 In the previous video for lecture eight, we learned how we can construct a truth table to help us evaluate logical expressions. And we mentioned how in algebra, you can have algebraic equations and algebraic expressions, things like x plus y squared. And in the previous video, we may have mentioned this is not the same thing as x squared plus y squared. You can actually plug in values for x and y, like if x is three and y is four, you can see that these two things aren't equal to each other. On the other hand, if I were to take x plus y squared and I wrote x squared plus two xy plus y squared, then certainly if you plug in x equals three and y equals four, you're gonna see that both sides of the equation agree with each other. And in fact, it doesn't matter what you choose for x or you choose for y. This equation holds for any choice of the variables. In algebra, this is often when we refer to as an identity, an equation for which any assignment of the variables is satisfied. And this follows, of course, from the distributive laws, that is the typical FOIL method here. That's what justifies this. Now, in logic, we have something similar to an algebraic identity, and this is what's referred to as a tautology. It's a statement for which every possible assignment of truth values to the primitives will always make the statement true. It doesn't matter what you plug into it, it's always true. And so one can actually show that a statement, a logical statement is a totality by constructing a truth table. And you would show that the column consists only of truths. Now, one of the simplest tautologies is the following. P implies P. If P is true, then P is true. That kind of seems self-evident, but let's just for the sake of practice draw it here. Now, with this statement, P implies P, there's only one primitive in play here, and that's the primitive P itself. So we need to look at the truth values for P. I'm only gonna have two rows because there's only one primitive here, and then compare the statement P implies P. Well, P could be true, P could be false. Now, if P is true, then we're looking at the statement T implies T. That's a true statement for conditionals. So that's true. And then if you look at the next one, the next row here, you're looking at P implies P, that's false implies false. That's actually a true statement. It's vacuously true, but still true, like so. And so when you look at this table, the last column, this statement P implies P gives you true. It doesn't matter what you assign to the variable. There's only one variable in this case, but it doesn't matter what you assign to the variables, the statement is always a true statement. So this is exactly what one means by a totality. It's always true that even though there might be some variability, some opened ended parts of the statement, it doesn't matter, fill in the blank, whatever you want, you have a true statement. Totalities are always true. Therefore, you can imagine why in logic we love them and we would talk about them as soon as we can. Let's do a slightly more complicated one. This one actually is gonna be foreshadowing to something we're gonna talk about in the future. Take the statement P and P implies Q, that compound statement implies Q, like so. This one's a little bit more involved, so let's take our time to draw it. There's only two primitives in play. There is P and Q. So we're gonna have four rows for that. For the first one, the second one, the third and fourth one right there. And so our two primitives, remember, are P and Q. So some things I'm gonna consider. I wanna put in P implies Q. I want P and P implies Q. And then we're gonna take the whole enchilada there. P and P implies Q implies Q. Okay, let me finish out this table. Align there, align there. Let's start doing some truth values. So there's four possibilities with two primitives, true, true, true false, false, true, and then false, false. Now it doesn't matter which order you list the rows in. You can put them in any order you want. This is the one I always put them in. Make sure I always get them, but you could do other things as well. So now let's look at the conditional here. P implies Q, the first row. True implies true, that's true. True implies false, that is false. False implies true, that is true. And false implies false, that is true for a conditional. Now for the next one, P and P implies Q. So we're gonna look at this column and this one, the first and third columns there. True and true is true. True and false is false. False and true is false. And false and true is in fact false, all right? Now for the final statement, we want P and P implies Q. All of that implies Q. So we're looking at the second column, that's our conclusion, and the fourth column is our hypothesis. So the order does matter here. So we have true implies true, that is a true statement. We have false implies false, that is also true for conditionals. False implies true and false implies false. Remember, when the premise is false, doesn't matter what the conclusion is. So I want you to examine this final column here. We have another example of a totality. That is no matter how you evaluate the truth values, you always end up with true. This is always a true statement, regardless of the values here. Now, this last one, the second one we did was for a purpose here. Now, when one thinks about a totality, we don't typically think of it as a single statement. We actually think of it instead as an argument where you say things like the following. Oh, it is true that P implies Q. It is true, P. Therefore, Q is then the conclusion. This is often known as the modus pinus argument, which will do some more arguments like this in the future. But this is one of the most common, one of the most common arguments that you use. And we will see as we learn about logical arguments in the future, logical arguments coincide with totalities and logical fallacies, well, those ones aren't gonna be totalities. So totalities are important because you can take these complicated looking totalities and rewrite them into a logical argument, aka a proof. And so this is just the embryo of how proof writing is gonna work in the future. And so remember this very important word at totality. These are logical statements that are always true, independent of how you evaluate the primitives there. And so that brings us to the end of lecture eight. Thanks for watching. If you learned anything about truth tables, our Venn diagrams in this lecture, please like these videos, subscribe to the channel to see more videos like this in the future. And as always, if you have any questions, please post them in the comments below and I'll be glad to answer them as soon as I can.