1 0:00:00,000 --> 0:00:04,706 In the last video we talked about relational logic and did ... 2 0:00:04,706 --> 0:00:07,843 some translations with quantifiers. 3 0:00:07,843 --> 0:00:12,353 In this video we'll see how relational logic can give us ... 4 0:00:12,353 --> 0:00:16,078 translations with more than one quantifier. 5 0:00:16,080 --> 0:00:21,760 Let's start by reviewing translations with single quantifiers. 6 0:00:21,760 --> 0:00:25,530 Remember, a relation is a capital letter ... 7 0:00:25,530 --> 0:00:28,856 followed by two lowercase letters; 8 0:00:28,856 --> 0:00:33,402 here Lxy for x loves y. And we'll use r as a constant ... 9 0:00:33,402 --> 0:00:35,398 to stand for Raymond. 10 0:00:35,400 --> 0:00:39,760 To translate Everybody loves Raymond 11 0:00:39,760 --> 0:00:45,075 we first put the sentence halfway into notation, 12 0:00:45,075 --> 0:00:49,094 as Ax x loves Raymond. The x went where ... 13 0:00:49,094 --> 0:00:51,038 "Everybody" was. 14 0:00:51,040 --> 0:00:58,520 Since x loves Raymond is Lxr, we get AxLxr. 15 0:00:58,520 --> 0:01:02,336 On the other hand, if we have Raymond loves ... 16 0:01:02,336 --> 0:01:06,479 everybody, we put that halfway into notation 17 0:01:06,480 --> 0:01:11,569 as Ax Raymond loves x; now the x is in the second ... 18 0:01:11,569 --> 0:01:15,319 place, where the "everybody" was 19 0:01:15,320 --> 0:01:19,960 and that's AxLrx. 20 0:01:19,960 --> 0:01:25,600 So the variable can wind up in either place after the relation letter. 21 0:01:25,600 --> 0:01:31,120 Now let's see how we can wind up with double quantifiers. 22 0:01:31,120 --> 0:01:34,684 If we have "somebody loves somebody," 23 0:01:34,684 --> 0:01:38,359 first we put that partway into notation 24 0:01:38,360 --> 0:01:43,321 and we do that step by step, using variables and leaving ... 25 0:01:43,321 --> 0:01:46,879 the rest in English until the last step. 26 0:01:46,880 --> 0:01:51,079 So first we turn the first "somebody" 27 0:01:51,079 --> 0:01:55,007 into Ex, and put the x in to make it "x ... 28 0:01:55,007 --> 0:01:57,038 loves somebody." 29 0:01:57,040 --> 0:02:01,570 Now we have to figure out what "x loves somebody" 30 0:02:01,570 --> 0:02:04,627 is. That has a quantifier that we ... 31 0:02:04,627 --> 0:02:06,439 need to translate. 32 0:02:06,440 --> 0:02:12,534 It'll be an existential, but it can't be Ex since we're ... 33 0:02:12,534 --> 0:02:16,596 already using x. So we start with Ey. 34 0:02:16,596 --> 0:02:20,658 THen we put y in where the "somebody" 35 0:02:20,658 --> 0:02:23,637 was, and we get Ey x loves y. 36 0:02:23,640 --> 0:02:30,040 So "x loves somebody" is EyLxy. 37 0:02:30,040 --> 0:02:39,840 We put that after the Ex, and the final answer is ExEyLxy. 38 0:02:39,840 --> 0:02:43,243 Now, the order in which quantifiers ... 39 0:02:43,243 --> 0:02:46,079 appear can make a difference. 40 0:02:46,080 --> 0:02:52,760 Let's say we have AxEyLxy. 41 0:02:52,760 --> 0:02:58,186 That means that for every x, there is a y that that x loves. 42 0:02:58,186 --> 0:03:01,534 Pick an x, and you can always find a y ... 