Introduction to Philosophy & Logic (1999)
Lecture 36: The Four Forms of Hypothetical Syllogisms

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Nobody wants to learn logic anymore? No, no, there are people who are teaching logic. But first I was chairman, and that wasn't even my schedule for logic. And then, now, let me fill that too. Okay, I'm going to have copies here. One, two, three, four, how many copies do you need? One, two, four, five, six, seven, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight, eight Okay? And I talk about the forms of the either-or sojism and the forms of the if-then sojism. Okay? Because they're fairly brief, right? Okay? But first we'll be finishing the form and matter of sojism, right? We're going into those forms. Okay? Now, we may, I was next to wondering, if I can copy it again, this is the form of the simple sojism, the regular. There's what I just call the sojism period. That's going to be involved now. Okay? So, four, five, six, seven, eight. I'll give you now, but maybe we shouldn't read this the next time. Let's read the first one I passed out. Okay? But, uh, okay? And, uh, I mean, eventually we'll look at a few simple theorems of Euclid. And you'll see, like, like, for example, in the, uh, sixth theorem of Euclid, in the first form, that uses the, the, the sojism and uses the hypothetical, uh, either-or sojism, right? Sometimes they call them categorical sojism, like that, right? Categorical sojism, the, uh, the, uh, hypothetical condition of the if-then sojism, and the disjunctive with either-or sojism, right? Okay? But even in a simple theorem, like theorem six, you know, the first one, as far as you see, using all three, right? In the first, uh, theorem, say, in Euclid, he uses just one kind of sojism, right? So the first theorem in Euclid is, is the one where, on a straight line, you're going to construct a, what, equilado triangle. And so he takes this end as the center of a circle, and this is a radius, and he draws a circle like that, right? And he takes this as the center of a circle, and this is a radius, and he draws another circle, right? And when they cut, he draws lines from there to there, right? And he's got to be equilado, right? Well, he's got to prove that these two lines are equal because they are what? Let's get numbers here, A, B, and C. He's going to prove that A, B, and A, C are equal, and what's the statement which is one of the reasons that? A, B, and A, C are radii of the same circle, and all radii of the same circle are equal, right? They have two of the cell to there, right? All radii of the same circle are equal, right? A, B, and B, C are radii of the same circle, therefore, they're equal, right? It's very simple, right? Then he wants to prove that A, C, and A, B, C are equal, right? Okay? And it's exactly the same sojism, because they, A, C, and B, C are radii of the same circle, and radii of the same circle equal, right? That's what Aristotle just called the sojism period. It needs to be proved. A, B, and C are. Yeah, you have to be able to declare everything, right? Now, how do you have to prove that A, B, and B, and C? You can't prove that in the same way. And then the second statement, which is saying that A is so, right? Well, then necessarily you must say what? B is so, right? So this form here is a syllogism, and it's obvious, huh? I want you to understand what you're saying, right? Okay? You have to go back to something obvious, because you can't prove everything, right? You have to prove everything, you can prove nothing. Okay? Now, what about if A is so, then B is so? A is not so. Does it foul necessarily that B is not so? See, now a lot of times students will think that fouls and B is not so, right? And as far as your imagination, you know, there's something which we call in philosophy false imagination, right? And false imagination is the main cause of deception on the side of the knowing powers, huh? Now notice, huh? If A is so, then B is so, A is so, B is so. If A is so, then B is so, A is not so, then B is not so. It's imagination, it seems to, what? Exactly the same, right? Doesn't it? Yeah. That's false imagination, right? Now, how do we show that this form here is not a syllogism? You know, that contradicts it. Okay. You've got to find examples where, when you, for A and B, right? Such that when you substitute them in, right? You get a false. Well, first of all, that these statements will be true, right? Okay. And you need one example where B is, in fact, so, and one example where B is, what? Not so, right? Okay. Now, why do you need that, see? Well, if something followed necessarily about B, and that's what syllogism must require, right? It's in the very definition of syllogism. The conclusion followed necessarily. If something followed necessarily about B, it would have to be always so, right? Okay? In a sense, you see the connection there between necessary and always, right? And you can see that in simpler things than the syllogism. If I say that two is necessarily half of four, right? Then two is going to be always, what? Half of four, right? But if I say that man is necessarily, let's say, white, then, if man was necessarily white, man would always be white, then. So if you produce one man who is, what? Not white, a black man, you've shown that man is not always, right? Necessarily, what? He's not always