 Let's look at some practice exercises using truth tables to determine whether an argument is valid or invalid. Now remember, an argument is invalid only when you have all true premises and a false conclusion. That's the only situation in which an argument is invalid. Otherwise it's valid. If you have all false premises and a true conclusion, it's still valid. If you have all false premises except for one, like maybe all false and one true premise, and a true conclusion, it's still valid. In order to find an invalid argument, you must have all true premises and a false conclusion. And we use truth tables to determine this. So just briefly, the way you do it is you chart out all the possible combinations of the truth assignments for the atomic propositions, and this determines the truth values for the complex propositions and the conclusion. You just look for a row where the conclusion is false, and yet all the premises are true. So the first kind of exercise that we'll deal with this is you'll determine whether an argument is valid or invalid. And if it's valid, you'll mark valid. If it's invalid, you'll tell me which rows are invalid. So here we have our sequent, it's not the case that PNQ, therefore not P. So by rule five, we've got our two variables. Since we have two variables, we've got four rows. So then by rule six, we give our every possible combination of truth values. Then we put in our formula. Notice we've got that solid, dark line between the premise, in this case there's only one premise and the conclusion. Got that solid, dark line indicating the difference between those. And now by rule seven, we copy and paste our truth values from P and Q over, I'm sorry, first over to P, now to Q. And so here we can actually take a little, I would say necessarily shortcut, a little bit of shortcut. And we look at the conclusion and we could just fill out all the truth values and find the rest and that's fine, right? I'm not knocking it, you go for it. If you wanna do that. But maybe a tiny little bit of shortcut we could do here is we could just simply look towards those rows where we have a false conclusion, right? So where P is true, since it's a negation, we look for those rows where P is true. And that complex proposition and not P, the conclusion that that's false. So we've got two rows for that, but just to offer kicks and grins, let's just note that the other two rows are true. They're not gonna be useful in helping us find invalidity, right? We're looking for the rows where it's a false conclusion and true premises, but like I said, for kicks and grins, let's put it in there. And then we start looking over at the premise. And work from the inside the parentheses out, right? Work from the inner most parentheses out, okay? So conjunctions are really easy. You just look for either a time proposition that's false. And if at least one of them is false, you know that's false, that conjunction is false. So the bottom three work that way. With row two, Q is false, with row three, P is false, with row four, both P and Q are false. So those three are false, just leaving the top one as true, right? That top one is true, okay? Well, since that conjuncts to the top row is true and the entire conjunction is negated, right, that negation is false. Okay, well this doesn't help us determine invalidity, right? We need a true premise, all true premises, and a false conclusion. Well, we get that with number two, right? We get that with row two. That premise is true, and the conclusion is false. So this argument is invalid, it's invalid. And you'd mark, you know, set row two, you'd mark that row with, now over here in your answer column, right? Over here in the answer column. All right, let's try another one, all right? We've got a conditional if P then Q. Therefore, if Q then P. All right, let's see if this is valid. So by rule five, we have our two variables raised to the power of two. That gives us four rows. Then we put in our first premise, or first and only premise really. If P then Q, then we have, of course, a solid dark line there to indicate that after that, that's our conclusion, and a conclusion if Q then P. Now by rule seven, we copy and paste the true values for P into the columns for P and do the same thing for Q. Okay, next step, again, this is a tiny little bit of a shortcut, right? Cuz we're looking for invalidity. That means we're looking for those rows where the conclusion is false. And that's really only row three, right? With conditionals, a conditional is false only when the antecedent is true and the consequent is false, and it gives us row three. So we take a look at the rest of row three, and sure enough, right? The first, that premise there, if P then Q, well, that's true. Because it's a conditional and conditionals with false antecedents are always true. So that means our row three is invalid, right? Our row three is invalid. Now, if four kicks, just for the sake of showing how to fill out the rest of the truth table, you can simply look for the, if you're doing a conditional, just look for those rows where the antecedent is