 Well, we've already seen one kind of complex proposition, that's negations. Negations could deal with something as small as a single atomic proposition. Well, now we're going to move on to conditionals. Conditionals need at least two atomic propositions. And the conditional, what it does is it expresses a truth relationship between these propositions. A truth relationship is a relationship between propositions such that the truth or error of one proposition means that another proposition is true or false, right? Or that the truth value of one proposition impacts the truth value of another proposition. Now, since we've got, and we're only going to deal with two propositions at a time. And since we're only dealing with two propositions at a time, now we've only got two truth values, right? That just gives us four possibilities for simple truth relations. Simple meaning that these are the smallest ones, and that we can build relationships out of combining these four kinds. So there's four possibilities, right? We start with a true proposition, and then the relationship means that another proposition is true or true makes true. That's one possibility, second possibility, that the proposition is true. And that makes another proposition false, true makes false. Well, those are the relationships when you got just truth. We're dealing with truth, let's try false. Where the error of one proposition means that another proposition is true or false makes true. Then we have the error of one proposition means that another proposition is false, or false makes false. So that's four possibilities. We've got true makes true, true makes false, false makes true, and false makes false. True makes true, it's called sufficiency. True makes false is contrary, false makes true is subcontrary, and false makes false is necessity. We're going to take these one at a time through the course of this book. First, for now, let's just deal with sufficiency. True makes true. So there are lots of propositions that make lots of other propositions true. So here's a proposition. Dr. Haugen is in Guadalupe River State Park. That's a proposition. If that proposition is true, that means that another proposition is true. Dr. Haugen is in Texas. Because Guadalupe River State Park is located in Texas. So Dr. Haugen is in Guadalupe River State Park, is sufficient for Dr. Haugen is in Texas. All right. How about this one? My pet is a dog. There's a proposition. My pet is a dog. That's sufficient for another proposition that makes another proposition true. My pet is a mammal. My pet is a mammal. So my pet as a dog is sufficient for my pet as a mammal. Let's think of one more example, right? That organism is a tree. I want to any one of these. I don't want these behind me. That organism is a tree. That proposition is sufficient for another proposition. That organism is a plant. So these proposition, you know, that one proposition is sufficient for another. That means that it first proposition is true. The second proposition must also be true. Now I want to warn you, sufficient, sufficiency does not necessarily go both directions. It can, but it doesn't necessarily go both directions. And what I mean by that is this, right? You know, Dr. Haugen is in Guadalupe River State Park. That's sufficient for Dr. Haugen is in Texas. Okay. But Dr. Haugen being in Texas does not necessitate. You know, that's true. But that does not necessitate that Dr. Haugen is in Guadalupe River State Park. So for instance, I'm going to leave here in a little while. I'll still be in Texas, but it won't be in Guadalupe River State Park. So sufficiency does not necessarily run in both directions. It can, but it doesn't necessarily run in both directions. That an animal is a mammal does not necessitate that as a dog. There's plenty of mammals that are not dogs. That an organism is a plant does not necessitate that as a tree. There's plenty of plants that are not trees. It can run both directions, but it doesn't necessarily run both directions. So sufficiency. The truth of one proposition means that another proposition is true. So I said that we'll express truth relations with these complex propositions. In English, we express sufficiency with a logical connective. If proposition, comma, then proposition. So I mentioned Dr. Haugen is sufficient. Dr. Haugen being in Guadalupe River State Park is sufficient for Dr. Haugen being in Texas. That's true. And we'd express that with if Dr. Haugen is in Guadalupe River State Park, then Dr. Haugen is in Texas. There are other ways to express sufficiency in English with these logical connectives. We have other logical connectives, kind of variations of them. So, you know, you could just use the word if. So when we have if, proposition, comma, then proposition, what follows the if, that's the antecedent, and what follows the then, that's the consequence. We could just have if. If my pet is a dog, comma, my pet is a mammal. And the if still represents, it still indicates the antecedent. We could just use, we can use only if and leave out the then. So, my pet is a dog, only if my pet is a mammal. The only if indicates the consequence. So you want to make sure you don't confuse these two. If you have just the word if that indicates the antecedent. If you have only if that indicates the consequence. So we express sufficiency with a conditional. We express sufficiency with a conditional. We have if, proposition, comma, then proposition. And then, so noticing, we link those two together. And again, this is called a conditional. All right, let's keep going. So we're still going to symbolize conditionals. Now before, when we symbolized negations, right, we just had the atomic proposition and a little minus sign right before it, no space in between them. For conditionals, we're going to have a space between the atomic propositions and the symbol. So, suppose we have a proposition, you've been using before, if my pet is a dog, then my pet is a mammal. All right, so follow the rules that we have so far. We've got two atomic propositions, my pet is a dog. So we symbolize that with P. And we'll symbolize my pet is a mammal with Q. Again, following the rules, right, that's the order in which they appeared. All right, the symbol we're going to use for conditionals is the greater than symbol. It's that little arrow looking sort of thing. And in the book, I gave instructions on how to find that on your keyboard. So to symbolize if my pet is a dog, then my pet is a mammal. We'll use P and Q. We'll provide P first, have P, space, the greater than symbol, space, one and only one space, and then Q. So P, greater than symbol, Q. And that's going to symbolize if P, then