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Fundamentals: More About Necessary and Sufficient Conditions

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(intro music) Hi, my name is Kelley Schiffman. I'm a graduate student at Yale University, and today I'd like to talk a bit more about necessary and sufficient conditions. Now, this video builds somewhat on the discussion of necessary and sufficient conditions in my previous video on the topic. That said, let's review quickly. To say that p is a necessary condition for q is to say that q is true only if p is true. We can put this in more ordinary terms, as one of our viewers did, by saying that p is necessary for q because p being true is needed for q to be true. To say that r is a sufficient condition for s is to say that if r is true, then s is true. In other words, r is sufficient for s because r is all you need to get s. r is enough to get s. Now, there are four possible conditions of necessity and sufficiency. We can have necessity and sufficiency together. We can have necessity but not sufficiency. We can have sufficiency but not necessity. And we can have a case where we have neither necessity nor sufficiency. Let's look at an example of each. Here's an example where we have a necessary and sufficient condition for some outcome. I walk into a fast food restaurant. I place an order for some french fries, and pay for those fries. Now, my placing that order and paying for the fries is both necessary and sufficient for receiving them. Necessary, because they're not going to give me the french fries unless I order them and pay for them. And sufficient, because that's all I have to do. I just have to order them and pay for them, and I'll get them. Of course, there's a caveat here. In saying that ordering and paying for the fries is both necessary and sufficient, I'm making certain background assumptions. I'm assuming, for example, that there's not going to be a meteor hitting the restaurant in two seconds, because if that were the case, then my ordering the french fries and paying for them would not be sufficient for my receiving them, since the restaurant will be destroyed in two seconds, before I can ever receive them. Similarly, I'm assuming that the restaurant isn't giving away free fries today. If they were, then my paying for the fries would not, in fact, be necessary for receiving them. Now, there's an important lesson here, which is that any time we assess the necessary and sufficient conditions for an outcome, we make these sorts of background assumptions. We take certain things for granted. Consider, now, a condition that's necessary, but not sufficient. It's relatively easy to come up with these sorts of examples. All we have to do is come up with something that's needed for something else, but not enough for it. A moderate degree of exercise, for example, is necessary for staying healthy. It's not sufficient, because some other things are necessary - a good diet, for example. What, now, about sufficient, but not necessary, conditions? Consider my dog Teddy, who's really easy to make happy. Giving him a treat is sufficient to make him happy. It's not necessary, though, since there's lots of other ways to make him happy. Throwing a ball for him is also enough to make him happy. Or I could scratch his belly. All of these activities are sufficient for Teddy's happiness, since each of them is enough to make him happy. But none is necessary, since there are so many different ways to make him happy. Finally, let's consider conditions that are neither necessary nor sufficient. Here's an example. I want a piece of cake. So I go into the kitchen, and I mix a bunch of ingredients for the cake together in a bowl. Now, that alone is not sufficient for my getting a piece of cake, because I still have to stir those ingredients, put them in a pan, put them in the oven, bake the cake. Mixing those ingredients is also not necessary for my getting a piece of cake. Why not? Well, because I could just as easily walk down the street and buy a piece of cake. Let's turn, now, to consider the question of why necessary and sufficient conditions are important. Why is it important that we be clear on this distinction? Well, one answer is that we tend to confuse necessary and sufficient conditions all the time, sometimes with harmful results. Here's an example pulled from real life. Sarah, Tom, and Bobby are all found at the scene of a crime, along with an illegal gun. Sarah is charged with possession of the gun, and brought to trial on that charge. Her attorney argues that Sarah's innocent, because the gun actually belongs to Tom. Now, before the jurors deliberate, the judge tells them the following: "If you decide that the gun belongs to "Tom, then you must judge Sarah to be innocent." The jurors deliberate, they decide that the gun does not belong to Tom, and on that basis alone, conclude that Sarah is, in fact, guilty. That is her illegal gun. Now, if we pause, we see the jurors made a huge mistake. They confused a sufficient condition with a necessary one. If the gun belongs to Tom, that's a sufficient condition for Sarah's innocence. It's not necessary, though. Sarah could be innocent even if the gun does not belong to Tom. There could be another reason for her innocence. For example, if the gun belongs to some third person - Bobby, for example. Because the jurors confused a sufficient condition with a necessary one, they overlook this possibility, and so hastily convicted Sarah. If you're interested in this sort of mistake, I highly recommend that you check out Matthew Harris's video on affirming the consequent. Now, a very different sort of reason that we're interested in necessary and sufficient conditions is that they can help us figure out what things are. To see how this works, consider a simple case: what sort of thing is a square? We can begin to answer this question by articulating all the necessary conditions for being a square. One necessary condition is that it's a plane figure. Another is that it have four sides. A third necessary condition is that those sides are equal and straight. And finally, it's necessary that it has four right angles. When we put all these necessary conditions together, we get the sufficient condition for being a square. Indeed, we get the definition of a square: a square is a plane figure with four equal, straight sides and four right angles. Now, of course, not all concepts are as easy to get a handle on as the concept of a square. Consider the concepts of fairness, or equality, or happiness. These concepts, unlike the concept of a square, are a lot fuzzier, or at least more controversial. We disagree over what fairness is and over what happiness is, and having a good grasp on the difference between necessary and sufficient conditions allows us to start making progress toward figuring out what they are. I'll leave you, then, with the task of trying to come up with the necessary and sufficient conditions for happiness. Subtitles by the community