true. Let's do it one step at a time. Those rows where the antecedent is true, excuse me, antecedent is false. Excuse me, antecedent is false. Well, we know that that conditional is true then. And then we could look for the consequent where the consequent is true. We could also mark that as true. And so not to forget our original notation there that the conditional is false with the true antecedent and the false consequent. And then you could just do the same thing for the premises. If you just want to fill it all out, you look for those rows where the antecedent is false or with conditionals and you know that that conditional is true. Then you look for where the consequent is true. That's that top row and the remaining one is false, right? And that's just for this, just want to show you that just for the sake of showing you a completed truth table. You can do it in a couple of different ways. But anyway, just to reiterate our row here, three has a true premise. So it has only one premise, a true premise and a false conclusion. That tells us that this argument is invalid, all right? Now, again, from the sake of illustration, I want to show you how to do this with an abbreviated truth table. An abbreviated truth table, you know, the nice thing about having the, doing the full truth table is there's just almost no room for error. Unless you just make a simple little mistake somewhere. There's no room for error. The abbreviated truth table, you kind of have to know what you're doing in order to get to work right. But it can be a faster way of finding out whether an argument is invalid. So you start out, since you're looking for invalidity, you're looking for all true premises and a false conclusion. Well, you start out with just supposing, right, that your conclusion is false. And if so, you have to figure out all the different possibilities where all the different possible truth assignments within the conclusion, that would give you a false conclusion. Now, we're lucky in this case, because we've got a conditional. There's only one way that a conditional is false, and that's where the antecedent is true and the consequent is false. Now this also gives us our truth assignments for P and Q. So we go back and we fill in our truth assignment for P and Q. And, sorry, we fill the truth assignment for P and Q, right? Then you fill in those remaining truth assignments. Since we got our truth assignments for our time propositions, by rule seven, we go in for the premise and put in our truth assignment for P and Q. And this tells us that that premise is true. So once again, doing it another way, right? Once again, we find out that the conclusion is, I'm sorry, that this is an invalid argument by row three, right, row three. So you go back and you probably look back and you see your row assignments, maybe in the text or some of your other truth tables to find out which row gives you true premises and a false conclusion. So let's try another one. This time we've got three variables, right? So three variables. And since we've seen a full truth table, let's try this one though, just by an abbreviated truth table. Let's be adventurous, yes, adventures and logic. So by rule five, right, sorry, but so we put in our three, not by rule five, yes, so the full truth table. So we put in our three variables, P, Q and R, right? Then we put in our first premise, not P and not Q. We put a gray line there to indicate that that's a premise. Then we put in our second premise followed by a solid black line to indicate that what follows is the condition, excuse me, conclusion, right? So we got if R then the conjunction of P and Q, that's our second premise. And finally, our conclusion is not R, okay? So immediately we wanna put in our false conclusion. Wait, we've made a mistake, right? We're not looking for the atomic proposition being false. We need the complex proposition as false, right? So the complex proposition here is false. So that means that R is true, okay? So we put in true for R and then we go back to our variables over here. And we've got a T in for R, okay? Well now we take that truth assignment by rule seven, go through the rest of the premises and put that same truth value in for R wherever we find it in that row. And we only got one place. Okay, so what do we do now? Got a leak. Well, remember, we want to find a row where the premises are true and the conclusion is false. Well, we can't really do anything with, I mean, we could start playing around the truth values for P and Q and maybe hit on something, but that would take way too long, okay? And so just looking at what we have here, all right? We want that second premise to be true in order to find an invalid argument. So we'll just label it as true. Okay, what would it be like for it to be true? All right, well, since it's a conditional and the antecedent is true, there's only one possible way that the conditional can now be true. And that's where the consequent is true, okay? That's consequent is true. So we're getting kind of lucky here. We don't have any other possibilities happening, all right? So