Today we're going to talk about a new propositional connective, a truth functional connective that I'll call the biconditional. Now what's a biconditional? In order to explain what a biconditional, let me start off my telling you the following story, and this is a true story, I should add. When I was in eighth grade I remember my first class in the morning was a Math class. And there was a boy who sat next to me in Math class. The teacher called him Bob. My third class later in the morning was a Science class and there was a boy who sat next to me in the Science class and the teacher called him George. And I was struck by the fact that Bob and George, who sat next to me in Math and Science class respectively, had the same last name and they also bore an uncanny resemblance to each other. Well one day, I asked Bob if he had a brother named George and it turned out that he didn't have a brother named George. In fact, Bob was George. It was one and the same boy and they just went by two different names to the two different teachers. But suppose that all you knew from my description of the situation was that there was a boy who sat next to me in Math class named bob and a boy who sat next to me in Science class named George. And you asked me one day, was Bob born in the United States? Now I might say to you, well Bob was born in the United States if, and only if, George was. Now, there I'm using the phrase if and only if. Now remember how the phrase if works. If I say if to connect two propositions, P if Q, that's equivalent to saying if Q then P. But now remember how only if works, if I say P only if Q, that's equivalent to saying if P than Q. So when I use the phrase if and only if, I'm conjoining the conditional if P than Q. And the conditional if Q then P, and so I get what I'll call a biconditional. A biconditional is a propositional connector that connects two propositions into a larger proposition. And the larger proposition is true just in case the two propositions. That are part of it, have the same truth value. In other words, the larger proposition, P if and only if Q is going to be true just in case P and Q are both true or P and Q are both false. So. I could say George was born in the United States if and only if Bob was. Under what circumstances is that proposition going to be proved? Well, it will be true if George and Bob were both born in the United States. Or, well, I guess it's misleading to say both because really there's only one boy that we're talking about. George was born in the United States and Bob was born in the United States. It'll be true when both of those propositions are true. But it'll also be true when George was not born in the United States and Bob was not born in the United States. In other words, it'll be true when both of those two propositions are false. What makes true the proposition George was born in the United States if and only if Bob was. Is simply that the two propositions that are part of it George was born in the United States and Bob was born in the United States. Have the same truth value whatever that true value is weather it's the truth value true or the truth value false. As long as those two propositions have the same truth value It's going to be true that George was born in the United States if and only if Bob was. And so the biconditional connecting those two propositions, George was born in the U.S. and Bob was in the US, that biconditional is going to be true. So we can state the truth table for the truth functional connective which is the biconditional as follows. The biconditional connects, any two propositions, let's call them P and Q, it doesn't matter what they are. When P is true and Q is true, then the biconditional, P if and only if Q is going to be true. When P is true and Q is false, then the biconditional P if and only if Q is going to be false. When P is false and Q is true then the biconditional P if and only if Q is going to be false. And finally, if P is false and Q is false then the biconditional P if and only if Q is going to be true. So, that's the truth table for the biconditional. Now, in the last couple of lectures I described both the conditional and the bi-conditional as truth functional connectives. And I've given some reason to think that they are truth functional connectives. But you might worry that there's some examples that suggest that both the conditional and the biconditional are not truth functional connectives. For instance, consider the sentence, if 2 + 2 = 4 then Pierre is the capital of South Dakota. Now there I am using if then to express the conditional. Right? I am saying if it's true that 2 +2 = 4, then it's true that Pierre is the capital of South Dakota. Now according to the truth table for the conditional that conditional has gotta be true. Because the first proposition what we'll call the antecedent, the proposition that occurs right after the if, that antecedent is true, it's true that 2+2=4. And also the consequent, the proposition that occurs right after the event, the consequent is also true. It's true that Pierre is the capital of South Dakota. So, according to the truth table for the conditional, it's going to be true that if 2+2 = 4, then Pierre is the capital of South Dakota. Now, you might worry wait a second, this is a very strange consequence of the truth table for the conditional. Is it really true that if 2+2=4, then Pierre is the capital of South Dakota? And that's a very baffling thing to say. Now I want to say I completely agree with this objection. It is a baffling thing to say that if 2+2=4 then Pierre is the capital of South Dakota. But just because it's a baffling thing to say, doesn't mean it's not true. Look, it's baffling that Pierre is the capital of South Dakota. And baffling as that may be, it's still true that Pierre is the capital of South Dakota. Some things are baffling even though they are true. And this is another example of that general kind of thing. It's baffling, nonetheless true.
