 Welcome back to one more screencast from chapter two, section one. And in this video, we're going to talk about something that came up a little earlier in another video. We looked at this particular statement here, if p then either q or not q. And the kind of interesting thing about this statement is that it's never false. This is an implication that no matter what the situation is with p or q, that conditional statement's always true. And it's pretty easy to understand why if you kind of disengage from the truth table and just think about what this statement is saying. The conclusion of this statement here, the q or not q, well, that's always true, right? It's either raining or it's not. It's either Monday or it's not. The conclusion of that statement's always true. And so there's no way the conditional statement's ever going to be false. Hence, it's always true. Now, statements that are always true are pretty helpful to us. They're sort of the ultimate in dependability and they tell us something important about the statements. And we're going to see a little bit more detail later on. But for now, we're going to attach a special word to this kind of statement. It's called a tautology. That's a statement that is always true, no matter what the individual truth values of the many statements that make it up are. And on the flip side, a statement that's always false, and we haven't seen one of these yet, is called a contradiction. And contradictions are going to be especially important to us later on. So one of the things we want to be able to do at the present is to take a statement and decide whether or not it is a tautology or is a contradiction or something else. So let's look at a couple of examples where we're going to take a statement and determine whether it's a tautology or contradiction based on the outcome of a truth table. So this is perhaps a poorly titled slide because I'm calling it another tautology and it gives the answer away here. But here's a, looks like a fairly complicated statement. Let's show that this is actually a tautology. And what this is saying is that if there's an and statement here, it says that if p implies q and not q, then not p. This is going to be actually a very important form of argumentation and we're going to pick up a little bit later in this course here. So lots of moving parts here. I think it's helpful to label the parts of the statement before we start building the truth table. So if we kind of dive into the innermost pieces of the statement, here's maybe the first thing we ought to make a column for. I've already gone and set up the columns for p and q. Here's another one. Now once I have a column that lays out the truth values for p implies q and not q, I'm going to need a third one to do the entire and statement. So another column for that whole thing. That would, once I create that column three, that'll be the entire hypothesis for this conditional statement. Then I have to take care of the conclusion. This is a not p, so I'm going to need to make a fourth column for not p. And then finally, I'll be able to make a gigantic fifth column for the entire statement. So I'm going to make five columns here, see if I can cram all this in. Also, I'll make a column for p implies q. Just breaking it down to the simplest possible form. I need to make a column for not q. Those seem like they should be pretty easy. Then I'll need a column for the entire hypothesis. That would be p implies q and not q. Okay, that takes care of the whole hypothesis. Then I'm going to move over to the conclusion and make a column for not p. And then finally, I need to make a column that pulls all this together. I don't have enough room to write all this. So I'm just going to write in three circle implies four. Okay, again, what that's going to be is the circle three here that encompasses this entire hypothesis is what I'm calling three. And then four is not p, the conclusion. So let's go through and set all this up. And remember, this is the final answer here that I want to see what the results of that conditional statement are. Now, each of these individual columns is actually pretty simple. And that was the whole point. Let's work on them one by one. So p implies q, we know how that works. P implies q is always true except in one situation, that one, when the hypothesis is true and the conclusion is false. Not q is just the logical opposite of q. So this would be false, true, false, true. And now I'm just going to take these two statements, the p implies q and the not q and join them with and. And if you remember how and works, in order to register a true, it really means and like we would normally use it in English. Both statements have to be true. And you notice that almost never happens. It doesn't happen here. It doesn't happen here because that one's false. It doesn't happen here because that one's false. And then this last one here is true. Okay, that's the hypothesis of the big conditional statement we're working with. The conclusion of the conditional statement is not p. So I want to just kind of ignore all else and just go over to p and reverse the logical value. So this would be false, false, true, true. Okay, now for the grand finale here, I want to have three implies four. P implies q and not q implies not p. Now this is a conditional statement. So let me just remind you that the hypothesis of this conditional statement is all this stuff or it's or better yet all this stuff is the hypothesis. And this right here is the conclusion. Okay, so just remind yourself when is a conditional statement true? It's always true except when you have a true hypothesis and a false conclusion. And you notice that never happens here, does it? The hypothesis is false three fourths of the time. So I'm just going to automatically put true. And then the one situation where the hypothesis is true, the conclusion is also true. So we have ourselves a tautology, a statement that is always true. So I would be remiss in talking about tautologies if I didn't mention this great XKCD comic that really gets to the heart, in my opinion, of what a tautology really is. For example, the Facebook group, if a million people join this group, it will have a million people in it. Or the first rule of the tautology club is the first rule of the tautology club. Just a statement that's always true that restates sort of the obvious here. And it's important to know when you're dealing with a statement that looks complicated, like it might have some wrinkles in it. Knowing that it's a tautology says that really there aren't any wrinkles of this, that the statement is actually pretty obvious. But sometimes it takes a lot of work to figure out whether something is obvious. So since this video is getting a little bit long, we're going to save the next example for the next video. And we'll go through a process like this again to determine whether a given statement is a tautology or perhaps a contradiction, something that's never true, or something that's in between that's sometimes true or sometimes false. So see you there.