the consequent of this conditional is true and it's a conjunction. And the only way for conjunction to be true is if both conjuncts are true. Okay, so then that means P and Q are both true. Great, we figured out another step in the way. We figured out another step in the puzzle. So we go back to our truth assignments for our variables and we fill out, we put in that P and Q are both true. And now we can fill out our first premise. So then by rule seven, you get both P and Q is true there. And since they're negated, their negations are false. And then, uh-oh, that first premise is false. And this is the only way, this is the only possibility where that second premise can be true. And if this is the only way that second premise can be true, is with the first premise is false, then we know that this is the only possible way for the conclusion to be false. And then the second premise, okay, so that means that this argument is valid. And that means this argument is valid. Because we can't find a row where the premises are true and the conclusion is false. Okay, well, now maybe you don't believe me, so we'll do it the long way. So we've got by rule five, we've got eight rows. By rule six, we put in our truth values for P, Q and R. By rule seven, we copy and paste in for P, then we copy and paste in for Q, then we copy and paste in for R. Notice, there's just no need to be creative. This is a purely mechanical way of finding the solution. And that's just fine, right? You don't need to be creative with truth tables. That's part of what makes them really neat is you just follow the steps and you gotta fill that, you don't need to be creative. Okay, so let's just take a look at the conclusion. Just because we're looking for invalidity and we want those rows where the conclusion is false. Well, we've got a lot of rows here to choose from. So we've got was it one, three, five, and seven? Okay, well now we'll take a look at those rows then where the conclusion is false. And we'll see if we can't figure out how the premises can be true, right? So, sorry, then we, sorry, we fill in the truth this time. And you work with the innermost parentheses outward, right? Work with the innermost parentheses outward. Well, with row three, so rows one, three, five, and seven, we fill out the truth assignment for the conjunction, which is the consequence of the second premise. This is a mouthful, right? The conjunction, which is the consequence of the second premise. So we've got that TFF and F. And then we do fill out the rest for the conditional, right? So while we got T, a true antecedent and a false consequence, well that means that three, five, and seven, the second premise is false. So that's not gonna, so we don't need to really worry about those rows anymore. We only need to worry about row one. Well, row one, the second premise is true, and the second premise is true. All right, well now let's go to the first premise. We've got our two truth values for P and Q. Then we apply the rule for the negation. So we've got false and false. Well then that means that that first premise is false. So again, there shouldn't be a surprise by now. We've proven it one way already. But if you wanted to fill out the full table, this is the way to do it. And like I said, just for kicks and grins, just for the sake of illustration, we already know that it's valid, right? Because we can't find a row where the premises are true and the conclusion is false. But just to fill it out, you can see the steps that I'm taking here to complete the truth table. And we could just keep going with this and find out that there is indeed not, the only rows where the first premise is true is in rows seven and eight. And in row seven, the second premise is false. So we don't look at that anymore. Row eight, both the first premise and the second premise are true. But in row eight, the conclusion is also true. So we don't have a single row where it's all true premises and a false conclusion that means this argument is valid. It's valid. All right, that went to death. Really worked on the tables there for that problem. Well, let's try another one, right? And again, we'll just do an abbreviated table just to show you how it's done. So we put in our variables p, q, and r, and we've got our first premise. If p then q, separate with a gray line to indicate that's a premise. If q then r, that's our second premise. I mean now we've got a solid black line to show our conclusion and we've got a conclusion if p then r. Okay, remember, we're looking for invalidity. So to determine whether it's invalid, we assume, right? We try to figure out a way that the conclusion is false, right? And since it's a conditional, there's only one way for a conditional to be false and that's with a true antecedent and a false consequent. Well, this gives us the truth values for p and r. So we go back to our assignments and put t for p and true for p and false for r. Then we assign that to the rest of the premises within the argument following rule seven, we got a truth assignment for p in the first premise and the truth assignment