In the last lecture, we saw how the word and in English can sometimes be used to express the truth functional connective conjunction. And we defined conjunction by means of the truth table for conjunction. Now you'll notice in the last version of the truth table that was presented in the last lecture. The propositions that were conjoined in that truth table were called p and q. Now you might wonder, what's up with calling propositions p and q? Why are we using letters to refer to propositions? Is this some kind of strange algebra? Well today's lecture answers that question. Today's lecture is about variables like the variables that occur in algebra. And similar variables that occur in deductive logic, in the study of the rules for valid argument, which is exactly what unit two is about. So, let's begin by talking about rules and why we use variables when we state rules. The first point to make about rules is that rules are general. We're also designed to apply to many different possible cases. So, for instance, if I say to one of my kids, stop hitting your brother right now. Well that's not a rule, that's a command, right? I'm telling one of my kids, stop hitting your brother right now. But it's not a rule. It's not general. It doesn't apply to many different possible people. It doesn't apply to many different possible situations. I'm just telling someone what to do, or in this case what not to do. But if I said you should never under any circumstances hit another person, well now that's a rule. It's inherently general. It applies to many different possible situations and many different possible people. So is, you should never hit your brother. That's also a rule, because again, it applies to many different possible situations. So rules are general. But commands may or may not be general. So how can we express the generality of a rule? Well notice, if I say something that's not general. If I say to one of my kids, stop hitting your brother right now, well, I'm using the expression right now to indicate that my order, my command applies to a particular situation, namely the situation that we're in right now when I issued the order. But if I say you should never under any circumstances hit another person. I'm using the pronoun you and the expression never under any circumstances to indicate that the rule that I'm stating applies to a multitude of possible situations. In fact, the pronoun you, although it's a second person pronoun, it needn't be understood as referring merely to the person I'm addressing. It could be understood quite generally. Think about the ten commandments, thou shalt not kill. Thou is an archaic form of the second person singular, you. Or in French, tu. So, in those commandments, the second person pronoun is used not to refer to the specific person, to some specific person addressed by the commandment, but to refer to anyone at all. You, whoever you are should not kill. And notice, generality comes in degrees. And rules can be more or less general. So I could say, you should never under any circumstances hit another person. Well that's a general rule. But I could state a rule that's even more general than that. I could say, you should never under any circumstances do violence to another person. Now that's more general, because I'm not just prohibiting hitting, I'm prohibiting any kind of violence, whether it's hitting, kicking, biting, smacking, slapping. I could make a rule that's even more general than that. I could say you should never under any circumstances do something unkind to another person, whether the unkind thing in question is a form of violence or merely an insulting action, a demeaning action, a condescending action. There are lots of different ways of doing things that are unkind, even if they're not violent. So here, we have rules of different levels of generality. If I say, Walter should not force his dog to kill his cat. I'm saying something that has some generality to it. Presumably, what I'm saying doesn't simply apply to the present moment, but it applies more generally to a multitude of moments, not just the present, but maybe the recent past, maybe the near future. But of course, I can say something even more general. I could say people should not force creatures to kill innocent creatures. That's more general. I'm not just talking about Walter. I'm talking about people quite generally. I'm not just talking about Walter's dog. But I'm talking about creatures quite generally. And I'm not just talking about Walter's cat, I'm talking about innocent creatures, quite generally. And I'm saying that a certain relationship between those three things should not hold. People should not force creatures to kill innocent creatures. But now notice, that last sentence is unclear. In particular it has two interpretations, there are two ways of understanding that last sentence. We could label them the plural understanding or the distributive understanding. On the plural understanding what I'm saying is that people should not force a whole bunch of creatures to kill a bunch of innocent creatures. On the distributive interpretation, I'm saying that for any creature that you pick, people should not force that creature to kill an innocent creature. So on the first interpretation of the statement, I'm talking about groups of creatures. And on the second interpretation of the statement, I'm talking about each one of a whole bunch of particular creatures. I'm making a statement about each one of those particular creatures, whereas on the first interpretation I'm making a statement about the group of creatures. Now in ordinary language it's not easy to separate out those two interpretations. You can see just in the last minute how hard I had to work in order to distinguish those two interpretations. But if we use variables then we can distinguish the two interpretations pretty straightforwardly. We could say in order to specify the distributive interpretation of that statement, we could say where x stand for any creature at all and y stands for any innocent creature at all, people should not force x to kill y. So here we're using the variables x and y in order to express the distributive interpretation of people should not force creatures to kill innocent creatures. So variables can be useful to us in clearly expressing certain kinds of distributively interpreted rules. Now, in propositional logic variables play an equally useful role. So consider an inference that we looked at in the last lecture from the premise, I'm holding my binoculars and looking through them. It follows that I'm holding my binoculars. If it's true that I'm holding my binoculars and looking through them, if that premise is true, then the conclusion, I'm holding my binoculars, must be true. So that right there is a valid deductive argument. There's no possible way for the premise to be true without the conclusion also being true. But notice, in that deductive argument, there's nothing special about the proposition I'm holding my binoculars or about the proposition I'm looking through my binoculars. In fact for any argument where the premise is the conjunction of two propositions, and the conclusion is one of those two conjoined propositions that argument is also going to be valid, there's also going to be no possible way for the conjunction of the two propositions. That is the premise of that argument, to be true, without the conjoined proposition that is the conclusion of that argument, also being true. In other words, there's not going to be any possible way for the premise to be true while the conclusion is false. So any argument of this form right here is going to be valid. Now notice in the example I just used as well as in the truth table from last lecture, I was using Roman letters like x, or y, or P, or Q as variables. But there's nothing sacrosanct about using Roman letters as variables. I could have used anything else so long as it's easy enough to recognize and easy enough to reproduce. Here are some examples. I could have used a dot, or a percentage sign, or a plus sign, or a happy face, or a dollar sign. The problem with all of those is that while they're easy enough to produce, they're not that easy to recognize as variables in propositional logic for the simple reason that they're all used to express other things besides variables and proposition logic. So whereas Roman letters like x, y, P, Q typically aren't used by themselves to express anything at all other than variables. So we can use them safely as variables in propositional logic without running the risk that people won't recognize them for what they are. That people will think that they're expressing something else. Notice just as we have notation, Roman letters serving as notation for variables, we also have notation to express truth functions. For instance, in the last lecture I introduced the truth functional connective conjunction. And normally, we express conjunction using this ampersand right here. And there are other truth functions as well that we're going to learn about in upcoming lectures. One is disjunction which we normally express using an uppercase V. And another one is negation which we can express using either one of these two signs that looks a little bit like a minus sign, from arithmetic, but not exactly the same. These are two different ways of expressing negation. Now, again, there's nothing sacrosanct about this notation. There have been different notational systems that people have used, there is nothing better about this notational system. We just have to pick one and stick to it for the sake of consistency. So this is the one we are picking, not for any good reason, just because we have to pick one. So in upcoming lectures, you're going to learn about disjunction and about negation. And when we talk about those truth functions we're going to use the notation that you see here. See you next time.
