1 00:00:04,669 --> 00:00:09,971 We're going to do something a little different today, something a little out 2 00:00:09,971 --> 00:00:13,948 of the ordinary. I'd like to start off today's lecture by 3 00:00:13,948 --> 00:00:19,390 speaking to possibly the most important issue in philosophy, that is of course, 4 00:00:19,390 --> 00:00:22,808 myself. I have done a lot of incredibly important 5 00:00:22,808 --> 00:00:28,443 things in a very short period of time. One of the most interesting and profound 6 00:00:28,443 --> 00:00:32,565 topics in philosophy is the self, and there's no self worth talking about as 7 00:00:32,565 --> 00:00:35,603 much as myself. So I thought I would talk about my self 8 00:00:35,603 --> 00:00:39,720 during this lecture, to give you an idea of what other selves are like. 9 00:00:39,720 --> 00:00:47,040 Someone that you are now seeing on this screen is a philosophy professor. 10 00:00:47,040 --> 00:00:53,344 And what follows from that? Well, what follows from that, is that 11 00:00:53,344 --> 00:00:58,988 it's not the case. That no one that you're seeing on this 12 00:00:58,988 --> 00:01:05,820 screen is a philosophy professor. Now notice, the Venn diagram for the 13 00:01:05,820 --> 00:01:13,940 premise of that inference consists of two circles, with an x in their intersection. 14 00:01:15,400 --> 00:01:20,762 And, the conclusion of that inference. Is the negation. 15 00:01:20,762 --> 00:01:26,529 Of a statement. Whose Venn diagram consists of those same 16 00:01:26,529 --> 00:01:30,879 two circles, with their intersection shaded. 17 00:01:30,879 --> 00:01:38,809 In general, when you want to, negate. A statement that can be represented using 18 00:01:38,809 --> 00:01:45,457 a Venn diagram, when you want to negate a statement of the form A, E, I, or O, you 19 00:01:45,457 --> 00:01:52,276 can do it by replacing an X everywhere it occurs with shading or with replacing 20 00:01:52,276 --> 00:01:59,440 shading everywhere it occurs with an X. Here's another example of the same point. 21 00:01:59,440 --> 00:02:05,945 Someone that you're seeing right now is not a philosophy professor. 22 00:02:05,945 --> 00:02:13,324 And from that it follows that it's not the case that everyone you're seeing 23 00:02:13,324 --> 00:02:21,253 right now is a philosophy professor. The Venn diagram for the premise of that 24 00:02:21,253 --> 00:02:29,380 inference contains two circles with an x in one of them but not in the other. 25 00:02:29,380 --> 00:02:35,141 The Venn diagram for the conc, conclusion of that inference is the negation of a 26 00:02:35,141 --> 00:02:40,471 statement, the Venn diagram for which contains those same two circles, with 27 00:02:40,471 --> 00:02:44,380 shading in one of them but not in the other. 28 00:02:44,380 --> 00:02:53,463 So once again, you negate one statement. By another statement by replacing shading 29 00:02:53,463 --> 00:02:57,050 in the area where there was originally an x. 30 00:02:57,050 --> 00:03:03,026 We've just discussed how we can use venn diagrams to establish the validity or 31 00:03:03,026 --> 00:03:06,582 invalidity of categorical immediate inferences. 32 00:03:06,582 --> 00:03:12,558 Now, I'd like to show how venn diagrams could be used to establish the validity 33 00:03:12,558 --> 00:03:16,114 or invalidity of a different kind of inference. 34 00:03:16,114 --> 00:03:20,200 A kind of inference that we're going to call a syllogism. 35 00:03:20,200 --> 00:03:25,820 Now syllogisms are different from categorical immediate inferences in a 36 00:03:25,820 --> 00:03:30,114 couple of ways. First, instead of one premise syllogisms 37 00:03:30,114 --> 00:03:36,590 have two premises. Second, the two premises share. 