 Welcome back to Deductive Logic. I'm Professor Matthew Brown. Today we're talking about truth tables again. And all we're gonna do today is walk through some additional practice problems. So if that feels like something that'd be useful to you, go ahead and watch along and work along with us. Otherwise, you can move directly to the practice exercises and do some for yourself. Let's start with this problem, which is evaluating a sentence of SL to see whether it is a tautology, a contradiction, or a contingent sentence. Please pause the video and work this out for yourself before moving on. Okay, let's see how you did. First, we're gonna copy over the single atomic sentences, A and B. The negation of B and negation of A are simple enough. You can check that that's right. If you need to, by pausing. Similarly, we can just copy over the, the, we can simply copy over the characteristic truth table for if A, then B here. Now let's walk through the second half of the expression a little bit more carefully. So you see we've got if zero, then zero, that's a one. Whenever you've got a zero in the antecedent, the conditional is a one. The second line is a one and a zero, so that's a zero. Then we've got a zero, one, so that's a one. And then we've got a one, one, which is also a one. So now we're gonna go ahead and compute the main connective, which is this biconditional. There we have to compare the two sub expressions on the left and right, which are both conditionals. And let's step through this carefully. We've got a one and a one. That's a one on the biconditional, a zero and a zero on the next line. Also a one, a one and a one on the biconditional is also a one, and then one, one is a one. So it's a one in every line. And that tells us we've got a tautology that we're working with. So that's the first problem. Very similar problem here, slightly different expression, but go ahead and pause the video and see if you can work it out. Okay, so again, we're gonna copy over the atomic sentences. The negations are simple enough. We've got the regular characteristic truth table for the conditional, if A then B on the left. The conditional on the right is not, it's different from the last problem, but not more complicated. So you can go ahead and pause and check that if you need to, but it's one, one, zero, one. And then again, to get our main connective, we have to compare the two sub-expressions, both conditionals, again, in this problem. So we've got a one and a one, that's a one. We've got a zero and a one, also a one. We're disjunction here, not biconditional, keep that in mind. We've got a one and a zero, also a one, and a one and a one, also a one. The disjunction is true whenever either disjunct is true, or both. So that's also gonna be a tautology because it is a one or a true on every line. Let's look at a third example. Here we've got if A then B, and if B then A, okay. You might recognize this as a, well, spoilers. Pause the video, try to work it out. Okay, let's see how you did. Copying over our atomic sentences. These are both conditionals, they're very simple, right? So the one is one, zero, one, one, that's normal. The one on the left, the terms are flipped. You can see it's one, one, zero, one. Now for our main connective, which is a conjunction, we know the conjunction is true whenever both conjuncts are true. So on the first line, we've got a one and a one, so that's true. Our second line, we've got a zero and a one, so that's false. Our third line is a one, zero, so that's false, that's zero. And our last line is a one and a one, one, zero, one, one. That's the characteristic truth table. Also for a biconditional, not surprising, this expression is equivalent to a biconditional. And like any simple biconditional, it's a contingent sentence. But notice that in order to tell that it's contingent, we didn't have to go through all of the lines of the truth table. It would have been sufficient to just go through these first two lines, right? The reason is that in order to show a sentence is contingent, all you need to show is that it is not a tautology. The second line does that and show that it is not a contradiction. The first line does that. So those two lines together give you that it is a contingent sentence. Let's do an example where we compare two sentences to see if they're equivalent or and to see if they're consistent. So pause the video and work this out on your own. Okay, let's see how you did. So again, we're gonna copy over our atomic sentences A and B. We've got a negation in each case. That's relatively simple, right? We just flipped the truth value. And then we've got the conditional on the left. Zero in the antecedent makes it a one. Zero in the antecedent makes it a one. You got a one in the consequent and the antecedent. So that's a one. And here on the last line, we've got a one in the antecedent and a zero in the consequent. So that's a zero, okay? We step through the second one similarly. Zero in the antecedent is a one. One and one is a one. Zero and zero is a one. And then one and zero is the only case where the conditional is false. So that's a zero. Now to tell whether they're equivalent, we have to compare the truth values on every line, right? And here we can see they are, in fact, logically equivalent, right? We also wanna know if they're consistent. All that requires is one line where they're both true, right? We've got three lines where they're both true. So these are consistent. The only time you're gonna have logically equivalent and inconsistent is when your expressions are all contradictions. We'll keep that in mind, okay? Now to show that two expressions are inequivalent, you again don't need to go through the entire truth table. You can simply show that there are two lines where they differ in value to show that they're consistent when you're not worried about also being equivalent. You just have to show one line where they're all true. So that's something to keep in mind for efficiency's sake. Last, I want us to work through some more simple problems where we assess arguments for validity or invalidity, right? I've got three simple arguments here at the top of the screen. I want you to pause the video long enough for you to work through all three of these, right? You're trying to figure out whether they are valid or invalid in much the same way we did before. Okay, are you done? Let's check it out and see how you did. So this first argument, A or B, B, therefore not A is it a valid or invalid argument. These are all very simple truth tables. So I'm not gonna step through them but you can pause the video and check if you need to. And if we see on this first line, both of our premises are true but our conclusion is false. That's enough to tell you that our argument is invalid, right? And in fact, again, to show that an argument is invalid you just need that one line as a counter example. You don't have to go through the entire truth table. Now, of course, you don't know if it's valid or invalid so it's simple and easy to go through the whole truth table but you can also try sort of guessing and checking if you have an intuition about it being invalid. Let's look at the next argument. A or B, not B, therefore A. Again, these are simple expressions so you can pause and check if you need to. Here we see there's only one line in which both premises are true and in that line the conclusion is true and that tells us by our definition of validity that the argument is valid. Okay, last argument. Okay, here's the truth table. Again, pause and verify if you need to. Here we see, again, there's only one line where both the premises are true and on that line the conclusion is also true. So therefore, according to our definition of validity this argument is valid. And I might suggest you pause before the video concludes and think about what's that second premise doing for me here? Is that necessary for this argument or is it kind of irrelevant? So think about whether you can figure that out looking at just this truth table. That's all the examples I wanted to show you today. I hope that's helpful. Best way to do this, to learn this is just to practice it. So if you wanna gain mastery over truth tables and how to use them for evaluating these expressions and arguments and sets of sentences the best way to do it is to work on more of the practice problems which you should be able to find now on D2L. Best of luck with it and with exam three and I'll see you in our next unit which is proofs and SL. Bye.