1 0:00:00,000 --> 0:00:03,967 In this video I'm going to introduce conditional ... 2 0:00:03,967 --> 0:00:07,159 derivation, which is a new kind of rule. 3 0:00:07,160 --> 0:00:10,206 Conditional Derivation captures a kind of ... 4 0:00:10,206 --> 0:00:13,421 hypothetical reasoning. To see why it works, 5 0:00:13,421 --> 0:00:15,959 let's look at an example in English. 6 0:00:15,960 --> 0:00:19,693 Let's say that I have bread. And if I have bread and ... 7 0:00:19,693 --> 0:00:21,969 cheese, I can make a sandwich. 8 0:00:21,969 --> 0:00:24,518 But we don't know if I have cheese. 9 0:00:24,520 --> 0:00:29,320 I want to show that if I have cheese I can make a sandwich. 10 0:00:29,320 --> 0:00:33,167 So I make a thought balloon, and in that thought balloon, 11 0:00:33,167 --> 0:00:36,359 I think about what would happen if I had cheese. 12 0:00:36,360 --> 0:00:38,991 Well, I'd have bread and cheese! 13 0:00:38,991 --> 0:00:43,377 I can take the bread I actually have and use it inside my ... 14 0:00:43,377 --> 0:00:44,839 thought balloon. 15 0:00:44,840 --> 0:00:49,240 And it follows that I could make a sandwich. 16 0:00:49,240 --> 0:00:51,409 Now we exit the thought balloon. 17 0:00:51,409 --> 0:00:53,899 We don't know that, in the real world, 18 0:00:53,899 --> 0:00:56,631 I can make a sandwich. But we gamed out the ... 19 0:00:56,631 --> 0:00:59,443 scenario of what happens if I have cheese, 20 0:00:59,443 --> 0:01:02,817 and we figured out that in that scenario I can make a ... 21 0:01:02,817 --> 0:01:05,066 sandwich. So we know that IF I have ... 22 0:01:05,066 --> 0:01:07,396 cheese, THEN I can make a sandwich. 23 0:01:07,400 --> 0:01:12,280 Here's how that looks in our symbolic notation. 24 0:01:12,280 --> 0:01:17,280 Our premises are B and (B & C) -> S. 25 0:01:17,280 --> 0:01:20,211 The thing we want to show is C->S. 26 0:01:20,211 --> 0:01:25,639 We're going to make a line under it and show C -> S on that line. 27 0:01:25,640 --> 0:01:29,798 So we assume that C--this corresponds to entering the ... 28 0:01:29,798 --> 0:01:33,679 thought balloon and saying, what if I have cheese? 29 0:01:33,680 --> 0:01:37,956 Now we want to show S. If we can assume C and show ... 30 0:01:37,956 --> 0:01:42,119 S is true, then we know that if C then S is true. 31 0:01:42,120 --> 0:01:46,724 We make a second line--we will show S on this line. 32 0:01:46,724 --> 0:01:50,205 And we start doing rules on that line. 33 0:01:50,205 --> 0:01:53,798 So we can get B&C from lines 1 and 4 by &I. 34 0:01:53,800 --> 0:02:00,200 Now that we have B&C, we can use MP to get S from line 2. 35 0:02:00,200 --> 0:02:04,143 That's what we needed to show on this line! 36 0:02:04,143 --> 0:02:06,847 So we cancel out that SHOW for 37 0:02:06,960 --> 0:02:10,942 and label it Direct Derivation--since we got S, 38 0:02:10,942 --> 0:02:14,028 the thing that we were trying to show, 39 0:02:14,028 --> 0:02:16,318 right under that SHOW line. 40 0:02:16,320 --> 0:02:19,924 Now we get to cancel out the show line for C->S! 41 0:02:19,924 --> 0:02:24,192 We assumed that C was true, and we successfully showed ... 42 0:02:24,192 --> 0:02:28,460 S given that assumption, so that means that if C is true ... 43 0:02:28,460 --> 0:02:29,598 then S is true. 44 0:02:29,600 --> 0:02:34,840 This is the rule of conditional derivation, or CD. 45 0:02:34,840 --> 0:02:39,200 Here's how the rule looks written out formally. 46 0:02:39,200 --> 0:02:42,414 We can only apply this rule when we have a SHOW line ... 47 0:02:42,414 --> 0:02:44,608 whose major operator is an arrow. 48 0:02:44,608 --> 0:02:47,822 Notice that here, having the conclusion written ... 49 0:02:47,822 --> 0:02:50,565 on a special SHOW line makes a difference; 50 0:02:50,565 --> 0:02:54,798 if we didn't do that, we couldn't apply conditional derivation. 