Last time, we ended with a bit of a mystery. We concluded that fish still might be necessary and not sufficient for death, and red wine might be necessary, but it's definitely not sufficient for death. The question is, well, if neither of those is sufficient, what is sufficient? To answer that question, we've got to look at complex conditions. Now, complex conditions are just conditions, or features, or candidates, that are combinations, of other candidates, or features or conditions. It might be a negation; not having fish. It 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 a 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 a couple of lectures ago, that x is not a sufficient condition of y, that is of a certain target, like death, if there's any case where x is present and y is absent. Now 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 will 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? Ann does, because Ann had tomato soup and chicken but she didn't die. What about red wine and cake? Is red wine and cake together, as a combination, as a conjunction, is that sufficient for causing death? Is leek soup and fish necessary for death? No. Well who shows that? Barney, because Barney died 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. Does that work? No, because of Fred. Fred dies without having tomato soup and fish because he has leek soup and fish. So Fred shows us that the 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. [SOUND] Or here's one, fish and red wine. 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. But 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 test for conjunctions, because it's going to be just like the positive necessary condition test and the positive sufficient condition test that we went through in previous lectures. You just substitute conjunctions for variables and you get those tests. But it looks like for the data so far we've got at least some reason to belief 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 died of bad taste. The chef, who worked all afternoon on the fish, can't stand the fact that people might eat his beautiful fish And have red wine which is going to destroy the taste of the dish. So anybody who ordered red wine with the fish, he poisoned them. 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 believe that somebody back there in the kitchen, probably the chef, or someone 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 defeasable. 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 that 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 wine course or what each of them had in the dessert course. Or change which ones lived and which Which ones died. And you can create more examples to practice the necessary condition test and the sufficient condition test. And 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 set up that you created. So if all you students out there in Coursera land practice, practice, practice by changing the banquet to meet your own specifications, then 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.
