 Well, all right, we're finally getting to arguments. It'll take a little while, but we got there. So in this video, we're going to look at what it means to be an argument and to find the different parts of the argument, and we're going to take our first steps to evaluating arguments. You remember how we were talking about not every sentence is a proposition. There are interrogatives, there are imperatives, there are exclamations, but only at most declarative sentences are propositions. Well, not just every collection of sentences is an argument. Not just any collection of propositions is an argument. The propositions have to have the right kind of relationship, and it's a relationship between premises and conclusion. Premises necessitate the conclusion, or another way of saying this is, premises are sufficient for the conclusion. Let's just sound familiar because this sounds like conditionals. And really, a conditional is just kind of the model for all arguments. And a conditional, the antecedent is sufficient for the consequent. Well, in arguments, the premises are sufficient for the conclusion. Now, if you remember, sufficiency runs one direction. It can run both, but it primarily just runs one direction. Same thing here with an argument. The premises are sufficient for the conditional in rare cases, excuse me, sufficient for the conclusion. In rare cases, the conclusion is also sufficient for the premises, but that's, don't worry about that now. And this is important to realize when we're looking at some collection of propositions. We have to first determine whether it's an argument. And if it is an argument, what are the premises and what's the conclusion? What are the premises and what's the conclusion? And this is the difference. The premises necessitate the conclusion. The conclusion is inferred from the premises, but not necessarily vice versa. Now there are certain clues to help you figure this out. But probably one of the best ways to figure this out is to understand the meanings of the propositions. To determine the truth relations between the propositions, the atomic propositions as expressed with the logical connectives and the relationship between the propositions themselves to figure out what necessitates what. Now let's take practice and you'll get there. You'll get there. The more you work with these arguments, the more you work with propositions, you'll figure it out. But in the meantime, we're going to take a look at some non-meaning clues, right? Regardless of what the meaning of the sentences are, we're going to take a look at some clues independent of the meaning of the sentences to try to help us figure out what's the conclusion and what are the premises. All right, let's start with some obvious little tricks, little indications of which is the conclusion and which is the premises. In most cases, the conclusion is going to be the first sentence or the last sentence. In most cases. When the conclusion is the last sentence, there'll be, you know, an indicator word showing what the conclusion is. All right, therefore, thus, it follows that consequently. For these reasons, it is the case that these are supposed to indicate that this proposition is necessitated by the others. If the conclusion is first, and sometimes it is, or sometimes for a rhetorical effect, the conclusion is given first. The conclusion will have pretty much no indicator word, an indicator phrase, but the sentence is immediately subsequent to the conclusion will. So, for the reason that, since it's the case that, given that for these indicator phrases telling us that, you know, the first thing is true because these following things are true is the idea for these indicator phrases. Now, you know, for the homework and everything, that's pretty much all you really have to worry about. But, you know, in common English and common paragraphs, or you can see out there, there's going to be a whole host of other little problems. So, I mean, it's really, really bad writing to put the conclusion in the middle of the premises. It just is. Yeah, sometimes I've seen it happen. But that may not be the only time when you see the conclusion within the middle of a paragraph. Sometimes paragraphs have multiple arguments. Sometimes paragraphs have multiple arguments. So, to figure out what, you know, where the different arguments are, and what's the conclusion, what's the premises, is an indicator word should be a big help for that. Also, you know, just let me re-emphasize. It really, really helps to understand the meanings of the connectives and the relationship between the propositions to figure out what is sufficient for what, right, what necessitates what. There's one last little problem that sometimes pops up. Pops up, it's called the enthamine. Now, enthamine is a really nice, cool word, meaning an argument that does not list all the premises. And sometimes premises are not included because the author considers them to be so abundantly obvious that you don't need to list them. Or sometimes maybe it's for rhetorical effect, right. So, if I say something like this, what, I am human. So, I am a mammal. Okay. I mean, there's nothing really wrong with rhetorical flourish. But if you're going to understand what's happening with the rhetorical flourish and try to understand what's really important here, you need to be able to spot the missing premise. So, in this