1 0:00:00,000 --> 0:00:03,700 In this video, we're going to start learning ... 2 0:00:03,700 --> 0:00:08,200 about relations and how to do translations with them. 3 0:00:08,200 --> 0:00:11,598 To begin with, let's talk about what relations ... 4 0:00:11,598 --> 0:00:13,950 are. Another name for the kind of ... 5 0:00:13,950 --> 0:00:17,958 relation we'll talk about is two-place propositional ... 6 0:00:17,958 --> 0:00:20,920 functions. (There can be relations with ... 7 0:00:20,920 --> 0:00:24,318 more than two places, but we won't cover them.) 8 0:00:24,320 --> 0:00:28,522 You might remember that "one-place propositional ... 9 0:00:28,522 --> 0:00:32,039 function" was a fancy name for predicates. 10 0:00:32,040 --> 0:00:35,455 So a sentence like "Pooh is a bear," 11 0:00:35,455 --> 0:00:38,399 which we would translate as Bp 12 0:00:38,400 --> 0:00:42,513 is built around this structure "[blank] is a bear," 13 0:00:42,513 --> 0:00:47,679 which becomes a sentence if you fill in that one blank with a name. 14 0:00:47,680 --> 0:00:52,699 This is called a one-place propositional function because ... 15 0:00:52,699 --> 0:00:58,119 there's one place where you put a name in order to make a sentence. 16 0:00:58,120 --> 0:01:03,560 Now let's look at a sentence like "Joanie loves Chachi." 17 0:01:03,560 --> 0:01:07,651 This is built around the structure "[blank] loves ... 18 0:01:07,651 --> 0:01:10,573 [blank]," which has two places that ... 19 0:01:10,573 --> 0:01:14,079 both need names in order to make a sentence. 20 0:01:14,080 --> 0:01:19,680 So it's called a two-place propositional function, 21 0:01:19,680 --> 0:01:23,400 or a relation, because it expresses a ... 22 0:01:23,400 --> 0:01:27,120 relation between Joanie and Chachi. 23 0:01:27,120 --> 0:01:32,932 We would represent this with a capital L followed by two ... 24 0:01:32,932 --> 0:01:38,239 lower-case letters. So Lxy stands for "x loves y." 25 0:01:38,240 --> 0:01:42,448 Something about relations is that when you have the two ... 26 0:01:42,448 --> 0:01:46,839 letters after the relation letter, their order matters. 27 0:01:46,840 --> 0:01:51,226 So if we have Lxy for the relation x loves y, 28 0:01:51,226 --> 0:01:56,239 and constants representing Chachi and Joanie, 29 0:01:56,240 --> 0:02:00,160 Joanie loves Chachi 30 0:02:00,160 --> 0:02:05,847 would be represented by Ljc. Joanie is the one who loves, 31 0:02:05,847 --> 0:02:09,719 so j for Joanie comes first after the L. 32 0:02:09,720 --> 0:02:14,240 On the other hand, Chachi loves Joanie 33 0:02:14,240 --> 0:02:19,327 would be represented by Lcj. Now Chachi is the one who ... 34 0:02:19,327 --> 0:02:22,795 loves, so c for Chachi is first after ... 35 0:02:22,795 --> 0:02:25,916 the L; and Joanie is the one who is ... 36 0:02:25,916 --> 0:02:29,037 loved, so j for Joanie is second. 37 0:02:29,040 --> 0:02:32,446 And these two sentences aren't the same! 38 0:02:32,446 --> 0:02:35,451 Sadly enough, when one person loves ... 39 0:02:35,451 --> 0:02:38,656 another, the other one doesn't always ... 40 0:02:38,656 --> 0:02:39,958 love them back. 41 0:02:39,960 --> 0:02:44,227 We can also have the names in a different order. 42 0:02:44,227 --> 0:02:47,399 If we say Chachi is loved by Joanie, 43 0:02:47,400 --> 0:02:53,040 that's a passive voice way of saying Joanie loves Chachi. 44 0:02:53,040 --> 0:02:57,420 Which is Ljc again. So we have to untwist our ... 