so, right? He's not necessarily not white either, because you've got a white man, right? Mm-hmm. Okay. So you need that, huh? Okay? So, you think of examples for both, huh? If Socrates is a dog, then Socrates is an animal. Yeah. That's true, right? If Socrates is a dog, then Socrates is an animal. And it's also true that Socrates is what? Not a dog. In this case, Socrates is a what? But if Socrates is a, what, mother, I'd say, then Socrates is a woman. True. Socrates is not a mother, but he's also not a, what, woman. Woman, right? So it might be that B is so, when these are true, by Socrates and Anna. It might be that B is not so, like Socrates is not a woman, right? Now, that ain't enough for a, what, for a syllogism, yeah. Because a syllogism means that something's necessarily so, and therefore always so, right? So that sometimes so, sometimes not so, that isn't enough for necessity, right? If I say that man is necessarily white, and some men are white and some are not, forget about saying that man is necessarily white, you see that? Okay? Now, you can't show that it's foreign to see syllogism by examples. Because examples wouldn't show even that it's always so, would it? But you can't draw examples, that alone is necessarily so, right? In the same way with simpler things. If I say that numbers are necessarily even, see, 2, 4, 6, 8, 10, 12, 14, 15, 15, 15, 15 examples, all the sense, where numbers are even, right? But that doesn't show that they're always even, no matter how many I give, right? Let alone the numbers are necessarily, no matter how many I give, right? So you cannot show that they form, if then, speeches of syllogism by examples, but you can show by, what? Examples that it's not. You see the reason for that? But I can't show by examples that every 2 is necessarily half a 4, always half a 4, right? They can't look at every 2, right? But if I find one 2 that isn't, that would be enough to just prove it, right? Yeah, right, okay. So you see that this, then, is, what? Not syllogism, right? And often people are taken in by that, huh? I'll say to the students, putting in material, I'll say, that Socrates is a mother, and Socrates is a woman. But Socrates is not a mother, therefore it's not a woman. That seems to them make sense, right? Because they know that Socrates is not a mother is true. They know it's true that Socrates is not a woman, right? And it's not being a mother, and it's not being a woman. It has something to do with each other, right? But does it follow, right? Because he's not a mother, then he's not a woman? But then the woman who, in my class with students, you know, who are, you know, not mothers either, it was not the woman, right? Very strange conclusion, right? But if you give this, you know, with statements they know are true, Socrates is a mother, then he's a woman. Socrates is not a mother, Socrates is not a woman. But sometimes they think it follows, right? Aristotle compares that, well, he put that down. So you see the point? Okay. Now, this form up here, if A is so, then B is so, B is so, this is the one that deceives people the most. Most people on the first sight will think that what? It follows that what? And in the book called The Poetics, Aristotle says, this is the way that Homer taught the other poets how to tell anybody. Good lie, right? Okay. And we take that expected consequence being true as a sign, right? Of the truth of what went out before, right? If he's losing the argument, then you'll get angry. Well, that sounds highly probable, right? If he got angry, he must therefore, what? Be losing the argument, right? We could get angry for more than one reason, couldn't we? Not because you're losing the argument, but because the other guy is, you know, not seeing the obvious or denying the obvious or, you know, confusing the issue or something, right? Many reasons to get angry at somebody, besides the fact that you might be losing the argument, right? But often we'll do that, right? You have to disprove this to show that it's invalid, not a syllogism. You have to do that again to examine it, right? I know it's not. If this number is 2, then this number is half of 4. This number is half of 4, therefore this number is 2. Ain't that good arm done there? Now in terms of the matter there, you could say what? Well, half of 4 and 2 are convertible, right? Half of 4 is a property in a strict sense. It belongs only to 2, to every 2 and always. So if this number is 2, this number is half of 4. If this number is half of 4, it's 2. And you put it back in this form here, right? But if you just consider the form itself, this could follow from that, and yet that not be so. And there's infinite examples, but any kind of example B is more general. And A is like, what, species, let's say, B is a genus. If this is a dog, this is a mammal. This is a cat, this is an animal, therefore this is a dog. Because this pet at home, Tabitha, is a cat, or she's a cat. If Tabitha is a dog, then Tabitha is an animal. True. Tabitha is an animal. True. Therefore Tabitha is a dog. So, it might be that A is so. If Tabitha is a cat, Tabitha is an animal. Tabitha is an animal. Tabitha is a cat. Okay? If Tabitha is a dog, Tabitha is an animal. True. Tabitha is an animal. Tabitha is not a dog. So A might be so. It might not be so. When these are what? True, right? I'll give an example of each. So, there's nothing