case, right, I got this statement, I am human. Okay. And the conclusion is I am mammal. All right. But then you need the connection from I am human to I am mammal, namely, and conditional, right. If I'm human, then I'm mammal. You're probably going to have to learn how to spot enthamemes. Well, in a lot of common, say, discourse, you're going to have to learn how to spot enthamemes and what's the missing proposition. But, you know, for this course, we're probably not going to really work on spotting the enthamemes until later on, right. I want you guys to be more familiar with the arguments before we get there. So, when you're looking for the conclusion, it's either going to be the first sentence or the last. And with the last sentence, there'll be some kind of indicator phrase and thus, therefore, so, it follows that, right. Some phrase indicating that this proposition is made true by the previous ones. If the conclusion is the first proposition, there's going to be some indicator phrase saying, you know, kind of saying, you know, four, or since it's the case, given that, right, where, you know, the conclusion is listed first, maybe, you know, again, for rhetorical effects, something like this, and then this indicator phrase is this, these are the reasons why this first proposition is the conclusion, or this first proposition is true. It's for these reasons. All right, so that's the difference between premises and conclusion, not only in terms of, you know, meaning, because premises make the conclusion true, right, are sufficient for the conclusion, but there's going to be some indicator phrases. Let's, what we're going to look at next is how to formalize the argument to give, you know, to separate the premises from the conclusion using our symbols. Let's take a look at sequence. A sequence is a formalization, symbolization of an argument listed in the premises and the conclusion. So we're going to have two rules for this, rule eight and rule nine. Rule eight says that you symbolize the conclusion last, you'll list the conclusion last, right, so whether it's an atomic proposition or a complex proposition, you know, you'll list it last. It's preceded by a space, two vertical bars, a space, and then the conclusion. It doesn't matter whether the conclusion is provided first in the English argument or last in the English argument. In a sequence, you always list the conclusion last. Rule nine tells us to list the premises in order as they appear in the English argument separated by a comma and a space. The exception being the last premise. The last premise does not have a comma, it just merely has the space and the two bars, right, the space and then the conclusion. So this is pretty straightforward, it's not complicated, conceptually speaking, but sometimes it can be tricky, right. It can be tricky in these cases where the conclusion is listed first. Because following our rules from earlier, at the conclusion is listed first, it gets the atomic proposition P, right, it gets the sign the atomic, the letter for the time proposition P. So let's just look at this, right. So if my pet is a dog, then my pet's a mammal. My pet is a dog, therefore, my pet's a mammal. Now this is straightforward. My pet is a dog is a sign P, my pet is a mammal is a sign Q. So following the rules, I have the conclusion last. My pet is a mammal, that's Q. It's preceded by a space in the two vertical bars and space. So we list the premises, right, with our symbolization for if P then Q, which is that P greater than symbol Q, comma P, those are the two premises, right, the two vertical bars and then Q. Okay, that's straightforward. But suppose the conclusions are listed first. My pet is a mammal, for it is the case that if my pet is a dog, then my pet's a mammal and my pet is a dog. Okay, well, now since my pet is a mammal is listed first, in that instance, following our rules, right, it gets assigned P. So the conclusion is P, following rule eight, we got the space, the two vertical bars of space in P, right. Following rule nine, we have Q greater than P or if Q then P, and they search in the Q, right. Q greater than P, comma Q, vertical bar, vertical bar, vertical bar, space P. So conceptually, these are not complicated rules, but you have to spot the conclusion right away, figure out where it's occurring in the argument, in English, and that it gets assigned a proper letter. And then you list that properly in the sequence, in the sequence. Okay, so these are our two rules for sequence. And this is going to formalize the premises and the conclusion. It tells us what necessitates what. Well, now we get to at least part of the whole point of this course. And logic is they tend to answer the question, what is a good inference? And we're going to evaluate arguments in this way to answer the question. We're going to evaluate arguments in terms of whether they are deductively valid. Now deductive validity just means, now I think I said this before, but it's worth repeating, an argument deductively valid just in case the truth of the premises necessitates the truth of the conclusion. Or in other words, if the premises are true, the conclusion must be true. Now arguments can be valid without true premises. If Dr. Hogan is a dog, then Dr. Hogan breathes water. Dr. Hogan is a dog, therefore Dr. Hogan breathes water. Not a single proposition there is true, and yet the argument is