1 00:00:00,000 --> 00:00:05,586 In this video I'll talk about doing categorical translations ... 2 00:00:05,586 --> 00:00:07,519 in relational logic. 3 00:00:07,520 --> 00:00:10,800 We can combine categorical statements, 4 00:00:10,800 --> 00:00:13,598 like statements about "some cat," 5 00:00:13,598 --> 00:00:16,106 with relations, like "loves." 6 00:00:16,106 --> 00:00:20,158 So we get a statement like "Some cat loves Johnny." 7 00:00:20,160 --> 00:00:24,235 But with relational logic, the category we're talking ... 8 00:00:24,235 --> 00:00:28,487 about doesn't have to be at the beginning of the sentence. 9 00:00:28,487 --> 00:00:32,119 It can be at the end, as in "Johnny loves all cats." 10 00:00:32,120 --> 00:00:34,897 We can even relate two categories, 11 00:00:34,897 --> 00:00:38,631 as in "Every person was born in some location"; 12 00:00:38,631 --> 00:00:41,503 "every person" and "some location" 13 00:00:41,503 --> 00:00:46,482 are both categories that will wind up getting translated with ... 14 00:00:46,482 --> 00:00:48,397 separate quantifiers. 15 00:00:48,400 --> 00:00:53,407 All these translations will be built on the old categorical ... 16 00:00:53,407 --> 00:00:57,913 forms that we used to translate sentences with "all," 17 00:00:57,913 --> 00:00:59,114 "some," "no," 18 00:00:59,114 --> 00:01:01,517 and so on in predicate logic, 19 00:01:01,520 --> 00:01:05,832 things like the universal combined with the arrow for ... 20 00:01:05,832 --> 00:01:08,802 "all," and the existential combined ... 21 00:01:08,802 --> 00:01:10,718 with the and for "some." 22 00:01:10,720 --> 00:01:14,651 But when we're doing relational categoricals, 23 00:01:14,651 --> 00:01:19,959 we'll have to break these sentences down carefully step by step. 24 00:01:19,960 --> 00:01:24,081 The categorical translations we'll do here will depend on ... 25 00:01:24,081 --> 00:01:28,370 the four main frameworks that we used for doing categoricals ... 26 00:01:28,370 --> 00:01:29,799 in predicate logic, 27 00:01:29,800 --> 00:01:31,074 for all, some, 28 00:01:31,074 --> 00:01:33,729 no, and not all or some are not. 29 00:01:33,729 --> 00:01:38,190 They each have a quantifier and a symbol to join the ... 30 00:01:38,190 --> 00:01:42,545 subject and predicate, and some have a negation. 31 00:01:42,545 --> 00:01:46,156 (You should review these if you need to!) 32 00:01:46,160 --> 00:01:49,979 But when we're doing relational categoricals, 33 00:01:49,979 --> 00:01:54,085 the category may not be the subject of the sentence; 34 00:01:54,085 --> 00:01:56,185 it may not be first! Still, 35 00:01:56,185 --> 00:01:59,718 the category will go out in front of the form. 36 00:01:59,720 --> 00:02:04,600 Here's an example with one categorical phrase. 37 00:02:04,600 --> 00:02:08,640 Using this vocabulary 38 00:02:08,640 --> 00:02:13,320 we translate "Some cat loves Johnny." 39 00:02:13,320 --> 00:02:19,000 Since this is a categorical that begins with "Some," 40 00:02:19,000 --> 00:02:22,743 "some cat," we use the existential ... 41 00:02:22,743 --> 00:02:24,679 categorical form 42 00:02:24,680 --> 00:02:28,756 which starts with an existential quantifier and has an & 43 00:02:28,756 --> 00:02:32,399 between the category and the rest of the sentence. 44 00:02:32,400 --> 00:02:37,481 We translate the two parts of the sentence into half-English, 45 00:02:37,481 --> 00:02:40,217 English but using the variable x. 46 00:02:40,217 --> 00:02:44,516 So the category is "x is a cat," and what the rest of the ... 