1 0:00:00,000 --> 0:00:03,271 In this video I want to talk about an advanced technique ... 2 0:00:03,271 --> 0:00:06,613 using new SHOW lines that you don't get from a particular ... 3 0:00:06,613 --> 0:00:09,457 derivation rule. I want to start by emphasizing ... 4 0:00:09,457 --> 0:00:12,230 that I'm not going to test you on this material; 5 0:00:12,230 --> 0:00:15,998 if you find it confusing, concentrate on the earlier material. 6 0:00:16,000 --> 0:00:18,180 I call this spontaneous SHOW lines. 7 0:00:18,180 --> 0:00:20,142 Usually, when we write down a new ... 8 0:00:20,142 --> 0:00:22,686 SHOW line, it's because we're setting up a ... 9 0:00:22,686 --> 0:00:26,175 particular derivation strategy--showing what's after ... 10 0:00:26,175 --> 0:00:28,646 the arrow for a conditional derivation, 11 0:00:28,646 --> 0:00:31,917 showing a contradiction for an indirect derivation. 12 0:00:31,920 --> 0:00:34,786 But you can actually make a new SHOW line anytime you ... 13 0:00:34,786 --> 0:00:36,252 want, if you need a certain ... 14 0:00:36,252 --> 0:00:37,718 statement for your proof. 15 0:00:37,720 --> 0:00:42,200 Then you draw a vertical line under it where you show the statement 16 0:00:42,200 --> 0:00:44,137 and when you've shown the statement, 17 0:00:44,137 --> 0:00:47,199 you cross out the SHOW line and end the vertical proof line. 18 0:00:47,200 --> 0:00:50,144 Now that the spontaneous SHOW line is cancelled, 19 0:00:50,144 --> 0:00:53,879 you can use the statement on it as an input to other statements! 20 0:00:53,880 --> 0:00:56,603 Crossing out one of these spontaneous SHOW lines ... 21 0:00:56,603 --> 0:00:58,861 does *not* let you cross out any previous ... 22 0:00:58,861 --> 0:01:01,850 SHOW lines you've written down--that's only for SHOW ... 23 0:01:01,850 --> 0:01:04,706 lines that come from particular derivation rules. 24 0:01:04,706 --> 0:01:06,499 And once you finish a proof line, 25 0:01:06,499 --> 0:01:09,753 you're not allowed to use any of the statements within that ... 26 0:01:09,753 --> 0:01:11,812 proof line. The point of writing SHOW ... 27 0:01:11,812 --> 0:01:15,066 and drawing the vertical proof line is to get a statement to ... 28 0:01:15,066 --> 0:01:15,995 use in the proof. 29 0:01:16,000 --> 0:01:18,890 One reason to do spontaneous SHOWS is if ... 30 0:01:18,890 --> 0:01:22,919 we're trying to show an & statement and we need to use ID. 31 0:01:22,920 --> 0:01:26,318 It's usually difficult to get an & statement directly by ID; 32 0:01:26,318 --> 0:01:27,949 you'd be assuming a negated & 33 0:01:27,949 --> 0:01:30,056 statement, which turns into an arrow ... 34 0:01:30,056 --> 0:01:32,299 statement, which doesn't give you that ... 35 0:01:32,299 --> 0:01:33,318 much to work with. 36 0:01:33,320 --> 0:01:36,755 A better technique is to put one side of the & 37 0:01:36,755 --> 0:01:38,759 statement on a SHOW line. 38 0:01:38,760 --> 0:01:43,880 Then you should be able to use that SHOW line to set up an ID. 39 0:01:43,880 --> 0:01:46,841 Once you've finished that, you might have to repeat the ... 40 0:01:46,841 --> 0:01:49,351 technique with the other side of the SHOW line, 41 0:01:49,351 --> 0:01:52,248 or you might be able to use the statement you just got to ... 42 0:01:52,248 --> 0:01:53,278 get the other side. 43 0:01:53,280 --> 0:01:56,130 This can also work if you're showing an & 44 0:01:56,130 --> 0:01:59,239 statement where one half of the & is an arrow 45 0:01:59,240 --> 0:02:02,838 You write down the arrow statement as a SHOW line, 46 0:02:02,838 --> 0:02:05,119 and then you can use CD to get it. 47 0:02:05,120 --> 0:02:08,713 Here's an example of how to use ID to get half of an & 48 0:02:08,713 --> 0:02:11,079 statement that you need to show. 49 0:02:11,080 --> 0:02:16,440 Here's our premises, and we need to SHOW P & Q. 50 0:02:16,440 --> 0:02:20,160 So we start by SHOWING P. 51 0:02:20,160 --> 0:02:26,080 Now we set up an indirect derivation to SHOW P; assume ~P 52 0:02:26,080 --> 0:02:29,280 and SHOW a contradiction. Note that we started a new ... 