38 00:03:36,590 --> 00:03:41,429 One particular term. A term that we're going to call the 39 00:03:41,429 --> 00:03:45,994 middle term. So, one of the premises is going to be an 40 00:03:45,994 --> 00:03:52,751 a, e, I, or o statement, with two terms. We can call them a subject term and a 41 00:03:52,751 --> 00:03:57,316 middle term. The other premise will be another a, e, 42 00:03:57,316 --> 00:04:03,890 I, or o statement, with two terms. We'll call them the middle term and the 43 00:04:03,890 --> 00:04:08,409 predicate term. The conclusion is going to be an A, E, I, 44 00:04:08,409 --> 00:04:14,868 or O statement, without the middle term. It'll just have the subject term and the 45 00:04:14,868 --> 00:04:20,004 predicate term. Now because categorical inferences 46 00:04:20,004 --> 00:04:27,190 involve three terms, that is to say, they involve three categories, three kinds of 47 00:04:27,190 --> 00:04:30,341 thing. In order to represent them in order to 48 00:04:30,341 --> 00:04:35,630 represent what it is that they say and to figure out what they say is valid or 49 00:04:35,630 --> 00:04:40,384 invalid we're going to need a venn diagram that consists of three circles 50 00:04:40,384 --> 00:04:45,607 rather than just two circles one circle corresponding to the subject term one 51 00:04:45,607 --> 00:04:50,695 circle corresponding to the middle term and one circle corresponding to the 52 00:04:50,695 --> 00:04:55,148 predicate term. Let me give you some examples of how this 53 00:04:55,148 --> 00:04:58,842 works. Here's an example of a syllogism and a 54 00:04:58,842 --> 00:05:05,738 Venn diagram that we can use to represent what the syllogism is saying and why it's 55 00:05:05,738 --> 00:05:09,514 valid. So this syllogism has two premises, both 56 00:05:09,514 --> 00:05:15,097 of which are of the form a. The first premise says all Duke students 57 00:05:15,097 --> 00:05:17,642 are humans. In fact, that's true. 58 00:05:17,642 --> 00:05:24,699 There are no non-human Duke students. So, how do we represent that using this 59 00:05:24,699 --> 00:05:29,852 Venn diagram? Well, we shade out the circle of Duke 60 00:05:29,852 --> 00:05:34,490 students that's outside the circle of humans. 61 00:05:34,490 --> 00:05:38,195 Right? This is the circle representing all the 62 00:05:38,195 --> 00:05:42,546 Duke students. This is the circle representing all the 63 00:05:42,546 --> 00:05:48,186 humans, and the first premise tell us that all Duke students are human. 64 00:05:48,186 --> 00:05:54,550 In other words, whatever Duke students there are, they fall into the category of 65 00:05:54,550 --> 00:05:58,176 human. So, the circle of Duke students outside 66 00:05:58,176 --> 00:06:04,460 the circle of humans gets shaded out. The second premise is also of the form A. 67 00:06:04,460 --> 00:06:10,979 It tells us all humans are animals. Alright so how do we represent that? 68 00:06:10,979 --> 00:06:17,309 Well the way we represent that is by shading out the circle of humans that's 69 00:06:17,309 --> 00:06:20,268 outside the circle of animals. Right? 70 00:06:20,268 --> 00:06:25,119 Because all of the humans are inside the circle of animals. 71 00:06:25,119 --> 00:06:31,531 So, here's what we do, we shade out the part of the, human circle that's outside 72 00:06:31,531 --> 00:06:38,619 the circle of animals. And, what does that tell us? 73 00:06:38,619 --> 00:06:45,576 Well, according to the syllogism all Duke students are animals and looking at the 74 00:06:45,576 --> 00:06:49,184 Venn diagram you can see why that's right. 75 00:06:49,184 --> 00:06:56,142 It's right because if there are no Duke students in this region and there are no 76 00:06:56,142 --> 00:07:02,498 Duke students in this region, the only place where there could be any Duke 77 00:07:02,498 --> 00:07:08,135 students is this region. And that's inside the circle of animals. 78 00:07:08,135 --> 00:07:13,410 So if there are any Duke students they've gotta be animals. 79 00:07:13,410 --> 00:07:18,185 And that's what we at Carolina have been saying for over 80 years. 