 Okay, so we've dealt with definitions, we've dealt with terms, and that's great. We've gotten a good start, but we can't stick with terms for the rest of the semester. In fact, I mean, we could, and it's easy to talk about definitions for an entire semester, but we're not going to. Instead, we're going to move on to propositions, and propositions are what are true or false, or what are true or false, and terms compose propositions. Terms define or don't define, right? They define well or they don't. Propositions are what are true or false, and then arguments, as we'll see a little while, arguments are what are valid or invalid. So let's move on to propositions. So propositions. Propositions are what are true or false. What I think about this is to think about propositions compared to other sentences. So there's lots of sentences that are neither true nor false. Not every sentence is what is true or false. So how is the weather today? That's an interrogative. It's a sentence. It's a question. What is neither true nor false? It is weird to say it is false that how was the weather today? That's a nonsensical statement right there. You know, if I responded the weather was beautiful, well, that's something that's true or false. The weather was sunny. That's something that's true or false. Go stand over there. That is not a sentence that is either true or false. That's a command, an imperative. It would be weird. Again, it would be weird to say it is false too. Go stand over there. No. That's not something that's true or false. So propositions, we can think about it as sentences if you like, but that might be a little bit too narrow. But propositions are what are true or false. There are trees behind me. That's something that's true. There's an elephant behind me. That's something that's false. So propositions are what are true. These things that are either true or false. After this, we have to start thinking about atomic propositions. All right. So we've got propositions, but we're not done yet. There's actually lots of different kinds of propositions. Propositions are what are true or false. Well, let's start with atomic propositions. Now, atomic is your smallest, most basic proposition. It has a subject and a predicate. A subject is what's described. A predicate is what's describing. So what? This tree is leaning over. That's an atomic proposition. That tree, tree, that's the subject. Leaning over is the predicate. If you take just either one of those two, that tree, that's not a proposition. I might be pointing to something. You might understand that I'm trying to point to something, but that's not a proposition. That's neither true nor false. It's leaning over. That is also not true or false. It isn't until I combine the subject and the predicate together that I get an atomic proposition. It's the smallest proposition. You can't break it down any further and still have something that's either true or false. So that tree is leaning over. That is an atomic proposition. Now, I want to warn you that atomic propositions don't necessarily mean small. They just mean simplest. What that means is you can't break it down any further and still have something that's true or false. So what? Let's think of a lengthy atomic proposition. The sparse landscape located within the state park conveyed a sense of beauty and desolation at once. That's still an atomic proposition. The sparse landscape located within the state park, that whole thing is the subject. That whole thing is the subject. Conveyed a sense of what I say desolation and beauty. That whole thing is the predicate. It's proof of this, right? Just try and take one or the other. The sparse landscape located within the state park. That's not something that's rather true or false. Or if I just said, conveyed a sense of loneliness and beauty. I keep changing the predicate. Conveyed a sense of loneliness and beauty. That is also not something that is either true or false. So when you break that apart, you don't have a proposition. It's when only you take the whole thing, even though it's long, right? When you break it, simplify it, the area over there conveyed loneliness and beauty. That's a smaller proposition. But still, the original example that I gave, the sparse landscape located within the state park, even though it's got some modifiers in there, that's still just a subject. So when you put it all together, that's when you had the atomic proposition. So this is the smallest proposition that we're going to deal with. And this is what you have to keep an eye out for. When you're reading these sentences, you have to look out for the atomic propositions. You have to spot subjects and predicates. You have to understand the meaning of the sentence. This is not merely a mechanical affair, right? Google can't do this. You can. So you've got to get in there and comprehend what these terms mean. In order to spot the subject and the predicate. You know, I say Google can't do this, but I bet they're not far. In the meantime, however, you need to do this work. You need to read the sentence instead of having the computer think for you. Think for yourself. And comprehend, understand what's the subject, what's the predicate, what's being described, and what is doing the describing. You spot that and you've got your atomic proposition. All right. Well, we've compared atomic propositions or compared propositions to other kinds of sentences. Propositions to which true or false. And we talked about atomic propositions. And these are the smallest kinds of propositions. Well, we've also got complex propositions beyond that. And complex propositions join up atomic propositions in a variety of ways. Or they kind of modify atomic propositions in a variety of ways. So one obvious way is when you put two atomic propositions together. The weather was beautiful and the grass was green. The weather is beautiful is an atomic proposition. The grass is green is another type of proposition. You put them together with the word and it's called a conjunction. This is a complex proposition. It's taken two atomic propositions put them together. Now this is not good or bad, right? I'm not saying this is a bad thing, but you have to be aware of this. The word