Today, we're going to talk about Refutation by Parallel Reasoning.
Now, what does that mean? In order to explain what we mean by parallel
reasoning. I first have to go back and point out
something that we've already learned throughout this course.
What we've learned throughout this course is that, good arguments follow certain
rules. There are certain rules that an argument
has to follow in order to be a good argument.
Some of those rules we studied when we were studying Propositional Logic.
For instance you can validly infer a proposition, P, from its conjunction P &
Q. You can validly infer a disjunction, P or
Q, from either one of its disjuncts. You can validly infer the negation of the
negation of a proposition from that proposition itself.
So a lot of those rules we learned when we studied propositional logic.
Other rules we learned when we studied categorical logic.
For instance, if all F's are G's and no F's are G's, it follows, that there are
no F's at all. If some F's are G's, and no F's are not
G's, then it follows that all F's are G's.
So we learn something about those rules, of valid argumentation, when we studied
categorical logic. Then we learned about rules of good
argumentation concerning probabilities or concerning choices.
All of these different kinds of arguments follow rules and they need to follow
rules in order to be good arguments but, since arguments follow rules That opens
up to us, the possibility of refuting an argument, by showing that the rule that
that argument follows is not a good rule of argument.
In other words, if a particular argument follows a rule, we can show that that
particular rule is not a good rule and thereby call into question the argument
that follows that rule. Let me give you some examples to
illustrate how we could do that. So, now let me give you an example of
refutation by parallel reasoning. Consider the following argument.
Premise 1, most solar cars have motors. In fact I think that's an understatement.
I think it's true that all solar cars have motors.
But even if that's true, it's at least true that most solar cars have motors.
If all of them do, then most of them do. Premise two.
Most cars with motors need gas. Well, that's certainly true.
Most of the motorized cars in the world. At this point in time or any point in the
previous time need gas and suppose you tried to persuade me on the basis of
those 2 premesis. Well since most solar cars have motors
and most cars with motors need gas, you conclude therefore most solar cars need
gas. Now, what's wrong with that argument?
Well, one way to see what's wrong with that argument is to consider the rule
that that argument seems to follow. Right? For solar cars, [SOUND].
You think about solar cars as a particular category.
A, call it. Then there's the category of cars that
have motors. That's a second category.
And to say most solar cars have motors is an example of saying most of the things
that fall into one category, the As, also fall into a second category.
The b's. Most a's are b's.
Most solar cars have motors. But then, most of the cars with motors,
most of the b's, need gas. That's to say, they fall into a third
category. Things that need gas called that third
category, the C's. Well, if solar cars are the category of
A's and things that need gas are the category of C's.
Then we can represent the conclusion this way.
Most A's are C's. And so now we see that the rule that our
argument follows is this. Most A's are B's.
Most B's are C's. Therefore, we can conclude, most A's are
C's. Now is that rule a good rule of
argumentation? No, it's not. And I can prove that it's not a good rule
of argumentation by considering the following example that's very obviously a
bad argument. So suppose for the category of As, we use
the example of, humans. Humans.
Right here. And right here and suppose for the
category of these, we use the example animals.
Alright, cause, most humans clearly are animals.
In fact, not just most humans are animals, all humans are animals.
But if all humans are animals, then certainly most humans are animals.
[SOUND] And for the category of C, we use the example of microscopic beings.
[SOUND] See, while it's true that most of the animals that we normally think about
are animals big enough for us to see Animals that are roughly our size or
animals much larger, like a blue whale or animals much smaller than we are like
mice or rabbits or foxes. Still the animals that we see comprise a
very small portion of the animal kingdom. Most of the animal kingdom consists of
animals that are too small for us to see. They're microscopic beings.
So, most humans are animals. Well, that's certainly true.
Most animals are microscopic. That's also true.
So, on the basis of those two premises, should we then believe most humans are
microscopic? Obviously not. So this is an obviously bad argument.
But if this is a bad argument, then the rule that this follows, from most A's are
B's and most B's are C's Draw the conclusion most A's are Cs.
That rule is a bad rule of argument. In effect rule is a bad rule of argument
then our original argument concerning solar cars is also a bad argument.
Its's a bad argument becaus the rule that it follows.
