1 00:00:00,012 --> 00:00:05,745 Remember I said that deductive arguments are most common and most useful in 2 00:00:05,745 --> 00:00:11,751 disciplines where we have to reason with great precision, like mathematics or 3 00:00:11,751 --> 00:00:15,982 computer science. Now I want to consider an example of a 4 00:00:15,982 --> 00:00:19,560 sound deductive argument that was submitted. 5 00:00:19,560 --> 00:00:24,980 This one by Nathaniel Krueger. And the argument is for the conclusion 6 00:00:24,980 --> 00:00:28,977 that point nine repeating is exactly equal to one. 7 00:00:28,977 --> 00:00:35,093 One nice thing about this argument is not just that it's sound, but also that it 8 00:00:35,093 --> 00:00:41,673 shows us how sound deductive reasoning can sometimes lead us to knowledge of truths 9 00:00:41,673 --> 00:00:45,365 that we wouldn't have suspected heretofore. 10 00:00:45,365 --> 00:00:49,026 Deductive reasoning can teach us new truths. 11 00:00:49,026 --> 00:00:54,261 So, how does the argument go. Let's start off with a stipulation. 12 00:00:54,261 --> 00:00:59,331 The stipulation is going to be that we're going to call .9 repeating. 13 00:00:59,331 --> 00:01:03,886 We're going to call that x. Now, that's just a stipulation. 14 00:01:03,886 --> 00:01:08,496 We're just naming that. A number we could name it anything we want 15 00:01:08,496 --> 00:01:13,208 as long as we use that name consistently throughout our argument, we don't 16 00:01:13,208 --> 00:01:18,652 introduce any problems into our argument. Okay we're just going to stipulate that x 17 00:01:18,652 --> 00:01:23,690 is going to be our name for .9 repeating. Now, what happens if we multiply both 18 00:01:23,690 --> 00:01:29,768 sides of this equation by 10? Well if we multiply x by 10 then of course 19 00:01:29,768 --> 00:01:35,299 we get 10x. What happens if we multiply 0.9 repreating 20 00:01:35,299 --> 00:01:40,136 by 10? Well we move the decimal place over 1, and 21 00:01:40,136 --> 00:01:45,944 we get 9.9 repeating. Now since this equation Is just the same 22 00:01:45,944 --> 00:01:53,006 as the first equation multiplied by ten, it follows deductively from the first 23 00:01:53,006 --> 00:01:59,522 equation that the second one is true. Since this second equation is jut the 24 00:01:59,522 --> 00:02:05,522 first equation multiplied by 10, the argument from this premise to this 25 00:02:05,522 --> 00:02:11,459 conclusion is a valid argument. And since the stipulation can't possibly 26 00:02:11,459 --> 00:02:17,233 be false, the argument is also sound. So from the premise that x is identical, 27 00:02:17,233 --> 00:02:22,993 is equal to point nine repeating, it follows deductively that 10x is equal to 28 00:02:22,993 --> 00:02:25,236 9.9 repeating. Repeating. 29 00:02:25,236 --> 00:02:32,311 But now what happens if we subtract the first equation from the second equation? 30 00:02:32,311 --> 00:02:36,561 Well, if we subtract 1x from 10x, what do we get? 31 00:02:36,561 --> 00:02:41,212 We get 9x. And if we subtract point 9 repeating, from 32 00:02:41,212 --> 00:02:46,646 9 point 9 repeating, what do we get? We get nine. 33 00:02:46,646 --> 00:03:00,701 So from these two statements right here it follows that nine is equal to 9x. 34 00:03:00,701 --> 00:03:05,611 That follows from these two statements. There's no possible way that these two 35 00:03:05,611 --> 00:03:09,182 statements could be true while this conclusion is false. 36 00:03:09,182 --> 00:03:14,274 So this is a deductively valid argument, from these two steps to this conclusion. 37 00:03:14,274 --> 00:03:18,300 And this is a deductive, deductively valid argument right here. 38 00:03:18,300 --> 00:03:24,761 From this first step to the second step. But notice if 9 is equal to 9x. 39 00:03:24,761 --> 00:03:35,063 Then it follows once again deductively that x is equal to 1, dividing both sides 40 00:03:35,063 --> 00:03:40,292 by 9. So here we have a deductive argument, that 41 00:03:40,292 --> 00:03:46,741 actually consists Of three smaller deductive arguments. 42 00:03:46,741 --> 00:03:53,026 The first deductive argument is from the premise one that 0.9 repeating is equal to 43 00:03:53,026 --> 00:03:57,326 x to the conclusion that 9.9 repeating is equal to ten x. 44 00:03:57,326 --> 00:04:02,037 That argument is sound. The second deductive argument is from 45 00:04:02,037 --> 00:04:06,856 those two premises, to the conclusion that nine equals nine x. 46 00:04:06,856 --> 00:04:11,837 That argument is sound. And the third deductive argument is from 47 00:04:11,837 --> 00:04:16,276 that conclusion to the further conclusion that 1 equals x. 48 00:04:16,276 --> 00:04:20,766 That argument is sound. We just divided both sides by nine. 49 00:04:20,766 --> 00:04:26,687 So we have a series of three deductive arguments, all of them sound, that prove 50 00:04:26,687 --> 00:04:32,163 that x equals 1. But now remember x also equals point 9 51 00:04:32,163 --> 00:04:37,478 repeating. So what follows from x equals point 9 52 00:04:37,478 --> 00:04:41,220 repeating and x equals 1? Simple. 53 00:04:41,220 --> 00:04:46,636 Point 9 repeating equals 1. That's what follows. 54 00:04:46,636 --> 00:04:54,146 And so now, we have a deductive argument in four steps that proves as simple as day 55 00:04:54,146 --> 00:05:00,284 that .9 repeating is equal to 1. And so Nathaniel Kruger is right, and he 56 00:05:00,284 --> 00:05:06,686 gives an example of a sound deductive argument, that can give us surprising new 57 00:05:06,686 --> 00:05:09,505 knowledge. Thank you Nathaniel.