. Do you know any particular kind of dog? Yeah. That's what I was thinking, yeah. So I'll just pick Carpus Spaniel, right? Okay. So Carpus Spaniel, every Carpus Spaniel is a dog as an animal, right? That's true. And every Carpus Spaniel is a dog, right? Now I need an example, one other example here for a C that is always an animal but never a dog. So, now, I make stupid mistakes, right? You all make stupid mistakes. So I'm going to check my homework, right? Every A is a B. Every dog is an animal. Is that true? Every C is a B. Every Carpus Spaniel is an animal. Every cat is an animal. Yeah. I have satisfied condition number one, Mr. Berkowitz. Okay. Now you get the kids up on the board, you know. They got to go through all this stuff. Now what's the second condition I'd ask? Yeah, yeah. Now I'll circle the every C is a, Carpus Spaniel, right? And that marks out any, what, negative statement, right? And then one example, which I'll underline, where no C is an A. No cat is a dog, right? That marks out any, what, affirmative statement from being always true. So this shows that when these are true, even, right? Nothing negative could always be the case, because sometimes universal affirmative is the case. And no, what, affirmative statement could be true always, right? Because sometimes no C is a thing, right? And all I need is one example of each, right? To show that there's nothing affirmative or negative that is always, what, so. And that's a complete disjunction, right? It's got to be affirmative or negative, right? Universal or particular. But the universal affirmative knocks out both negatives, right? Until two birds and one stone, right? Universal does it a lot, right? And the universal negative knocks out both affirmatives, right? Kills two birds and one stone. And I could have three different examples for A, B, C, you know. So nothing follows from this case of the second figure, right? And what about this mess down here? Can anything come from the two negative parents, huh? But I want to make sure, right? So I want to find examples for A, A, B, and C that satisfy what two conditions. Well, when you substitute them again to this case, both parents say they're true, right? And that shows that you have no difficulty with your matter, right? It's not because your statements are false, right? She says nothing follows from them, right? Okay. And then what's the other condition that is satisfied? Yeah. At some time, every C could be A, which meant no negative can be always true. And another example where no C is A, which means affirmative statement can be A. So there's nothing, affirmative, negative, I think. Notice this is a little bit like addition, subtraction, multiplication, and division, right? It's one thing to have the right numbers, right? To add, subtract, multiply, or divide, right? And the other is to what? Is to multiply or to add or subtract, right? Now I always take the example of one's checkbook, right? And sometimes your numbers don't agree with the bank's numbers and so on. And sometimes you go back over your checkbook and you find out you made a mistake, right? Sometimes you misread the number you would have written in there because you did it kind of sloppily, right? Now you've discovered that, you know? But sometimes you didn't add or subtract, what? Correct it, right? Sometimes you might have made both mistakes, right? So your numbers you get by adding, subtracting, multiplying, or dividing can be the wrong number, right? Either because you have the wrong numbers or one wrong number at least, right? To add, subtract, multiply, or divide. Or because you didn't add, subtract, multiply, or divide. But if you're really stupid, you could make both mistakes, right? I had, you know? And, you know, when Aristotle, if you remember in the first book of an actual hearing, he takes a, what, a list, right? He says, well, his premises are false, right? And his inclusion doesn't follow. So, you know, he finishes the guy off, right? I mean, if there were two, it wouldn't fall, right? So, you know, what they say is the bad happens in many ways, the good in one way, right? So, in order for your citizen to be good, the premises have to be good, that is to say true, right? And the conclusion has to follow, right? Just like in order for my addition, you know, we're out in the ocean there and we're going to buy ice cream cones for the grandchildren. That's what I've got to have to, you know? So, I've got to know how many grandchildren I have, which is 19, and what an ice cream cone costs, right? And then I have to multiply correctly, right? Yeah. Yeah. So, if I have the wrong price for the ice cream, right, or the wrong number of grandchildren, I can be mistaken with them cost to me. Or if I don't multiply correctly, right? So, there's a likeness there between the two, right? The Greeks called the art of calculating, you know, logisticae, right? We're going to have logistics, right? But the one for reasoning is logical, right? But they're both taken from reason, right? Because reason does both of those things, right? So, if you think of an A and a B such that no A is a B, nothing wrong with the same example, no Rory. What? No stone is an animal. No stone is an animal. It's easy to think of that, right? I've got a little more thought, you know. I've got a little more thought. Now we've got to find two examples for sea. One such that no sea is an animal, right? Now a tree is not an animal and it's not a, what, kind of hard to follow, right? No kinds of stones. So that's going to be an example that would make you that kind of stupid about knowing the names of particular kinds of stone, right? But you could if you had no particular kind of stone, right? So, it's stone for our middle term here, right? A lot of things are