47 00:02:44,516 --> 00:02:48,718 sentence is saying about that x is "x loves Johnny." 48 00:02:48,720 --> 00:02:53,920 "x is a cat" translates to Cx 49 00:02:53,920 --> 00:02:59,120 "x loves Johnny translates to Lxj 50 00:02:59,120 --> 00:03:05,200 and we plug those back in to the existential categorical form. 51 00:03:05,200 --> 00:03:08,696 When we're doing translations with a single category, 52 00:03:08,696 --> 00:03:12,116 it's important to put the category at the beginning of ... 53 00:03:12,116 --> 00:03:15,080 the translation, no matter where it appears in ... 54 00:03:15,080 --> 00:03:16,600 the original sentence. 55 00:03:16,600 --> 00:03:19,945 Let's look at "Johnny loves all cats." 56 00:03:19,945 --> 00:03:23,290 Notice that the category, "all cats," 57 00:03:23,290 --> 00:03:25,799 is at the end of the sentence. 58 00:03:25,800 --> 00:03:29,391 We use the "All" form for a categorical, 59 00:03:29,391 --> 00:03:33,199 with a universal quantifier and an arrow. 60 00:03:33,200 --> 00:03:36,636 But the category goes before the arrow. 61 00:03:36,636 --> 00:03:40,072 The category will always go on the left, 62 00:03:40,072 --> 00:03:44,238 even if it's at the end of the original sentence. 63 00:03:44,240 --> 00:03:47,558 So we get "x is a cat" before the arrow, 64 00:03:47,558 --> 00:03:52,802 and after the arrow we put the thing the rest of the sentence ... 65 00:03:52,802 --> 00:03:55,799 says about x: That Johnny loves x. 66 00:03:55,800 --> 00:03:58,500 Notice where the x goes there. 67 00:03:58,500 --> 00:04:04,440 We put it into the sentence where the category "all cats" had been. 68 00:04:04,440 --> 00:04:08,364 Now we can translate each piece. 69 00:04:08,364 --> 00:04:10,399 "x is a cat" is Cx. 70 00:04:10,400 --> 00:04:15,600 "Johnny loves x" is Ljx. 71 00:04:15,600 --> 00:04:21,840 So we have universal x Cx arrow Ljx. 72 00:04:21,840 --> 00:04:26,219 When we're doing relational logic we can also wind up ... 73 00:04:26,219 --> 00:04:30,199 stacking categoricals one on top of each other. 74 00:04:30,200 --> 00:04:33,062 So in "Some people love every cat," 75 00:04:33,062 --> 00:04:37,405 we have category terms in two different places in the ... 76 00:04:37,405 --> 00:04:40,958 sentence, one on each side of the relation. 77 00:04:40,960 --> 00:04:44,702 To do a sentence like this, we have to go step by step. 78 00:04:44,702 --> 00:04:48,096 First we do the first categorical expression, 79 00:04:48,096 --> 00:04:51,838 leaving the second half of the sentence in English. 80 00:04:51,838 --> 00:04:56,798 Then we translate the second half of the sentence and plug it back in. 81 00:04:56,800 --> 00:05:01,260 So our first categorical expression is "some people." 82 00:05:01,260 --> 00:05:04,169 We use the "some" categorical form, 83 00:05:04,169 --> 00:05:08,726 with an existential and an &. The category will go before ... 84 00:05:08,726 --> 00:05:11,441 the &, and the rest of the sentence ... 85 00:05:11,441 --> 00:05:12,798 will go after it. 86 00:05:12,800 --> 00:05:15,534 The category is "x is a person." 87 00:05:15,534 --> 00:05:18,268 To get the rest of the sentence, 88 00:05:18,268 --> 00:05:21,108 we put the x in for "every person" 89 00:05:21,108 --> 00:05:24,473 and leave everything else in English, 90 00:05:24,473 --> 00:05:26,997 so we get "x loves every cat." 91 00:05:27,000 --> 00:05:31,700 Now "x loves every cat" is itself a categorical, 92 00:05:31,700 --> 00:05:34,520 so we have to translate that. 93 00:05:34,520 --> 00:05:36,305 We use the "all" form. 94 00:05:36,305 --> 00:05:40,868 Since we've already used x, we use y as the variable this ... 