80 00:07:18,185 --> 00:07:22,816 Here's another example. Here's a syllogism where one premise has 81 00:07:22,816 --> 00:07:28,460 the form A and another premise has the form I, and the conclusion has the form 82 00:07:28,460 --> 00:07:31,137 I. So one premise says all humans are 83 00:07:31,137 --> 00:07:34,321 animals. So to represent that using our venn 84 00:07:34,321 --> 00:07:39,531 diagram, what we do is shade out, the part of the circle of humans that's 85 00:07:39,531 --> 00:07:42,860 outside the circle of animals. And there we go. 86 00:07:42,860 --> 00:07:49,802 Okay so we're showing that if there are any humans they are in that circle of 87 00:07:49,802 --> 00:07:54,253 animals. The other premise says some Duke students 88 00:07:54,253 --> 00:08:00,929 are humans okay so we represent that by putting an X in the circle of Duke 89 00:08:00,929 --> 00:08:04,934 students that's also in the circle of humans. 90 00:08:04,934 --> 00:08:08,457 Well. The only place it could go is right 91 00:08:08,457 --> 00:08:11,371 there. It couldn't go there, because that's 92 00:08:11,371 --> 00:08:14,700 shaded out. So the only place that X could go is 93 00:08:14,700 --> 00:08:19,348 right there in that region. Okay, but if there's an X in that region, 94 00:08:19,348 --> 00:08:24,205 then what does that tell us? That tells us that some Duke students are 95 00:08:24,205 --> 00:08:26,356 animals. Right? 96 00:08:26,356 --> 00:08:32,953 Because there is something that's in the category of Duke students and that's also 97 00:08:32,953 --> 00:08:38,093 in the category of animals. So, using this Venn diagram to represent 98 00:08:38,093 --> 00:08:43,631 the information that we get from these two premises, we can show that this 99 00:08:43,631 --> 00:08:47,448 conclusion follows validly from those two premises. 100 00:08:47,448 --> 00:08:52,687 If these two premises are true, then this conclusion has gotta be true. 101 00:08:52,687 --> 00:08:55,830 And that's what the Venn diagram shows us. 102 00:08:55,830 --> 00:09:01,661 Now let's consider another example. So this syllogism tells us, first, that 103 00:09:01,661 --> 00:09:08,167 some Duke students are humans, and second, that all humans are animals. 104 00:09:08,167 --> 00:09:14,590 But it concludes from those two premises, that all Duke students are animals. 105 00:09:14,590 --> 00:09:20,174 Now is that inference valid? Let's use the venn diagram to try to find 106 00:09:20,174 --> 00:09:23,262 out. Well, this premise says all humans are 107 00:09:23,262 --> 00:09:26,802 animals. So how do we represent that on the Venn 108 00:09:26,802 --> 00:09:30,415 diagram? By shading out the portion of the circle 109 00:09:30,415 --> 00:09:33,881 of humans that's outside the circle of animals. 110 00:09:33,881 --> 00:09:37,569 Alright? So we shade out that portion to show that 111 00:09:37,569 --> 00:09:43,542 there's nothing in the category of humans that's outside the category of animals. 112 00:09:43,542 --> 00:09:46,640 All humans, if they're already, are animals. 113 00:09:46,640 --> 00:09:50,590 Some Duke students are humans, the other premise tells us. 114 00:09:50,590 --> 00:09:56,066 Now how do we represent that information? Well, by drawing an axe that is inside 115 00:09:56,066 --> 00:10:00,571 the circle of Duke students, and also inside the circle of humans. 116 00:10:00,571 --> 00:10:04,937 Now where could that go? Only one place for it to go and that's 117 00:10:04,937 --> 00:10:11,517 right here. Now, from that information, can we 118 00:10:11,517 --> 00:10:16,180 conclude that all Duke students are animals? 119 00:10:17,200 --> 00:10:21,230 No. We can't and here's why not. 120 00:10:21,230 --> 00:10:28,221 We have not shaded out the region of Duke students that's outside the circle of 121 00:10:28,221 --> 00:10:32,242 animals. We don't know if there's something in 122 00:10:32,242 --> 00:10:37,224 that region or not. There could very well be something in 123 00:10:37,224 --> 00:10:42,205 that region for all that we've been told by the premises. 124 00:10:42,205 --> 00:10:48,848 The premises leave it open that maybe most Duke students are not animals at 125 00:10:48,848 --> 00:10:53,830 all, but rather plants or microbes or inorganic materials. 