Today we're going to talk about a second way in which you can refute an argument.
The second way of refuting an argument is something that we would call reductio ad
absurdum. What is reductio ad absurdum? Let me give
you an example. Contrary to what most people believe, it
is actually impossible for a person to walk from 1 location to another location.
Let me explain why. Imagine that this is the distance between
any two locations. Now suppose a person starts at this
location right here and they want to walk over to the location over here.
How are they going to do that? Well in order to do that they first have
to traverse half of that distance. But, in order to traverse half of that
distance. They first have to traverse half of that
distance or a quarter of the total. But in order to do that, they first have
to traverse half of that distance, or an eighth of the total.
And in order to do that they first have to transverse half of that distance or a
16th of the total. Now notice, this process could go on and
on indefinitely before they can pass through any distance at all.
They have to pass through half that distance, and that goes on forever, so no
one can ever get started. Since no one can ever get started, no one
can ever traverse any distance at all, and so, it's impossible, for any person,
to walk from any one location, to any other location.
Quid est demonstrandum. What you just witnessed, was an example,
of a reducteo ad absurdam. I was giving an argument for a particular
conclusion. The conclusion that it's impossible, for
any person to walk from one location to another location.
And then, someone proved, simply by walking from one location to another
location, that the conclusion of my argument was false.
The conclusion of my argument had to be false since someone actually did walk
from one location to another location. In a reductio ad absurdum you refute
someones argument by showing that the conclusion of that argument is false.
But what does that tell you that the conclusion of someone's argument is
false. Well it could tell you one of two things.
Either the argument is not valid or if the argument is valid then 1 or more of
the premises must be false. Here's why.
If the argument. What that means is that there's no
possible way for all of the premises of that argument to be true while the
conclusion is false. But if the conclusion is false, and the
argument is valid than that means that there's no possible way for all of the
premises of that argument to be true. So one or more of the premises.
Must be false. Now in the argument that I was giving a
moment ago for the false conclusion that it's impossible for any person to walk
from one location to another location, is it clear that one or more of the premises
I was using was false? Well, no, it's not.
Because I wasn't entirely explicit about what premises I was using.
I was sneaky about that. So until I make explicit what premises I
was using just the fact that you can do a reductio ad absurdum of my argument
doesn't tell you whether any of the premises I was using was false.
Maybe all of the premises were true, but the argument wasn't valid.
The argument afforded some way of having all the premises be true, even when the
conclusion is false. So, until we see the argument laid out
more carefully, we can't tell from the fact that the conclusion of the argument
is false, whether that shows that one or more of the premises of the argument is
false. One of the useful things about laying out
arguments carefully, is that then, when you find out that the conclusion is
false, that can tell you what lessons you should learn.
About the truth or false hood of the premises.
Let's try to be more careful in spelling out the argument that I gave for the
conclusion that it's impossible for anyone to traverse any distance by
walking. So, one way of spelling out that argument
is like this, premise, there is no distance.
That can be traversed first before traversing any other distance.
And it might seem that from that premise I concluded there is no distance that can
be traversed at all. Now, stated that way the argument seems
clearly invalid. So, from the fact that the conclusion is
false. We don't learn anything, about the truth
or falsehood, of the premise. The premise could be true, or it could be
false. In fact, as it happens, the premise is
true. There is no distance that can be
traversed, before traversing any other distance.
because no matter what distance you traverse, you first have to traverse half
of it. Alright, so the premise is true.
But the conclusion is false, and that shows that the argument is invalid.
So the fact that someone was able to do a reductio ad absurdem of the conclusion of
this argument doesn't tell us anything about the truth or falsehood of the
premise that I used in this argument. But there are other arguments that are
valid And so when someone is able to do a reductio ad absurdum of their conclusion,
that does show that one or more of the premises is false.
Let me give you an example of those. Suppose someone says to you, I am lying
to you right now. Now, you might make the following
argument. Premise.
Every statement is either true or false. And from that generalization.
It follows that the particular statement that you just heard is either.
True or false. But the particular statement that you
just heard is the statement, I am lying to you right now.
Now, this is a valid argument. The argument from the premise 1 to the
conclusion 2. But is the conclusion 2 true? Well
consider again the statement you just heard: I'm lying to you right now.
Is that statement either true or false? Well, suppose it's true.
So when the person said, I'm lying to you right now, what they were saying to you
Was true, but if what they were saying to you was true, then they weren't lying to
you right now. But if they weren't lying to you right
now, then what they were saying to you, was false.
But if what they were saying to you was false.
Then the rule line to you right now and so what they were saying to you was true.
So if that statement is true then it must be false, and if its false then it must
be true So, it can't be either true or false, because if it's one, it has to be
the other as well, and that's impossible. A statement can't be both true and false.
So, their statement, I am lying to you right now, can't be either true or false.
It's got to be neither. But if their statement, I am lying to you
right now is neither true nor false, then the conclusion of this argument right
here conclusion two is false and, if conclusion two is false, then since the
argument from one to two is a valid argument, the premise one Must also be
false because there's no way for a valid argument to be one in which the premise
is false and the conclusion true. So premise 1 is false and we just proved
it's false by doing the reductio ad adsurdum on the conclusion 2.
Now so far in this lecture we've looked at two different reductio ad absurdum
arguments. The first one was very simple.
It was a reductio ad absurdum that consisted in someone's walking from one
point to another, thereby proving That the conclusion of the argument that I was
giving was false. They proved it just by walking from one
point to another. That's all they needed to do to prove it.
But the other reductio ad absurdum argument, that we just considered a
moment ago. Was much more elaborate.
It consisted in an argument to the effect that, if a statement that you just heard,
I'm lying to you right now, is true, then it must be false.
And if it's false, then it must be true. So it can't be either true or false,
because if it were either one, it would have to.
To be both, and that's impossible. Now that's a more complicated argument.
That isn't as simple as just walking from one place to another.
So sometimes giving a reductio ad absurdem argument can be very simple.
It can be as simple as walking from one place to another.
Other times, it can be very complicated. It can involve lots of steps Also,
Reductio ad absurdum arguments sometimes don't tell us anything about the truth of
the premises of the arguments, whose conclusions they prove to be false.
That was the case in our first argument. We didn't learn anything about the truth
of the premises I was using in my argument, because the argument, itself,
wasn't valid. All we learned was that the conclusion,
itself, was false. But we didn't learn whether the premises
were false. But sometimes reductio out of certain
arguments can teach us that the premises of the argument whose conclusion was
proven to be false are themselves false. That's a case where we can prove that the
conclusion of a valid argument is false. If the argument is valid, then a reductio
of its conclusion. Discredits its premises because there's
no way that the premises could be true if the conclusion is false and the argument
is valid. So reductio ad absurdum arguments come in
different kinds. Some are simple, some are complicated,
some tell us something about the truth of the premises of the argument that they're
targeting and some don't. But in all cases, reductio ad absurdum,
can be an effective tool for refuting someone's argument, for showing that
someone's argument doesn't succeed, doesn't succeed in demonstrating the
truth of its conclusion. In the next lecture, we'll talk about
straw men and false dichotomies. These are pitfalls.
That can befall any attempt at reductio ad abusrdum.
And these are also ways of refuting someone else's argument.
We'll talk about those next time.