Another kind of inductive argument concerns causes or causal reasoning. Now, we think about causes all the time in everyday life. Suppose that all of a sudden your computer screen goes black. Whoa. What made that happen? Was it the Coursera website going down, or is there some problem with your computer, or was I just playing a trick on you? Well, you need to know that in order to figure out whether or not you need to get your computer fixed. Or suppose that, make a cup of coffee. I love a good cup of coffee. that's disgusting. What went wrong? What made it taste so bad? There must be something in the coffee that made it taste bad. But what was it? What caused that horrible taste? Or think about a court room. A lawsuit. Someone's found guilty of murder. Well, to be found guilty of murder, you have to show that they are the one that caused the death. That their action, pulling the trigger, putting the poison in the food, whatever, caused the death. And whether you send them to prison for a long time depends on that causal judgement. So we gotta get causal judgement straight. The causal judgements cannot be certain. We can't know, with absolute certainty, what causes what. It's just not like that in life. That's why we need inductive arguments, in order to support these causal judgments that play such a big role in all of our lives. Okay. So we need inductive arguments to support causal claims. But how does that work? To get straight on that, we need to think a little bit about causation. Causal claims are about individual events. One event causes another event. For example, my phone is off but if I push this button on the bottom, then it turns on. Great. And if I push this button on the top, then it turns off. That particular event of me pushing the button on the bottom turned it on and the particular event of me turning, pushing the button on the top turned it off. But behind that connection between the particular events, lies a general rule. Every time I push the button on the bottom, it turns on. And every time I push the button on the top, it turns off. If it didn't hold as a general rule, it wouldn't be a causal relation. Right? So, how can we test general principles. That's going to be the topic for this whole week. The big questions going to be, which cases in the data, do we need to look at? And, how do we know, when we have enough data to really believe in a general principle? But the first thing we need to do is to draw an important distinction between two different types of general principles. Sometimes we say that one thing is a sufficient condition for another thing. And that means, basically, that whenever you have the first thing, you're also going to have the second thing. But sometimes we're going to claim that something is a necessary condition for the second thing. And, basically, again that means that if you don't have the first thing you're not going to have the second thing. And it's going to be crucial to distinguish sufficient conditions from necessary conditions because people confuse them all the time. And we need to get them straight, get them straight. So, let's start with a very simple example. First, being a whale is sufficient for being a mammal. Well, that's because every whale is a mammal, so if you've got something that's a whale, you know you've got something that's a mammal. Okay. But being a whale is necessary for being a sperm whale. If you're not a whale, you're not a sperm whale. You can't be a sperm whale without being a whale. So being a whale is necessary for being a sperm whale, and being a whale is sufficient for being a mammal. One way to illustrate this is to draw them into circles. So, if you know that there's a large group of mammals, that's a big circle. But within that is this set of whales, which is smaller than the mammals. But even within the whale's circle, there's a set of sperm whales. Then, you know that, being a whale, is sufficient for being a mammal because everything in the whale circle is in this mammal circle. And being a whale is necessary for being a sperm whale because everything that's not in the whale circle is also not in the sperm whale circle, because the sperm whale [INAUDIBLE] circle is totally inside the whale circle. Okay? Now let's try to formulate a definition of necessary and sufficient conditions a little more precisely. The definition for sufficient conditions and necessary conditions is a little different with events than it is with features. So we're going to give you two different versions and you can really use the one that applies to the case at hand. We can define a sufficient condition by saying that F is a sufficient condition for G, if and only if just in case. Whenever an event of type F happens, then an event of type G also happens. And to put it more appropriately for features, we can say that anything that has the feature F will also have the feature G. So in our whale example, anything that's a whale is also a mammal. Anything that has the feature of being a whale, also has the feature of being a mammal. So being a whale is sufficient for being a mammal. And we can do the same thing with necessary conditions. But you gotta put the negations in there. That's the only difference. F is a necessary condition for G if and only if, that is, just in case. Whenever an event of type F does not happen, then an event of type G also does not happen. By putting the negation on both sides, you turn into a necessary condition. And for features, we can say anything that does not have [INAUDIBLE] the feature F does not have the feature G. So back to our whale example, anything that is not a whale is not a sperm whale So being a whale is necessary for being a sperm whale. And that's how our definitions work. Now, since you took Propositional Logic a few years, a few year, a few weeks ago with [UNKNOWN] it might help to think of this as equivalent to the propositional forms. So you can think of F is sufficient for G, is like F, if F then G, or F horseshoe G. It's not really quite the same. It's different because you need