126 00:10:53,830 --> 00:11:00,889 So, this inference is not valid, because there is a possible way for the premises 127 00:11:00,889 --> 00:11:07,948 to be true, as is represented by this diagram, but the conclusion to be false, 128 00:11:07,948 --> 00:11:12,500 because maybe there are things that are out here. 129 00:11:13,900 --> 00:11:19,968 Now, let's look at another syllogism. Where both of the premises and the 130 00:11:19,968 --> 00:11:25,353 conclusion are all of the I form. Some Duke students are humans. 131 00:11:25,353 --> 00:11:30,824 Some humans are animals. Therefore, the syllogisms concludes, some 132 00:11:30,824 --> 00:11:35,440 Duke students are animals. Now, it that syllogism valid? 133 00:11:35,440 --> 00:11:39,210 Let's see if we can use our Venn Diagram to find out. 134 00:11:39,210 --> 00:11:42,767 So, one premise says some Duke students are humans. 135 00:11:42,767 --> 00:11:48,601 Let's try to represent that information. In order to represent that information we 136 00:11:48,601 --> 00:11:54,293 need to make a little mark a little X that's inside the circle of Duke students 137 00:11:54,293 --> 00:11:58,420 but also inside the circle of humans. So, we make that mark. 138 00:11:58,420 --> 00:12:02,307 Wait a second. It looks like there are two different 139 00:12:02,307 --> 00:12:07,915 places where we could make that mark. We could make it in this region or in 140 00:12:07,915 --> 00:12:11,878 this region. Because both of these regions are in the 141 00:12:11,878 --> 00:12:16,140 circle of Duke students, and also in the circle of humans. 142 00:12:16,140 --> 00:12:25,098 So where should we make the mark? Let's hedge our bets and make it between 143 00:12:25,098 --> 00:12:31,226 those two regions, right here. Okay. 144 00:12:31,226 --> 00:12:36,060 The second premise is some humans are animals. 145 00:12:36,060 --> 00:12:42,939 So we need to make a mark that's inside the circle of humans but also inside the 146 00:12:42,939 --> 00:12:45,232 circle of animals. But wait. 147 00:12:45,232 --> 00:12:50,020 Do we make that mark right here or right here? 148 00:12:50,020 --> 00:12:54,336 The premise doesn't give us enough information to decide. 149 00:12:54,336 --> 00:12:59,789 So once again, let's hedge our bets and make that mark, right here on the 150 00:12:59,789 --> 00:13:07,920 boundary between those two regions. So. 151 00:13:07,920 --> 00:13:14,620 Finally, the conclusion says, some Duke students are animals. 152 00:13:14,620 --> 00:13:21,516 Is that right? Well, you might think it is, because this 153 00:13:21,516 --> 00:13:29,528 x and this x both look like they're in the circle of Duke students and also in 154 00:13:29,528 --> 00:13:35,121 the circle of animals. But that's not right. 155 00:13:35,121 --> 00:13:39,517 See this X is on the border of the circle of animals. 156 00:13:39,517 --> 00:13:44,577 Maybe it's inside that circle. Maybe it's outside that circle. 157 00:13:44,577 --> 00:13:49,139 We don't know. So this X right here might not be inside 158 00:13:49,139 --> 00:13:54,447 the circle of animals. This X right here is on the border of the 159 00:13:54,447 --> 00:13:59,590 circle of the Duke's students. So maybe it's inside the circle. 160 00:13:59,590 --> 00:14:03,240 Maybe it's outside the circle. We don't know. 161 00:14:04,880 --> 00:14:11,952 So this x, while it's certainly in the circle of Duke students might not be in 162 00:14:11,952 --> 00:14:17,483 the circle of animals. And this x, while it's certainly in the 163 00:14:17,483 --> 00:14:23,104 circle of animals might not be in the circle of Duke students. 