In the last lecture, we learned about the truth functional connective that we call conjunction. The truth functional connective that, in English at least, is sometimes expressed by certain uses of the word and. And we learned about the truth table for conjunction. Now, one thing I'd like to do in today's lecture is show how we can use the truth table for conjunction. To show that certain kinds of inferences that use conjunction are going to be valid. Let me give you an example. Consider an inference that starts off with two premises, doesn't matter what the premises are, just any two premises, call them P and Q. because it doesn't matter what they are. And then the conclusion of the inference is simply going to be the conjunction of those two premises, whatever they were. So the inference can be sketched as follows, P, Q, conclusion P and Q. Now we can call that a conjunction introduction inference, because the conclusion of the inference uses a conjunction. It introduces a conjunction that wasn't present in either of the two premisses of the inference. Now, notice if you look at the first line of the truth table for conjunction, you can see that any conjunction introduction inference is going to have to be valid. Here's how you can see that. If you look at the the first line of the truth table for conjunction, that considers the situation in which P is true and Q is true. And in any such situation, according to that truth table, the conjunction P and Q is going to have to be true. Now, let's take that point and apply it to the inference from the premise P and the premise Q, to the conclusion P and Q. What does that first line of the truth table for conjunction tell you about that inference? Well, it tells you that in any situation in which the premises of that inference, P and Q, are both true in any situation in which those premises are both true, the conclusion, P and Q, is going to have to be true. But that's just what it is for the inference to be valid. Recall that for an inference to be valid is just for it to be such that there is no possible situation, where the premises are true and the conclusion is false. So from the first line of the truth table for conjunction, you can see that conjunction introduction inferences are all going to be valid. You could also see, by looking at the truth table for conjunction that another kind of inference is always going to be valid. So, consider an inference that starts with only one premise. A premise that conjoin's two propositions. Again, never mind what the two propositions are. They could be anything. Call them P & Q. because it doesn't matter what they are. So consider an argument that starts with that one premise, P&Q, and it moves to a conclusion that consists simply of one of those two propositions that is conjoined in the premise. So the conclusion of the argument will be either the proposition P or the proposition Q. Now an argument like that we can call an conjunction elimination argument, because the conclusion of the argument eliminates a conjunction that occurs in the premise of the argument. Right? The premise is a conjunction of two propositions, P & Q. And the conclusion is simply one of those two propositions, not conjoint to anything else. So the conclusion is either the proposition P or it's a proposition Q. Now, is that argument going to be valid? Well, if you look at the truth table for conjunction, you'll see that it is going to be valid. In any possible situation, in which of the premise of that argument P and Q is true. Both of the two propositions that are conjoined in that premise, both the proposition P and the proposition Q, are also going to be true. >> So any situation in which the premise of a conjunctional elimination argument is true, is going to be a situation in which the conclusion of that conjunctional elimination argument is true. >> Therefore, all conjunction elimination arguments are valid, no matter what they're about. And we can see that just by looking at the truth table for conjunction. Now, conjunction introduction and conjunction elimination arguments are not the most interesting kinds of arguments there are to be sure. But I just wanted to give a simple example for now of how we can use the truth table for a truth functional connective. In this case the truth functional connective conjunction. How we can use the truth table for that connective to discover that certain kinds of arguments are going to be valid. In the next lecture, we'll show how we can use other kinds of truth tables for other connectives, for other true functional connectives to show that certain other kinds of arguments are valid. See you next time.