43 0:03:01,534 --> 0:03:02,919 for it to love. 44 0:03:02,920 --> 0:03:07,840 But the x's could all love different y's. 45 0:03:07,840 --> 0:03:10,954 We could represent this with this graph, 46 0:03:10,954 --> 0:03:14,618 where the dots are people and the arrows are love ... 47 0:03:14,618 --> 0:03:18,007 relationships. Notice that every dot has an ... 48 0:03:18,007 --> 0:03:19,839 arrow going somewhere. 49 0:03:19,840 --> 0:03:27,836 What if we switch the quantifiers so the Ey is first? 50 0:03:27,836 --> 0:03:30,439 We have EyAxLxy. 51 0:03:30,440 --> 0:03:35,680 This means that there is a y that every x loves. 52 0:03:35,680 --> 0:03:41,920 Here it's the same y, beloved universally by every x. 53 0:03:41,920 --> 0:03:44,884 That could be represented by this diagram, 54 0:03:44,884 --> 0:03:48,425 where every dot has an arrow going to the upper right ... 55 0:03:48,425 --> 0:03:50,977 corner. That's the one that everyone ... 56 0:03:50,977 --> 0:03:53,118 loves. (Even that person loves ... 57 0:03:53,118 --> 0:03:56,494 themself.) So these two sentences don't mean the ... 58 0:03:56,494 --> 0:03:58,717 same thing, even though the only ... 59 0:03:58,717 --> 0:04:02,917 difference is that we switched the order of the quantifiers. 60 0:04:02,920 --> 0:04:07,108 So we need some way of telling which order the ... 61 0:04:07,108 --> 0:04:08,919 quantifiers go in. 62 0:04:08,920 --> 0:04:12,221 One thing is if we see the words "There is" 63 0:04:12,221 --> 0:04:15,813 or "The same," that will get translated as an ... 64 0:04:15,813 --> 0:04:19,308 existential, and it will go first no matter ... 65 0:04:19,308 --> 0:04:21,638 where it was in the sentence. 66 0:04:21,640 --> 0:04:25,140 So if we have "There is someone who loves ... 67 0:04:25,140 --> 0:04:27,473 everyone," the "There is" 68 0:04:27,473 --> 0:04:31,398 is an existential and we translate it first. 69 0:04:31,400 --> 0:04:35,203 So we get Ex, x loves everyone. 70 0:04:35,203 --> 0:04:39,919 And now we can translate "everyone." 71 0:04:39,920 --> 0:04:44,868 That becomes Ay, and we put y in for ... 72 0:04:44,868 --> 0:04:49,639 "everyone," so it's Ay x loves y. 73 0:04:49,640 --> 0:04:58,520 That's Lxy, so we wind up with ExAyLxy. 74 0:04:58,520 --> 0:05:01,889 Now if we have "everyone loves the same person," 75 0:05:01,889 --> 0:05:04,331 "the same" will be an existential, 76 0:05:04,331 --> 0:05:07,615 and we have to translate it before "everyone," 77 0:05:07,615 --> 0:05:10,478 even though it's second in the sentence. 78 0:05:10,480 --> 0:05:14,273 So we make it Ex, and we put the x where "the ... 79 0:05:14,273 --> 0:05:15,946 same person" was, 80 0:05:15,946 --> 0:05:19,181 so that part is "Everyone loves x." 81 0:05:19,181 --> 0:05:22,639 Now we have to translate "everyone." 82 0:05:22,640 --> 0:05:26,560 That'll be Ay, and we put y in where ... 83 0:05:26,560 --> 0:05:28,520 "everyone" was, 84 0:05:28,520 --> 0:05:30,760 so that's y loves x, 85 0:05:30,760 --> 0:05:39,920 which becomes Lyx, so the final answer is ExAyLyx. 86 0:05:39,920 --> 0:05:45,880 There are some more cues as to what order the quantifiers go in. 87 0:05:45,880 --> 0:05:50,614 If you see "some or other," that's an existential that goes ... 