Last time we ended with a bit of mystery. 
We concluded that, fish might still be 
necessary and not sufficient for death, 
and red wine might be necessary but it's 
certainly not sufficient for death. 
And the question is well, if neither of 
those is sufficient, what is sufficient. 
To answer that question, we've gotta look 
at complex conditions. 
Now, complex conditions are just, 
conditions or features or candidates, 
that are combinations of other candidates 
or features or conditions. 
They might be, a negation, not having 
fish. 
They might be a disjunction. 
Having either fish or beef. 
But we're going to look at conjunctions, 
that is having some combination of two or 
more of the food and drink at the 
banquet. 
And we're going to understand these in 
exactly the same way we understood the 
simple or atomic, conditions or features, 
or candidates. 
So for example, we're going to use the 
same definition of the sufficient 
condition. 
F is a sufficient condition for G. 
Well what that means is that whenever F 
is present so is G and then we're 
going to have a negative test for a 
sufficient condition. 
This is just reminding you of couple of 
lectures ago that X is not a sufficient 
condition of Y there is of a certain 
target like death if there is any case 
where X is present and Y is absent. 
And notice since we used X and Y as 
variables, you can stick anything in for 
them. 
So, to get a conjunctive, condition, 
we're going to simply substitute in the 
conjunction W and X. 
So, W and X, that combined conjunctive 
candidate, is not a sufficient condition 
for Y, if there's any case where both W 
and X are present, and Y is absent. 
Just like in the other cases, but we're 
just using a conjunction to replace one 
of the variables. 
And we can look at our data from before 
to determine which conjunctions are and 
are not sufficient conditions for death. 
So are there any conjunctions that still 
might be sufficient conditions for death. 
What about tomato soup and chicken. 
Maybe when you have tomato soup and 
chicken together that'll kill you. 
Nope, that can't be. 
Why not? 
Which case rules out the conjunction of 
tomato soup and chicken as a sufficient 
condition of death? 
Anne does, because Ann had tomato soup 
and chicken but she didn't die. 
What about. 
Red wine and cake. 
There's red wine and cake together as a 
combination, as a conjunction. 
Is that sufficient for causing death? 
Well it caused death, or it was related 
to death, in the case of Barney. 
But is there another case where you have 
red wine and cake without death? 
Look down the list and you'll see, it's 
Doug. 
Doug shows us that red wine and cake are 
not sufficient for death. 
Now, which conjunctions are not rule out 
as sufficient conditions for death? 
Wow what about fish and red wine? 
Could that be sufficient? 
Well Barney has fish and red wine and he 
dies. 
And Emily has fish and red wine and she 
dies. 
And Fred has fish and red wine and he 
dies. 
And there's nobody else that has fish and 
red wine who doesn't die.'Cause there's 
nobody else who has fish and red wine 
other than those three. 
So it looks like fish and red wine might 
be a sufficient condition for death. 
Okay what about fish and cake? 
Could that be sufficient? 
Yeah so far that could be sufficient as 
well. 
Fish and cake. 
Barney has them and dies. 
Fish and cake. 
Fred has them and dies. 
So fish and cake might be a sufficient 
condition, of death, 
according to the data that we have so 
far. 
So let's look next in what's necessary, 
because the same points are going to 
apply. 
We're going to take the definition of a 
necessary condition. 
F is a necessary condition for g, if 
whenever, f is absent, g is absent. 
We're going to use the same old negative 
test of a necessary condition. 
X is not a necessary condition of y. 
If there's any case where x is absent but 
y is present, because that shows that x 
is not necessary for y. 
And all we have to do is to substitute a 
conjunction in for the variables X and Y, 
in order to apply this test to 
conjunctive cases. 
So, a conjunction W end X is not 
necessary for Y. 
If there's any case, where you have that 
conjunction both w and x. 
As being absent and yet y is present. 
So now we have a negative test for 
conjunctive necessary conditions. 
Let's go back to our data. 
Is leek soup and fish necessary for 
death? 
