1 00:00:04,700 --> 00:00:10,146 Last week, we talked about propositional logic which is the study of inferences 2 00:00:10,146 --> 00:00:15,110 that are valid, because of the true functional connectives that they use. 3 00:00:15,110 --> 00:00:19,452 This week, we're going to be talking about a different kind of logic, called 4 00:00:19,452 --> 00:00:22,829 categorical logic. Which is also the study of inferences 5 00:00:22,829 --> 00:00:26,267 that are valid. But they're not valid because of the true 6 00:00:26,267 --> 00:00:30,911 functional connectives that they use. They're valid because of something else 7 00:00:30,911 --> 00:00:35,012 that I'd like to describe now. In order to describe what makes these 8 00:00:35,012 --> 00:00:38,390 inferences valid, I'd like you to consider some examples. 9 00:00:38,390 --> 00:00:44,168 So first, consider the inference. All squares are rectangles. 10 00:00:44,168 --> 00:00:52,660 All rectangles have parallel, therefore, all squares have parallel sides. 11 00:00:52,660 --> 00:00:57,465 That inference is obviously valid. There's no possible way for the premises 12 00:00:57,465 --> 00:01:02,741 to be true if the conclusion is false. But notice that even if we replace 13 00:01:02,741 --> 00:01:08,993 squares with another category like, let's say, squirrels, we replace rectangles 14 00:01:08,993 --> 00:01:15,163 with another category like, let's say, mammals, and we replace the category of 15 00:01:15,163 --> 00:01:20,846 having parallel sides with another category like, let's say animals, we 16 00:01:20,846 --> 00:01:24,987 still end up with a valid inference. Let's try that. 17 00:01:24,987 --> 00:01:29,047 All squirrels are mammals, all mammals are animals. 18 00:01:29,047 --> 00:01:34,440 Therefore, all squirrels are animals, still valid. 19 00:01:34,440 --> 00:01:40,455 So, both of these inferences are valid. And they're valid because of some common 20 00:01:40,455 --> 00:01:44,720 feature that they share but what is that common feature? 21 00:01:44,720 --> 00:01:49,072 Is it a truth functional connective that they both use? 22 00:01:49,072 --> 00:01:53,863 Well, no, because neither inference uses any truth functional connective. 23 00:01:53,863 --> 00:01:58,923 In fact, neither inference uses any propositional connective, so it can't be 24 00:01:58,923 --> 00:02:03,782 some because of some truth functional connective that they use that the 25 00:02:03,782 --> 00:02:08,033 inferences are valid. It must be for some other reason that the 26 00:02:08,033 --> 00:02:12,256 inferences are valid. And in the study of categorical logic, 27 00:02:12,256 --> 00:02:17,732 that we'll be doing this week, we'll try to understand what that other reason is. 28 00:02:17,732 --> 00:02:22,523 Let me say something first about the notation that we're going to use. 29 00:02:22,523 --> 00:02:28,136 Last week when we did propositional logic we used lower case letters like p and q 30 00:02:28,136 --> 00:02:33,407 to represent propositions that were being connected, by some truth functional 31 00:02:33,407 --> 00:02:37,059 connective. This week in categorical logic, we're 32 00:02:37,059 --> 00:02:43,092 going to use, upper case letters from a different part of the alphabet, we'll use 33 00:02:43,092 --> 00:02:49,047 upper case F, upper case J and upper case H, to represent categories, that are 34 00:02:49,047 --> 00:02:54,923 being connected together into a single propasition and then as, both of the 35 00:02:54,923 --> 00:02:59,546 inferences that we just discussed, have this form in common. 36 00:02:59,546 --> 00:03:05,110 The first prement says, all of the things that fall into one category, F. 37 00:03:05,110 --> 00:03:13,491 Also fall into a second category, G. For instance, all things that fall into 38 00:03:13,491 --> 00:03:22,211 the category square also fall into the category rectangle, or all things that 39 00:03:22,211 --> 00:03:30,540 fall into the category squirrel also fall into the category mammal. 40 00:03:30,540 --> 00:03:37,508 The second premise, said that all things, that fall into that second category over 41 00:03:37,508 --> 00:03:42,756 here also fall into a third category. So for instance, 42 00:03:42,756 --> 00:03:49,380 all things that fall into the category, rectangle also fall into the category, 43 00:03:49,380 --> 00:03:55,488 having parallel sides or all things that fall into the category, mammal. 