 Hi, welcome back to Fill 320, Deductive Logic. I'm Matt Brown, your professor and today we're going to be talking about symbolization in SL. We learned last time about the basic connectors of SL, including how they can be used to translate or symbolize various kinds of sentences. In this lecture, I'll show you a variety of example and talk about how to think through the translation process. Let's start with one we saw last time, but didn't talk very much about, which is unless, and this is described in detail in Section 2.3 of the textbook. Unless you wear a jacket, you will catch a cold. You will catch a cold unless you wear a jacket. We might say either of these sentences is a version of, unless JD. So how do we symbolize this? Well, one of the things the sentence says is that if you don't wear a jacket, then you're going to catch a cold. We can think about it this way. If not J, then D. But unless also means that if you do not catch a cold, then you must have worn a jacket. It also means this. What it doesn't mean is something like this. If you wear a jacket, then you're not going to catch a cold. You might still catch a cold. You don't know. Could other things could happen. So that's not going to do it. Actually, what is useful is that these other two conditional statements are logically equivalent to each other, and to a simple or, a simple disjunction. Now, as Magnus says in the textbook, you might worry that the or here should be an exclusive or. But the sentences actually don't exclude the possibility that you might have both. That is, you might wear a jacket and catch a cold. So jackets don't always protect you from all the ways you might catch a cold. So it doesn't need to be an exclusive or. Inclusive or will do it. So this is the easiest way to remember an unless statement. You don't have to get worried about the order of the statements as you do when you are dealing with your conditional forms. And since they're all equivalent, it's a fine way to translate it. So that's some of the thought process we might go through as we think about how to do these translations. I'm going to take you through an example now from the practice exercises. This is from part A. I'm going to take you through a few examples here just to, again, think a little bit more about how we use these, how we do this translation, symbolization process, right? First off, let me say this part here at the beginning of the question, this is what we call a symbolization key. This is what allows us to move from English to SL or vice versa. It tells us what sentence letters we use to represent our atomic sentences, right? So let's look at this problem. We have M stands for those creatures are men in suits. C, those creatures are chimpanzees. G, those creatures are gorillas, right? So our first one, number one, those creatures are not men in suits. That's a pretty straightforward negation of M, right? So we use not M, right? Question two, those creatures are men in suits or they are not, that is those creatures are men in suits or those creatures are not men in suits, right? Here we've got a simple kind of M or not M statement, right? So the disjunction and the negation are sufficient for those. So you have to think as you do the translation process, how does the they are not relate to our atomic sentences? We've got some, you know, we've got some pronouns here we gotta connect up, right? Let's look at number three. The either or here, right, is equivalent to those creatures are gorillas or those creatures are chimpanzees, right? It doesn't look like two sentences, right? You just got two nouns, gorillas or chimpanzees, but we have to sort of expand it out into those creatures are either, those creatures are gorillas or those creatures are chimpanzees. And then we see we can translate it in this way. Let's look at another example, not from our practice exercises, our homework problems, right? These are some examples from Hurley's concise introduction to Logic textbook. So credit to Hurley. I want you to pause the video and look at these three questions and try to work them out on your own. Create your symbol key and try to try to give the translations for these three sentences. Let's see how you did. Let's start with number one. How are we gonna create our symbolization key for this statement, this sentence? Well, we've got a simple sentence here. Pharmaceutical makers conceal test results. Let's use C to represent that, right? And then they are subject to substantial fines, right? We say pharmaceutical makers are subject to substantial fines. Cash out our pronoun there. We'll use the sentence letter F to represent that atomic sentence. And then we've got a simple if then, if C, then F, right? Let's look at number two. Here we've got a both and, right? Psychologists and psychiatrists do not both prescribe antidepressant drugs. So we've got the both and there. So these conjunctions are pretty safe bet. We also have this not, right? So which do you think is the main connective? That's what I want you to think about. Let's create a single symbolization key. Psychologists prescribe antidepressant drugs, right? O, we use O for that. And psychiatrists prescribe antidepressant drugs. We'll use I for that, right? The not remember is a negation, right? So we don't wanna put that into our atomic sentences. We wanna pull that structure out with the logical connective. So, and then let's go back to the question. What is the main connective in the sentence? I think what the sentence is trying to tell us here is that it's not the case that both psychologists and psychiatrists do this thing, right? Prescribe antidepressant drugs. Rather than saying it is the case that both of them do not do it, right? So that is to say, I don't think it's, I don't think our main connective is an and. I think our main connective here is the negation. Negation goes outside of the parentheses here, right? That's