deductively valid. It has the right kind of relationship. The premises aren't true, but if they were, the conclusion would necessarily follow. So an argument is deductively valid just in case the truth of the premises necessitates the truth of the conclusion. Another way of saying this is, it's impossible that there's all true premises and a false conclusion. It's impossible that there's all true premises and a false conclusion. And again, at the risk of repeating myself again, this should sound familiar. This should look like a conditional and in a way it is, right? If we take all the premises together, that's the metaphorically speaking, the antecedent. And the conclusion, metaphorically speaking, is the consequent. A similar sort of relationship applies between the antecedent and the consequent with a conditional. And the premises then the conclusion and an argument, a deductively valid argument. So a sound argument is a deductively valid argument with all true premises. A sound argument is the deductively valid argument with all true premises. So when you're evaluating arguments, there's two ways that you can evaluate. You can ask first whether it's valid. And we're going to have ways of determining whether it's valid. The second question you can ask is, are all the premises true? All right, if the answer is yes, that is valid and the answer is yes, that all the premises is true, then not only is it a good argument, but the conclusion is also true. Now, I want to be careful here. Just because an argument has all true premises and a true conclusion, that does not mean the argument is sound. You can have a quote unquote argument. It's not really an argument in this case, but a quote unquote argument with premises that are all true and a conclusion that is true, but the argument is not valid. So for instance, if Dr. Haugen is a human, then Dr. Haugen is a mammal. Dr. Haugen is a mammal, therefore Dr. Haugen is human. All true premises, true conclusion, not sound. It's not deductively valid. So you can have all true premises and a true conclusion not be valid, so therefore it's not sound. Just because an argument is valid doesn't mean that it's sound. You have to have the true premises. Okay, so we've got deductive validity. We've got soundness. Now the question is, how do we determine whether an argument is valid? And in order to do that, we're going to use the truth tables. So what we'll do is we'll lay out, we'll take our sequence and put it in our truth table. Then we're going to start, you know, carrying out the truth assignments for the atomic propositions on over, determine the truth value of the premises, determine the truth value of the conclusion. And if we have a row on the truth table where we have a false conclusion and all true premises, that argument is not valid. If there's a row where the false conclusion and all true premises, that argument is not valid. If we don't have a row like that, if every false row, if every row that has a false conclusion has at least one false premise, the argument is valid. Okay, let's see how this works. Okay, so the question is, how do we determine validity using the truth tables? Or remember the truth tables do, they give us every possible combination of truth assignments for the atomic propositions. This tells us every possible, or what the truth values of the compound propositions will be. And this will tell us the truth values of the premises and the conclusion. So let's start with a, let's start with a real simple argument. You already know this is valid, but it's worth looking at. If P then Q, we assert P, conclude Q. So this is our sequence, P greater than Q's, the greater than symbol Q, comma P again, double vertical line Q. So we've got two atomic propositions, that means we have four rows in our truth table. The first two rows for P are true, the bottom two for P are false, Q alternates true false, true false. All right, we take our sequence, F, P then Q, assert P, conclude Q, we put that across the top of our truth table. Then we take the truth values of P, copy, paste them into the truth table wherever P occurs. And I take the truth values of Q, copy, paste those into the truth table wherever Q occurs. Now you notice P is a standalone premise. Since P is a standalone premise, it gets abbreviated. All right, let's truth values get abbreviated. Q is the conclusion, it's a standalone conclusion as an atomic proposition. So its truth values also get not abbreviated, they get enclosed in parentheses, get enclosed in parentheses. So where we have P as the premise, the truth values enclosed in parentheses, and we have Q as the as the conclusion, we enclose Q in parentheses. Okay, well now what we need to do is to fill in the truth values then for our conditional, that's what's left. So the first row P is true, Q is true. So the conditional is true there, we enclose that, we enclose the truth value there in parentheses. Second row P is true, Q is false. So that means the conditional is false on row two, we enclose the truth value in parentheses there. Rows three and four, the conditional, the antecedent is false. So the conditional is true in rows three and four. Okay, so now we filled out our truth table, it's pretty simple, we filled out our truth table. And we're going to look at the conclusion and the conclusion is Q. Conclusion is true at rows one and row three. Now you might think