44 00:03:55,488 --> 00:04:02,551 Also fall into the category, animal. And then the conclusion said that all 45 00:04:02,551 --> 00:04:09,135 things that fall into that first category also fall into the third category. 46 00:04:09,135 --> 00:04:16,065 For instance all of the things that fall into the category square fall into the 47 00:04:16,065 --> 00:04:22,388 category having parallel sides or all things that fall into the category 48 00:04:22,388 --> 00:04:26,580 squirrel also fall into the category animal. 49 00:04:26,580 --> 00:04:34,089 We can use a certain kind of diagram called a Venn diagram, after it's 50 00:04:34,089 --> 00:04:42,027 inventor John Venn, to represent the relationships that these premises and 51 00:04:42,027 --> 00:04:49,107 this conclusion describe. And using Venn diagrams we can show that 52 00:04:49,107 --> 00:04:58,180 this inference, any particular example of this inference is valid, here's how. 53 00:04:58,180 --> 00:05:04,150 Remember the first premise says, all things that fall into a first category 54 00:05:04,150 --> 00:05:09,961 also fall into a second category. So what we do with a Venn diagram, is we 55 00:05:09,961 --> 00:05:14,180 use a circle. to represent, the first category. 56 00:05:14,180 --> 00:05:20,231 So all the things that fall into that first category, are inside this circle. 57 00:05:20,231 --> 00:05:26,900 And all the things that fall outside that first category, are outside the circle. 58 00:05:26,900 --> 00:05:33,375 So this first category we've called F [SOUND] and it's represented by that 59 00:05:33,375 --> 00:05:37,273 circle. Now if, everything in that first category 60 00:05:37,273 --> 00:05:43,620 is also in the second category, then if we use a circle to represent the second 61 00:05:43,620 --> 00:05:49,645 category, the circle that we use to represent the second category has to be 62 00:05:49,645 --> 00:05:55,911 at least as large as the circle we use to represent the first category because it 63 00:05:55,911 --> 00:06:01,936 has to include everything in the circle that we use to represent the first 64 00:06:01,936 --> 00:06:03,160 category. So, 65 00:06:03,160 --> 00:06:10,240 we could draw a second circle to represent the second category, G. 66 00:06:12,100 --> 00:06:19,760 So if F is the category of being a square then this interior circle represents all 67 00:06:19,760 --> 00:06:24,929 of the squares. If F is the category of being a squirrel 68 00:06:24,929 --> 00:06:30,190 then the interior circle represents all of the squirrels. 69 00:06:30,190 --> 00:06:35,411 If G is the category of being a rectangle, then this exterior circle 70 00:06:35,411 --> 00:06:38,943 represents all the things that are rectangles. 71 00:06:38,943 --> 00:06:45,009 If G is the category of being a mammal, then the exterior circle represents all 72 00:06:45,009 --> 00:06:51,501 the things that are mammals. So the first premise tells us that all 73 00:06:51,501 --> 00:06:58,773 the things in the first category, all the Fs are also Gs, they're also in second 74 00:06:58,773 --> 00:07:03,262 category. But the second premise tells us that all 75 00:07:03,262 --> 00:07:10,175 the things that are in that second category, all the things that are Gs also 76 00:07:10,175 --> 00:07:17,178 fall into a third category, they're Hs. So, we'll represent the category of Hs 77 00:07:17,178 --> 00:07:24,257 using a circle that includes inside it. Everything that the g circle includes 78 00:07:24,257 --> 00:07:38,998 inside it like this. So, the second premise says, all the 79 00:07:38,998 --> 00:07:44,699 things that are in the category of G are also in the category of H. 80 00:07:44,699 --> 00:07:50,996 We can represent that in the Venn diagram, by having the circle of all the 81 00:07:50,996 --> 00:07:57,804 Gs be inside the circle of all the Hs. So, if the category of G is the category 82 00:07:57,804 --> 00:08:03,663 of rectangles, and the category of H is the category of things that have parallel 83 00:08:03,663 --> 00:08:09,411 lines, then the second premise says that all the things that are rectangles also 84 00:08:09,411 --> 00:08:13,722 have parallel lines. If the category of G is the category of 85 00:08:13,722 --> 00:08:18,823 mammal, and the category of H is the category of animal, then the second 86 00:08:18,823 --> 00:08:24,140 premise tells us that all the mammals are also in the category of animals. 87 00:08:25,680 --> 00:08:31,655 So these three circles right here represent the information contained in 88 00:08:31,655 --> 00:08:38,040 premise one and premise two together. The information contained in premise one 89 00:08:38,040 --> 00:08:42,788 is represented by having the F circle inside the G circle. 