88 0:05:50,614 --> 0:05:51,773 last. The "or" 89 0:05:51,773 --> 0:05:55,830 here isn't a wedge, it's just something that tells ... 90 0:05:55,830 --> 0:05:58,438 you where to put the quantifier. 91 0:05:58,440 --> 0:06:02,301 So if it's "someone or other loves everyone," 92 0:06:02,301 --> 0:06:05,044 we have to do the "someone" last, 93 0:06:05,044 --> 0:06:07,279 and the "everyone" first. 94 0:06:07,280 --> 0:06:09,825 The "everyone" is Ax, 95 0:06:09,825 --> 0:06:13,643 and we put the x where "everyone" 96 0:06:13,643 --> 0:06:17,319 was, so it's Ax someone loves x. 97 0:06:17,320 --> 0:06:21,371 Now we can translate "someone" 98 0:06:21,371 --> 0:06:24,487 as Ey, and put the y where ... 99 0:06:24,487 --> 0:06:26,512 "someone" was, 100 0:06:26,512 --> 0:06:29,317 so it's Ax Ey y loves x. 101 0:06:29,320 --> 0:06:40,800 y loves x is Lyx--notice that y comes first!--so we get AxEyLyx. 102 0:06:40,800 --> 0:06:44,877 If we have "everyone loves someone or other," 103 0:06:44,877 --> 0:06:48,310 we still translate "someone or other" 104 0:06:48,310 --> 0:06:50,563 last, so we do "everyone" 105 0:06:50,563 --> 0:06:52,279 first again, as Ax. 106 0:06:52,280 --> 0:06:58,023 So we get Ax and put x where "everyone" 107 0:06:58,023 --> 0:07:02,839 was, so it's Ax x loves someone. 108 0:07:02,840 --> 0:07:08,922 Now we turn "someone" into Ey and put y where ... 109 0:07:08,922 --> 0:07:11,118 "someone" was, 110 0:07:11,118 --> 0:07:14,159 so it's Ax Ey x loves y. 111 0:07:14,160 --> 0:07:20,200 Which becomes AxEyLxy. 112 0:07:20,200 --> 0:07:24,426 Sometimes we have sentences with no explicit cue ... 113 0:07:24,426 --> 0:07:28,239 that tells you which quantifier goes first. 114 0:07:28,240 --> 0:07:32,404 The general idea is that if there isn't one of those cues, 115 0:07:32,404 --> 0:07:37,719 you translate the quantifiers in the order they appear in the sentence. 116 0:07:37,720 --> 0:07:41,347 So if we have "everybody loves somebody," 117 0:07:41,347 --> 0:07:44,559 we translate the "everybody" first. 118 0:07:44,560 --> 0:07:48,151 That becomes Ax x loves somebody, 119 0:07:48,151 --> 0:07:51,999 so now we translate the "somebody." 120 0:07:52,000 --> 0:07:57,440 which becomes Ey x loves y. 121 0:07:57,440 --> 0:08:05,400 And that's Lxy, so we get AxEyLxy. 122 0:08:05,400 --> 0:08:09,499 But some of these sentences can be ambiguous. 123 0:08:09,499 --> 0:08:12,519 Take "somebody loves everyone." 124 0:08:12,520 --> 0:08:16,775 Does that mean "there is some one person who loves ... 125 0:08:16,775 --> 0:08:20,199 everyone"? A universal source of love? 126 0:08:20,200 --> 0:08:23,391 Or does it mean that somebody or other loves ... 127 0:08:23,391 --> 0:08:25,961 everyone? So everyone has someone ... 128 0:08:25,961 --> 0:08:30,038 who loves them, but they might all be different people? 129 0:08:30,040 --> 0:08:33,376 We don't know. This sentence is ambiguous. 130 0:08:33,376 --> 0:08:37,732 We need to rely on context to figure out which of these two ... 131 0:08:37,732 --> 0:08:41,995 translations is accurate. So I'll try not to assign you ... 132 0:08:41,995 --> 0:08:45,238 sentences that are ambiguous in this way.