Today’s lecture is about conditionals. Conditionals are a kind of truth functional connective that we haven’t discussed yet. But today we're going to introduce this new kind of truth functional connective, discuss the truth table for it, and say why it’s important. It's especially important in understanding the rules by which to assess the validity of deductive arguments. Now in order to introduce this new truth functional connective, the conditional, I want to start by telling you a story. Imagine that Walter has been breaking into my office and stealing my stuff. So, each day I come into my office, and I find that more and more of my stuff is gone. One day, he steals my electronics, and another day, he steals my coffee cup, and then another day, he steals my clothing. And then yet another day, he steals my glasses. And before long, there's very little stuff in my office. Now as this happens, I start to get suspicious, and I decide to have Walter followed by a private investigator, just to figure out exactly whether or not Walter is stealing my stuff. So I have the private investigator follow Walter everywhere he goes, right? He follows him to his house at night. Then he follows him to the bar early in the morning. He follows him to the golf course later in the morning. Then he follows him to the water polo tournament in the afternoon, follows him everywhere he goes. Now, suppose you asked me, is the private investigator having lunch at that Cuban restaurant, New Havana? Now, I might say to you, well, if Walter is having lunch there, then the private investigator is having lunch there. Okay, now notice what I just said. I said if Walter is having lunch there, then the private investigator is having lunch there. That phrase, if, then, can be used to connect two propositions, the proposition, Walter is having lunch at New Havana, with the proposition, the private investigator is having lunch in New Havana. So, that phrase if, then works as a propositional connective. It connects two propositions to make a larger proposition. But does it work as a truth functional connective? Now, I'd like to argue that it does. Okay, so how are we going to argue that if, then is not just a propositional connective, but it's also a truth functional connective? Well, I'd like to begin by considering the following truth functional construction. Consider the negation of the conjunction of P and the negation of Q. All right, I've written that out here, the negation of the conjunction of P and the negation of Q. Now suppose I say, that whole proposition is true. Well, if I say that whole proposition is true, then what I'm saying is just that the negation of that proposition, in other words, just the conjunction of P and the negation of Q, that that proposition is false. But what does it mean when I say that the conjunction of P and the negation of Q is false? Well, what I'm saying there is that if P is true, then, the negation of Q has to be false, right? Because if the conjunction of P and the negation of Q is false, then you can't have both of those conjuncts being true. At least one of them has to be false. So if P is true, then the other conjunct, the negation of Q has to be false. So you're saying there that if P is true, then the negation of Q is false. But remember, for the negation of Q to be false is just the same thing as for Q to be true because the negation of Q has just the opposite truth value of whatever Q has. So to say if P is true, then the negation of Q is false is just the same as to say if P is true, then Q is true. But that's exactly what you're doing whenever you use if, then as a propositional connective. You're saying, if one proposition is true, like let's say, Walter is eating lunch at New Havana, then another proposition is true. Like let's say, the private investigator is eating lunch at New Havana. If P, then Q, that proposition follows from the negation of the conjunction P and the negation of Q. But while if P, then Q follows from the negation of that conjunction, is it equivalent to it? Now I just argued that if P, then Q follows from the negation of the conjunction P and the negation of Q. But now what I'd like to do is argue that latter proposition, the negation of the conjunction of P and the negation of Q, that that proposition follows from if P, then Q. Consider for a moment what you're saying when you say if P, then Q. You're saying if P is true, then Q has gotta be true. So you're ruling out a certain option. You're ruling out the possibility that P is true while Q is false. When you say, if P, then Q, you're ruling out the possibility that P is true and Q is false. In other words, you were ruling out the possibility that P is true, and the negation of Q is true. In other words, you're saying P and the negation of Q, that conjunction right there has gotta be false. In other words, you're saying that the negation of the conjunction P and the negation of Q is true. So you see the negation of the conjunction P and the negation of Q follows from if P, then Q. And if P, then Q follows from the negation of the conjunction of P and the negation of Q. So what that tells us is that the proposition if P, then Q is going to be true in precisely the same situations as the proposition which is the negation of the conjunction P in the negation of Q. So if they're true in all the same situations, that just means that they're going to have the same truth table. In other words, if P, then Q is going to have a truth table. And it's going to have precisely the same truth table as the negation of the