90 00:08:42,788 --> 00:08:49,582 And the information contained in premise two is represented by having the G circle 91 00:08:49,582 --> 00:08:53,511 inside the H circle. So having these three circles 92 00:08:53,511 --> 00:08:59,849 concentrically arranged as they are right now, represents the information contained 93 00:08:59,849 --> 00:09:03,091 in these two premises. But notice something, 94 00:09:03,091 --> 00:09:08,443 if we put the F circle inside the G circle, and then we put the G circle 95 00:09:08,443 --> 00:09:14,436 inside the H circle. Then, the F circle is going to have to be 96 00:09:14,436 --> 00:09:20,592 inside the H circle. And if the F circle is inside the h 97 00:09:20,592 --> 00:09:28,616 circle, what does that represent? That represents the fact that everything 98 00:09:28,616 --> 00:09:34,332 that is in the F category is also in the h category. 99 00:09:34,332 --> 00:09:42,389 In other words, all F are H. And so, when we construct a Venn diagram, 100 00:09:42,389 --> 00:09:48,200 to represent the information contained in these two premises. 101 00:09:48,200 --> 00:09:54,891 That Venn diagram will have to also represent the information contained in 102 00:09:54,891 --> 00:10:00,117 the conclusion. And so the Venn diagram shows us that 103 00:10:00,117 --> 00:10:05,486 there is no way for the premises to be true and the conclusion to be false. 104 00:10:05,486 --> 00:10:11,070 If the premises are both true, then the conclusion is going to have to be true. 105 00:10:11,070 --> 00:10:16,367 And the venn diagram shows us that. Using venn diagrams, there are a number 106 00:10:16,367 --> 00:10:21,880 of different relationships that we can visually represent between categories. 107 00:10:21,880 --> 00:10:28,119 I just showed you how we can represent the relationship that we're claiming to 108 00:10:28,119 --> 00:10:33,331 hold when we say, all F or G. We're saying all the things that fall 109 00:10:33,331 --> 00:10:39,018 into the F category, whatever that may happen to be, also fall into the G 110 00:10:39,018 --> 00:10:45,178 category, whatever that may happen to be. And we represent that by putting an F 111 00:10:45,178 --> 00:10:52,260 circle inside a G circle to show that all the things that are Fs are also Gs. 112 00:10:52,260 --> 00:10:57,791 Here's how we would represent the claim that some F is a G. 113 00:10:57,791 --> 00:11:04,822 We would draw an F circle and a G circle that overlaps it and then mark a spot 114 00:11:04,822 --> 00:11:11,853 that's in the intersection of the F circle and the G circle and that shows 115 00:11:11,853 --> 00:11:19,513 that there is something whatever it is. That falls into the F category, so it's 116 00:11:19,513 --> 00:11:24,235 an F. And it also falls into the G category, so 117 00:11:24,235 --> 00:11:29,368 it's also a G. So we are showing there that some F 118 00:11:29,368 --> 00:11:37,903 whatever it is, is also a G. If we want to represent that no F f is G 119 00:11:37,903 --> 00:11:45,396 that nothing in the F category is in the G category, then we can draw the F circle 120 00:11:45,396 --> 00:11:52,158 and the G circle overlapping and shade out the area where they overlap to 121 00:11:52,158 --> 00:11:56,910 indicate that there is nothing in that intersection. 122 00:11:56,910 --> 00:12:04,699 There is nothing that is both an F and a G and that would represent no F is G. 123 00:12:04,699 --> 00:12:10,422 Finally, if we want to represent the claim that some F is not G. 124 00:12:10,422 --> 00:12:16,690 Then we could draw the F circle, and draw the G circle overlapping it. 125 00:12:16,690 --> 00:12:22,814 But mark a spot inside the F circle, but outside the G circle. 126 00:12:22,814 --> 00:12:30,344 That way we're showing that there is something, whatever exactly it is, that 127 00:12:30,344 --> 00:12:35,665 is in the F category, but it's outside the G category. 128 00:12:35,665 --> 00:12:43,017 So some F, whatever it is, is not a G. And that's how we can use Venn diagrams 129 00:12:43,017 --> 00:12:48,633 to represent different relationships that we could claim to hold between 130 00:12:48,633 --> 00:12:52,418 categories. Now let's see how we can represent the 131 00:12:52,418 --> 00:12:58,294 validity of an inference that uses claims like this by means of Venn diagrams. 