saying it's not the case that both psychologists prescribe antidepressant drugs and psychiatrists prescribe antidepressant drugs, right? And so the not is our main connective. It applies to the conjunction, which is inside. The other way we might translate is we might say not O and not I, right? But I don't think that's what the sentence is saying, right? It's possible that one or the other could under this translation, and I think that's the right way to go. Let's look at number three. Human life will not perish unless either we poison ourselves with pollution or a large asteroid collides with the earth, right? We've got an unless here and remember with unless the easiest translation is just an or. I think this is the main connective of this sentence, right? And then we've got a basic atomic sentence here. Human life will not perish. Let's use L to represent that. We poison ourselves with pollution, right? That's another atomic sentence. Let's use P for that. And a large asteroid collides with the earth, right? We'll use A for that atomic sentence, right? So we got three sentence letters. We've got an unless as I think our main connective. We've got another or here as well, right? But that or is gonna be connecting these two things on the one side of the unless, right? I think we finish it out like this. Human life will not perish or we poison ourselves with pollution or a large asteroid collides with the earth. Because of the way that or works as we'll talk about in future units, it doesn't matter too much which way we translate where we put our parentheses or even if we have parentheses. But for now, I think this is the best way to think about how to translate this. And if we had some other connectives in here like a conjunction or a conditional that would matter, the order would matter more. Let's look at this argument. We can translate or symbolize not only sentences but arguments, right? And that's gonna make a big difference to us as we think about the use of formal logic, right? So if Socrates is a man, then he is mortal. Socrates is a man, therefore Socrates is mortal. Very standard kind of argument, common example, right? Let's look at it here. So let's create a symbolization key. Let's use S for Socrates is a man. Let's use M for Socrates is mortal. We've got a simple if then, if Socrates is a man, then Socrates is mortal. Socrates is a man, therefore he is mortal. That's all it involves. That's all symbolizing an argument involves. Let's look at a more complicated one. There'll be rain only if it is cloudy. If it is cloudy, then there is moisture in the air. There is moisture in the air if the wind is blowing. Therefore the wind is blowing if and only if there will be rain. There's several different kinds of conditional statements here, ifs and only ifs. So I want you to pause the video for a second and see if you can figure this out on your own. All right, have you got it? Let's see how you did. So first we need our symbolization key, right? Let's use R to stand for there will be rain. Let's use C to stand for it is cloudy. We use M for there is moisture in the air. We use W for the wind is blowing. I believe that colors all of our atomic sentences that are embedded in this argument. Now how do we symbolize the argument? Well, we start with there'll be rain only if it is cloudy. Remember that the only if goes in this direction. If it is cloudy, then there is moisture in the air. That's pretty straightforward. There is moisture in the air if the wind is blowing. There's no only in that statement. So the if goes in this direction, right? It reverses the order from the way it is in the English sentence. And then therefore the wind is blowing if and only if there will be rain. That's a by conditional. So that goes like that. Now this is not a good argument by the way. If you think about how the argument works, it doesn't actually really do the job, but it's a good translation of this bad argument into SL. So those are some examples, but the best way to learn this process is to try it yourself. So what I want you to do is I want you to give the practice problems a go and let me know if you have any questions. But before I end the lecture, let me show you how this process works in Carnap. Just give you a few little examples and let you go from there. So here we're looking at the chapter two, part A, practice exercise. And I've put some general instructions here just to give you a sense of how this works and including the different ways you can use your keyboard to represent our main connectives. So we talked about these problems already at the beginning of the lecture. So you wanna give it a go, right? What you need to do here is you need to first, if there's text in the box, you need to delete that and then you try out some symbolizations. So those creatures are meant in suits. Let's put the M in there. You can hit enter at any time to check and it'll say, okay, well, not quite. Let's try again. I forgot the negation. Those creatures are not meant in suits. Let me scroll back up and remind myself how negation works. I can use a minus or a tilde or a not. So let's use a tilde and then check it. Perfect match it says. So then, now don't forget for every question you need to hit submit. Oh, I already did it before. So, but that's my go. How about this one? Those creatures are meant in suits or they are not. So those creatures are meant in suits. That's M or they are not. How do we do the disjunction? Oh, we can either do, we can do like this. The using the slash and the backslash on our keyboard or they are not. That is a negation and M. Let's check it. Correct, Karnap says. And then don't forget to submit and that says I submitted it. Okay, so that's how we do problems in Karnap. So give it a go on your own. Let me know if you have any questions or any issues as you try to work on it. And I will see you in the next lecture. Bye.