not a stone, right? Now can you think of a sea, an A that is not a stone? Or you could say animal, right? No animal is a stone. Now can you think of a sea that is not a stone but is always an animal? Go off, right? Now I've got to think of another example where instead of every sea being an A, no sea is an A. But it's still not a, what, stone either. You can use the same stones, you know, to kill two birds, do so, right? Now I'm stupid and I make stupid mistakes sometimes, right? And I think I've shown something ahead of me. So I just check my work here. With these examples, are these statements, when you substitute them in, are they true? Well, no A is B. No animal is a stone. That seems to be quite true, right? No sea is a B. No dog is a stone. That seems to be true. And no tree is a stone, right? I have satisfied condition one, but that's not enough. Now do I have one example, or the second condition, right? Of every sea is A, and one example where no sea is A? Yeah. And I'll circle the one where every sea is A. Every dog is an animal, right? And I'll draw a line under the one where no sea is an A, right? I have satisfied both conditions, right? So dog and animal show that when these premises are true, no negative statement is true always, right? And these examples, animal and tree, show that when these statements are true, no affirmative statement is always true. So nothing affirmative, nothing negative is always the case, right? And if nothing is always the case, then nothing is what? Necessarily so. And if nothing is necessarily so, then you've got a syllogism. What's the definition of syllogism? Well, it's speech, right, in which some statements lay down, another follows necessarily because of those laid down, right? Shakespeare says it must follow the night of the day, if you're true to yourself and it can't be false to any man, right? But is that a syllogism, right, that might follow the day, right? Not because of the day of the day of the night, right? It's something else, right? It's the sun going around the Earth, or the Earth going around the Earth. Yeah, I don't play the Earth too, that's actually the sequence here. Okay. What? He's here. Okay. The two conditions, huh? When these premises are true, I want to show, it could be that every C is A, or it could be that no C is A. And if it could be that every C is A, then no negative could be always true when these two are true. And if some time no C is A, who is it, no affirmative could always be the case. So there's nothing affirmative or negative, that's all there is. That's a complete division, right, of simple statements, right? Either affirmative or negative. So neither, any affirmative. Dog and animal shows, what, no negative statement is always the case. This shows that no affirmative statement is always the case. Therefore, the ball game's over, right? This is not a syllogism, right? So this one is what? Same thing, wasn't it? Yeah. So these, neither one of these is. Now, how about down here now? Thank God, you can convert a universal negative, and it remains universal. That's why I can get a universal negative in the second figure. So I convert to no B is A, right? Which shows you can do that always. And then just bring this over, right? You get every C is B. Let's just abbreviate here, here, here. To behold, the first figure. See why you call that the first figure? All you can do in the second figure is show the universal negative, right? But to do so, you've got to convert, and you run back into the first figure, right? Powerful figure, and it's more clear. It's all right. You put it first, right? So, by conversion, I see, then, that it follows that no C is A. So this is another way I can prove the universal negative, right? You get the best propositions in that form. And what about up here? What? You get the universal negative in the bottom. Yeah, yeah. So I could, you know, I could have... But could I have any conclusion with C as a subject and A as a predicate? That's the question I'm asking, right? Because no B is C would... If you put that here, no B is C, and every A is B, well then, no A is what? Is C, right? No A is C, right? I can convert that around to no C is A. So I left this to the last because you have to convert twice, right? Again, it's a universal negative, right? And you might suspect you can't get a universal affirmative conclusion even out of two universal affirmative statements. How are you going to get so if you don't get a negative statement or you have particular statements, right? It's falling off the power that's going down, right? You can get a universal conclusion but only universal negative in the second figure. In the first figure you could have universal affirmative and universal negative, right? So that's a more powerful figure, right? And if it's a syllogism in the first figure, you can see it right away. Here you've got to convert, right? Here you've got to convert twice. See, here you just have to convert the major premise and you're all set back in the first figure. Here you've got it if the A and C are reversed and you can turn it around. Convert shapes. Well, that was fun, wasn't it? You think that I got you. Well, that's an extra time. Yeah, I can no A and C and I turn around and I don't see it right away. I want to see where that thing I can lose and C is a subject that has a predicate, right? And you can invite two conversions, right? Yeah. Messy here, huh? I take my magic. Oh, did you take my magic? I took my magic, but I wasn't very... One, I wasn't entirely awake and... You never get anything in any figure with two negatives, huh? Well, they're affirmative and negative because you always have a, what? Negative conclusion, huh? Two affirmatives together. Affirmative conclusions. Now, in the third figure, your