for r in the second premise. All right, now what do we do? Well, remember, we're looking for a row with all true premises and a false conclusion. All right, well, look at that second premise there. All right, we got a false consequent, okay? Well, we want this premise to be true and with a false consequent. There's only one way, right? Remember for the truth conditions for conditional, either antecedent is false or the consequent is true. Well, the consequent ain't true. So we need a false antecedent, right? We need the false antecedent. Okay, so this gives us our truth assignment for q. So we go back, we fill in q for our truth assignments. And then by rule seven, we apply that to the first premise. And now we have a conditional with the antecedent is true and the consequent is false. Well, that means that this first premise is false, all right? So we can't construct a row where it's all true premises and a false consequent. And that's because of the way the truth conditions worked out, working backwards from the conclusion to the premises. So far, we've been real lucky. It's been kind of simple, but this one is also valid. This is also valid. And if you want, you can construct a full truth table to show that is valid as well. Well, let's try one more and this one looks a little complicated. And since it's so complicated, right, I'm feeling lazy and I want to be adventurous and crazy. So I'm just gonna do an abbreviated truth table. So I've got my P, Q and R for my truth assignments. And I put in the first premise, not P or not Q. And I have that gray line to indicate the supremacist follows. And I've got my second premise, if R then P or Q. Now I've got that solid black line to show that what follows is the conclusion. And so I have my conclusion, not R. All right, well, I want a false conclusion and all true premises. So it's a negation in the conclusion. So I assign the negation false by the truth conditions of a negation. That means that R is true. So I go back to my truth assignments and I fill in T for true for R. Then by rule seven, I go to the second premise where R appears. I'm gonna put in true for R. Okay, let's get us into work with. Now we want this second premise to be true and it's got a true antecedent. The only way for the second premise to be true with a true antecedent, since it's a conditional, is for the consequent to also be true. Okay, but this raises an issue because there are three ways for a disjunction to be true. Now you can see how the abbreviated truth table could be a little less abbreviated than you might have thought, okay? So remember, the consequent here, we want neither consequent to be true. And it's a disjunction. The only way disjunction, excuse me, there's three ways a disjunction is true. When both disjuncts are true, when one disjunct is true or the other disjunct is true, okay? So that's when both p and q are true, when only p is true or if only q is true. And those, these three ways that a consequent is true. Okay, well, since we've got now three different rows we're working with, let's go back to our truth, to our truth assignments. Fill in the rest for r, r now has all true, okay? Now we take the truth assignments for p and we fill that in. We got the truth assignments for p, that's there. Then we got the truth assignments for q, okay, great. Now, we take these truth assignments then for p, q and r and by rule seven, finish out, you know, finish fleshing out the conclusion there. It's for r, then we fill in the truth assignments for p and we got the truth assignments for q, okay? And we apply the conditions for a negation to p. And so now we got our, our, our truth assignments for the, for that negation for p, we do the same thing for q, okay? Well, that first row, both negations are false. So that's not gonna help us out. It's the bottom two rows that we have here where one, at least one of the disjuncts is true. So we've got two rows now where the, the premises are true. And the conclusion is false. All right, we've shown that this argument is invalid. Well, now we gotta identify the rows. Well, you can do a couple of different things, right? You can either construct the full table and figure out which rows are, are there, or you can go in the text and take a look at the truth tables that I've signed there. Since we're always filling out the truth assignments the same way, by rows five, six, and seven, we will always have the same rows. Well, if you go back and look, you find out that that's, those are the truth assignments for row three. And these, these are the truth assignments for row five. So then you go over here to your answer and mark rows, mark rows three and five as invalid, as invalid. All right, whoo, that got a little involved. You've got a little surprise ending there with the disjunction. All right, the next kind of problem is a two-stepper. Cuz now you've got an argument in English. You've got to identify the premises. You've got to identify the conclusion. You've got to translate them into a sequence and then determine whether the sequence is valid or invalid. So you kind of