53 0:02:29,280 --> 0:02:32,480 vertical line right under the SHOW statement for P. 54 0:02:32,480 --> 0:02:36,061 Now we start another vertical line right under the SHOW ... 55 0:02:36,061 --> 0:02:38,318 statement for the contradiction; 56 0:02:38,318 --> 0:02:41,665 we're going to show the contradiction on this line. 57 0:02:41,665 --> 0:02:44,078 We start by getting Q by vO from 1 and 5. 58 0:02:44,080 --> 0:02:47,039 Now we need to do something with the double arrow. 59 0:02:47,039 --> 0:02:49,349 Since we have Q, which is one side of the ... 60 0:02:49,349 --> 0:02:51,586 double arrow, we make sure that Q is in ... 61 0:02:51,586 --> 0:02:53,318 front of the arrow statement 62 0:02:53,320 --> 0:02:57,240 so we can do MP to get P. 63 0:02:57,240 --> 0:03:01,520 This gives us a contradiction with P and ~P. 64 0:03:01,520 --> 0:03:04,920 So we've shown the contradiction 65 0:03:04,920 --> 0:03:08,560 and since we've assumed ~P and shown a contradiction, 66 0:03:08,560 --> 0:03:11,148 we've shown P by indirect derivation. 67 0:03:11,148 --> 0:03:13,089 But we haven't shown P&Q yet! 68 0:03:13,089 --> 0:03:17,296 We don't get to cross that out; there's no derivation rule that ... 69 0:03:17,296 --> 0:03:19,642 says that if you show one half of an & 70 0:03:19,642 --> 0:03:22,797 statement, we get to cross out the whole thing. 71 0:03:22,800 --> 0:03:25,372 Now we go ahead and finish the proof, 72 0:03:25,372 --> 0:03:29,401 using the P that we've shown. Notice that we've ended the ... 73 0:03:29,401 --> 0:03:32,230 proof line under SHOW: contradiction, 74 0:03:32,230 --> 0:03:34,373 and the proof line under SHOW: 75 0:03:34,373 --> 0:03:36,430 P, because we did SHOW those. 76 0:03:36,430 --> 0:03:39,516 So those lines have served their purposes. 77 0:03:39,516 --> 0:03:43,717 And every statement on those lines is something that we got ... 78 0:03:43,717 --> 0:03:47,661 using an assumption that ~P so we can't use those lines 5 ... 79 0:03:47,661 --> 0:03:49,804 through 10 as inputs anymore. 80 0:03:49,804 --> 0:03:52,290 We can only use our premises 1 and 2, 81 0:03:52,290 --> 0:03:56,148 and the thing we showed in 4. So we start by doing double ... 82 0:03:56,148 --> 0:03:59,663 arrow out on 2 to get P -> Q. Since we have P in line 4, 83 0:03:59,663 --> 0:04:02,235 we want to get P in front of the arrow. 84 0:04:02,240 --> 0:04:06,553 So now we can do MP with 4--which we can use now, 85 0:04:06,553 --> 0:04:10,639 since we cancelled the SHOW line--to get Q. 86 0:04:10,640 --> 0:04:13,869 Since we have P in 4, and Q in 12, 87 0:04:13,869 --> 0:04:18,519 we can use &I to put them together and get P&Q 88 0:04:18,520 --> 0:04:23,200 and we've shown P&Q through direct derivation. 89 0:04:23,200 --> 0:04:24,161 So, again, 90 0:04:24,161 --> 0:04:27,046 once we cancel the SHOW line in 4, 91 0:04:27,046 --> 0:04:30,358 we can use it as an input to more rules. 92 0:04:30,360 --> 0:04:33,135 But once we've ended a vertical proof line, 93 0:04:33,135 --> 0:04:35,832 we can't use the statements on it anymore, 94 0:04:35,832 --> 0:04:39,146 since they may depend on assumptions that aren't in ... 95 0:04:39,146 --> 0:04:41,766 effect any longer. So we can't use lines 5 ... 96 0:04:41,766 --> 0:04:45,080 through 10 outside the proof lines that they're on. 97 0:04:45,080 --> 0:04:49,396 I call this the Vegas rule: what happens in a proof line stays there. 