middle term is the subject in both cases, right? So, every V is A, every V is C, right? So, the middle term, the common term, is the subject in both premises, right? In the second figure, it was the predicate in both. In the first, it was that slant, right? It was the subject in the first major premise and the predicate. Or middle term should be in between, right? And then you have, down here, these two negatives, right? No V is A. And then you have the two ones where we have a mixed, one universal affirmative. Every V is A, and no is the reverse. No mistakes. Now, the question is, with premises that are true in this arrangement, does anything follow necessarily, right? Now, every V is an A. Now, if I could find out that something was a V, I could say it's an A, right? But, I'm not saying anything is a V over here, am I? You can convert a, what, universal affirmative partial, right? So, I can convert this to some C is a V, and keep on top here, every V is A, and lo and behold, by magic. And what figure are we now? And I can see right away that the set of all, right, applies. If every B is an A, then whatever is a B is an A. Now, you only know that some C is a B, right? Yeah. So, this is what you're going to see in the third figure. You only have particular conclusions, right? So, even on two universal affirmative, that's the most affirmative affirmation you can get, right? You can only get some C is A, right? I mean, it gets me to find out the power is dying quickly. He's squeezing as hard as he can. See, if I can reason out that some C is A, right? Reasoning out, right? Or thinking out of conclusion, right? The ultimate thing, right? So, I mean, episteme, you know, reasoned out knowledge sometimes, right? Geometry is a reasoned out knowledge of lines and angles and figures, right? And arithmetic is a reasoned out knowledge of numbers and so on. Now, how about no B is A and every B is C? Can I get the set of none there? Well, you got a universal negative, no B is A. Convert. Because it didn't seem to be a B. No. But I can convert it. It would be a C to every, all of this power, yeah. Some C is B and no B is A. Lo and behold, by magic, I am back in the figure. First figure, it's not the figure, right? You have the set of none, right? If no B is an A, then whatever is a B is not an A. But you know only that some C is a B. So, again, it's a particular conclusion, right? So those are the kind of obvious that you're making on. Now B is A and no B is C. Well, two negative parents, you know, an enemy offspring, right? Can't get the set of all. You don't even have a universal affirmative, right? Can't get the set of none. Because that requires something affirmative, isn't it? If no B is A, whatever is a B is not an A. So you've got to have an affirmative part there, right? So two negatives, you just expect nothing. So we're going to find examples for A, B, and C. It says, sorry, the two, what? Additions, right? And so, can you think of a B and an A such that no B is an A? Okay, now I've got to find two examples for C. One where no stone is it, right? Or both where no stone is it. But one is a, what? Always an A. Do you think of something that is never a stone but always an animal? Oh, yes. So, no stone is an animal and no stone is a dog. Is it true, right? Do you think of something else that no stone is it and is not ever an animal? Okay, true, right? Now, just so to avoid a stupid mistake, right? Have I satisfied both conditions? Condition number one. When I substitute what I have for A, B, and C as examples, these premises will be true. So I have perfect matter. The numbers I'm multiplying are cracked without any doubt. So, no B is A, no stone is an animal. Looks true to me, right? No B is C, no stone is a dog. No stone is a tree. I have satisfied condition number one. With those examples, the premises are both true. There's no defect whatsoever in the matter, right? Because I'm just concerned with the form. Now, if I have one example where every C is A. So, let's circle the example of the universal affirmative, right, example. Now, I've knocked out two birds, right? Two universal negative, and particularly negative. They're not always true, are they? And I have one example where no C is A. Yeah, I'll underline that example, right? So, sometimes universal negative is the case, right? Knocks out the two affirmative sounds. So, two stones, I've killed four birds. A jato is not a soldier, right? Now, how about every B is A, and no B is C, right? Can I get any conclusion with C as a subject and A as a predicate, right? Well, I'm going to have to try from a set of none, right? Now, if I convert every B as A, this would be sum A as B, right? Sum A as B, and no B as C, right? I could conclude that sum A is not C, right? But, does that give any conclusion with C as a subject? No, because you can't convert a particular negative, can you? If sum A is not C, then you can't turn around and say sum C is not A. So, there's no conclusion with C as a subject today as a predicate, right? Question you're asking, right? So, let's see now if I can get examples to prove this on A, B. Can you think of a B and an A such here as B as an A? Question you're asking, right? Question you're asking, right? Question you're asking, right? Question you're asking, right? Question you're asking, right? Question you're asking, right? That's the easy part, right? Every dog is an animal. Now, can you think of a sea such that no dog is a sea, but every sea is an animal? Can you think of an example of a sea such that no dog is it, and it's never an animal either? Yeah. Now, if I satisfied the two conditions, are the premises in fact true, as a matter, perfect? Every B is an A, every dog is an animal. No B is a sea. No dog is a cat. No dog is a stone. I have