that is always true when those premises are true. You can't say that B is always so. You can't say that A is always so. You can't always say that A is not so, right? Tabitha is a cat. Tabitha is not a dog, right? And if nothing is always so, then nothing is, what? Necessarily so, right? And therefore, you're going to have syllogism. Because syllogism means something's necessarily so, as it is, right? Now, what about this fourth form here? If A is so, then B is so. B is not so, right? Is anything followed necessarily about A? That's not so clear as up here. So, we show that A is not so, necessarily, through the more obvious case. You show what is not obvious to what is obvious, right? And we say that either A is so or A is not so, right? Now, if A were so by the first case, then B would have to be so, right? Necessarily. What you're given is B is not so, right? So, the statement A is so, B is not so, and A is so. Those three statements, are they compatible? No. Because by the first case, statement one and statement three contradict statement, what? Two. So, you can't hold on to those three statements, can you? And just like in Calc today, you know, if I say that the length, let's say, is 10, and the width is 2, and the area is, let's say, 40, you say, I don't know which of those numbers is wrong. But they can't all be right, right? You know? Because two of them would contradict the, what, third? Two times 10 would not be 40, or 40 divided by 2 would not be 10, right? Okay? So, statement one, if A is so, then B is so, and statement A is so, they necessarily, by the first case, the conclusion B is so, which goes against B is not so. So, if you're given that B is not so to start with, that's laid down. Then B is so must be eliminated, right? You're also given that if A is so didn't B is so, so you can't hold on to what? A is so. You must reject that, and therefore you must say that A is not so. Either A is or not so. You see that? Okay? And often we reason this way, right? We'll take a statement we know is so, and a statement that is false, and at least it's something we know is false, and then we know that the one statement is unknown. So, it must be what? False, right? So, it must be what we know is false, and it must be what we know is false, and it must be what we know is false, and it must be false, and it must be false, and it must be false, and it must be false, and it must be false, and it must be false, and it must be false, and it must be false, and it must be false, and it must be false, and it must be false, and it must be false, and it must be false, and it must be false, and it must be false, and it must be false, and it must be false. So if you're given it's true, if A is so, then B is so, and you're given that B is not so, then A is so could not also be true. If A is so, and the if-then statement would have led to B is so, but that's false. So the cause of the false would have to be that A is so. You see that then? So there's two forms of if-then speech that are syllogisms, and two that are what? Not. And so you have to become very familiar with those four forms, and you're going to find them repeated, right? Infinity at times. If you read the first book of Natural Hearing, what they call the Physics of Aristotle, and he examines the argument of what? Melises, right? And he argues thus, if being had a beginning, then being had an end. If being doesn't have a beginning, therefore it doesn't have an end. This is the way Melises argues, right? If being had a beginning, then it has an end. There's some probability to that. If it doesn't have a beginning, therefore it has an end. He's using this form, which is not a, what, syllogism, but his conclusion doesn't follow, right? This matter is bad to Aristotle. But either a defect is enough to overcome somebody's argument, right? Either his conclusion doesn't follow, or the premises and uses aren't true or improbable, right? And Melises eventually is shown to have both defects, right? Poor pieces. But it shows the need for logic, right? The philosophers need logic, right? It's the last part of philosophy to be discovered, right? But we can't deceive the need for it this time or not. Now, when I first explain this, and most times I go just using the letters A and B there, right? Now, it might be, if you want them to follow the order of letters and the order of, what, the unseated and consequent, right? Then you could state this a little differently, right? You could say, reason looks before and after, right? Now, if you have a simple statement, which we'll call B, if you have a simple statement called B, if you want to prove that B is so by an instant syllogism, right? Would you look before B or after B? If you look for some statement that comes before B that B follows upon, right? Like A, or would you look after B for a statement that follows upon A, C? Well, if you wanted to establish that B is so, you'd look before, right? You'd have something from which B follows, right? Now, if you knew or could prove that A is in fact so, then you could, what, syllogize that, what, B is so, right? Okay? If you want to overthrow B, right? We're going to try to overthrow B. Then you want to look for something that follows upon B, right? Okay? So, you look after B and say, if B is so, then C is so, right? Now, if you know that C is not so, or you can prove that C is not