Another kind of inductive argument concerns causes or causal reasoning.
Well, we think about causes all the time in everyday life.
Suppose that all of a sudden, your computer screen goes black.
Wow, what made that happen? Was it the Coursera website going down or
is there some problem with your computer? Or was I just playing a trick on you?
Well, you need to know that in order to figure out whether or not you need to get
your computer fixed. Or suppose that, make a cup of coffee,
I love a good cup of coffee. [SOUND] That's disgusting.
What went wrong? What made it taste so bad?
There must be something in the coffee that made it taste bad.
But what was it? What caused that horrible taste?
Or think about a court room, a law suit. Someone is found guilty of murder.
Well, to be found guilty of murder, you have to show they are the one that caused
the death. That their action, pulling the trigger,
putting the poison in the food, whatever caused the death.
And whether you send them to prison for a long time depends on that causal
judgement. So, we got to get causal judgement
straight. The causal judgements cannot be certain.
We can't know with absolute certainty what causes what.
It's just not like that in life. That's why we need inductive arguments in
order to support these causal judgements that play such a big role in all of our
lives. Okay.
So, we need inductive arguments to support causal claims.
But how does that work? To get straight on that, we need to think
a little bit about causation. Causal claims are about individual
events. One event causes another event.
For example, my phone is off. But if I push this button on the bottom,
then it turns on. Great. And if I push this button on the
top, then it turns off. That particular event of me pushing the
button on the bottom, turned it on. And the particular event of me turning,
pushing the button on the top, turned it off.
But behind that connection between the particular events lies a general rule.
Every time I push a button on the bottom, it turns on and every time I push the
button on the top, it turns off. If it didn't hold, as a general rule, it
wouldn't be a causal relation, right?
Every time I push the button on the bottom, it turns on.
Now, not really. because look,
if I push it now, something else happens. It doesn't turn on.
I can keep pushing it. If it's already on, that won't turn it
on. So, it's not true that every time I push
the button on the bottom, it turns on. And it's not true that every time I push
the button on the top, it turns off. It might be broken, for example.
There might be something wrong with the button.
So, these generalizations behind causal reasoning only hold in certain
circumstances, in a particularly restricted set of
circumstances. And we'll have to think about that in
order to get causation straight. But first, we want to look at general
rules and how we're going to determine which general rules hold and which
general rules don't hold. So, how can we test general principles?
That's going to be the topic for this whole week.
The big question is going to be which cases in the data do we need to look at
and how do we know when we have enough data to really believe in a general
principle? Well, the first thing we need to do is to
draw an important distinction between two different types of general principles.
Sometimes, we say that one thing is a sufficient condition for another thing.
And that means basically that whenever you have the first thing, you're also
going to have the second thing. But sometimes, we're going to claim that
something is a necessary condition for the second thing.
And basically again, that means that if you don't have the first thing, you're
not going to have the second thing. And it's going to be crucial to
distinguish sufficient conditions from necessary conditions because people
confuse them all the time and we need to get them straight,
get them straight. So, let's start with a very simple
example. First, being a whale is sufficient for
being a mammal. Well, that's because every whale is a
mammal so if you got something that's a whale, you know you've got something
that's a mammal, okay?
But being a whale is necessary for being a sperm whale.
If you're not a whale, you're not a sperm whale.
You can't be a sperm whale without being a whale.
So, being a whale is necessary for being a sperm whale but being a whale is
sufficient for being a mammal. One way to illustrate this is to draw
them into circles. So, if you know that there's a large
group of mammals, that's a big circle? But within that is the set of whales,
which is smaller than the mammals. But even within the whales circle,
there's a set of sperm whales, then you know that being a whale is
sufficient for being a mammal, because everything in the whale circle is in this
mammal circle. And being a, a whale is necessary for
being a sperm whale because everything that's not in the whale circle is also
not in the sperm whale circle because the sperm whale circle is totally inside the
whale circle, okay?
Now, let's trying to formulate a definition of necessary and sufficient
conditions a little more precisely. The definition for sufficient conditions
and necessary conditions is a little different with events than it is with
features. So, we're going to give you two different
versions and you can really use the one that applies to the case in hand.
We can define a sufficient condition by saying that F is a sufficient condition
for G. If and only if, that is, just in case,
whenever an event of type F happens, then an event of type G also happens.
And to put it more appropriately for features, we can say that anything that
has the feature F will also have the feature G.
So, in our whale example, anything that's a whale is also a mammal.
Anything that has the feature of being a whale also has the feature of being a
mammal. So, being a whale is sufficient for being
a mammal. We can do the same thing with necessary
conditions, but you got to put the negations in there.
That's the only difference. F is a necessary condition for G, if and
only if, that is just in case, whenever an event of type f F does not
happen, then an event of type G also does not happen.
By putting a negation on both sides, you turn it into a necessary condition.
And for features, we can say, anything that does not have F, the feature F, does
not have the feature G." So, back to our whale example, anything that is not a
whale is not a sperm whale. So, being a whale is necessary for being
a sperm whale. And that's how our definitions work.
