Now that we understand the distinction between sufficient conditions and
necessary conditions, we need to ask how we can tell which conditions are
necessary, and which conditions are sufficient.
So we're going to have one pair of test for sufficient conditions, one pairs of
tests for necessary conditions. For each of these we're going to have a
positive test and a negative test. So for example, the negative test for
sufficient conditions is going to tell you what's not a sufficient condition.
And the positive test for sufficient conditions will tell you what is a
sufficient conditions. One of these is going to be deductive and
the other's going to be inductive. So let's go through these various tests
one by one. First of all, you get the negative
version of a sufficient condition test, which basically says.
That something x is not a sufficient condition of something else.
Why? If there is any case where x is present
and y is absent. There don't have to be a lot of cases,
just one. If there's any one case where x is
present and y is absent. Then that shows you that x is not
sufficient for y. This test is negative because it tells
you what is not a sufficient condition for y.
It does not tell you what is a sufficient condition for y.
Basically it says if there's one case where X is present and Y is absent then
that's a counter example of the generalization that whenever X is present
Y is always present, because if Y is absent it's not present.
So, we've now got basically an deductive arguement against the generalization that
whenever X is present Y is present as well.
And because it's deducted that means it's not defeasible.
If you add other cases it doesn't matter. More information in the premises, the
argument is still going to be valid. Because as long as you have one
counterexample that shows that x is not sufficient for y because it's not the
case that whenever x is present y is present.
To see how this negative sufficient condition works, let's go through an
example. Now the example we're going to talk
about. Derives from the great comedy classic,
airplane, with Lesly Neilson. So we have a link, that you can check
out, if you want, and I warn you, there's some offensive parts of this comedy, and
you all be ready for that. Partly, to avoid those essential parts,
it also say that we can really get all the cases, straight, and specify them in
the way we need to. We are going to use our own example.
We are going to call it the banquet. So what happens, is that there are group
of people who are eating together, and they eat a full meal.
They have a soup course. They have a main course.
They have some wine. Then they have dessert.
And the meal goes pretty well until the end.
Oh, that was great. I love the tomato soup.
Yeah. This fish is to die for.
Yeah and this wine, oh my gosh it's so good.
I love the combination of both the fish and the red wine together.
Yeah, it's so sad that the dinner is over.
[SOUND] Oh man when I stood up, my stomach started hurting.
Mine did too. Ohhhh, oh man.
[SOUND] Oh, really? Yeah I think I'm dying.
[COUGH] Ohhh, [COUGH] What's happening? You okay?
Oh man what's making them all feel this way?
The question is, what caused them to die? So we checked a number of different
things, they weren't shot, they weren't stabbed.
They were poisoned. Where was the poison?
We checked their water glasses, and we checked for injection points, if somebody
shot the poison through a seringe. None of that.
So we figured the poison must have gotten to them through the food.
And now, we need to look at all the different people who were at the banquet,
and what each particular person ate at the banquet, in order to figure out which
of these different food items was the one that killed them.
So we have a little chart. Ann went to the banquet, Barney went to
the banquet, and Cathy went to the banquet.
So each of the rows on this chart indicates a different diner, a different
person who ate at the banquet. And each of the columns indicates what
they ate or drank. So one column is for the soup course.
One column's for the main course. One column's for the wine, one's for the
dessert. And then, of course, the last column to
the right tells you whether they lived or died.
And what we need to figure out is which of those food items or drinks is what
caused their death. We're going to use the letter, y, to
refer to the target feature, or the effect, the thing that is caused.
And we're going to use the letter X to refer to the different candidates for the
cause. And our question is, which candidate X is
sufficient. For, the target Y.
And the table with all the diners and all the courses, represents all of the data
that we have available to us, to answer that question.
So let's look first, at the very first course.
The soup course. What does the soup course show us?
Does it show us that anything is not sufficient for death?
Yes. Shows tomato soup is not sufficient for
death. Why?
Because Anne had tomato soup and she didn't die.
So it's not true that whenever you have tomato soup you die, since Anne had
tomato soup and did not die. So that candidate X, tomato soup, is not
sufficient for the target Y, death. Cathy shows the same thing, she shows the
tomato soup is not sufficient for death. No.
What do we learn from the main course? Do we learn that anything is not
sufficient for death? Yes.