quantifiers, but if you think of it that way, it won't hurt. And then, to say that F is necessary for G is kind of like saying, if not F, then not G. Or in symbols, not F horseshoe not G. And thinking of it along those lines is, is again, not quite right, because there are no quantifiers. But it's close enough and it might help you understand the distinction between sufficient conditions and necessary conditions. The same concepts apply in causal examples, but they're a little bit trickier to apply. And the main reason is that word whenever, because it's a quantifier and it applies to a restricted domain of discourse. Now, you don't know what that means because we haven't studied that yet. But the point is that it only holds whenever in a certain range of cases. Not just any old case. So to get to clear about that let's, let's look at an example. Striking this match. [SOUND] This match right here, right, is sufficient for lighting it. [SOUND] So, when I strike it, I light it. Okay? And striking the match on a rough surface is necessary for lighting it because if I strike it [SOUND] on a surface that's not rough, it doesn't light. [SOUND] So now we know that striking it is sufficient and striking it on a rough surface is necessary. But wait a minute. That can't be right. It can't be true that whenever you strike the match it lights because we know that when you strike it on a surface that's not rough, it doesn't light. So how do we get that striking it is sufficient for lighting it when we know that if you strike it on the wrong place [SOUND] it's not going to light. And the answer is that when we say whenever we strike it, we're taking it for granted, that we're talking about a normal match, maybe just this one, with the right chemicals on the end. And we're talking about striking it, we're talking about striking it in a certain way, in a certain place. So when we say whenever, we're saying whenever we strike it [SOUND] on that surface, then it will light. I mean, for example, if you were to take [SOUND] the match and you were to start with a regular match, but then dip it in the coffee, that still tastes horrible, then [SOUND] it won't light when you strike it because it's wet. So in causal cases you're always saying that within a certain range of circumstances it will hold. Whenever, in those circumstances, you strike the match, it will light. And the same point holds for necessary conditions. So, this match, this box, rough surface, it's not lighting. Well, why not? Because if you don't strike the match, it won't light. Right? No, that's not right either [SOUND] because if you take another match [SOUND] and you light it, then look, I can light this match without striking it. [SOUND] So it turns out that necessary conditions and sufficient conditions in causal cases have to be understood as holding as a general rule within appropriate circumstances. And that means that we have to change our definition of necessary conditions and sufficient conditions to make that clear. To be strictly true then, we need to, qualify the definition with the phrase in normal circumstances. And the, in normal circumstances tells us how the term whenever, applies. So then, F is a sufficient condition for G, if and only if, in normal circumstances whenever F happens, G also happens, or in normal circumstances, anything that's an F is also a G. And the same holds for necessary conditions F is a necessary condition for G, if and only if, just in case, or that's what this means, is that in normal circumstances, whenever F does not happen, G does not happen Or for features, in normal circumstances for normal cases, anything that's not an F is also not a G. Now of course what counts as normal circumstances is going to be a tricky term that the changes from case to case and this definition, applies in a lot of different kinds of cases. We already saw one case. That's the conceptual case. So let's look at that first. Being a whale is sufficient for being a mammal. Now, notice being a whale is not necessary for being a mammal because there's lots of mammals that aren't whales, like sea otters. And being a whale is necessary for being a sperm whale, bu it's not sufficient for being a sperm whale because there are lots of whales that aren't sperm whales. Okay? So that's a conceptual case. Then, we saw a causal case. Striking this match is sufficient for lighting it. And, striking this match on a rough surface is necessary for lighting it. Okay? But it also has the same distinction in the, in other kinds of cases. Here's a moral example. Torturing someone just for fun is sufficient for doing something wrong. Because torturing somebody just for fun is always wrong. You shouldn't do it, period. Or, at least, so someone might claim. I think so, at least. And doing something wrong is necessary for being punishable. Now, notice that torturing for fun is not necessary for doing something wrong because you can do wrong without torturing for fun. Okay? And doing something wrong is not sufficient for being punishable because you ought to get a fair trial before you're punished. If you do something wrong and didn't get a fair trial, you're not ready to be punished yet. but notice, I've now made some controversial claims here. Some people might say that doing something wrong is not really necessary for being punishable. Sometimes, we ought to punish people even when they didn't do something wrong. So when you get into the moral realm, it's going to be a little bit more controversial, and that shouldn't be surprising. In any case, the main point here is that we try to understand the claims that are being made, and in particular to distinguish the claim that something is sufficient from the claim that the thing is necessary. Okay? Distinguishing sufficient conditions from necessary conditions is crucial because if you mess these up you'll make all kinds of mistakes and arguments. So let's do a few examples, a few exercises, in order to be sure that you understand that before we look at the tests for sufficient conditions and necessary conditions.