Now, since you took propositional logic a few years, [LAUGH] few years, a few weeks
ago with Ram it might help to think of this as equivalent to the propositional
forms. So, you can think of F is sufficient for
G, is like if F then G, or F horseshoe G. It's not really quite the same.
It's different because you need quantifiers.
But if you think of it that way, it won't hurt.
And then, to say the F is necessary for G, it's kind of like saying, if not F,
then not G. Or in symbols, not F horseshoe not G.
And thinking of it along those lines is, is again, not quite right because there
are no quantifiers, but it's close enough and it might help you understand the
distinction between sufficient conditions and necessary conditions.
The same concepts apply in causal examples but they're a little bit
trickier to apply. And the main reason is that word,
whenever because it's a quantifier and it applies to a restricted domain of
discourse. And you don't know what that means
because we haven't studied that yet. But the point is that it only holds
whenever in a certain range of cases, not just any old case.
So, to get the clear about that let's, let's look at an example.
Striking this match, [SOUND] this match right here,
right, is sufficient for lighting it.
[SOUND] So, when I strike it, I light it, okay?
And striking the match on a rough surface is necessary for lighting it because if I
strike it [SOUND] on a surface that's not rough, it doesn't light.
So now, we know that striking it is sufficient and striking on a rough
surface is necessary but wait a minute, that can't be right.
It can't be true that whenever you strike the match,
it lights because we know that when you strike it on a surface that's not rough,
it doesn't light. So, how do we get that striking [SOUND]
it is sufficient for lighting it when we know that if you strike it on the wrong
place [SOUND] it's not going to light? And the answer is that when we say
whenever we strike it, we're taking for granted that we're talking about a normal
match, maybe just this one, with the right chemicals on the end, and we're
talking about striking it, we're talking about striking it in a certain way at a
certain place. So, when we're saying whenever, we're
saying, whenever we strike it [SOUND] on that surface then, it will light.
I mean, for example, if you were to take [SOUND] the match, and you were to start
with a regular match but then dip it in the coffee,
[SOUND] that still tastes horrible, then it won't light when you strike it
because it's wet. So, in causal cases, you're always saying
that within the certain range of circumstances it will hold.
Whenever in those circumstances you strike the match, it will light and the
same point holds for necessary conditions. So,
this match, this box, rough surface, it's not lighting.
Well, why not? Because if you don't strike the match, it
won't light, right?
No, that's not right either.
because if you take another match and you light it,
then look, I can light this match without striking
it. So, it turns out that necessary
conditions and sufficient conditions in causal cases have to be understood as
holding, as a general rule within appropriate circumstances.
And that means that we have to change our definition of necessary conditions and
sufficient conditions to make that clear. To be strictly true then, we need to
qualify the definition with the phrase in normal circumstances.
And the in normal circumstances tell us how the term, whenever, applies.
So then, F is a sufficient condition for G, if and only if,
in normal circumstances, whenever G happens, G also happens.
Or, in normal circumstances, anything that's an F is also an G.
And the same hold for the necessary condition.
F is a necessary condition for G, if and only if, just in case, or that's what
this means is that in normal circumstances, whenever F does not
happen, G does not happen. Or for features,
in normal circumstances, for normal cases, anything that's not an F is also
not a G. Now, of course, what counts as normal
circumstances is going to be a tricky term that, that changes from case to
case. And this definition applies in a lot of
different kinds of cases. We already saw one case.
That's the conceptual case. So, let's look at that first.
Being a whale is sufficient for being a mammal.
Now notice, being a whale is not necessary for being a mammal because
there are lots of mammals that aren't whales,
like sea otters. And being a whale is necessary for being
a sperm whale, but it's not sufficient for being a sperm whale because there are
lots of whale that are not sperm whales, okay?
So, that's a conceptual case. Now, we saw a causal case.
Striking this match is sufficient for lighting it.
And striking this match on a rough surface is necessary for lighting it,
okay? But it also have the same distinction in
the, in other kinds of cases. Here's a moral example.
Torturing someone just for fun is sufficient for doing something wrong
because torturing somebody just for fun is always wrong.
You shouldn't do it, period or at least so, someone, my client, I think so at
least. And doing something wrong is necessary
for being punishable. And notice that torturing for fun is not
necessary for doing something wrong because you can do wrong without
torturing for fun, okay?
And doing something wrong is not sufficient for being punishable because
you ought to get a fair trial before you're punished.
If you do something wrong and didn't get a fair trial, you're not ready to be
punished yet. but notice, I've now make some
controversial claims here. Some people might say that doing
something wrong is not really necessary for being punishable.
Sometimes, we ought to punish people even when they didn't do something wrong.
So, when you get into the moral realm, it's going to be a little more
controversial and that shouldn't be surprising.
In any case, the main point here is that we're trying to understand the claims
that are being made and in particular to distinguish the claim that something is
sufficient from the claim that, that thing is necessary,
okay? Distinguishing sufficient conditions from
necessary conditions is crucial because if you mess these up, you'll make all
kinds of mistakes in arguments. So, let's do a few examples,
a few exercises in order to be sure that you understand that before we look at the
test for sufficient conditions and necessary conditions.