We learned that chicken is not sufficient for death.
Because Ann has chicken and doesn't die. And what does Cathy tell us is not
sufficient for death? The case of Cathy tells us that beef is
not sufficient for death. So, now we have a whole list of things
that have been ruled out as sufficient conditions.
Tomato soup, chicken and beef. And what about the wine course?
What does Ann tell us in the wine course? Tells us that white wine is not
sufficient for death. And what does Kathy tell us in the wine
course? Red wine is not sufficient for death.
And notice that it's Ann and Kathy that are giving us all this information about
what's not sufficient, and that's because they didn't die, so anything that they
ate or drank cannot be sufficient for death.
But Barney did die. So, the things that he ate and drank
still might be sufficient for death. They're not ruled out by the negative
sufficient condition test. Of course, tomato soup was ruled out by
the other cases. And red wine was ruled out by Kathy.
But still, what about fish? Could fish be sufficient for death?
It's not ruled out, right? It's not ruled out by the negative
sufficient condition test because there's no case in this data, where somebody eats
fish and doesn't die. But does that mean that eating fish is
sufficient for death? No, cause this is just the negative
sufficient condition test. It tells you whats not sufficient, it
doesn't tell you what is sufficient, okay.
But still, you want to look at the rest of the chart and figure, is there
anything else is not ruled out sufficient condition.
Well pie is ruled out as a sufficient condition because Anne had pie, is still
alive. Kathy shows that ice cream is not a
sufficient condition for death. But cake still might be a sufficient
condition for death, for all we've read. So now we've got two candidates, fish and
cake still might be sufficient conditions for death.
They're not ruled out by the negative sufficient condition test but that
doesn't mean they are sufficient for death.
In order to conclude that they are sufficient conditions for death, we need
to apply not the negative sufficient conditions but instead the positive
sufficient condition test. The positive sufficient condition test
tells us when we have a good reason to believe that X is a sufficient condition.
What makes its positive is it tells you that you have a reason to believe that X
positively is a sufficient condition, unlike the negative test.
It says that you have a good reason to believe that X is a sufficient condition
of Y, if all of the following conditions are met.
Now I want to warn you, there are going to be four conditions and they have to
all be met in order for you to reach a positive conclusion, or be justified in
reaching a positive conclusion that X is a sufficient condition of Y.
Now, the first condition is pretty simple.
It simply says, we have not found any case where x is present, and y is absent.
Basically, it says, you've already passed the negative part of the sufficient
condition test. You can't rule out x as a sufficient
condition on the basis of any cases. The second part says that we've tested a
wide variety of cases. Including cases where x is present, and y
is absent. And the reason that we have to have this
condition, is that, without it, we could reach really silly conclusions much too
easily. For example, on the basis of the data
that we looked at before. We could reach the conclusion that pea
soup is sufficient. Because there wasn't a single person who
had pea soup and didn't die. There can't be a case where somebody had
pea soup and didn't die, if there isn't a case where somebody had pea soup.
That's why we have to check cases where X is present.
Similarly, imagine that the only case that we knew was Barney.
Right? And that means that nobody lived.
Everybody died. Well, if everybody dies then we don't
have any cases where somebody had tomato soup and didn't die, or where somebody
had fish and didn't die or where somebody had beef and didn't die or where somebody
had cake and didn't die. Whatever they had they died because
everybody died. So if everybody dies, it's really hard to
tell which of the different candidates is the one that's really causing the death.
So that's why we have to have some cases where X is present, that is where the
candidate that we're testing is present. And other cases where the target feature,
Y, is absent. And that's what the second part of the
positive sufficient condition test tells you that you need.
Those two clauses are pretty tricky, but the next ones more important and its also
a little bit trickier. What it says is that, if there are any
other features that are never present or y as absent, then we've tested cases
where those other features are absent but x is present.
this can be confusing but the basic idea is that if there are two competing
hypothesis, two competing candidates for sufficient condition, and neither one is
ruled out by the negative sufficient condition test, then need to look at
cases that will decide between those hypothesis.
Now in our example we saw that neither fish nor cake is ruled out as the
sufficient condition. And if we want to decide whether it's
fish or cake, this really is a sufficient condition.