so, right? Then you can syllogize that B is not so, right? Okay? And you can see how well the phrase of Shakespeare fits this, right? Looking before and after, right? Notice, you know, the kind of obvious before and after is in the syllogism. The premises are before the conclusion, right? That's one before and after, obviously. Okay? The word premise indicates that, pre, before, right? But then, in the if-then statement, you have to be before and after. Because the antecedent is before the, what, consequent, right? Okay? But then you see that you look before, if you want to establish the statement, you look after, if you want to, what, overthrow it, right? And so it represents the fact that looking before and after B, and so it represents the fact that looking before and after B, and so it represents the fact that looking before and after B, and so it represents the fact that looking before and after B, and so it represents the fact that looking before and after B, and so it represents the fact that looking before and after B, and so it represents the fact that looking before and after B, and so it represents the fact that looking before and after B, and so it represents the fact that looking before and after B, and so it represents the fact that looking before and after B, and so it represents the fact that looking before and after B, I use A for what is before B, and C for what is after, right? Okay? But that's to just tie it up with where you look. So these are the two ways you can, what? Reason, right? Now, let's take an example of this. Aristotle, in the metaphysics there, he says that Socrates was trying to syllogize him, and he gives us a sign that Socrates is trying to syllogize, that Socrates is trying to define. The definition is very much the beginning of syllogism, especially what we call the syllogism theory, right? The simple category of syllogism. Now, if you look at the dialogue called the Mino, in the Mino, Socrates is asked by Mino in the beginning of the dialogue, can virtue be taught, right? And Socrates says, I don't know. Further more, I don't even know what virtue is, right? Further more, I've never met anybody who does know what virtue is. And Mino says, well, I know what virtue is, right? And Socrates says, well, come tell me what it is then, right? And so Mino tries to say what virtue is, and first he gives examples, and Socrates says, well, you don't really tell me what virtue is, right? I ask you what water is, you say, well, there's the rain, and there's the lake, and there's a faucet, and so on. You know, he gives me examples, but not telling me what water really is. So then finally, Mino tries to say what virtue is, and Socrates points out that his definition of virtue doesn't even separate the blue man from the bad man. So by the end of that first conversation, which is an examination conversation, it's clear that Mino doesn't know what virtue is. Then begins the second part of the dialogue, where Socrates says, now, let's put our two heads together and find out if we can what virtue is, right? And at that point, Mino gives the sophisticated objection, you can't look for what you don't know. And Socrates tries to show how you can, by trying to show that learning is, what, in a way, recalling what you do already know. He has some difficulties there, too, because on a close examination, it may appear that the slave boy is not really recalling how double-squared, but coming to know it, do things he already knows. But coming to know it for the first time, right? In fact, he's so far from recalling it in the beginning that he's actually mistaken as to how it's done, right? But then, in the third part of the dialogue, Mino still wants to know where the virtue can be taught, right? Even though they don't know what virtue it is. And another way to discuss the virtue it is, right? My brother Mark says, what do you do with a guy like that, huh? You're going to find people in daily life, in the academic world, or just like Mino, right? They want to know this, but you have to know this before you can know that. And you explain that, and then you show them that they don't know this that you have to know beforehand. Okay, okay, okay, okay. I still want to know this. It's like a student came to me and he said, I've heard the Stategorian Theorem, you know? And I said, well, I know the Stategorian Theorem. I said, well, all I know is Euclid's proof in Proposition 47. I've got to do 46. I've got to do 46. Oh, I don't want to do that. 46, right? You know? I'm sorry, you know? It's illogical, right? But Socrates says, well, if you, you know, twist my arm, right, I'll at least I'll discuss it, right? But since we don't really know very well what virtue is, let's look on both sides and see if there's any reason to think that virtue can be taught, right? Any reason to think that virtue cannot be taught, right? So you're going to proceed dialectically, right? Reasoning and probable opinions, right? Even to contradictory conclusions. And the demonstrator never does that because he knows what things are. But the dialectician is on the way to mind. So, the dialectician very often and naturally adopts one of these two forms, right? It's sort of appropriate to his matter, right? So, Socrates' first reasons that virtue can be taught And he uses, what, this form here. And then he reasons that virtue cannot be taught, and