that you need to look at row one and row three, since we have a true conclusion, and make sure all the premises are true. No. All validity says is that if the premise is true, the conclusion must be true. That doesn't mean if the conclusion is true, all the premises are true. The inference doesn't work both ways. So for rows one and three, we can pretty much ignore, right? We don't determine validity there. We determine validity by looking wherever the conclusion is false and seeing whether all the premises are true. If we have a row where the conclusion is false and all the premises are true, the argument is invalid. Because validity means the truth of the premises guarantees the truth of the conclusion. Okay. So rows two and four. That's where Q is our conclusion. Rows two and four. Q is false. So we check the rows to see whether the premises are true. All right. Well, row two, P, one of the premises, it's true. Oh, this is starting to look bad. Starting to look bad because I got one true premise now. Well, look over the conditional. Well, the conditional is false. If you remember. In that row, the conditional false. So we have at least one false premise with a false conclusion. So this doesn't count against invalidity. Or it just doesn't count against validity. Okay. So row four, Q is true. And we look over our premises. Well, the conditional is true this time. Uh-oh. It seems like it's bad news. But P is false and it's a premise. P is false and it's a premise. So with this row, we only have two rows where P is false. Or the conclusion is false. Row two and row four. But in row two, one of the premises is false. Namely the conditional. In row four, one of the premises is false. Namely P. So this has no row where we have a false conclusion and all true premises. So this argument is valid. Is valid, right? Okay. So that tells us that that argument structure is valid. Every time we have a conditional and the assertion and the antecedent, we can infer the consequence. And that will be valid inference every single time. All right. Well, that, you know, so we figured out that argument is valid, but we aren't always going to be so lucky. So here's the question. All right. You know, what's it look like when our argument is invalid? Okay. Well, let's take something similar. Let's take the, let's take a different argument this time, but we'll take something similar. We'll have the conditional if P then Q. But this time we'll assert Q and conclude P. All right. This time we'll assert Q and conclude P. So we have our sequence P greater than symbol Q comma. We have Q double vertical line P. All right. That's our sequence. So let's put this into the truth table. Once again, we only have four rows. Since we have two truth values or two atomic propositions, first two rows of P are trues, bottom two rows of P are false. Q alternates truth values, true false, true false. Okay. We copy and paste our truth values for P everywhere in the argument. That's the antecedent to the conditional, and then the conclusion. Then we take the truth values of Q, copy and paste them over to every instance of Q in the truth table. All right. Now remember, we got one of our premises Q. One of our premises is just Q of the atomic proposition. So we can close that in parentheses. The conclusion is just the atomic proposition P. We can close that in parentheses. Okay. Now we look over at the conditional, same truth values as before. First row is true. Second row is false. Bottom two are true. We can close those in parentheses. Okay. So now we got our truth table figured out. Let's determine whether this argument is valid or invalid. Well, the conclusion is P. And rows three and four are false. Rows one and two are true for P in our conclusion. Remember, with the true rows, we really don't care about. That's not going to help us determine whether the argument is valid or invalid. We'll look at rows three and four. Okay. P is false there. So the question is, do we have all true premises? Well, we look at row four and Q as a premise is false there. Right. And Q as a premise is false. So the conditional is true, but Q as a premise is false. So that doesn't help us determine whether it's valid or invalid. We got one false premise that doesn't do it. Go to row three. Row three. P is false. The conclusion is false. Q is true. Uh-oh. And the conditional is true. Uh-oh. Right. So we have a row where we have all true premises and a false conclusion. That means this argument is invalid. Right. The truth of the premises does not guarantee the truth of the conclusion. The truth of the premises does not guarantee the truth of conclusion. And an instance of this argument would be something like this. If Dr. Haugen is a dog, then Dr. Haugen is a mammal. True premise. Right. Being a dog necessitates being a mammal. Dr. Haugen is a mammal. True premise. True premise. I am a mammal. But it's, you know, if we're going to follow the inference as given in the a sequence, the conclusion is Dr. Haugen is a dog. Right. So the premises are true. The conclusion is false. The conclusion is false. So this argument form is invalid. It's a classical, it's a classic fallacy and deductive inference that's called affirming the consequent. Affirming the consequent. All right. Now that, you know, so we have an instance where we use the truth tables to determine validity. An instance where we determine