Today, we're going to talk about the truth functional connective that we'll call disjunction. And disjunction is a truth functional connective that, in English, is usually expressed by means of the word or, O-R. But there are a couple different ways of using that word, or. Let me give you some examples. Supposed you ask me, who won the game last night? And I say, well, either Manchester won it or Barcelona won it. But there, I'm clearly trying to indicate that, while Manchester may have won it, and Barcelona may have won it, they didn't both win the game, if Manchester was playing Barcelona last night, then they couldn't both have won, right, soccer doesn't work like that. Either one team wins or the other team wins, but they can't both win. So if I say, either Manchester won it or Barcelona won it, what I mean is that either one of two possibilities occurred, either Manchester won it or Barcelona won it. But they couldn't both have occurred. But sometimes when I use the word, or, I don't mean to convey anything like that, for instance, suppose I say to you, this is breakfast or lunch. Now I'm not suggesting when I say this is breakfast or lunch. I'm not suggesting it can't be both It could be breakfast, it could be lunch or it could be both. And when I say this is breakfast or lunch, I mean it could be one, it could be the other it could be both. So there in that second usage I'll say, or is inclusive. It includes both of the options and the possibility of they're both being true. In the first usage where I said either Manchester won or Barcelona won, I'll say or is exclusive. In other words, either one is true or the other is true, but they can't both be true. So or can be used exclusively to mean that either one of two options is true, but they can't both be true. Or or could be used inclusively. To mean one of two options could be true or they could both be true. Now, the truth functional connective disjunction is expressed in English by the use of the inclusive or. The or that leaves open the possibility that both of the two options could be true. So what's the truth-table for the truth functional connective disjunction going to look like? Well, we're going to start with two possibilities. Two propositions that are connected by the truth functional connective disjunction. Doesn't matter what the propositions are. Just call them P & Q because it doesn't matter what they are so There's P, there's Q and then, there's the disjunction, a P and Q, which will symbolizes as follows. P disjunction, which looks like a V, Q. Now, when is P disjunction Q going to be true? Remember, disjunction is expressed by the inclusive or so it's going to be true whenever P is true. And it's also going to be true whenever Q is true and of course since it's expressed by the inclusive or it's going to be true whenever P and Q are both true. So the truth table for disjunction is going to look like this, right? The first three lines, the lines that consider the possibility of p and q's, both being true, or p's being true q's being false, or p's being false and q's being true. And all three of those lines the disjunction p or q will be true. The only scenario in which P or Q is false is the scenario in which P and Q are both false. In which neither P nor Q is true. Now that we've learned the truth table for the truth functional connective, this junction, we can use the truth table to figure out when certain arguments that use this junction Are going to be valid. For example, consider this simple argument. Premise one, I'm going to tickle you with my right hand. Premise two, I'm going to tickle you with my left hand. Conclusion, I'm going to tickle you with either my right or my left hand. Is that argument valid well it obviously is valid but you can use the truth table for disjunction to explain why it's valid. Look at the truth table for disjunction again In this argument we have premise one being the proposition, I'm going to tickle you with my right hand. Premise two being the proposition, I'm going to tickle you with my left hand and the conclusion that's the disjunction of those two propositions. Well, what can you tell from the truth-table for disjunction? What you can tell is that in a situation in which P is true and Q is true, in which both of two propositions are going to be true. The disjunction of those two propositions is also going to be true. You see, that's what you can read off from the first line of the truth table. In a situation in which each of the two propositions is true, their disjunction has to be true. So in the argument that I just gave you which has as premise one, I'm going to tickle you with my right hand. And as premise two, I'm going to tickle you with my left hand. That argument has to be valid, because the conclusion of that argument is simply the dysfunction of the two premises. So there's no possible way for the premises of that argument to be true, while the conclusion is false. If the premises of that argument are true, then the conclusion has to be true. We could think of that argument as a kind of disjunction introduction argument, because just as in the case of a conjunction introduction argument where the conclusions introduces a conjunction that wasn't there in the premises, in this argument, the conclusion introduces a disjunction that wasn't there in the premises. Right, there were two premises. I'm going to tickle you with my right hand, and I'm going to tickle you with my left hand. The conclusion introduces the disjunction of those two premises. The conclusion just is the disjunction of those two premises. But notice, even though that disjunction introduction argument is valid, there are even simpler disjunction introduction arguments that are valid. Consider, for instance, the argument that starts with just one premise. Could be anything, call it P. And that draws as a conclusion the disjunction of P with anything else. So let's say, P is the premise I'm going to tickle you with my right hand. And then the conclusion is that it's just the disjunction of that premise, I'm going to tickle you with my right hand, with any other proposition, like say, I'm going to tickle you with my left hand. That argument is going to have to be valid, and you can read that off the truth table for disjunction. Because you can see that in any situation in which one of the disjunct of a disjunction is true, the disjunction is going to have to be true. So, any disjunction introduction argument that starts with just one premise and that concludes the disjunction of that premise, if anything else, any such argument is going to have to be valid. There's no possible way for the premise of that argument to be true while the conclusion is false. So, all this junction and introduction arguments are valid. But what about disjunctional elimination arguments? Well, remember conjuctional elimination arguments are valid, because if you start off with a premise that states the conjunction of two propositions, then the conclusion that is either one of those two conjoint propositions, either one of those propositions, the conclusion is going to have to be true whenever that conjunction is true. You saw that from the truth table for conjunction. But what about with disjunction? Does it work that way with disjunction? Well consider an argument that starts from the premise, I'm going to tickle you with either my right or my left hand, and then it draws the conclusion that it is simply one of those two disjoint propositions. Let's say it draws the conclusion, I'm going to tickle you with my right hand. Is that argument valid? No it's not. Because there's a possible situation in which the premise, I'm going to tickle you with either my right or my left hand is true. While the conclusion, I'm going to tickle you with my right hand is false. Namely, the situation in which I'm going to tickle you with my left hand. In that situation, the premise would be true but the conclusion would be false. So that disjunctional elimination argument is not valid. And you can also see that by looking at the truth table for disjunction. If you look at the truth table for disjunction you'll see that there are three possible scenarios in which the premise, I'm going to tickle you with either my right hand or my left hand, is true. The three possible scenarios are the three scenarios in which both of the disjuncts are true, the first disjunct is true and the second one false, or the first disjunct is false and the second one is true. Well there are three possible scenarios in which that disjunction is true. When I say I'm not going to tickle you with my right hand, I'm ruling out two of those scenarios, namely the two scenarios in which it's true that I tickle you with my right hand. But that still leaves open a scenario in which it's true, that I'm going to tickle you with either my right hand or my left hand, but for the disjunction elimination argument to be valid, it would have to be the case that in every scenario in which the disjunction, I'm going to tickle you with either my right or my left hand is true. Each of those disjunct's is true. But that's not the case, and you can see that just by looking at the truth table for disjunction. So, the truth table for disjunction, shows you why it is that disjunction introduction arguments are all valid. But symbol disjunction elimination arguments that start with a premise that's just a disjunction of two propositions and end with a conclusion that is simply one of those disjoint propositions by itself, those arguments are not valid. You can see that by the truth table from disjunction.