and is neither a subject nor predicate. We don't say things like, well, the tree and. The tree was ending. The tree ended very well. No, this is all nonsense. And doesn't have meaning as either a subject or predicate. It does something else. It joins up to atomic propositions or to, you know, propositions, right? Could be atomic and complex, right? Could be other one. It joins up propositions and says both of these are true. Both of these are true. So you've got to be able to spot atomic propositions within complex propositions. So I already give you one clue. I got ant. Other kinds of complex propositions would be disjunctions either or. Well, either I'm in a state park or I'm. Was it either I'm on public land or I'm on private land? Right? Those are the two possibilities. Disjunction says at least one of these are true. So one atomic proposition. I'm on public land. Here's another atomic proposition. I'm on private land. These are both atomic propositions and they're kind of, you know, connected together with either or. By the way, I'm on public land. Just so nobody freaks out, right? I'm not trespassing on somebody's property. So when you're looking at propositions, you've got to find the atomic proposition within the sentence. And again, either or, or even just the word or, it is not a subject or predicate. Right? The tree was oaring. No, that doesn't mean that way. Or was beautiful today. No, that has no meaning. That's a nonsense statement. Nonsense statement. Other kinds of connectives. These are called logical connections, by the way. We had conjunctions, both and. Disjunctions, either or. We have conditionals, if then. If I'm on public land, then I'm in a park. There we go. There's a, well, that's actually not a true conditional. There's lots of ways to be on public land, but not in a pain of a park. If I'm in a state park, then I'm on public land. There we go. If I'm in a state park, then I'm on public land. That's a conditional. That's a conditional. What the conditional says is if the first proposition is true, then the second is also true. Right? So if I'm in a park, if that proposition is true, the second proposition I am on public land is also true. The last one is not really a connective in the sense that it, you know, joins up to different propositions. It really just modifies a single proposition. It's negation. Right? Negation. And this is when you, when you're stating in some way or another form that the subject is not described by the predicate. Is not described by the predicate. I am not on private land. That is a negation. I am not on private land. Now here's the kicker, right? You might think that the subject is me and not on private land is the predicate. No. No. You have to learn to read the atomic proposition within that negation. Okay? The negation is actually on private land. I am the subject. On private land is the predicate. The negation says I'm not on private land. Notice that it doesn't say where I am. So, so we want to be careful, right? When you have a negation, the predicate is not the negated predicate and the predicate is what is negated. So, if we're, if we say for example, I am not on private land, subject is I or me. The predicate is on private land. But then it's negated, making this a complex proposition. It's negated, right? I'll try another example. I am not climbing a tree, right? What's the subject? Me. What's the predicate? Not climbing a tree. That's good. See, I just messed up. What's the, what's the subject? Me. What's the predicate? Climbing a tree, right? And then the negation is added on to that atomic proposition and the negation says, well, it's not, the predicate does not describe the subject in that way. So you want to be careful when you're reading, right? You want to be careful when you're reading. You want to make sure that you're able to spot the predicate, the subject and the predicate within a complex proposition. This is easier when you've got both and either or and if then. It's easier that way. It gets more complicated when dealing with negation. So just remember the first thing, right? If you spot a negation, the negation of the predicate is not the predicate. The predicate is the predicate, not the negation of the predicate. So here are some other negations. I am not eight feet tall. Subject is I. Eight feet tall is the predicate and not makes it the complex proposition, right? Elephants are not reptiles. Elephant is the subject. Reptiles is the predicate and then the negation is it. Not reptiles is not the predicate. Not reptiles is not the predicate. Reptiles of the predicate not makes it a complex proposition. So we've got atomic propositions, right? Or we've got propositions, propositions of what's true or false. When you spot the atomic propositions, this is the subject and the predicate. And we compare atomic propositions to complex, spotting complex. Most of the complexes is easy. You've got both and you've got either or. When you look at the negation, negated predicate is not the predicate, right? The predicate is the predicate. The negation makes that atomic proposition then a complex proposition. All right, now that we've got all that taken care of, let's move on to how we're going to symbolize these things. Truth tables, take one. Okay, so we're going to symbolize our arguments, right? When we're doing a formal logic, we symbolize our arguments. And that just means that we use letters to represent atomic propositions. And we use other symbols to represent basically complex propositions, the relationships between atomic propositions. And then we're going to have a way to symbolize the whole argument. What distinguishes premises from the conclusion. And the inferences that we make along the way to reach the conclusion. Okay, now this can probably get a little bit tricky. And usually teaching this course and dive right in, it confuses everybody. So let me just explain why we're doing this. So when we're, we've talked about terms, terms are either defined or they don't define. And you can evaluate terms by the kinds and the rules. If it's not one of the kinds, well, maybe you can find another kind of defining. Okay, but you'd better, it better at least adhere to those rules, right? Those