132 00:12:58,294 --> 00:13:04,019 Notice that these Venn diagrams already tell us something about the logical 133 00:13:04,019 --> 00:13:07,710 relations between the claims that they represent. 134 00:13:07,710 --> 00:13:11,778 For instance, look at the Venn diagram for some f is g. 135 00:13:11,778 --> 00:13:17,343 It shows us that there is a thing. Don't mind what it is. 136 00:13:17,343 --> 00:13:23,910 But there's a thing that's in the F category and that's also in the G 137 00:13:23,910 --> 00:13:28,320 category. Now consider the Venn diagram for the 138 00:13:28,320 --> 00:13:33,761 claim, no F is G. what it shows us is that there is nothing 139 00:13:33,761 --> 00:13:38,264 that is in the F category and in the G category. 140 00:13:38,264 --> 00:13:45,394 So this claim right here no F is G, is the negation of this claim right here, 141 00:13:45,394 --> 00:13:49,817 some F is G. If this claim is true, if some F is G, 142 00:13:49,817 --> 00:13:54,898 then this claim is not true, it's not true that no F is G. 143 00:13:54,898 --> 00:14:01,939 And similarly if this claim is true, no F is G, then this claim is not true, it's 144 00:14:01,939 --> 00:14:07,198 not true that some F is G. So this claim and this claim are 145 00:14:07,198 --> 00:14:12,015 negations and you can tell that by looking at the Venn diagram that 146 00:14:12,015 --> 00:14:15,807 represent them. This Venn Diagram shows you that there is 147 00:14:15,807 --> 00:14:20,930 something in a particular area and this Venn diagram shows you that there is 148 00:14:20,930 --> 00:14:28,797 nothing in that same area. But, now notice the sum F is not G is 149 00:14:28,797 --> 00:14:35,620 also the negation of the claim all F are G. 150 00:14:35,620 --> 00:14:44,061 If all F are G then it can't possibly be true that some F is not G, but if some F 151 00:14:44,061 --> 00:14:52,398 is not G then it can't possibly be true that all F are G, so this claim and this 152 00:14:52,398 --> 00:15:01,049 claim are negations and yet the way we've done the Venn diagram for this claim, all 153 00:15:01,049 --> 00:15:08,240 F or G, it doesn't show that this claim is the negation of this claim. 154 00:15:08,240 --> 00:15:16,701 So we're going to change the Venn diagram for all F or G so that it indicates that 155 00:15:16,701 --> 00:15:20,356 all F or G is the negation of sum F is not G. 156 00:15:20,356 --> 00:15:30,146 And here's how we're going to do it. We're going to draw an F circle and a G 157 00:15:30,146 --> 00:15:38,063 circle that overlaps it . And we're going to shade out, the area of 158 00:15:38,063 --> 00:15:46,486 the F circle that's outside the G circle to indicate that there's nothing that is 159 00:15:46,486 --> 00:15:51,520 in the F circle that is not also in the G circle. 160 00:15:53,840 --> 00:16:01,587 Now, with this Venn diagram, you can easily see that all F or G, is the 161 00:16:01,587 --> 00:16:06,608 negation, of sum F is not G. The Venn diagram for sum F is not G, 162 00:16:06,608 --> 00:16:12,586 shows you that there's a thing in the F circle, that's outside the G circle. 163 00:16:12,586 --> 00:16:18,723 And the Venn diagram for all F or G, shows you that there's nothing that's in 164 00:16:18,723 --> 00:16:25,100 the F circle that's outside the G circle. And so the Venn diagrams now represent. 165 00:16:25,100 --> 00:16:29,090 The relation between sum F is not G and all F are not G. 166 00:16:29,090 --> 00:16:34,647 The same way that the Venn diagrams represent the relation the sum F is G and 167 00:16:34,647 --> 00:16:38,779 no F is G. Venn diagrams are useful in helping us to 168 00:16:38,779 --> 00:16:44,885 see what the logical relationships are between different claims concerning 169 00:16:44,885 --> 00:16:51,642 what's true about all things of a certain kind, or some things of a certain kind or 170 00:16:51,642 --> 00:16:56,120 no things of a certain kind. Let me give you an example. 171 00:16:56,120 --> 00:17:01,982 Consider a claim of the form all f or g and a claim of a form no f or g. 172 00:17:01,982 --> 00:17:08,170 For instance, all of the sheep are in the field or no sheep are in the field. 173 00:17:08,170 --> 00:17:14,341 Now, those two claims might at first sound inconsistent with each other, like 174 00:17:14,341 --> 00:17:19,230 they can't both be true. But if you look at the Venn Diagram for 175 00:17:19,230 --> 00:17:25,037 all of the sheep are in the field and the Venn Diagram for no sheep are in the 176 00:17:25,037 --> 00:17:29,815 field, you realize actually that both of those claims can be true. 177 00:17:29,815 --> 00:17:35,843 The Venn Diagram for all of the sheep are in the field shows that what that claim 178 00:17:35,843 --> 00:17:40,680 means is that there are no sheep that are not in the field. 