satisfied condition number one. Now, second condition, right? I have one example where the universal affirmative is so, so I can knock out the two negatives. Yeah. Every cat is an animal, right? So, I have one example where every C is A, which means that no C is A, and some C is not A, can't be true always. It's never true necessarily, right? And now I have one example where no C is what? A. Yeah. On the line of that example. No stone is an animal, right? So, I have one example where the universal negative is the case, no C is A, which means no affirmative statement is true always. So, nothing affirmative or negative is true always, and therefore, huh? Nothing is necessarily, therefore, you don't have a syllogism, right, huh? The syllogism is a speech in which some statements lay down, another follows necessarily because of those laid down. So, nothing follows here necessarily. So, notice you have two cases here, you have a conclusion, but they're both particular, right? But you have to convert, right, to see that, right? That's where the first figure comes first, but also because it's more powerful, right? You have both universal affirmative and universal negative in the first figure. Second figure used to have universal negative. Here you've got no uresality, right? This is really, you know, feeble. So, we put that third, right, huh? You know? Now, now it's thinking about God, right, huh? You use these forms, right? But, of course, God is not universal, right? Not strictly speaking. Of course, in the philosophy, we don't know about the Trinity, but even there, you've got to be careful. The universal, in the strict sense, is set of many that have, what, a separate existence, right? The existence of the Father and the Son and the Holy Spirit is the same. I am who am God. But, I noticed that with a universal, with a negative, with a singular, you can convert it, huh? Let me show you what I mean here. Socrates is not a woman. Can you convert that and say, no woman is Socrates? If some woman could be Socrates, we'll call that person X, and X is both Socrates and a woman, and therefore, some, Socrates is some woman. But we said, you know, Socrates is not a woman, right? So you can convert that, right, huh? That's important to know, because if you're syllogizing with God as one of the terms, right, you can treat Socrates as not a woman the same way as no man is a woman, right? You can convert it, right? Say no woman is Socrates. Tabitha is not a dog. That's my daughter's cat, Tabitha. Tabitha comes into my, you know, awful example. Tabitha is not a dog. Well then no dog is what? Tabitha, right? Kind of obvious, right? So you can syllogize, you know, and convert with that just like you do with universal negative, huh? That's what we syllogize. God is pure act. Being composed is pure. We can go into this. That's, logic is not studied for its own sake, right? Thomas says it's reduced to looking philosophy as a tool, but not as a chief part, right? Kind of amazing though, you know, Aristotle takes up being, you know. He speaks of the two main distinctions of being as act and ability, which goes back to what? The natural philosophy, and then being according to the figures of what? Predication, right? Substance and quantity and so on, right? So you borrow one, it comes out of natural philosophy that you understand act and ability, which is one of the main divisions of being. And then the other main division is being according to the figures of predication. The figures are being said of, right? Being said of individual substances like you and me. But it's interesting when Thomas is talking about Aristotle's leading us by the hand, you know, from sometimes from this science and this other lore science. So he leads us by the hand from logic, like in the eighth book on substance, right? He leads us by the hand from logic. And then the ninth book, he leads us by the hand from natural philosophy, right? And Thomas says, with the brevity of wisdom, that natural philosophy proceeds by way of motion. That's how you first learn act and ability, right? When you're in grade school and you get to cumple clay and you can mold it into a sphere and mold it into a cube. You just have to get the idea of ability and act, right? But then in logic, it proceeds by way of predication, Thomas says, you know? By the way that something is said of something, right? So the isagoic is about the five predicables, right? And the category is about the ten predicaments, right? And you've got the said of all and the said of none, right? So I mean, logic is about the way something is said of something, right? Because something is said of something in reason, you know? That's the only place where something is really said of something. We use language, too, but that's signifying, you know, something in the mind is said of something. So it's kind of interesting, the two main distinctions of being. One is gotten by the way something can be said of individual substances. And that gives you substance, quantity, and so on, right? What are you? How big are you? How are you? Towards what are you? You know? And so on, right? Where are you? And the other is from act and ability. My two greatest teachers, you know, graduate schooler Charles de Connick in Monsignor, right? I would see Jan would be doing a scene from logic and kind of from natural philosophy, right? And they could both do both, but I mean, that's kind of… But I mean, you know, they wouldn't teach metaphysics, right? Because then you have to get a little natural philosophy first in logic, right? And it's kind of interesting. You've got the secondary guys who teach a little metaphysics. You've got the secondary guys. You've got the secondary guys. You've got the secondary guys. You've got the secondary guys.