pulling together all the skills that we've learned so far. Okay, by rule one, remember way back with rule one, you signed the first atomic proposition P, the second atomic proposition Q, and so on. So that first proposition there, it is false that we know that any of our perceptual beliefs are true. Well, the atomic proposition there is we know that any of our perceptual beliefs are true, so that gets P, okay? The next atomic proposition, we are sometimes mistaken in our perceptual beliefs, so that gets Q. The next proposition is either it's false that we are sometimes mistaken in our perceptual beliefs or it's false that we know that any of our perceptual beliefs are true. Well, that's a disjunction of negations, we should figure that out by now. Well, the atomic proposition there is we are sometimes mistaken in our perceptual beliefs, well, we just saw that, right? So that's Q. By rule two, it gets the same assignment of an atomic, same letter for that atomic proposition. And the same thing happens for we know that any of our perceptual beliefs are true, again, by rule two. Okay, now we're gonna find the conclusions. Well, looking at these sentences, I really can't think of an example of where the conclusion is stated somewhere in the middle. I want to say this is true, I can't promise it, but I want to say it's true. It's at least the best bet, right? If you have no idea what's going on, the best bet is the conclusion is either going to be the first sentence or it's going to be the last sentence, all right? Well, looking at that first sentence, it is false that we know that any of our perceptual beliefs are true and the last sentence is that disjunction. And we don't really have any indicator words except for the second sentence. Look at that second sentence, we got that word for. Well, that indicates that here's a set of premises, right? So what's going on here is this paragraph is stated in the conclusion and this is for the following reasons, right? It gives those reasons. So the conclusion, right, is actually that first sentence. That conclusion is actually that first sentence. Okay, well remember our rules for this, following rule three, right? The conclusion is always listed last in our sequence. And our conclusion here is indication. So it's not P and then rule four, we list the premises in order, right? In order with a comma separating them. So that first premise is we are sometimes mistaken in our perceptual beliefs. That's our first premise, remember they got Q. And our second premise is that disjunction of negations. So here we go. We've got our argument symbolized as a sequence, all right? Now we take the sequence and determine whether it's valid or invalid. And if it's invalid, which row demonstrates invalidity? All right, well, I'm feeling lazy, so I'm just gonna do an abbreviated table. We've got our truth, the sign is P, we've got our variables, two variables P and Q. We've got our first premise Q. So one of the first times we've seen this, right? Where the premise is just a single atomic proposition. And that can happen, right? That can happen. The second premise is our disjunction of negations. And I've got a solid dark line there to indicate the conclusion and our conclusion is a negation, all right? So the first thing we do is we want to demonstrate invalidity. So we assume that the conclusion is false, right? We put that false there. Well, it's a negation, so we apply the conditions for a negation that gives us a true value of true for P. So we go back to our truth assignments and we fill in true for P. And then we go back, following rule seven, we go back to the argument. And everywhere P appears in the argument, we assign it as true, all right? It's only one place in this case, but hey, there it is, all right? Well, we want this premise here to be true, right? Sorry, now we apply the conditions for P. So P is the negation for P, so we've got F. Well, we want this second premise to be true, all right? And we've got a false disjunct. Now remember, a disjunction is true just in case at least one of the disjuncts is true. Well, one of them is false here, right? So we need that the other one to be true. And the only way for this disjunction to be true is for the negation, not Q to be true. So we assign it as true and using the truth conditions for negation, well, that means that Q is false. Okay, so our second premise is true. And now we've got our truth value for Q. So we go back to our, then we place that in for our, we'll put in the truth assignment for Q, and we fill it out for the rest of the arguments. Well, gosh, if Q is false, that means that there's no row where all the premises are true and the conclusion is false. That means this argument is valid, is valid. And you mark that as you'd have the same thing where you have the rows available to you and you have to mark the rows. So this argument is valid, okay? Well, let's try another one. So we use our rule one to start assigning atomic propositions. So we got that first atomic proposition, morality is justified by cultural belief. Then we keep going to the next