 Hello, in this screencast we're going to take a look at this proposition with three variables in it and try to construct a truth table for it. So the proposition is P implies not Q or R. First thing we need to do is make three columns, one for P, one for Q, and one for R. And what we're going to do first is list all possible truth combinations of these three variables. So there are eight of those, and we're going to go with true, true, true, false, false, false, true, false, false, and set R equal to true to get our first four rows. And then the next four rows, we're going to copy down the same first four rows for P and Q. True, true, true, false, false, true, false, false, and set R equal to false for those. That's going to give us all eight distinct rows for this truth table. Okay, so just as an algebra, we're going to begin to build the truth table for this proposition from the inside out going into the innermost parentheses first. So let's start with not Q. We're going to make a quick column for not Q. To do that, we're just going to look over at Q and then the negation of Q, not Q, is going to have the opposite truth value. So if that's true, this is now going to be false. This is false. This is now going to be true. And so on, false, true, false, true, false, true. Okay, now the next level of parentheses out would be to create this or statement, not Q or R. So we're going to make a column for it yet. Not Q parentheses or R. Now this is an inclusive or a regular disjunction. So we're going to look at not Q. I'm just going to make a little star with this. Not Q and the column for R. And if either of the truth values in this column or both are true, we're going to put true here. That's how a disjunction works. And if both of these two statements are false, we're going to put false. So here we're going to have true because the R is true. Here we're going to have true because both statements are true. This is going to be true as well. And this is going to be true as well. In this row, both R and not Q are false. So we're going to put false there. This would be true, false, false again. So that's false, false, true, and we have true. So that's just a simple disjunction. Okay, now the last column to put in is for, we've gotten all of this stuff in a column. Now we're just going to make a column for the main proposition. P implies not Q or R. All right, now to do that, let me, let's see, let me get rid of, let's see, we don't need to do that. I'm just going to highlight what we're going to look at here. We're going to look at this column and P. Now this is a, excuse me, this is an implication, a conditional statement. Here is the hypothesis over here, the hypothesis, and the conclusion is right here. Okay, so we can ignore all the other columns. Those are just used to build up the, this column and the last one. So we're looking for, at a, at a conditional statement, that is true in all cases except where the hypothesis is true, but the conclusion is false. Okay, so we're just going to look in those two columns. True and true, that would give me a true conditional statement. True and true, again true here. In these two columns, the third and fourth one, the hypothesis is false, and so in that case we say the entire conditional statement is true. In the fifth row, I've got a true hypothesis and a false conclusion, and there is where we pick up a false for the conditional statement. In the sixth row, I have true hypothesis and a true conclusion. In that case the conditional statement is true. In the last two rows, the hypothesis is false and so the entire conditional statement is true. And that is the truth table for this statement. There is the final column that tells us when this large conditional statement is true. It's almost always true except in this particular circumstance.