45 0:02:57,420 --> 0:03:01,800 original sentence before we translate it. 46 0:03:01,800 --> 0:03:06,323 When we're doing relational translations like this, 47 0:03:06,323 --> 0:03:09,439 we have to keep the letters together. 48 0:03:09,440 --> 0:03:13,840 Let's look at this vocabulary. 49 0:03:13,840 --> 0:03:17,587 If we want to say "Amy doesn't love Ted," 50 0:03:17,587 --> 0:03:20,766 we treat "Amy loves Ted" as a unit, 51 0:03:20,766 --> 0:03:23,718 and then we put a negation on it. 52 0:03:23,720 --> 0:03:29,339 So we get ~Lat. The Lat stays together and ... 53 0:03:29,339 --> 0:03:32,479 the ~ goes out in front. 54 0:03:32,480 --> 0:03:35,363 If we have "Bobby and Debra love Amy," 55 0:03:35,363 --> 0:03:38,339 then the "and" has to join two complete ... 56 0:03:38,339 --> 0:03:40,850 sentences when we translate it. 57 0:03:40,850 --> 0:03:44,198 We can't just stick it between two letters. 58 0:03:44,200 --> 0:03:51,320 So we spell it out more fully: Bobby loves Amy and Debra loves Amy. 59 0:03:51,320 --> 0:03:57,180 Then we translate each half. Bobby loves Amy is Lba, 60 0:03:57,180 --> 0:04:03,040 Debra loves Amy is Lda, and we join it with the & sign. 61 0:04:03,040 --> 0:04:09,240 Now let's talk about how to combine relations with quantifiers. 62 0:04:09,240 --> 0:04:13,840 If we're saying Everybody loves Raymond, 63 0:04:13,840 --> 0:04:19,200 that means that for all x, x loves Raymond. 64 0:04:19,200 --> 0:04:26,347 For all x is Ax, and x loves Raymond is Lxr, 65 0:04:26,347 --> 0:04:29,079 so we get AxLxr. 66 0:04:29,080 --> 0:04:34,040 On the other hand, Raymond loves everybody 67 0:04:34,040 --> 0:04:37,949 is still a statement about for all x. 68 0:04:37,949 --> 0:04:42,119 It says that for all x, Raymond loves x. 69 0:04:42,120 --> 0:04:49,560 Raymond loves x is Lrx, so now we get AxLrx. 70 0:04:49,560 --> 0:04:53,504 Notice that the quantifier goes out front! 71 0:04:53,504 --> 0:04:57,119 We can't separate the L, the r, and the x. 72 0:04:57,120 --> 0:05:00,820 Notice also that it helps to translate our sentence ... 73 0:05:00,820 --> 0:05:03,229 halfway, using the variable x but ... 74 0:05:03,229 --> 0:05:06,412 keeping the rest of the sentence in English. 75 0:05:06,412 --> 0:05:10,456 This half-English translation will become especially ... 76 0:05:10,456 --> 0:05:14,758 important as we do more and more complicated translations. 77 0:05:14,760 --> 0:05:19,404 Now let's talk about how quantifiers interact with ... 78 0:05:19,404 --> 0:05:22,320 negations in relational logic. 79 0:05:22,320 --> 0:05:25,520 So for the sentence "Nobody loves Chris," 80 0:05:25,520 --> 0:05:28,720 first we translate it into half-English, 81 0:05:28,720 --> 0:05:32,285 turning "nobody" into the quantifier and x and ... 82 0:05:32,285 --> 0:05:34,479 leaving the rest in English. 83 0:05:34,480 --> 0:05:37,567 There are two ways to translate "nobody" 84 0:05:37,567 --> 0:05:40,563 sentences, either with a universal and ... 85 0:05:40,563 --> 0:05:44,649 then a negation or a negation and then an existential. 86 0:05:44,649 --> 0:05:47,554 Whichever way we represent "nobody," 87 0:05:47,554 --> 0:05:51,821 we follow it up with what the sentence is saying is true of ... 88 0:05:51,821 --> 0:05:53,273 no x: that x loves Chris. 89 0:05:53,640 --> 0:05:56,930 "x loves Chris" is Lxc, 90 0:05:56,930 --> 0:06:00,913 so we have Ax~Lxc or ~ExLxc, 91 0:06:00,913 --> 0:06:08,879 depending on which way we chose to translate "nobody." 