51 0:02:54,800 --> 0:02:58,680 So if we're showing A -> C 52 0:02:58,680 --> 0:03:04,040 we get to assume A, the thing before the arrow in the SHOW line 53 0:03:04,040 --> 0:03:09,560 and now we need to show C, the thing after the arrow in the SHOW line. 54 0:03:09,560 --> 0:03:14,000 Now we draw a new vertical line and show C on it. 55 0:03:14,000 --> 0:03:18,760 Once that's done, we get to cross out the SHOW for C, 56 0:03:18,760 --> 0:03:22,449 and when we do that, we immediately get to cross ... 57 0:03:22,449 --> 0:03:24,719 out the SHOW for A -> C as well. 58 0:03:24,720 --> 0:03:28,379 We label A->C as having been shown through ... 59 0:03:28,379 --> 0:03:30,679 conditional derivation. 60 0:03:30,680 --> 0:03:34,413 So we assume what's before the arrow of the SHOW: 61 0:03:34,413 --> 0:03:37,399 line, and SHOW what's after the arrow. 62 0:03:37,400 --> 0:03:40,280 Here's an example. 63 0:03:40,280 --> 0:03:43,608 Let's look at these premises and conclusion. 64 0:03:43,608 --> 0:03:46,236 The SHOW line is an arrow statement, 65 0:03:46,236 --> 0:03:49,039 so we can use conditional derivation. 66 0:03:49,040 --> 0:03:54,680 Our assumption will be what's before the arrow of the SHOW line. 67 0:03:54,680 --> 0:03:57,853 That's ~R, so we put that down on the ... 68 0:03:57,853 --> 0:04:00,479 next line as our assumption. 69 0:04:00,480 --> 0:04:06,120 Our new SHOW line will be what's after the arrow of the old SHOW line. 70 0:04:06,120 --> 0:04:11,440 That's Q, so we put down Q as the new SHOW line. 71 0:04:11,440 --> 0:04:15,540 Now we start a new vertical line under the new SHOW ... 72 0:04:15,540 --> 0:04:17,740 line, and we do rules on it. 73 0:04:17,740 --> 0:04:20,040 We can get P by vO from 2 and 4. 74 0:04:20,040 --> 0:04:25,240 We can get Q by MP from 1 and 6. 75 0:04:25,240 --> 0:04:27,611 Q was what we needed to show, 76 0:04:27,611 --> 0:04:31,839 so we cancel that SHOW out as a direct derivation. 77 0:04:31,840 --> 0:04:34,996 And when we cancel out that SHOW line, 78 0:04:34,996 --> 0:04:39,679 we get to cancel out ~R->Q as a conditional derivation. 79 0:04:39,680 --> 0:04:42,738 Notice that, to find the assumption and the ... 80 0:04:42,738 --> 0:04:45,626 new SHOW line, we only look at the old SHOW ... 81 0:04:45,626 --> 0:04:47,834 line! Whenever you need to show ... 82 0:04:47,834 --> 0:04:50,382 ~R->Q, you'll make that assumption ... 83 0:04:50,382 --> 0:04:53,780 and new SHOW line. Don't try to get an assumption ... 84 0:04:53,780 --> 0:04:56,753 out of the premises; we already know them, 85 0:04:56,753 --> 0:04:59,556 we don't need to assume any part of them. 86 0:04:59,560 --> 0:05:04,360 Here's another example with a more complicated SHOW line. 87 0:05:04,360 --> 0:05:08,240 Given these premises and conclusion 88 0:05:08,240 --> 0:05:12,760 we want to assume what's before the arrow of the SHOW line 89 0:05:12,760 --> 0:05:16,280 which is P & ~S. 90 0:05:16,280 --> 0:05:21,400 Our new SHOW line is what's after the arrow of the old SHOW line 91 0:05:21,400 --> 0:05:25,000 which is Q & ~R 92 0:05:25,000 --> 0:05:27,649 The assumption in line 4 is an & 93 0:05:27,649 --> 0:05:30,933 statement, so we can do &O to it and get 94 0:05:31,040 --> 0:05:33,800 and ~S. 95 0:05:33,800 --> 0:05:38,080 Now we can do MP with 1 and 6 to get Q 96 0:05:38,080 --> 0:05:42,280 and MT with 2 and 7 to get ~R 97 0:05:42,280 --> 0:05:46,600 and then put those together to get Q&~R. 98 0:05:46,600 --> 0:05:49,311 That's what we were trying to show, 99 0:05:49,311 --> 0:05:52,022 so we cancel out the SHOW line for Q & 100 0:05:52,022 --> 0:05:54,079 ~R as a direct derivation. 101 0:05:54,080 --> 0:05:57,370 And having shown that, we get to cancel out the ... 102 0:05:57,370 --> 0:06:00,919 original SHOW line as a conditional derivation. 103 0:06:00,920 --> 0:06:03,763 Notice that, when we're looking for inputs ... 