No. Well, who shows that? 
Barney, because Barney dies without 
having leek soup and fish. 
Is there any other case that shows that? 
Yeah, Emily shows that, because she died 
without having leek soup and fish. 
So that combination is not necessary for 
death. 
Okay, so Barney and Emily, show that leek 
soup and fish is not necessary for death. 
What about tomato soup and fish? 
They both had tomato soup and fish, so 
maybe tomato soup and fish is necessary 
for death? 
Because that work, no because if Fred, 
Fred dies without having tomato soup and 
fish because he has leek soup and fish. 
So Fred shows up a combination tomato 
soup and fish is not necessary for death. 
Because you can die without that 
combination if you have leek soup and 
fish instead. 
So neither of those combinations is 
necessary. 
What might be necessary? 
Which combination is not ruled out as 
necessary? 
Well, here is one. 
They should red want. 
Fish and red wine might be necessary for 
death, because everybody who died at this 
banquet had fish and red wine. 
There was nobody who died without having 
fish and red wine, so fish and red wine 
is not ruled out as a necessary condition 
of death. 
The wait a minute. 
Fish and red wine was also not ruled out 
as a sufficient condition for death, 
because everybody who had fish and red 
wine together, died. 
That means that this particular 
combination, fish and red wine, is both 
necessary and also sufficient for death, 
at least given the data that we have so 
far. 
I won't go through developing the 
positive tests for conjunctions because 
its going to be just like the positive 
necessary condition test at the positive 
sufficient condition test that we went 
through in previous lectures you just 
substitute conjunctions for the very well 
set you get those test. 
But it looks like for the data so far 
we've got at least some reason to believe 
that fish and red wine is necessary for 
death and also sufficient for death. 
Great, now we know which combination 
meets those tests, but what does that 
tell us. 
We don't know any mechanism. 
Right? 
Why would fish and red wine cause death. 
Well obviously these people die to bet 
taste, the chef could put all afternoon 
on the fish, kept stand effective people 
might eat his beautiful fish and have red 
wine which is going to destroy the taste 
of the dish. 
So anybody order the red wine with the 
fish poisoned him. 
That would be the mechanism for why 
everybody who had fish with red wine 
died. 
You know, there might be some other story 
about how some chemical in the fish 
interacted with the red wine. 
But if you can rule that out from 
background knowledge we've got a pretty 
good reason to belief that somebody back 
there in the kitchen probably the chef 
was the one who was mad at them, poisoned 
them. 
They died of bad taste. 
Now we need more research to be sure. 
We need lots more candidates. 
We need to test various combinations, 
but that's just an inductive argument. 
Inductive arguments never make you 
absolutely certain that the conclusion is 
true. 
they're defeasible. 
They're not valid. 
We know all that. 
But at least we have some evidence to 
believe that fish and red wine is 
necessary and sufficient for the death. 
That's what caused these people to die. 
And it's the chef, who ought to go to 
jail. 
We've gone through one example in some 
detail but like many other things in this 
course, the best way to learn this 
material is to practice, practice, 
practice. 
So let me give you a little hint. 
You can create all the exercises you 
would ever want just by taking the chart 
that we've looked at in the last couple 
of lectures that goes from Ann through 
Harold. 
And change what each of them had in the 
soup course, or what each of them had in 
the main course, or what each of them had 
in the line course, or what each of them 
had in the dessert course. 
Or changed which ones lived and which 
ones died. 
And you can create more examples to 
practice the necessary condition tests 
and the sufficient condition test. 
Hey, if you want to know whether you got 
it right, bring your chart and your 
answer to the discussion forums, 
and the other students in the course can 
help you out by telling you whether they 
agree about what's necessary and 
sufficient, in the setup that you 
created. 
So if all you students out there in 
coursairea land, practice, practice, 
practice by changing the banquet to 
change your own specifications. 
The you'll all learn better, how to 
distinguish necessary and sufficient 
conditions and how to test for necessary 
and sufficient conditions. 
So go off and have some fun with it