95 00:05:40,868 --> 00:05:44,141 time. So that's a universal quantifier ... 96 00:05:44,141 --> 00:05:47,712 at the beginning. Then we have the category ... 97 00:05:47,712 --> 00:05:49,398 *before* the arrow, 98 00:05:49,398 --> 00:05:52,473 y is a cat. After the arrow we have what ... 99 00:05:52,473 --> 00:05:56,837 the rest of the sentence is saying about it, x loves y. 100 00:05:56,840 --> 00:06:00,917 The category "y is a cat" has to come first, 101 00:06:00,917 --> 00:06:05,344 before the arrow. And we put the y in the rest of ... 102 00:06:05,344 --> 00:06:08,839 the sentence where "every cat" was. 103 00:06:08,840 --> 00:06:17,027 Translating those two parts, we get that "x loves every cat" 104 00:06:17,027 --> 00:06:19,319 is Ay(Cy -> Lxy). 105 00:06:19,320 --> 00:06:25,110 Then we plug that back in with the existential and Px to get ... 106 00:06:25,110 --> 00:06:26,919 our final answer. 107 00:06:26,920 --> 00:06:32,440 Let's look at another example of doubled up categoricals. 108 00:06:32,440 --> 00:06:37,760 Using this vocabulary, with the relation "born in" 109 00:06:37,760 --> 00:06:42,205 we translate "Every person was born in some location." 110 00:06:42,205 --> 00:06:44,814 Again we have two categoricals, 111 00:06:44,814 --> 00:06:48,679 and we start with the first one, "Every person." 112 00:06:48,680 --> 00:06:51,808 Since this is an "every" categorical, 113 00:06:51,808 --> 00:06:55,328 we use a universal quantifier and an arrow. 114 00:06:55,328 --> 00:06:57,772 The category, "x is a person," 115 00:06:57,772 --> 00:07:01,780 goes before the arrow. After the arrow is what the ... 116 00:07:01,780 --> 00:07:04,224 rest of the sentence is saying about x: 117 00:07:04,224 --> 00:07:06,472 x was born in some location. 118 00:07:07,160 --> 00:07:11,857 Now we have to translate "x was born in some location," 119 00:07:11,857 --> 00:07:15,719 which has the categorical "some location." 120 00:07:15,720 --> 00:07:17,823 We use the "some" form, 121 00:07:17,823 --> 00:07:23,246 with an existential and an &. The existential is Ey because ... 122 00:07:23,246 --> 00:07:26,013 we already used x. Before the & 123 00:07:26,013 --> 00:07:29,555 goes the category we're working with, 124 00:07:29,555 --> 00:07:31,990 y is a location. After the & 125 00:07:31,990 --> 00:07:36,749 goes what the rest of the sentence is saying about y, 126 00:07:36,749 --> 00:07:38,077 x was born in y. 127 00:07:38,080 --> 00:07:43,946 Now we translate those elements into notation: 128 00:07:43,946 --> 00:07:46,732 "y is a location" is Ly, 129 00:07:46,732 --> 00:07:49,518 "x was born in y" is Bxy. 130 00:07:49,520 --> 00:07:53,731 Plugging it back in, we get Ax(Px -> Ey(Ly & 131 00:07:53,731 --> 00:07:57,099 Bxy)). Make sure that you have all ... 132 00:07:57,099 --> 00:08:01,671 the parentheses. And it's a good idea to check ... 133 00:08:01,671 --> 00:08:06,844 that the x's and y's line up in a way that makes sense. 134 00:08:06,844 --> 00:08:10,212 x is the person, y is the location, 135 00:08:10,212 --> 00:08:15,145 so it makes sense that we're saying x was born in y; 136 00:08:15,145 --> 00:08:19,476 it'd be bad if we had it the other way around! 137 00:08:19,480 --> 00:08:31,520 [no narration, just cat picture]