104 0:06:03,763 --> 0:06:07,079 to our rules, we can use the assumption we've made. 105 0:06:07,080 --> 0:06:10,455 But we can't use the SHOW lines we've written down ... 106 0:06:10,455 --> 0:06:14,407 before we've cancelled them. You can't use line 3 or line 5 ... 107 0:06:14,407 --> 0:06:18,029 as inputs to rules when the SHOW lines aren't crossed ... 108 0:06:18,029 --> 0:06:20,416 out, because those are things that ... 109 0:06:20,416 --> 0:06:22,556 we haven't shown to be true yet. 110 0:06:22,560 --> 0:06:26,218 We can even have a conditional derivation ... 111 0:06:26,218 --> 0:06:29,039 inception, with CDs within CDs. 112 0:06:29,040 --> 0:06:32,800 Let's look at this premise and conclusion. 113 0:06:32,800 --> 0:06:34,791 The SHOW: line is an arrow, 114 0:06:34,791 --> 0:06:38,140 so we set up a CD by assuming what's in front of ... 115 0:06:38,140 --> 0:06:39,588 the arrow, which is 116 0:06:39,680 --> 0:06:42,631 and showing what's after the arrow, 117 0:06:42,631 --> 0:06:45,779 which is Q -> R. But now our new SHOW line ... 118 0:06:45,779 --> 0:06:48,238 itself is an arrow statement! 119 0:06:48,240 --> 0:06:52,368 So we set up a conditional derivation for it by assuming ... 120 0:06:52,368 --> 0:06:55,150 what's in front of the arrow, which is 121 0:06:55,240 --> 0:07:01,000 and making a new SHOW line with what's after the arrow, which is R. 122 0:07:01,000 --> 0:07:04,807 Now we're trying to SHOW R. Notice that each new SHOW ... 123 0:07:04,807 --> 0:07:08,083 statement gets a new vertical line under it, 124 0:07:08,083 --> 0:07:10,473 so we have three vertical lines. 125 0:07:10,473 --> 0:07:12,509 We do &I to put P&Q together. 126 0:07:12,509 --> 0:07:15,696 We're allowed to use lines 3 and 5 as inputs, 127 0:07:15,696 --> 0:07:17,998 because they're assumptions. 128 0:07:18,000 --> 0:07:22,560 Then we do MP with line 1 to get R. 129 0:07:22,560 --> 0:07:25,242 Now we have a crossout cascade. 130 0:07:25,242 --> 0:07:29,575 We showed R directly, so we cross out that show line ... 131 0:07:29,575 --> 0:07:31,638 as a direct derivation. 132 0:07:31,640 --> 0:07:33,883 We assumed Q and showed R, 133 0:07:33,883 --> 0:07:39,759 so we cross out the show line for Q -> R as a conditional derivation. 134 0:07:39,760 --> 0:07:42,009 We assumed P and showed Q->R, 135 0:07:42,009 --> 0:07:46,602 so we cross out the show line for P->(Q->R) as a conditional ... 136 0:07:46,602 --> 0:07:49,789 derivation. Since that was our original ... 137 0:07:49,789 --> 0:07:52,038 SHOW line, the proof is done. 138 0:07:52,040 --> 0:07:55,920 To sum up how we do CD: 139 0:07:55,920 --> 0:08:01,080 We do CD when our SHOW line is an arrow statement. 140 0:08:01,080 --> 0:08:03,458 This only applies to SHOW lines! 141 0:08:03,458 --> 0:08:06,981 If you have a premise that's an arrow statement, 142 0:08:06,981 --> 0:08:09,359 that doesn't mean you can do a CD. 143 0:08:09,360 --> 0:08:12,412 To set up the CD, you assume what's before the ... 144 0:08:12,412 --> 0:08:15,629 arrow of the SHOW line, and your new SHOW line is ... 145 0:08:15,629 --> 0:08:17,279 what's after the arrow. 146 0:08:17,280 --> 0:08:20,800 You only have to look at the SHOW line to figure out how ... 147 0:08:20,800 --> 0:08:24,000 to set up the proof. The premises don't play into ... 148 0:08:24,000 --> 0:08:25,920 this part of the proof at all! 149 0:08:25,920 --> 0:08:28,955 If the new SHOW line is an arrow statement, 150 0:08:28,955 --> 0:08:32,164 set up a new CD. You keep repeating this until ... 151 0:08:32,164 --> 0:08:35,806 you have a SHOW line that isn't an arrow statement. 152 0:08:35,806 --> 0:08:39,448 Then you do rules until you get this last SHOW line. 153 0:08:39,448 --> 0:08:42,570 And when you get it, you can cross out all the ... 154 0:08:42,570 --> 0:08:43,957 SHOW lines at once.