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Last time we ended with a bit of mystery. 
We concluded that, fish might still be 

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necessary and not sufficient for death, 
and red wine might be necessary but it's 

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certainly not sufficient for death. 
And the question is well, if neither of 

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those is sufficient, what is sufficient. 
To answer that question, we've gotta look 

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at complex conditions. 
Now, complex conditions are just, 

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conditions or features or candidates, 
that are combinations of other candidates 

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or features or conditions. 
They might be, a negation, not having 

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fish. 
They might be a disjunction. 

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Having either fish or beef. 
But we're going to look at conjunctions, 

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that is having some combination of two or 
more of the food and drink at the 

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banquet. 
And we're going to understand these in 

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exactly the same way we understood the 
simple or atomic, conditions or features, 

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or candidates. 
So for example, we're going to use the 

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same definition of the sufficient 
condition. 

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F is a sufficient condition for G. 
Well what that means is that whenever F 

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is present so is G and then we're 
going to have a negative test for a 

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sufficient condition. 
This is just reminding you of couple of 

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lectures ago that X is not a sufficient 
condition of Y there is of a certain 

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target like death if there is any case 
where X is present and Y is absent. 

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And notice since we used X and Y as 
variables, you can stick anything in for 

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them. 
So, to get a conjunctive, condition, 

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we're going to simply substitute in the 
conjunction W and X. 

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So, W and X, that combined conjunctive 
candidate, is not a sufficient condition 

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for Y, if there's any case where both W 
and X are present, and Y is absent. 

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Just like in the other cases, but we're 
just using a conjunction to replace one 

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of the variables. 
And we can look at our data from before 

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to determine which conjunctions are and 
are not sufficient conditions for death. 

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So are there any conjunctions that still 
might be sufficient conditions for death. 

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What about tomato soup and chicken. 
Maybe when you have tomato soup and 

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chicken together that'll kill you. 
Nope, that can't be. 

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Why not? 
Which case rules out the conjunction of 

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tomato soup and chicken as a sufficient 
condition of death? 

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Anne does, because Ann had tomato soup 
and chicken but she didn't die. 

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What about. 
Red wine and cake. 

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There's red wine and cake together as a 
combination, as a conjunction. 

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Is that sufficient for causing death? 
Well it caused death, or it was related 

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to death, in the case of Barney. 
But is there another case where you have 

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red wine and cake without death? 
Look down the list and you'll see, it's 

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Doug. 
Doug shows us that red wine and cake are 

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not sufficient for death. 
Now, which conjunctions are not rule out 

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as sufficient conditions for death? 
Wow what about fish and red wine? 

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Could that be sufficient? 
Well Barney has fish and red wine and he 

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dies. 
And Emily has fish and red wine and she 

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dies. 
And Fred has fish and red wine and he 

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dies. 
And there's nobody else that has fish and 

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red wine who doesn't die.'Cause there's 
nobody else who has fish and red wine 

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other than those three. 
So it looks like fish and red wine might 

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be a sufficient condition for death. 
Okay what about fish and cake? 

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Could that be sufficient? 
Yeah so far that could be sufficient as 

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well. 
Fish and cake. 

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Barney has them and dies. 
Fish and cake. 

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Fred has them and dies. 
So fish and cake might be a sufficient 

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condition, of death, 
according to the data that we have so 

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far. 
So let's look next in what's necessary, 

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because the same points are going to 
apply. 

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We're going to take the definition of a 
necessary condition. 

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F is a necessary condition for g, if 
whenever, f is absent, g is absent. 

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We're going to use the same old negative 
test of a necessary condition. 

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X is not a necessary condition of y. 
If there's any case where x is absent but 

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y is present, because that shows that x 
is not necessary for y. 

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And all we have to do is to substitute a 
conjunction in for the variables X and Y, 

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in order to apply this test to 
conjunctive cases. 

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So, a conjunction W end X is not 
necessary for Y. 

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If there's any case, where you have that 
conjunction both w and x. 

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As being absent and yet y is present. 
So now we have a negative test for 

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conjunctive necessary conditions. 
Let's go back to our data. 

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Is leek soup and fish necessary for 
death? 