Follows is a bad rule of argument. So even if the conclusion of our original
argument is true, even if it's true that most solar cars need gas, which, by the
way, it isn't, but even if that were true.
It would still be a bad argument for that conclusion.
Remember, you can have a bad argument that has a true conclusion.
And even if the conclusion of that argument were true, the argument itself
would still be a bad argument because the rule that it follows is not a good rule
of argument. As this example illustrates.
So that's an example of refutation by parallel reasoning.
We refute the original argument from most solar cars Have motors, and most
motorized cars need gas, to more solar cars need gas.
We refute that original argument by producing a parallel argument that's
obviously unsuccessful. Even though this argument is parallel,
that's to say it follows the same rule as the original argument.
Now let me give you another example of refutation by parallel reasoning.
Consider the following argument. It's similar to an argument that you
might hear for a worldwide economic stimulus program right now.
I might say, look. If I had more money, then I could buy
more stuff. Therefore, if everyone had more money,
everyone could buy more stuff. So I might conclude from that that
governments should somehow or other, maybe through tax mitigation, maybe
through rebates Somehow or other, get everyone to have more money so that
everyone can buy more stuff, so that there would be more demand for businesses
and so on. Okay.
Now, is this a good argument? Clearly, the premise is true.
If I had more money, then I could buy more stuff.
But, is the argument from the premises, from the premise to the conclusion a good
argument? That's the question. In order to answer that question, let's
consider what the form of that argument is.
You see. I could try to refute that earlier
argument by parallel reasoning as follows: consider the reasoning from the
premise. If I stand up let's say in a crowd of
people who were all watching a certain event.
Everyone in front of me is sitting down. If I stand up.
Then I'll get a better view, because I'll be able to see over their heads.
Now from that premise, can we draw the conclusion that if everyone stands up
then everyone will get a better view? Clearly not.
If everyone stands up, then no one will get a better view.
You only get a better view if you stand up while everyone else is sitting down.
If everyone stands up Then none of you gets a better view than you had before.
So from the premise that if I stand up I will get a better view, an obviously true
premise, we cannot draw the conclusion that if everyone stands up then everyone
will get a better view. But if this argument is no good.
Then it looks like our original argument is also no good because our original
argument has the same pattern as this argument.
Our original argument starts off with a premise about how things would be
different for me if I had more money. And then, it draws a conclusion to the
effect that things would be different in that same way for everyone if everyone
had more money. But again, that doesn't follow.
That doesn't follow. As this argument proves.
This argument is a refutation by parallel reasoning of our original argument.
Now let me show you a third example of refutation by parallel reasoning.
In this third example, what we're going to be doing is not refuting the
rule That an argument follows. Rather, we'll be refuting a particular
premise of the argument by means of parallel reasoning.
Let me show you what I mean. Consider the following reasoning.
Premise 1, if God had wanted us to fly He would have given us wings.
That's something I've heard people say. Premise 2, we don't have wings. So
conclusion, therefore, God did not want us to fly.
Now, there's nothing wrong with the rule that that argument follows.
The rule that that argument follows is just Modus Ponens, which is a perfectly
good argument, but there is something wrong with the first premise of that
argument, as I can indicate by considering the following parallel
reasoning. If God had wanted us to.
Swim, he would have given us, fins. We don't have fins, obviously, and
therefore, God did not want us to swim. Now, that argument seems pretty clearly
unacceptable. But, maybe you don't like swimming.
Maybe, more than not liking swimming, you think that swimming is an inappropriate
activity for human beings. So let me change the argument yet again,
and make it even more obviously unacceptable than this.
How about this? If God had wanted us to stay on the ground, He would have given
us roots. Well, we don't have roots.
And so God did not want us to stay on the ground.
Now, that Is an obvious unacceptable argument.
Why is it unacceptable? It's not unacceptable because of the rule it
follows, it's unacceptable because its first premise is obviously false.
But if this first premise is obviously false then the first premise of our
original argument if God wanted us to fly he would have given us wings.
Should be equally obviously false. So that's an example of reputation by
parallel reasoning in which what we refute is not the rule that the argument
follows, but rather one of the premises of the argument.
That concludes our lecture on Refutation by Parallel Reasoning.
Which concludes our week on Refutation. Next week, we're going to be applying
some of the skills we've learned. To consider various examples of arguments
that arise in everyday life.