rules better be followed. And if it doesn't follow those rules, well, it's not a good definition. We've got propositions, propositions are either true or false. And there's lots of ways to figure out whether propositions are true or false through investigation, through conceptual analysis. There's lots of ways to go about doing this. But basically a proposition is true just in case the subject is, when you're dealing with atomic propositions, right? When the subject is accurately predicated by the predicate. And it's false when it's not. When you deal with atomic propositions. And if you've got complex propositions, well, you know, whether they're true or not, that changes depending upon the relationship expressed by the complex proposition. But we'll just stick with the atomic proposition for right now. So terms either define or don't define. Propositions are either true or false. That's how we're evaluating propositions, how we're evaluating terms. And arguments are either valid or invalid. Arguments are either valid or invalid. That's how we are evaluating arguments. So valid means, and we're talking about deductive validity, right? I know we use valid in a lot of different ways in popular talk, you know, pop talk these days. Usually we mean something like you're entitled to that belief or it's well supported or maybe something like, or it's not offensive, right? That's how we talk about valid versus invalid. We're not dealing with that in deductive logic, right? With deductive logic, an argument is deductively valid just in case the truth of the premises and t's the truth of the conclusion. Or in other words, an argument is deductively valid just in case, you know, if the premises are true, then the conclusion must be true. Or another way of saying it's impossible for the premises to be true and the conclusion false. So with the deductively valid argument, if the premises are true, the conclusion must be true. Now we're not going to investigate whether arguments are valid or invalid by figuring out whether the premises and the conclusion are all true. No. In fact, you can have an argument where the premises are all true and the conclusion is true and yet the argument is not valid. So here's one. If a, what? If an organism is a tree, then an organism is a plant. That organism is a plant. Therefore, that organism is a tree. Both the premises and the conclusion are, both the premises are true and the conclusion is true. If an organism is a tree, then an organism is a plant. That's true. That organism is a plant. That is also true. That organism is a tree. That is also true. But the argument is not valid. Is not valid. And it instantiates a kind of fallacy called affirming the consequence. In case you're wondering. But we're getting a little bit ahead of ourselves. And the reason why it's not valid is because an argument of that form, an argument of that form, we have a conditional, you assert the consequent and therefore you infer the antecedent. That is invalid. You can make mistakes with that all the time. If you're enrolled as Senator Tony College, then you're a member of Senator Tony College. I'm a member of Senator Tony College. Therefore I'm enrolled as Senator Tony College. No. I'm not enrolled as Senator Tony College. I'm a faculty as Senator Tony College. I teach as Senator Tony College. So I'm a member of Senator Tony College. That's how I'm a member of Senator Tony College by being faculty. Not because I'm enrolled as a student. So that form of the argument, that's invalid. That form does not guarantee a true conclusion given true premises. So what we're trying to do is to figure out whether these arguments are valid or invalid. We're not going to figure that out simply by discovering whether the premises and the conclusion are all true. That's not going to do it. That's a mistaken idea about evaluating arguments. Rather, we are going to look at the relationship between the premises and the conclusion. Now in order to do that, we need to look at all the different ways or the possibilities of the truth of these atomic propositions. And to do that, we'll use a truth table. Now remember, propositions are either true or false. So we'll lay out, starting off with our truth tables, we'll lay out all the different possible combinations of our atomic propositions as either true or false. As either true or false. And this will help us determine whether the argument is valid or invalid. Now we had rules for defining. Well, now we're going to have rules for constructing our truth tables. And we're going to do this because we're all going to construct our truth tables in the same way. If five of you work on a truth table, I should be able to, you know, and you're working independently, all five following the rules should look exactly the same, right? Whole class should look exactly the same. Okay. So we're going to have some rules for constructing these truth tables. Let's take a look at those. Okay. Rule number one for constructing our truth tables. First, you have to identify all the atomic propositions, right? You're reading a passage, find those atomic propositions. Great. Rule one, symbolize atomic propositions with letters p, q, r, st, u, v, w, x, y, and z. I doubt we're going to get to an argument greater than u. All right. Don't worry about that. But this is what's going to symbolize atomic propositions, p, q, r, s, t, u, v, w, x, y, z. All right. That's rule one. So you read your passage. And I've even had students say they're going with a highlighter and they highlight the atomic proposition. Rule two, if you symbolize an atomic proposition with p, every instance of that atomic proposition in a passage also gets p. Okay. So if you find an atomic proposition, trees are plants. In a passage, every time that proposition pops up in that passage, it's also symbolized as p. Atomic propositions are going to pop up probably repeatedly in any given argument. And you're going to keep track of that atomic proposition with one and only one letter. By the way, keep in mind how negations could trip you up here. You want to make sure that if you have trees are plants, some on the argument, and some other place in the argument or passage that says trees are not plants, the