Then what we need to look at is a case where fish is present, but cake is not.
And a case where cake is present, but fish is not.
We can't be sure from the first three cases because we don't have that
combination of fish and cake. So we have to do a little more research.
And luckily for us, not for them of course because one of them dies.
So luckily for us Doug and Emily were also at the banquet.
And Doug had tomato soup, beef, red wine and cake and lived.
Emily had tomato soup, fish, red wine, pie and poor Emily died.
The crucial thing here is that Doug had cake but not fish, and Emily had fish but
not cake. Now, did these cases help us determine
whether cake is sufficient for death? Yes, because one of these new cases rules
out cake as a sufficient condition of death.
Which one does that? Well it's Doug, because by the negative
sufficient condition test, Doug had cake and didn't die, so that shows you the
cake is not sufficient for death. Now does Emily rule out anything as a
sufficient condition for death? No.
Why not? Well, because she died.
And if somebody dies they can't rule out something that's sufficient condition for
death. Cases of death can't rule out what's
sufficient for death, because to rule it out, you need a case where the candidates
present and the targets not present. Okay?
So the cases that rule out something as a sufficient condition of death are going
to be the cases without death. Anyway, fish still might be a sufficient
condition for death. So fish seems to be the only remaining
candidate because we ruled out cake. So fish is the only remaining candidate
for a sufficient condition, at least among those on the list.
Now, we can conclude fish is a sufficient condition for death.
Right? No, because remember, I told you, I
warned you, there could be four conditions.
We need all those conditions to be met in order for us to be, reasonably conclude
that something sufficient in this one more to come.
But why do we need more? The answer really is that.
We don't know whether all the different features that might of caused the death
are on our list. We've only looked at the soup course, the
main course, the wine and the dessert. It might be something else.
So we need to add one more clause. This last positive clause to our positive
sufficient condition test. Namely that we've tested enough cases of
various kinds that are likely to include a case where x is present and y is absent
if there is any such case. So this cause is obviously going to be
hard to apply. Right, it's not mechanical.
How do we know that whether we've tested enough cases of various kinds?
We have to know what kinds of things can cause death, and which kinds of things
that can't cause death. That's going to depend on background
conditions, background knowledge, right? We need to know something about potential
causes of death, what can and cannot cause death.
And we need to take that for granted in applying this last cause, so it's not
going to be simple at all. But if we do know that there has to be
some sufficient condition. I mean after all, people just don't die
for no reason. And if we know that nothing else could be
a sufficient condition, because maybe we checked the water and we looked for
syringe marks and there were no bullet holes and so on.
Then we have to have at least some reason to believe that the sufficient condition
must be somewhere among the features that we're testing.
Some people might say it's something else.
After all, could be, I've got an idea. It could be the fact that both Barney and
Emily both have the letter E in their name.
But, you know that's not going to cause their death.
That's just common sense, background knowledge.
We looked at their glasses. There was no poison in their glasses.
So, we have at least some reason to believe that the sufficient condition
must lie somewhere among the features that we're testing.
Now, this argument is not deductive. If something fails in negative sufficient
condition test, then it's valid to conclude that, that candidate is not a
sufficient condition for that target. But if something passes the positive
sufficient condition test, then the argument's not valid.
It's possible still that this is not sufficient and that argument, namely an
application of the positive sufficient condition test, can be undermined by
future data. It's defeasible like all inductive
arguments. For example, we haven't mentioned Fred,
Gertrude and Harold yet. And what do they show?
Well wait a minute, Gertrude. Shows that fish is not sufficient after
all. Notice that all the data before Fred,
seem to a point of conclusion, that fish, was sufficient for death, but it just
take ones more case, go through, to show that fish is not really sufficient for
death afterall. That shows that it is an inductive
argument, and that means that its going to be diffusible.
Still, of'course, the inductive argument, can be strong, if w have enough.
Data, we've looked at enough cases. And our background knowledge really is
reliable. So, we're not saying that inductive
arguments are no good, but the point is that their defeasible.
Their going to be better when we've got more data, when our background
assumptions are more reliable. And strength then is going to be a matter
of degree and can go from slightly strong to very strong.
In this case, because of Gertrude, we know that fish is not sufficient.
So I have to think more about what still could be sufficient.
But first, let's ask what's necessary.