he uses this form here, right? The other two forms that we saw were not syllogisms, he doesn't quite use, right? So, he says, without knowing, he says, what virtue is, what sort of a thing would virtue have to be in order to be taught? Well, it seems that we teach what we know, right? Okay, so, if virtue is knowledge, then it seems that it can be taught, right? Okay, and that's not saying that, in fact, virtue is knowledge. It's not saying that, in fact, virtue can be taught. But it's saying, if it is knowledge, then it seems it could be taught. That's other forms of knowledge, right? Okay, so, if virtue is knowledge, then it can be taught. Now, that statement there is a statement that seems to be kind of obvious on the very surface, right? Okay, it's knowledge that's taught, right? Okay, so, if virtue is knowledge, then it can be taught. So, the question is, is virtue knowledge, then it can be taught. Now, Socrates is going to eventually say, well, there's some reason to think that virtue is knowledge, right? We're going to have to show that, or back that statement up by an argument, right? The first statement is kind of standing by itself, right? Okay? Now, eventually, you're going to give a reason for thinking that virtue is knowledge. If you can give a reason for thinking that virtue is knowledge, statement of fact, then you can syllogize that virtue, what? Can be taught. Yeah. I think that virtue can be taught. Therefore, virtue can be taught. Now, a rule you have to do when you're examining somebody's reasoning. Very often, we start from their main conclusion, right? And then we see the statements that are next to the main conclusion, huh? These two statements here lead up to the main conclusion. Now, sometimes, one, sometimes, both of those statements are in need of, what? Some proof or manifestation, right? In this case, yes, then is pretty clear, because people know that virtue, that knowledge is something that you can teach, right? So it seems right away acceptable to say that if virtue is knowledge, then it can be taught. But Socrates has to give some kind of reason to think that virtue is knowledge, right? And he uses it, it seems, you know, what we'll call the soldiers in theory, right, then? He'll say that something like this. To back it up, he'll say, he'll take this kind of, at least a probable statement, huh? That virtue directs us to good things, right? Okay? Right, vice leads us to what? Bad things, huh? That's something you might sell on the surface, right? Virtue directs us to good things. It would also seem to be probable that it's some kind of knowledge that directs us to good things, right? It's knowledge that directs you, right, to good things. So, what directs us to good things is knowledge, right? So medical art is a form of knowledge that directs us to good things, like health, right? Economics directs us to wealth, the military art to victory, and so on, right? Okay? So, at least this probability here, right, that what directs us to good things is knowledge. And it also seems probable that virtue directs us to good things, right? We all think of that. Therefore, virtue is knowledge, right? Okay? That's another kind of syllogism backing up the, what, second premise of the main or chief syllogism, right? The main or chief syllogism is the one whose conclusion is the main conclusion, right? And then the back-up syllogism, they call it the pro-syllogism sometimes, but it's backing up the second premise, right? Thank you. okay and uh sometimes i have to back up one of the premises in the second solotism right but here both statements seem to have some probability right so notice what socrates is doing he's looking before the statement virtue can be taught right and what sort of thing about virtue from which it would follow that could be taught right well let's see the connection between knowledge and what teaching right okay we teach what we know right when we teach something it's a sign we know something about what we're teaching right then socrates syllogizes on the opposite side okay now here he has an if-then statement which is i think somewhat weaker than the if-then statement in the first argument but nevertheless one if you stop and think about it has some what probability probability right he's saying if virtue can be taught right and then there are teachers of it now why does that seem reasonable right well we might have done these we have kind of a very narrow understanding of virtue right and but the greeks would say wow look uh the most obvious virtue is what purge right okay okay okay in fact the word virtue is you know sometimes identified almost with courage virtue is a latin word meaning what mandarin right okay the greek word art is like that but this first of all is a sense of courage okay now notice if the uh citizens of the city are not courageous they're going to what be unable to defend the city the city will be what taken by enemies right what do they do they'll kill the men and they don't kill the wife and children women and children they'll enslave them right this is a horrible thing right to have a city what yeah so courage is extremely necessary when you go to the city right so if virtue can be taught you can be sure that there'll be people what teach me right and that makes sense right okay and temperance and justice i'm helping men live together without being