Q. That's going to symbolize our conditional. So we looked at how we're going to symbolize our conditionals. Now the question is, how are we going to handle these on the truth table? Remember last time we looked at under what conditions the negation is true or false, given that P is true or false, or P and Q are true, that sort of thing. Now we're going to look at this conditional if P, then Q. So following our rules so far, we've got two atomic prepositions. We've got P and Q. So that means that we've got four rows on our truth table. We've got four rows on our truth table. Again, following our rules for the truth assignments for P, the first two rows of P are true, the second two rows of P are false, and then Q alternates true, false, true, false. So we place our proposition, our complex proposition, if P, then Q. So we have P, we've got the greater than symbol, and then Q. Every atomic proposition and every symbol gets its own column. And this is pretty much true for everything that we're going to do with truth tables in this notation, the symbolic notation in the course. So again, following our rules that we looked at last time, we take the truth value of P and we copy and paste that same truth value in our truth table. And then we take the truth values for assigned truth values for Q and we copy and paste those under Q. All right, now what we have to do is figure out the truth values with the truth value of this complex proposition that expresses sufficiency from P to Q. So let's look at row one. Row one, P is true and Q is true. Okay, well this fits with the truth relation. The truth relation says if P is true, then Q must also be true. The truth of P makes Q true. Well then, in that case, in this row, the conditional is true. And if P is true, then Q is true. Then the Q is true, and that works. Okay. What about the second row? The second row of P is true, but Q is false. But our complex proposition says if P is true, P is sufficient for Q, that means if P is true, then Q must also be true. Well, Q is not true here. So at this row, our complex proposition, that's our conditional, is false. So at the first row, it's true. At the second row, it's false. And we enclose, since this proposition overall is a conditional, we enclose it within the parentheses. We enclose that within the parentheses. Okay. Well, that's the first two rows down. Let's look at the next two rows. Now the next two rows, P is false. Now you might be tempted to think then, well, if P is false, then the conditional is false. But that doesn't work. Because we have conditionals where the antecedent is false and the consequent is true. And we think that the conditional is still true. So for example, if Dr. Haugen is an enchanted rock state park, then Dr. Haugen is in Texas. Okay. Well, the antecedent is false. I'm not an enchanted rock state park. I'm in Guadalupe River state park. But the consequent is still true. So the antecedent is false and the consequent is true. Moreover, if I actually were an enchanted rock state park, yeah, that conditional would be right. Me being an enchanted rock state park is sufficient for me being in Texas. Okay. So we don't want to say that the antecedent, you know, that since the antecedent is false, the whole conditional is false. No, it doesn't work that way. It doesn't work that way. So we're going to keep that as true. If the antecedent is false and the consequent is true, the conditional is still true. Remember, the conditional only claims that the antecedent is sufficient for the consequent. It doesn't say anything about whether the antecedent is true or false. What about row four? Row four, we still have a false antecedent and we have a false consequent. You might think, okay, for sure this one has to be false. No, no. I mean, think about it. If Dr. Haugen is at Pike's Peak, then Dr. Haugen is at the top of Pike's Peak, then Dr. Haugen is at Colorado. Well, the antecedent is false and so is the consequent. I am currently in Texas. I'm currently in Guadalupe River State Park. But if that antecedent were true, being at Pike's Peak, I would be in Colorado. I would be in Colorado. So if the antecedent is false and the consequent is false, well, we don't want to say that that conditional is false. The only time a conditional is false is when the antecedent is true and the consequent is false because all the conditional claims is that the truth of the antecedent means that the consequent is true. So a consequent is going to be true at rows one, three, and four. At rows one, three, and four when we have P and Q. All right. Another way of saying this is a conditional is true. It looks weird to say it this way, but when you look at the truth table, this is how it works out. A conditional is true when either the consequent is true or the antecedent is false. I know that seems weird, but that's the way it works on the truth table. So a conditional is true when we just got P and Q. A conditional is true at rows one, three, and four. Or we say, or another way of saying this is a conditional is false when the antecedent is true and the consequent is false and true otherwise. That makes more sense. A conditional is false when the antecedent is true and the consequent is false and true otherwise. All right. So that's how we express a conditional on the truth table. Now, I kind of want to give you a little bit of warning here. So far, we just have P and Q, but we can have some pretty complex propositions as either the antecedent or the consequent. We've got atomic propositions in this case, but we could have complex propositions. We could have negations as the antecedent or the consequent. We could have what? Further conditionals, right? We could have a conditional on the antecedent and we can have a conditional on the consequent. We could have something like this. What? If Dr. Haugen is in Enchanted Rock, excuse me, if Dr. Haugen is in Guadalupe River State Park, then Dr. Haugen is north of San Antonio only if Dr. Haugen is in Texas. Right? That's a complex proposition where Dr. Haugen is in Guadalupe River State Park. That's the antecedent. The consequent is Dr. Haugen is north of San Antonio only if Dr. Haugen is in Texas. And that consequent there, that's a whole other conditional. And that's going to change the truth assignments. So now we've got P, Q, and R, so we've got eight rows. But that whole conditional is going to be true just in case the consequent is true or the antecedent is false. And that's how that's going to work on the truth tables. It seems a little counterintuitive, but when you figure out what sufficiency is, it maps out that way into the truth tables. And counterintuitive or not, that's what we got.