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 Krueger. And the argument is for the conclusion
that point nine repeating is exactly equal to one.
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 .9 repeating.
We're going to call that x. Now, that's just a stipulation.
We're just naming that. A 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 we're just going to stipulate that x
is going to be our name for .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 repreating
by 10? Well we move the decimal place over 1, and
we get 9.9 repeating. Now since this equation Is just the same
as the first equation multiplied by ten, it follows deductively from the first
equation that the second one is true. Since this second equation is jut 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 point nine repeating, it follows deductively that 10x is equal to
9.9 repeating. 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 subtract point 9 repeating, from
9 point 9 repeating, what do we get? We get nine.
So from these two statements right here it follows that nine 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 ten x.
That argument is sound. The second deductive argument is from
those two premises, to the conclusion that nine equals nine x.
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 nine.
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 point 9
repeating. So what follows from x equals point 9
repeating and x equals 1? Simple.
Point 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 .9 repeating is equal to 1. And so Nathaniel Kruger is right, and he
gives an example of a sound deductive argument, that can give us surprising new
knowledge. Thank you Nathaniel.