In the last couple of lectures, we've learned about the truth functional connective conjunction, and we've learned about the truth functional connective disjunction. Now both of these connectives can be used to take propositions and put them together to create new propositions whose truth depends solely on the truth of the propositions that you used as their ingredients. Now, in today's short lecture, I just want to make the point that just as you can combine propositions using conjunction, and you can combine propositions using disjunction, you can also use conjunction and disjunction together to string together a bunch of propositions into a larger proposition. Let me give you an example to illustrate this point. Suppose I say, I'm going to tickle you with this hand. And then I say, and I'm either going to tickle you with this hand, or with this hand. Okay? Now here, what I've done is string together three propositions. The proposition I'm going to tickle you with this hand, and the proposition, and I'm either going to tickle you with this hand or with this hand. So there, we've created a new proposition by stringing together three other propositions using conjunction and disjunction. Now notice, we can construct a truth table to show how the truth of the larger proposition, I'm going to tickle you with this hand and with either this hand or this hand, how the truth of that larger proposition depends on the truth of the propositions that go into creating it. Okay, so there are three propositions that go into creating it, and they're connected in different ways. So one of the propositions was, I'm going to tickle you with this hand. Let's find terms to distinguish the hand, so I don't have to keep waving around all my hands, let's call this hand number one. Now the proposition we can say is, I'm going to tickle you with hand number one, and with either hand number two or hand number three. Okay, so there are three propositions that go into creating that larger proposition. So the three propositions are these. First, I'm going to tickle you with hand number one. Second, I'm going to tickle you with hand number two. And third, I'm going to tickle you with hand number three. Now, we can use the truth table to show when precisely it would be true that I'm going to tickle you with hand number one, and with either hand number two or hand number three. How does the truth of that whole statement depend on the truth of the three propositions that go into creating it? Well, that whole statement, I'm going to tickle you with hand number one, and with either hand number two or hand number three, is a conjunction. It's a conjunction with two conjuncts. The first conjunct is, I'm going to tickle you with hand number one. And the second conjunct is, I'm going to tickle you with either hand number two or hand number three. So since it's a conjunction, we know that for it to be true, both of its conjuncts have to be true. So, it has to be true that I'm going to tickle you with hand number one. So that proposition, I'm going to tickle you with hand number one, that proposition has to be true. Does it have to be true that I'm going to tickle you with hand number two? No. All that has to be true is that I'm either going to tickle you with hand number two or with hand number three. So, that disjunction, I'm going to tickle you with hand number two or with hand number three, has to be true. But for the disjunction to be true, it doesn't have to be true that I'm going to tickle you with hand number two, after all, there's a scenario where I'm not going to tickle you with hand number two, and yet, it's still true that I'm going to tickle you with either hand number two or hand number three. And for the disjunction, I'm going to tickle you with hand number two or hand number three, to be true, it doesn't have to be true that I'm going to tickle you with hand number three, again, remember, there's a scenario where I'm not going to tickle you with hand number three, but it's still going to be the case that I'm going to tickle you with either hand number two or hand number three. But what does have to be true for the disjunction to be true is that at least one of those other two propositions are true. It's at least true that either I'm going to tickle you with hand number two, or I'm going to tickle you with hand number three, or both. But at least one of those has to be true. So, for the whole proposition, I'm going to tickle you with hand number one, and with hand number two or hand number three, for that whole proposition to be true, it's gotta be true that I'm going to tickle you with hand number one, and it's either gotta be true that I'm going to tickle you with hand number two, or it's gotta be true that I'm going to tickle you with hand number three. So if we look at the truth table, we can see precisely which lines are going to make the whole proposition true. So here's how we can construct a truth table for a complex proposition that combines conjunction and disjunction. And notice, by the way, when we're combining conjunction and disjunction, the order in which we combine them matters. We can say that conjunction and disjunction are not associative. What matters to determining the truth table of the proposition that they're used to create, is not just the occurrence of conjunction and disjunction, but what order they're taken in. And the proposition that I considered just a moment ago, I'm going to tickle you with hand number one, and with either hand number two or hand number three, that proposition has a different truth table, then the proposition, I'm going to tickle you with hand number one and hand number two, or I'm going to tickle you with hand number three. That second proposition has a very different truth table from the first. As we can see, if we construct the two truth tables and compare them. So, conjunction and disjunction are not associative. The truth table for the proposition that they create depends on the order in which we apply conjunction and disjunction.