atomic proposition there is still trees are plants. It's just negated. So that atomic proposition, trees are plants, is there even though the sentence says trees are not plants. All right. So that's something you got to be careful about. That's going to probably be the biggest trip up. You know, just have alarms go off whenever you see a negation. Ding, ding, ding, ding, ding. There's something going on here. And just recognize that you got an atomic proposition embedded in the negation. So rule one, atomic propositions are symbolized as p, q, r, s, t, and so on. Rule two, if an atomic proposition gets assigned a letter, it gets that same letter throughout the rest of the passage. All right. So you want to consistently label your atomic propositions. Okay. So the first two rules deal with finding the top of propositions and symbolizing them. The second part deals with the rows of our truth table. So remember, we're trying to give all possible combinations of true and false with these atomic propositions. Okay. So rule three, that deals with the number of rows assigned in a truth table. The number of rows is equal to two to the power of the number of atomic propositions. Now, I don't mean, you know, you're breaking up with the passage. I mean, you know, p, q, r, and s, right? You know, how many distinct atomic propositions there are. So if trees or plants is one atomic proposition and trees are green as another atomic proposition, that's just it. It doesn't matter if you've got a wide combination of atomic propositions within the passage embedded within complex propositions, right? So the number of rows in your truth table is equal to two to the power of the number of atomic propositions. So if you've got one atomic proposition, here's a complex proposition with one atomic proposition. Either I'm on public property or I am not on public property. That's a sentence with one atomic proposition. And the atomic proposition is I'm on public property. And it's a disjunction with a negation of it, right? So if an argument has one atomic proposition, it has two rows, right? It's assigned the letter p. It's following rule one and rule two. And it has two rows. And it would just be true or false. If we have two atomic propositions, right? The number of rows is two to the power of two propositions. Well, that's four rows. Four rows. If we've got three atomic propositions, right? That means we have eight rows in our truth table. You see how the number of rows can grow pretty quickly? If we've got four atomic propositions, that's 16 rows. If we've got five atomic propositions, that's 32 rows. If we've got six atomic propositions, that's 64 rows. And I'm not going to go further than that. You can see what's in the book, right? Figure it out for yourself, right? So the number of atomic propositions is going to increase the number of rows in our truth table. All right. So rule one, atomic propositions are symbolized with PQRSTUVWXYZ. Rule two, if an atomic proposition is signed a letter, it keeps that same letter throughout the rest of the passage. You always assign that atomic proposition with that same passage. And again, I've seen students going with a marker or digital highlighter and they get the first atomic, second, and they keep with that color through the rest of the passage. Rule three, the number of rows to the truth table is equal to two to the power of the number of atomic propositions. All right. Rule four, this is how you assign the truth values. This is how you assign the truth values for our truth table. You take the top half of the row. So you take the leftmost row, the leftmost row, right? You assign the top half true and the bottom half false. So this is really easy when you have one atomic proposition because you've got two rows. The first row is T, the second row is F. Boop! Suppose we have two atomic propositions. The leftmost row. If we have two atomic propositions, we've got four rows. You go to the leftmost column, the leftmost column. Rows one and two get true for P. For P, rows three and four get false. Moving over one row to the right, you take just that half that gets true. You take the top half of that and it gets true and the bottom half gets false and you follow that pattern for the rest of that column. So for four rows, rows one and two for P get true. Rows three and four get false. For Q, we've got true, false, true, false. Suppose we have three atomic propositions. P, Q and R. That's eight rows. P, you go to the leftmost column. P, rows one through four get true. Rows five through eight get false. Go over to Q. Just take those rows that have true for P and divide it up there. So that'll be just before rows. Well, then for Q, rows one and two get true. Rows three and four get false. And repeat it on the way down. Two true, two false. Then R gets true, false, true, false, true, false, true, false. Suppose we have four, four variables. P, Q, R and S. Now that is 16 rows. So you go over to P, the leftmost column, you go over to P. Rows one through eight get true. Rows nine through 16 get false. Now just stick with those first eight rows that you had for P. Go over to Q. Rows one through four get true. Rows five through eight get false. That's the pattern. Copy it, paste it all the way down. R. So Q had four rows, four. R, four rows is true at the top there. So R, rows one and two get true. Rows five, three and four get false. Copy that pattern all the way down. S then will just be true, false, true, false, true, false, true, false. So if you follow these rules, the leftmost column, the first half will be true, the bottom half will be false, the rightmost column will always be true, false, true, false, true, false, true, false. Now when you do it this way, you will find every, you will provide every possible combination of true and false atomic propositions. And that's how we're going to start assigning the truth value towards atomic propositions and our truth table. Let's get this skill down now. Sort of identifying the atomic propositions and how to symbolize, how to assign the truth, how to symbolize the atomic propositions and assign truth value to them and our truth table. And we'll get to filling out the rest of the truth table in later chapters.