invalidity. And this is really simple and straightforward when we only have two atomic propositions. But when you start having three atomic propositions, therefore eight rows, four atomic propositions, therefore 16 rows. It can get, you know, pretty complicated really fast and you may or may not want to deal with all that. So there's still a shortcut way to do some of this if you want to. Right. You don't have to, but if you want to, if you do it the long way, that's fine. And if you, you know, just kind of quick and efficient with this is no problem. But there is a shortcut way. It just requires a little bit more thought and just a little bit more thought. Okay. The shortcut way is called an abbreviated truth table. Let's take a look at that next. Okay. So we're determining validity using truth tables. And if you build a full truth table, that's fine. That's not a problem. You could do that. But sometimes it could be a lengthy truth table very quickly. If, you know, say if you have a lot of atomic variables or there's more than one way for the conclusion to be false, the sort of thing, it can get pretty lengthy. You know, if you have three variables and, you know, that's already eight rows, so that gets pretty long, pretty quick. So an abbreviated truth table kind of cut, you know, taking a, cuts out a few steps. So remember with, with validity, we're just trying to find a row that is where the conclusion is false and you have all true premises. If you have a row where the conclusion is false and all true premises, the argument is invalid. If you can't find that row, the argument is valid. You find a row with all true premises and false conclusion, that argument is invalid. If you don't, the argument is valid. All right. So an abbreviated truth table just pays attention to the rows where the conclusion is false. Okay, well let's try a different argument then. Now in this case, there's only one row where the argument, where the conclusion is false and so it's going to make things really simple, but things can't get complicated. All right. So for, so let's, let's try our argument then. If p then q, therefore if q then p. All right. If p then q, therefore if q then p. All right. So we got one premise, one conclusion. Now, so to do this, let's just, let's just create our little like skeleton, our little skeleton truth table here. Let's just have our two atomic variables, p and q. Let's have our premise at p then q and the conclusion q then p, but don't fill any truth values at. So we just need to figure out when the conclusion is false. Well, the conclusion is a conditional. That means the conclusion is false just in case q is true and p is false. So let's put that in there. Right. This is the one way our conclusion is false. We got p, q is true and p is false. All right. Well, that happens on only one row of our truth table. That's row three. That's row three. Okay. So we'll take, we'll go back over to the beginning of our abbreviated truth table here. We got p and q. We put in our row number three just to keep track of our rows. We got p is false and q is true. p is false and q is true. Okay. Well, then following our rules, we take those truth assignments and we plug them into the argument. It's only the one, only one row. We got the one premise, f, p then q. Well, since p is the antecedent of p is false. Right. That conditional is true. Okay. Well, our one premise is true and the conclusion is false. That means this argument is invalid. Right. We just need to find the one row where the argument is invalid. Now, that's again, that's pretty straightforward. It's almost too simple an example. But think about the one we discussed earlier. Right. If p then q, given q, therefore p. All right. Well, the conclusion p is false on two rows. All right. It's false on two rows. So if you had abbreviated a true table with that, you'd have rows three and four. Right. Rows three and four. Go back, you'll go back to the, to the first column, put in our truth values for p and q at rows three and four. And p is false at rows, those two rows. And then q is true at row three and false at row four. All right. Then we plug in our truth, sorry, we copy and paste the truth values on over for p and then q. Since, you know, q is a premise by itself, it gets enclosed in parentheses. And at row four, q is false. So that doesn't help us determine whether it's valid or invalid. But row three, right, we got row three where q is true. Go over to the conditional. All right. Go over to the conditional. And at the conditional, p is false and q is true. All right. Well, then that makes that conditional true. So then at row three, the conditional, the both premises are true and the consequent is false. Now, like I said, this can get complicated really fast when you have three variables in your sequence. It can get, you can have more rows that are false when you have different kinds of complex propositions. So for instance, you haven't seen them yet but conjunctions is like p and q. Conjunctions are going to be false when one of the conjunctions is false. So if it's just for just two atomic propositions, that's three rows that are false. It can get more complicated than what we've seen so far. But abbreviated true tables can also save you some time, right, some time. All right. That does it for evaluating our arguments using validity. Try the homework and I'll see you in class.