1 00:00:00,012 --> 00:00:03,167 . >> Remember, I said that deductive 2 00:00:03,167 --> 00:00:08,661 arguments are most common and most useful in disciplines where we have to reason 3 00:00:08,661 --> 00:00:13,056 with great precision, like mathematics or computer science. 4 00:00:13,056 --> 00:00:18,402 Now, I want to consider an example of a sound deductive argument that was 5 00:00:18,402 --> 00:00:21,866 submitted. This one by Nathaniel Kruger. 6 00:00:21,866 --> 00:00:28,346 And the argument is for the conclusion that 0.9 repeating is exactly equal to 1. 7 00:00:28,346 --> 00:00:34,657 One nice thing about this argument is not just that it's sound, but also that it 8 00:00:34,657 --> 00:00:41,447 shows us how sound deductive reasoning can sometimes lead us to knowledge of truths 9 00:00:41,447 --> 00:00:45,309 that we wouldn't have suspected heretofore. 10 00:00:45,309 --> 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 --> 00:01:00,109 The stipulation is going to be that we're going to call 0.9 repeating, we're going 13 00:01:00,109 --> 00:01:03,976 to call that x. Now, that's just a stipulation. 14 00:01:03,976 --> 00:01:08,598 We're just naming that number. We could name it anything we want. 15 00:01:08,598 --> 00:01:13,062 As long as we use that name consistently throughout our argument, we don't 16 00:01:13,062 --> 00:01:18,218 introduce any problems into our argument. Okay, so we're just going to stipulate 17 00:01:18,218 --> 00:01:21,209 that x is going to be our name for 0.9 repeating. 18 00:01:21,209 --> 00:01:25,613 Now, what happens if we multiply both sides of this equation by 10? 19 00:01:25,613 --> 00:01:30,700 Well, if we multiply x by 10, then of course, we get 10x. 20 00:01:30,700 --> 00:01:35,662 What happens if we multiply 0.9 repeating by 10? 21 00:01:35,662 --> 00:01:42,543 Well, we move the decimal place over one, and we get 9.9 repeating. 22 00:01:42,543 --> 00:01:49,689 Now since this equation is just the same as the first equation multiplied by 10, it 23 00:01:49,689 --> 00:01:56,220 follows deductively from the first equation that the second one is true. 24 00:01:56,220 --> 00:02:02,589 Since this second equation is just the first equation multiplied by 10, the 25 00:02:02,589 --> 00:02:08,345 argument from this premise to this conclusion is a valid argument. 26 00:02:08,345 --> 00:02:14,066 And since the stipulation can't possibly be false, the argument is also sound. 27 00:02:14,066 --> 00:02:20,247 So, from the premise that x is identical, is equal to 0.9 repeating, it follows 28 00:02:20,247 --> 00:02:24,171 deductively, that 10 x is equal to 9.9 repeating. 29 00:02:24,171 --> 00:02:32,132 But now, what happens if we subtract the first equation from the second equation? 30 00:02:32,132 --> 00:02:36,751 Well, if we subtract 1x from 10x, what do we get? 31 00:02:36,751 --> 00:02:41,622 We get 9x. And if we subtact 0.9 repeating from 9.9 32 00:02:41,622 --> 00:02:46,576 repeating, what do we get? We get 9. 33 00:02:46,576 --> 00:03:00,687 So, from these two statements right here, it follows that 9 is equal to 9x. 34 00:03:00,687 --> 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 --> 00:03:19,534 And this is a deductive, deductively valid argument right here, from this first step 38 00:03:19,534 --> 00:03:25,654 to the second step. But notice, if 9 is equal to 9x, then it 39 00:03:25,654 --> 00:03:35,747 follows, once again deductively, that x is equal to 1, dividing both sides by 9. 40 00:03:35,747 --> 00:03:44,008 So here, we have a deductive argument that actually consists of three smaller 41 00:03:44,008 --> 00:03:49,792 deductive arguments. The first deductive argument is from the 42 00:03:49,792 --> 00:03:56,485 premise one, that 0.9 repeating is equal to x, to the conclusion that 9.9 repeating 43 00:03:56,485 --> 00:03:59,708 is equal to 10x. That argument is sound. 44 00:03:59,708 --> 00:04:05,811 The second deductive argument is from those two premises to the conclusion that 45 00:04:05,811 --> 00:04:08,811 9 equals 9x. That argument is sound. 46 00:04:08,811 --> 00:04:15,257 And the third deductive argument is from that conclusion to the further conclusion 47 00:04:15,257 --> 00:04:18,347 that 1 equals x. That argument is sound. 48 00:04:18,347 --> 00:04:23,667 We just divided both sides by 9. So we have a series of three deductive 49 00:04:23,667 --> 00:04:28,064 arguments, all of them sound, that prove that x equals 1. 50 00:04:28,065 --> 00:04:33,636 But now, remember, x also equals 0.9 repeating. 51 00:04:33,636 --> 00:04:40,466 So, what follows from x equals 0.9 repeating and x equals 1? 52 00:04:40,466 --> 00:04:44,186 Simple. 0.9 repeating equals 1. 53 00:04:44,186 --> 00:04:50,020 That's what follows. And so now, we have a deductive argument 54 00:04:50,020 --> 00:04:56,431 in four steps that proves, as simple as day, that 0.9 repeating is equal to 1. 55 00:04:56,431 --> 00:05:02,214 And so, Nathaniel Krueger is right. And he gives an example of a sound 56 00:05:02,214 --> 00:05:07,651 deductive argument that can give us surprising new knowledge. 57 00:05:07,651 --> 00:05:09,575 Thank you, Nathaniel.