one, still following rule one. Morality is justified by cultural belief. Well, by rule two, that gets the same assignment. We've got the next atomic proposition. Our culture is morally impeccable. There it is, that gets Q. The next atomic proposition, our culture is morally impeccable. That gets Q as well. The next atomic proposition, something that needs to be changed in our culture. Okay, that's a new atomic proposition. So we get an R. And then finally, there is something that needs to be changed, that needs to change in our culture. Well, there it is, that gets R again. Okay, so we've assigned our variables for our atomic propositions. Which one's the conclusion? Well, again, try to think of it. Is it the first sentence or the last sentence? Well, that last sentence has that word however, right? Well, that's not a conclusion, right? That doesn't tell us a conclusion. However, would tell us something, so think about it, right? Try and test it in your head for something that's an obvious one. Say, if an animal's a dog, then an animal's a mammal. My dog, my pet is a dog. However, my pet is a mammal. Well, no, that just sounds weird, right? That wouldn't work. We'd say, therefore, my dog is a mammal. Okay, that would work. So that word, however, tells us not something that's a conclusion, but it's gonna be one of the premises. It might be surprising, right? It might be something surprising, and that's something we expected, but it's a premise nevertheless. So our conclusion is that first sentence again, right? It's that first sentence again. And it's a negation. It's false that morality is justified by the cultural belief. Okay. So that following rule three, we list the conclusion in our sequence. And by rule four, we list our premises. So we got that first premise is if morality is justified by cultural belief, then our culture is morally impeccable. So we got Q, if P, then Q. Second premise is a disjunction of negations, and it's not Q or not R. And the third premise, I got three premises now. Wow, big time. Third premise is simply the atomic proposition of R. Okay, so again, I'm feeling lazy, so I'm just gonna do the abbreviated truth table. So I've got my P, Q, and R set up for my variables, all right? I'll list my first premise, if P, then Q. I got the gray line to indicate another premise after that, and we got the disjunction of negations, not Q or not R. The third premise is simply the time of proposition R. We got that solid black line to indicate the conclusion follows, and conclusion is not P. All right. So we're looking for a row of all true premises and a false conclusion. So we get that conclusion, and we assign it false. And since it's a negation, that means that P is true. So then we go back over to our truth assignments to give P as assigned true, about rule seven, wherever P appears in the rest of the argument, we also assign it as true, all right? Now we, so you might think, well, what do we do for the next step? Well, probably the easiest thing here, look at that atomic proposition R. R is one of the premises, and it's kind of the easiest to look at next, because we need all true premises. There's only one way for R to be true, and that's for R to be true. So it's assigned true. We go back to our assignments. We assign R as true, then by rule seven, we apply it for the rest of the argument. We got R as true over there. And there's a couple of different things we could do here. We can either look at the first premise because it's only way for a true antecedent to be, excuse me, it's a conditional. There's another way for a conditional to be true when the antecedent is true, and that's for the consequence to also be true. We can go that way, or we could just simply stick with R. I'm feeling like we started with R, so let's keep going with R. So R is true, and so we apply the conditions for the negation, and that gives R, the negation of R as false, right? And then we got a disjunction here, and we want the disjunction to be true, right? Since we're looking for invalidity, and the only way for a disjunction to be true with the least one false disjunct is for the other disjunct to be true. It's a negation. So we put the negation there for true. We're gonna get the negation there for true. Well, then we apply the conditions for an negation to Q, right? And so that now means that Q is false, and we go back to our truth assignments. We've got the truth assignments for all the atomic propositions. We take that F or Q, and place it over in that first premise following rule seven. Well, now we got a conditional with a true antecedent and a false consequence. That means that this, and this is the only possibility, right? The only possibility, that means that first premise is false. That means that this argument is valid. It is valid. All right, you've seen some of these practice exercises. Go ahead and give it a shot with the rest. Keep working. Work through those practice exercises to get you really good and confident, to get you to run through them, and you're probably being good shape at that point. So good luck, and keep thinking.