98 0:04:49,400 --> 0:04:53,267 And we couldn't cancel the SHOW in line 3 right away ... 99 0:04:53,267 --> 0:04:56,121 when we cancel the SHOW line in line 4, 100 0:04:56,121 --> 0:05:00,540 because we didn't get 4 from a derivation rule for showing ... 101 0:05:00,540 --> 0:05:02,657 3. We had to wait until we get ... 102 0:05:02,657 --> 0:05:04,866 P&Q on the proof line under 3; 103 0:05:04,866 --> 0:05:08,917 then we could cancel it out using Direct Derivation. 104 0:05:08,920 --> 0:05:14,680 We can use the spontaneous SHOW technique to show double arrows. 105 0:05:14,680 --> 0:05:18,840 You need two single arrows to get a double arrow. 106 0:05:18,840 --> 0:05:23,880 So you write down each of the single arrows you need on a SHOW line. 107 0:05:23,880 --> 0:05:28,880 And then you can use CD to SHOW each of these single arrows. 108 0:05:28,880 --> 0:05:32,080 Here's an example of that. 109 0:05:32,080 --> 0:05:36,665 We have P double arrow Q, and we want to SHOW ~P ... 110 0:05:36,665 --> 0:05:38,399 double arrow ~Q. 111 0:05:38,400 --> 0:05:43,160 We know we're going to need ~P -> ~Q 112 0:05:43,160 --> 0:05:46,785 and ~Q -> ~P, so we write those down on ... 113 0:05:46,785 --> 0:05:48,960 separate SHOW lines. 114 0:05:48,960 --> 0:05:53,166 Now we set up CD to show line 3 by assuming what's ... 115 0:05:53,166 --> 0:05:54,999 before the arrow, ~P 116 0:05:55,000 --> 0:05:59,240 and SHOWing what's after the arrow, ~Q. 117 0:05:59,240 --> 0:06:02,984 Since we have ~P, the negation of one side of the ... 118 0:06:02,984 --> 0:06:05,960 double arrow, we put P after the arrow 119 0:06:05,960 --> 0:06:09,680 and do modus tollens to get ~Q. 120 0:06:09,680 --> 0:06:12,800 We've shown ~Q 121 0:06:12,800 --> 0:06:16,734 so we've finished the conditional derivation and we ... 122 0:06:16,734 --> 0:06:18,879 can cancel the SHOW line in 3. 123 0:06:18,880 --> 0:06:24,242 Now we set up the CD for 8; we assume what's before the ... 124 0:06:24,242 --> 0:06:25,239 arrow, ~Q 125 0:06:25,240 --> 0:06:28,280 and SHOW what's after the arrow, ~P. 126 0:06:28,280 --> 0:06:30,894 We have ~Q, so we make sure Q is after ... 127 0:06:30,894 --> 0:06:33,599 the arrow when we do double arrow out 128 0:06:33,600 --> 0:06:37,920 and we can get ~P by MT. 129 0:06:37,920 --> 0:06:40,520 Now we've shown ~P 130 0:06:40,520 --> 0:06:44,294 and we can cancel out the SHOW for ~Q->~P using ... 131 0:06:44,294 --> 0:06:49,359 conditional derivation. Now we can use lines 3 and 8 as inputs 132 0:06:49,360 --> 0:06:53,050 so we put them together and get ~P double arrow ~Q, 133 0:06:53,050 --> 0:06:55,120 by the double arrow in rule. 134 0:06:55,120 --> 0:07:00,520 So we've shown ~P double arrow ~Q. 135 0:07:00,520 --> 0:07:05,240 There are more things we can do with spontaneous SHOW lines. 136 0:07:05,240 --> 0:07:08,047 For some proofs, we need to show the negation ... 137 0:07:08,047 --> 0:07:10,399 of one side of a wedge in order to do vO. 138 0:07:10,400 --> 0:07:14,200 Those particular proofs are extra tricky to deal with. 139 0:07:14,200 --> 0:07:16,908 As you can see from the examples we worked with, 140 0:07:16,908 --> 0:07:19,824 all the proof involving spontaneous SHOWs tend to ... 141 0:07:19,824 --> 0:07:21,907 be long and involved. For instance, 142 0:07:21,907 --> 0:07:24,476 when you're getting a double arrow this way, 143 0:07:24,476 --> 0:07:26,489 you have to show two single arrows, 144 0:07:26,489 --> 0:07:29,683 so the proof is twice as long. That's why I'm not going to ... 145 0:07:29,683 --> 0:07:32,183 test you on these; I just wanted to make sure ... 146 0:07:32,183 --> 0:07:34,196 that you had the chance to see them.