179 00:17:40,680 --> 00:17:44,900 . But it could be true that there are no 180 00:17:44,900 --> 00:17:49,285 sheep that are not in the field, even when it's also true that there are no 181 00:17:49,285 --> 00:17:52,969 sheep that are in the field. That can be true, both of those two 182 00:17:52,969 --> 00:17:55,717 claims can be true whenever there are no sheep. 183 00:17:55,717 --> 00:18:00,512 If there are no sheep, then there are no sheep that are not in the field and there 184 00:18:00,512 --> 00:18:05,307 are also no sheep that are in the field. And the Venn Diagram for those two claims 185 00:18:05,307 --> 00:18:09,120 helps us to see that they are consistent with each other. 186 00:18:09,120 --> 00:18:17,350 Consider two claims of the form sum F is G and sum F is not G. 187 00:18:17,350 --> 00:18:24,510 For example, some politician is honest, and some politician is dishonest. 188 00:18:24,510 --> 00:18:28,435 Now. At first, you might hear those two claims 189 00:18:28,435 --> 00:18:30,278 and think. Well wait a second. 190 00:18:30,278 --> 00:18:33,710 Who is this, some politician that you're talking about? 191 00:18:33,710 --> 00:18:37,141 Whoever it is, they can't both be honest and dishonest. 192 00:18:37,141 --> 00:18:39,620 Right? Someone can't be both honest and 193 00:18:39,620 --> 00:18:42,606 dishonest. So whoever this, some politician that 194 00:18:42,606 --> 00:18:46,673 you're talking about is. One Or the other of the two things that 195 00:18:46,673 --> 00:18:49,470 you're saying about them has got to be false. 196 00:18:49,470 --> 00:18:54,098 So you might think that the two claims are just made are inconsistent with each 197 00:18:54,098 --> 00:18:56,586 other. But again the Venn diagram for those 198 00:18:56,586 --> 00:19:00,578 claims helps us to see that they're not inconsistent with each other. 199 00:19:00,578 --> 00:19:04,743 To say that some politician is honest is just to say that there is some 200 00:19:04,743 --> 00:19:07,290 politician. Nevermind who. 201 00:19:07,290 --> 00:19:11,342 Who is honest. And again to say that some politician is 202 00:19:11,342 --> 00:19:15,100 dishonest is just to that there is some politician. 203 00:19:15,100 --> 00:19:18,150 Never mind who. Who's dishonest. 204 00:19:18,150 --> 00:19:21,706 But of course, those two claims could both be true. 205 00:19:21,706 --> 00:19:26,827 There could be one politician who's honest, and another politician who's 206 00:19:26,827 --> 00:19:30,384 dishonest. So once again, the venn diagrams for the 207 00:19:30,384 --> 00:19:36,003 two claims help us to see that those two claims are consistent with each other. 208 00:19:36,003 --> 00:19:39,560 They can both be true under certain circumstances. 209 00:19:39,560 --> 00:19:45,307 It'll be useful for us to label the four kinds of propositions that we've been 210 00:19:45,307 --> 00:19:48,945 talking about. Propositions of the form all f or g, 211 00:19:48,945 --> 00:19:52,510 we're going to label with the uppercase letter A. 212 00:19:52,510 --> 00:19:59,817 Propositions of the form no f is g, we're going to label with the upper case letter 213 00:19:59,817 --> 00:20:03,454 E. Propositions of the form, sum F or G, 214 00:20:03,454 --> 00:20:07,580 we're going to label with an uppercase letter. 215 00:20:07,580 --> 00:20:14,441 I and propositions of the form sum f is not g we're going to label with an 216 00:20:14,441 --> 00:20:21,239 uppercase letter o. Using this labeling scheme, we can now 217 00:20:21,239 --> 00:20:26,477 describe immediate categorical inferences. 218 00:20:26,477 --> 00:20:34,460 As inferences that have a single premise, of the form A, E, I or O. 219 00:20:35,691 --> 00:20:41,568 These inferences occur very commonly in everyday life, and so we're going to 220 00:20:41,568 --> 00:20:45,589 discuss the conditions under which they're valid. 221 00:20:45,589 --> 00:20:47,600 Rahm, Rahm, wake up. . 222 00:20:48,808 --> 00:20:57,397 Oh, wow. I was just having the most boring dream. 223 00:20:57,397 --> 00:21:08,911 You know, some books are so boring. Yeah, but not all books. 224 00:21:08,911 --> 00:21:14,659 Oh. Not all books are boring, but does that 225 00:21:14,659 --> 00:21:18,500 mean that all books are not boring?