164 00:14:23,104 --> 00:14:30,177 So, this Venn diagram doesn't tell us whether or not there is anything that is 165 00:14:30,177 --> 00:14:37,385 both a Duke student and an animal. It doesn't tell us that there is anything 166 00:14:37,385 --> 00:14:43,753 in this region right here. And if it doesn't tell us whether there's 167 00:14:43,753 --> 00:14:49,458 anything in this region right here, that means that the information provided by 168 00:14:49,458 --> 00:14:54,370 the premises is not enough to guarantee that the conclusion is true. 169 00:14:54,370 --> 00:15:01,644 So using this venn diagram we can figure out that this syllogism right here is not 170 00:15:01,644 --> 00:15:04,878 valid. The premises don't guarantee the truth of 171 00:15:04,878 --> 00:15:08,360 the conclusion. Now notice since every syllogism has a 172 00:15:08,360 --> 00:15:12,938 subject term, a middle term and a predicate term, it's going to be about 173 00:15:12,938 --> 00:15:17,451 three different categories, a subject category, a middle category and a 174 00:15:17,451 --> 00:15:20,933 predicate category. And so you're going to need a Venn 175 00:15:20,933 --> 00:15:25,962 diagram with three different circles to represent the information conveyed by 176 00:15:25,962 --> 00:15:29,613 that inference. Now that Venn diagram is going to have 177 00:15:29,613 --> 00:15:34,154 eight different regions to it. There's going to be a region that's 178 00:15:34,154 --> 00:15:39,728 inside the S circle, inside the P circle and inside the M circle, inside all three 179 00:15:39,728 --> 00:15:42,755 circles. There's going to be a region that's 180 00:15:42,755 --> 00:15:46,608 inside both the S and P circle, but outside the M circle. 181 00:15:46,608 --> 00:15:51,975 There's going to be a region inside the S and the M circles but outside the P 182 00:15:51,975 --> 00:15:56,930 circle, a region inside the P and the M circles but outside the S circle. 183 00:15:56,930 --> 00:16:01,365 Then there's going to be a region that's in the S circle, but outside the M and 184 00:16:01,365 --> 00:16:04,458 the P circles. A region that's in the, in the M circle, 185 00:16:04,458 --> 00:16:08,893 but outside the S and the P circles. And a region that's in the P circle, but 186 00:16:08,893 --> 00:16:13,154 outside the S and the M circles. And finally, there'll be an eighth region 187 00:16:13,154 --> 00:16:17,224 that's outside all three circles. Now. 188 00:16:17,224 --> 00:16:23,765 Thinking about the fact that every Venn diagram representing the information 189 00:16:23,765 --> 00:16:29,117 conveyed in the syllogism is going to have those eight regions. 190 00:16:29,117 --> 00:16:35,828 I want you to think about why it is that there would be 256 possible different 191 00:16:35,828 --> 00:16:40,840 kinds of syllogism. That concludes our lecture on syllogism. 192 00:16:40,840 --> 00:16:46,935 I want to point something out about the logic that we've learned over the last 193 00:16:46,935 --> 00:16:50,561 two weeks. Both the propositional logic and the 194 00:16:50,561 --> 00:16:55,227 categorical logic. While both of them are useful tools in 195 00:16:55,227 --> 00:17:01,446 understanding why certain kinds of arguments are valid in virtue of their 196 00:17:01,446 --> 00:17:07,665 form, neither of them provides a complete treatment of all valid arguments. 197 00:17:07,665 --> 00:17:13,632 Not even of all valid arguments that are valid in virtue of their form. 198 00:17:13,632 --> 00:17:20,019 For instance, consider the argument, for every number, there is another number 199 00:17:20,019 --> 00:17:24,290 larger than it. Therefore, there is no number that is the 200 00:17:24,290 --> 00:17:27,962 largest. That argument is valid and it is valid in 201 00:17:27,962 --> 00:17:32,295 virtue of its form. If we replace the word number with some 202 00:17:32,295 --> 00:17:35,820 other word the argument is still going to be valid. 203 00:17:35,820 --> 00:17:40,503 . But nothing that we've studied so far in 204 00:17:40,503 --> 00:17:46,255 propositional logic or in syllogistic logic and categorical logic can help us 205 00:17:46,255 --> 00:17:50,679 understand why that argument is valid in virtue of its form. 206 00:17:50,679 --> 00:17:56,726 In order to understand why that argument is valid in virtue of its form we'd need 207 00:17:56,726 --> 00:18:02,548 to get into more advanced kinds of logic. There are lots more advanced kinds of 208 00:18:02,548 --> 00:18:07,187 logic, but we're not going to be discussing them in this course. 209 00:18:07,187 --> 00:18:11,900 If you're interested, I recommend another course in logic to you.