 Previously, we introduced the notion of a tautology, which is a statement that is true for every assignment of the primitives in that compound statement. In this video, we want to introduce the opposite notion that is the notion of a contradiction. A contradiction is a compound statement, which is false for every possible combination of truth values of contradictions and tautologies are opposites of each other. Now the method that we introduced to show that something is a tautology was to construct a truth table and then show that the column in the truth table involving the proposed tautology contains only true values. Well, to do that, as modifying that procedure, you can also show that a contradiction is in fact a contradiction from a truth table. You would construct a truth table and show that every value in the column is actually false. So I want to then demonstrate to you that the mother of all contradictions, p and not p, is in fact a contradiction. That is to say, p and not p cannot be a true statement. p and its negation cannot simultaneously be true. Now this is a very easy one to demonstrate because there's only one primitive involved in this one. So you have your primitive p, which means it takes on two values true and false. There's only two rows in this table. I'll use the negation as another column, which you switch the signs false and true, and then you take the conjunction of that, looking at these values here. A conjunction is only true if they're simultaneously true. True and false is false, and then false and true is likewise false. So this statement is always false, regardless of what the primitive statement p is. It doesn't matter. p and not p is always a false statement hint of a contradiction. Now I do want to mention that in the last video of this lecture, we introduced the notion of a logical equivalence, for which two statements are logically equivalent if they have always the same truth values. I want you to note that every contradiction is logically equivalent to every other contradiction. That is, you are a contradiction if you are logically equivalent to this statement right here. And similar things can be said for totalities, because a totality has t's in all of its columns. That is to say, in every row in its column. And therefore, all totalities are logically equivalent to each other as well. And so if you have one, you have basically all of them, because they're all logically equivalent to each other. Now, we're not going to place a lot of emphasis on tautologies, but we are going to put a lot of emphasis on contradictions. Because in the future, we're going to introduce a technique known as proof by contradiction. And the basic idea is we can prove something to be true by finding a contradiction. We'll make more sense of that in the future. And the contradiction we typically find is this one. But honestly, if we find any contradiction, it's logically equivalent to this one. So whether it looks like it has this form or not, a contradiction is always arrived by looking for this statement right here. And that brings us to the end of lecture nine in our lecture series. Thanks for watching. If you learned anything from these videos, please like them. If you want to see more videos like this in the future, subscribe to the channel. 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.