tempered right if a man was around briefing his neighbor's wife or his neighbor's daughter right then i'd turn around the city right now if men go around robbing from each other and so on stealing from each other then they have chaos in the city right now okay so it seems that you know things like justice and so on can be taught they would surely be like taught right okay now the second premise would be but there are no teachers now how does um socrates in the conversation with me know and so on and any of us gets in there gets very angry with socrates he's one of the guys who brings the uh court to adherence against socrates huh well when they discuss are there teachers of virtue there are two groups of men that are possible candidates who teaches the virtue right and one is those men called sophists right who are going around in greece right but they already have a somewhat unsavory reputation right especially to the nobility like you know and christian so right so in the conversation with mino minos would deny that the sophists would teach virtue right okay socrates maybe too but for the reasons huh okay um but what about the great men of athens right well now the great men of athens were known for their virtue for being courageous men and being just men and so on And it seems that if these things could be taught, they should surely have taught their sons, right? But here's great men of Athens who were courageous and their sons were cowards, right? Here's a great man of Athens who was just and his son is a, what, thief, right? So if I am a sober man, I don't want my son to be a drunkard, right? If I'm a just man, I don't want my son to be a thief. If I'm a courageous man, I don't want my son to be a coward, right? So it seems that if these could be taught, these men would have had their sons taught these, right? But here are these great men that taught their sons how to ride a horse, right? They didn't teach them how to be just! They hired somebody to teach them how to ride a horse, but they didn't hire anybody to teach them to be just. Is it because it's more important to be able to ride a horse than to be just? It's more important to ride a horse than to be courageous? No. So it seems that there are no, what, teachers of it, right? Okay. And, like I might say, you know, I can name the first teacher that I had for my children of the piano, and the second teacher who taught them the piano, right? Now if you ask me, and who did you hire to teach your children courage or temperance? Gee, I didn't hire anybody to do that. I could hire somebody to teach my children to play the piano. Is it more important that they play the piano and they be just and temperate and courageous, huh? Well, why did I pay money for a piano teacher and not a teacher of justice? Well, it seems like there's no, what, such teachers, right? Okay. So there's a certain probability, at least, to that statement that there are no teachers of virtue. Now I might mention that in the, I guess it's in the Protagoras, right? There you see a little bit, the other side, right, of that, huh? And I think Protagoras says, you know, well, it's like, who talks to Greek? Who talks to English, right? I could name maybe the first man I had as a teacher in French, but I can't really in English, right? Who taught me English? Well, to some extent, it was my mother, to some extent, my father, to some extent, my older brothers, maybe, right? And it was kind of diffused, right? Okay. And to some extent, my mother taught me justice and my father taught me justice and so on, right? Okay. But it's not, you know, like you go and get somebody, right? There are, in a sense, teachers, right? Okay. But Socrates here, in this dialogue, it just brings out the way that there seems to be no teachers, right? I had no teacher of justice for my children, only the piano. Is it because I, Berkowitz, think that it's more important to play the piano than to be just? No. No, I don't think that. I think it's important that my children be just. But why didn't I hire anybody? Apparently, there isn't anybody. Okay. So now you're saying, if A is so, or the second way is to be, if B is so, then C is so, but C is not so, therefore, what? B is not so. Okay. So in one case, you're affirming the antecedent, and then affirming the consequent. Other case, you're denying the consequent, and therefore denying the antecedent, right? But you can't do the other two ways, right? You can't deny the antecedent and necessarily deny the consequent, right? I'm sorry, I'm going to say, well, what about this, you know? If this number is two, then this number is half of four. This number is not two, therefore it's not half of four. Well, you might say that, right? Because you realize that two and half of four are what? Convertible, right? But from the form, it isn't necessarily so, right? If I'm a dog, I'm an animal. I'm not a dog, therefore I'm not an animal. No. It isn't necessarily so, right? And likewise, you can't reason from the, what, affirmation of the consequent to the affirmation of the, what, antecedent, right? Now, materially speaking, on the two most common examples, maybe, are where something, when the consequent is more general, it's in the antecedent, right? Okay. Or where the consequent is an effect, right, of a cause, but there are many, what, possible causes, right? For example, I was given class.