conjunction of P and the negation of Q. That's what we learn by noticing that if P, then Q follows from the negation of the conjunction of P and the negation of Q and that the negation of the conjunction of P and the negation of Q follows from if P, then Q, is that they have the same truth table. And since if P, then Q has a truth table, that proves that it's a truth functional connective. It's a propositional connective that create propositions whose truth value depends solely on the truth values of the propositions that go into it. So it's a truth functional connective. Now that we've talked about the conditional, and we've explained how the truth table for the conditional works, let's talk about some rules governing our use of the conditional. And then when we say something about why the conditional is an especially important truth functional connective. So first I want to talk about a rule called modus ponens. Modus ponens says from the premises P, whatever P is, and if P, then Q, whatever exactly P and Q are, infer the conclusion Q, that's what modus ponens says. Now notice, we can use the truth table for the conditional to show that modus ponens is a good rule of inference. Look here. So, suppose you know that P is true, and if P, then Q is true. So, what does that tell you? Well, since P is true, we've gotta be in one of these first two scenarios, one of the two top rows of the truth table. And since the conditional if P, then Q is true, we've gotta be in either the first, third, or fourth row of that truth table. But then, what does that tell us? Well, if we've gotta be in one of the two top rows and we've gotta be in either the first, third, or fourth row, then the only possible choice is that we've gotta be in the top row. In other words, we've gotta be in a situation in which P is true, if P, then Q is true, and Q is true. Another way of putting that is in any situation in which P is true and if P, then Q is true, Q has gotta be true. And so modus ponens is a good rule of inference. If we follow modus ponens, we'll never give an invalid argument. All right, no modus ponens argument can be invalid. Because whenever P is true and if P, then Q is true, Q has gotta be true. Okay, so modus ponens is a good rule of inference. There's another rule called modus tollens, which says the following. From the premises not Q, the negation of Q and the premise if P, then Q, infer the conclusion, not P, or the negation of P. Now, is that a good rule? Well once again, we can use the truth table to see that it is. So here is what the two premises tell us. They tell us if P, then Q is true, so we've gotta be in the first, third, or fourth rows of that truth table. And they also tell us that the negation of Q is true. In other words, that Q itself is false, right, because when the negation of Q is true, then Q is false. So that tells us we've gotta be in either the second or the fourth row of that truth table. Okay, so here's what the premises tell us. We're either in the second or fourth row of the truth table, and we're in either the first, third, or fourth row of the truth table. Well, the only option left is that we are in the fourth row of the truth table. In other words, if we're in a situation in which if P, then Q is true, and not Q is true, in other words, Q is false, then the only possible situation we could be in is a situation in which P is false. But of course, that's a situation in which not Q is true. So if we follow modus tollens and make arguments that follow that rule, all of our arguments will be valid. Any argument that has as its premises, two premises of the form if P, then Q and the negation of Q, and has as its conclusion the negation of P, any argument of that form will have to be valid. And that's what we can see, using the truth table for the condition. So modus ponens and modus tollens are two rules governing our use of the conditional. And we can see, using the truth table, that they're both valid rules of argument, just like conjunction elimination or disjunction introduction. Okay, now I've said something about what the conditional means, what its truth table is, and what some good rules for its usage are. Why is the conditional an important truth functional connective? It's pretty clear why disjunction and conjunction are important truth functional connectives. They're very intuitive, they seem to correspond to notions that we operate with in everyday life, but what about the conditional? Here's what makes the conditional especially important for propositional logic. Consider any argument whatsoever that has some premises and a conclusion. Let's call the premises of that argument P and the conclusion C. Now, whatever that argument is, if that argument from premises P to conclusion C is valid, then the conditional, if P, then C, has gotta be true. And here's why, if the argument from P to C is valid, what that tells us is that there's no possible way for P to be true while C is false. But what does it mean for the conditional if P, then C to be true? All it means is that if P is true, then C has gotta be true. It's not the case that P is true, and C is false. So, whenever the argument from P to C is valid, the conditional if P, then C is going to be true. And so we can use the conditional to express in the form of a single proposition the validity of an argument from premises P to conclusion C, whatever that argument is about. The conditional can be used to express the validity of that argument. And that's what makes the conditional especially useful in propositional logic.