92 0:06:08,880 --> 0:06:14,440 Now let's look at "Chris doesn't love everybody." 93 0:06:14,440 --> 0:06:16,974 Here, because the word "not" 94 0:06:16,974 --> 0:06:20,775 is in front of the quantifier "everybody," 95 0:06:20,775 --> 0:06:24,048 we translate it as ~ first and then Ax, 96 0:06:24,048 --> 0:06:29,222 all in front of what the rest of the sentence is saying is not ... 97 0:06:29,222 --> 0:06:31,756 true for all x: Chris loves x. 98 0:06:31,760 --> 0:06:36,826 "Chris loves x" is Lcx, 99 0:06:36,826 --> 0:06:40,559 so we get ~AxLcx. 100 0:06:40,560 --> 0:06:43,067 You can see in both these cases, 101 0:06:43,067 --> 0:06:47,309 when we translate into half-English the x went where ... 102 0:06:47,309 --> 0:06:50,780 the quantifier word was. In the first case, 103 0:06:50,780 --> 0:06:53,094 x went in where "nobody" was. 104 0:06:53,094 --> 0:06:54,926 In the second case, "x" 105 0:06:54,926 --> 0:06:57,433 went in where "everybody" was, 106 0:06:57,433 --> 0:07:00,615 and we'd already taken care of the "not" 107 0:07:00,615 --> 0:07:03,122 by putting a ~ in front of the Ax, 108 0:07:03,122 --> 0:07:05,918 so "Chris loves x" is what was left. 109 0:07:05,920 --> 0:07:09,983 There are some other ways of phrasing sentences with ... 110 0:07:09,983 --> 0:07:12,199 relations and quantifiers. 111 0:07:12,200 --> 0:07:17,040 Let's look at "There is someone who Raymond loves." 112 0:07:17,040 --> 0:07:20,282 "There is someone" is existential. 113 0:07:20,282 --> 0:07:22,984 Putting the x where "Someone" 114 0:07:22,984 --> 0:07:25,686 was, we get basically "x is who ... 115 0:07:25,686 --> 0:07:27,199 Raymond loves." 116 0:07:27,200 --> 0:07:32,800 which is to say, Raymond loves x. 117 0:07:32,800 --> 0:07:39,400 So this turns out as ExLrx. 118 0:07:39,400 --> 0:07:43,079 This sentence shows that, when we turn a sentence into ... 119 0:07:43,079 --> 0:07:45,695 half-English, we can't always put the ... 120 0:07:45,695 --> 0:07:49,047 variable exactly where the quantifier word was. 121 0:07:49,047 --> 0:07:52,563 Sometimes we have to untangle the sentence a little ... 122 0:07:52,563 --> 0:07:55,997 so the variable can go somewhere that makes sense. 123 0:07:56,000 --> 0:07:59,121 Finally, there are some sentences in ... 124 0:07:59,121 --> 0:08:04,559 English that are ambiguous; they can have more than one meaning. 125 0:08:04,560 --> 0:08:09,360 Take "Everybody doesn't love Ted." 126 0:08:09,360 --> 0:08:14,474 Does this mean "For all x, x doesn't love Ted"; 127 0:08:14,474 --> 0:08:17,839 Ted is universally not loved? 128 0:08:17,840 --> 0:08:24,360 That would be Ax~Lxt. 129 0:08:24,360 --> 0:08:29,225 Or does it mean that it isn't the case that everybody loves ... 130 0:08:29,225 --> 0:08:31,759 Ted? Not everybody loves Ted? 131 0:08:31,760 --> 0:08:37,880 That would be ~AxLxt. 132 0:08:37,880 --> 0:08:42,716 These are different meanings that original sentence can ... 133 0:08:42,716 --> 0:08:45,134 have, at least in some ways of ... 134 0:08:45,134 --> 0:08:48,358 speaking English. So it's ambiguous. 135 0:08:48,360 --> 0:08:53,720 You have to tell which meaning is meant from the context. 136 0:08:53,720 --> 0:08:59,053 I will try to avoid asking you to translate sentences that are ... 137 0:08:59,053 --> 0:09:01,039 ambiguous in this way.