. >> Remember, I said that deductive
arguments are most common and most useful in disciplines where we have to reason
with great precision, like mathematics or computer science.
Now, I want to consider an example of a sound deductive argument that was
submitted. This one by Nathaniel Kruger.
And the argument is for the conclusion that 0.9 repeating is exactly equal to 1.
One nice thing about this argument is not just that it's sound, but also that it
shows us how sound deductive reasoning can sometimes lead us to knowledge of truths
that we wouldn't have suspected heretofore.
Deductive reasoning can teach us new truths.
So, how does the argument go? Let's start off with a stipulation.
The stipulation is going to be that we're going to call 0.9 repeating, we're going
to call that x. Now, that's just a stipulation.
We're just naming that number. We could name it anything we want.
As long as we use that name consistently throughout our argument, we don't
introduce any problems into our argument. Okay, so we're just going to stipulate
that x is going to be our name for 0.9 repeating.
Now, what happens if we multiply both sides of this equation by 10?
Well, if we multiply x by 10, then of course, we get 10x.
What happens if we multiply 0.9 repeating by 10?
Well, we move the decimal place over one, and we get 9.9 repeating.
Now since this equation is just the same as the first equation multiplied by 10, it
follows deductively from the first equation that the second one is true.
Since this second equation is just the first equation multiplied by 10, the
argument from this premise to this conclusion is a valid argument.
And since the stipulation can't possibly be false, the argument is also sound.
So, from the premise that x is identical, is equal to 0.9 repeating, it follows
deductively, that 10 x is equal to 9.9 repeating.
But now, what happens if we subtract the first equation from the second equation?
Well, if we subtract 1x from 10x, what do we get?
We get 9x. And if we subtact 0.9 repeating from 9.9
repeating, what do we get? We get 9.
So, from these two statements right here, it follows that 9 is equal to 9x.
That follows from these two statements. There's no possible way that these two
statements could be true while this conclusion is false.
So this is a deductively valid argument, from these two steps to this conclusion.
And this is a deductive, deductively valid argument right here, from this first step
to the second step. But notice, if 9 is equal to 9x, then it
follows, once again deductively, that x is equal to 1, dividing both sides by 9.
So here, we have a deductive argument that actually consists of three smaller
deductive arguments. The first deductive argument is from the
premise one, that 0.9 repeating is equal to x, to the conclusion that 9.9 repeating
is equal to 10x. That argument is sound.
The second deductive argument is from those two premises to the conclusion that
9 equals 9x. That argument is sound.
And the third deductive argument is from that conclusion to the further conclusion
that 1 equals x. That argument is sound.
We just divided both sides by 9. So we have a series of three deductive
arguments, all of them sound, that prove that x equals 1.
But now, remember, x also equals 0.9 repeating.
So, what follows from x equals 0.9 repeating and x equals 1?
Simple. 0.9 repeating equals 1.
That's what follows. And so now, we have a deductive argument
in four steps that proves, as simple as day, that 0.9 repeating is equal to 1.
And so, Nathaniel Krueger is right. And he gives an example of a sound
deductive argument that can give us surprising new knowledge.
Thank you, Nathaniel.