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No. Well, who shows that? 
Barney, because Barney dies without 

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having leek soup and fish. 
Is there any other case that shows that? 

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Yeah, Emily shows that, because she died 
without having leek soup and fish. 

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So that combination is not necessary for 
death. 

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Okay, so Barney and Emily, show that leek 
soup and fish is not necessary for death. 

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What about tomato soup and fish? 
They both had tomato soup and fish, so 

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maybe tomato soup and fish is necessary 
for death? 

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Because that work, no because if Fred, 
Fred dies without having tomato soup and 

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fish because he has leek soup and fish. 
So Fred shows up a combination tomato 

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soup and fish is not necessary for death. 
Because you can die without that 

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combination if you have leek soup and 
fish instead. 

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So neither of those combinations is 
necessary. 

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What might be necessary? 
Which combination is not ruled out as 

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necessary? 
Well, here is one. 

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They should red want. 
Fish and red wine might be necessary for 

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death, because everybody who died at this 
banquet had fish and red wine. 

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There was nobody who died without having 
fish and red wine, so fish and red wine 

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is not ruled out as a necessary condition 
of death. 

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The wait a minute. 
Fish and red wine was also not ruled out 

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as a sufficient condition for death, 
because everybody who had fish and red 

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wine together, died. 
That means that this particular 

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combination, fish and red wine, is both 
necessary and also sufficient for death, 

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at least given the data that we have so 
far. 

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I won't go through developing the 
positive tests for conjunctions because 

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its going to be just like the positive 
necessary condition test at the positive 

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sufficient condition test that we went 
through in previous lectures you just 

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substitute conjunctions for the very well 
set you get those test. 

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But it looks like for the data so far 
we've got at least some reason to believe 

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that fish and red wine is necessary for 
death and also sufficient for death. 

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Great, now we know which combination 
meets those tests, but what does that 

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tell us. 
We don't know any mechanism. 

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Right? 
Why would fish and red wine cause death. 

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Well obviously these people die to bet 
taste, the chef could put all afternoon 

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on the fish, kept stand effective people 
might eat his beautiful fish and have red 

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wine which is going to destroy the taste 
of the dish. 

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So anybody order the red wine with the 
fish poisoned him. 

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That would be the mechanism for why 
everybody who had fish with red wine 

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died. 
You know, there might be some other story 

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about how some chemical in the fish 
interacted with the red wine. 

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But if you can rule that out from 
background knowledge we've got a pretty 

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good reason to belief that somebody back 
there in the kitchen probably the chef 

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was the one who was mad at them, poisoned 
them. 

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They died of bad taste. 
Now we need more research to be sure. 

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We need lots more candidates. 
We need to test various combinations, 

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but that's just an inductive argument. 
Inductive arguments never make you 

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absolutely certain that the conclusion is 
true. 

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they're defeasible. 
They're not valid. 

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We know all that. 
But at least we have some evidence to 

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believe that fish and red wine is 
necessary and sufficient for the death. 

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That's what caused these people to die. 
And it's the chef, who ought to go to 

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jail. 
We've gone through one example in some 

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detail but like many other things in this 
course, the best way to learn this 

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material is to practice, practice, 
practice. 

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So let me give you a little hint. 
You can create all the exercises you 

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would ever want just by taking the chart 
that we've looked at in the last couple 

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of lectures that goes from Ann through 
Harold. 

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And change what each of them had in the 
soup course, or what each of them had in 

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the main course, or what each of them had 
in the line course, or what each of them 

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had in the dessert course. 
Or changed which ones lived and which 

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ones died. 
And you can create more examples to 

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practice the necessary condition tests 
and the sufficient condition test. 

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Hey, if you want to know whether you got 
it right, bring your chart and your 

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answer to the discussion forums, 
and the other students in the course can 

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help you out by telling you whether they 
agree about what's necessary and 

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sufficient, in the setup that you 
created. 

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So if all you students out there in 
coursairea land, practice, practice, 

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practice by changing the banquet to 
change your own specifications. 

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The you'll all learn better, how to 
distinguish necessary and sufficient 

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conditions and how to test for necessary 
and sufficient conditions. 

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So go off and have some fun with it