 Hello, and welcome back to Fill 320, Deductive Logic. I'm Professor Matthew Brown, and today we're starting Unit 5. We'll be talking about Quantified Logic, or QL, and we're gonna talk about the basics of QL in this lecture and how we use it to symbolize and translate certain English statements into this new system. So let's start by talking about QL, what it is and why we need it. So there are certain logical structures in our language, structures including quantities like all, every, some, and none that are not captured by SL. There are also relations between objects and predicates, objects and properties or objects in relation to other objects that are not captured in a sentential level logic, which is why some accounts or some texts called this system predicate logic, right? And in yet other books and other discussions, it's called first order logic. And historically, it's a very important logical system. Here's an example of arguments that cannot be captured as well by SL as by QL. So this is perhaps a familiar one, Socrates is a man, all men are mortal, therefore Socrates is mortal. Seems like a valid argument, that is, if the premises are true, it seems like the conclusion must be true. But there are no logical connectives in any of the sentences, no if, then, no and, or none of those kinds of connectives that we used in SL. So these will be represented as atomic sentences of SL. Here's another example with the same issue, right? No person is an island, every person is mortal. So some persons are not islands. That may not be a great argument, but there's still structures, no, every, some here that are not captured by SL. So let's talk about the features of QL and its components in order to start to understand what QL does for us. Now, I should say from the get-go, everything that's part of SL, all of those connectives are also part of QL. The conjunction, disjunction, conditional negation, they're all already part of QL. Here are the other parts of QL, right? They're singular terms or what we call constants that we represent with the lower case letters, A through W, optionally with subscripts, right? So you could have them with or without subscripts. Singular terms refer to what we might call proper names in English, Socrates, Beyonce, the Eiffel Tower, Carbondale, these are all, we're gonna represent all of these sorts of proper names with constants, right? Lower case letters. And then we have variables. These are the lower case letters X, Y and Z with or without subscripts. They don't stand for specific things with proper names, but for unknown objects. We also have predicates, what we call predicates. These are capital letters, A through Z, with again, or without subscripts. These are properties or relations, right? They go before some number of singular terms or variables. Then we have the quantifiers, we have two in QL, right? One is the universal quantifier, which we represent with an upside down A. It means for all and the existential qualifier, which we represent with a backwards E, which stands for their exists. So let's look at some basic examples. If we have for all X, PX, that means that for all X, X is P or for everything that we're quantifying over, it is P, P is true of it. There exists an X, PX means that there exists an X, such that X is P. We also have this kind of, we have this exchange, right? This equivalency between the two quantifiers, right? For all X, A, where A is some arbitrary sentence is equivalent to, there does not exist an X that is not A. And also, it is not the case that for all X, A, means there is some X, such that not A. Another important feature of QL is that we have to specify a universe of discourse whenever we are making translations to and from English. That means that we have to specify what kinds of objects our variables are going to quantify over, right? We might, for example, have a universe of discourse that is animals, if we're just talking about animals or living things, right? We might, it really depends on what it is we're trying to translate and we'll look at some examples. Here are some common translations in QL. These are very common kinds of things that you're going to see. Whenever we have something like all SRP, all men are mortal in our earlier example, or every SSP, or SRP, men are mortal, where it's sort of understood that there aren't exceptions, we use universal quantifier for all X. And typically when we're relating two predicates with the universal quantifier, it's gonna be a conditional. In all of these cases, it's S first then P, if S then P. Remember our old friend only, flipping things around, it does it here too. Only SRP, we're gonna translate as for all X if PX then SX. No SRP, we could do two different ways, right? One is to say that for all X, if it's S then it's not P, right? Another is to say there does not exist an X that is both S and P. Some SRP is gonna be translated by the existential quantifier, right? There exists. There is just an X that is SX and it's PX. And note that when you use the universal quantifier, you're typically doing a conditional. When you're using the existential quantifier, you're usually doing a conjunction, very common. And then finally, some SR not P, you would translate in this way, right? There exists an S such that SX and not PX. Let's look at some examples, right? So here I've got a symbolization key at the top. First thing we start off with in our symbolization key is the universe of discourse. That's what our variables and quantifiers are going to quantify over. Then we have a number of predicates, right? AX stands for X as an organism, DX stands for X as a dog, CX for X as a cat, and FX for X as four legs. And so take a second to think about how would you translate these three sentences into English? You can pause the video if you need to. So here's the way that I would translate these. For all X, AX, I would translate as all animals are organisms. Now note, animal is not one of our predicates, but it is embedded in our universe of discourse, right? For all X in this case means for all animals, right? And so this sentence naturally translates as all animals are organisms. For all X, if DX then FX, I would translate as all dogs have four legs. D is dog, F is has four legs. So this translates to all dogs have four legs. And then this last sentence, there exists an X, CX and not FX, right? I would translate this as some cats don't have four legs, right? There are some X, right? Such that it is a cat and it does not have four legs. We don't know how many, but there are some. That's one example. Here is another series of examples. I'd like you to pause the video and try to come up with how you would translate these from English into QL. All right, let's see how you did. Zazzles is not a cat. So Zazzles is a proper name. We're gonna use a lowercase letter Z to represent Zazzles and we'll use a capital C for the predicate cat. X is a cat, right? Zazzles is not a cat. I would translate this way, not CZ. The predicate cat, capital C is applied to Zazzles, lowercase Z, and then it's the negation of that. It's not the case that Zazzles is a cat. Okay, number two, there are no new predicates. Just telling us something is a cat. I would use the existential quantifier here. There exists an X, CX. And likewise, something is not a cat. I would say there exists an X such that not CX. You might be tempted to put the negation in the front of the quantifier, not exists X, CX, but I think that's not right because that would say it is not the case that something is a cat, right? Or effectively nothing is a cat. If there's something that isn't a cat, it's best to say there exists an X such that not CX. It is not the case that something isn't a cat. We would translate it this way. It does not exist X such that not CX. You could also, this is equivalent to saying everything is a cat, right? Which would just be for all X, CX. So four and five are actually equivalent, right? Logically equivalent to each other. And we saw already why that is based on based on the equivalencies, the logical equivalencies we talked about earlier. If Ralph is a dog, then he is an animal. We gotta introduce some new symbols here for our symbol key. We'll use lower case R for the proper name Ralph. We'll use uppercase D for the predicate dog. And we'll use uppercase A for the predicate animal, right? And then I would translate this to if DR then AR, right? If Ralph is a dog, then Ralph is an animal. And then all dogs are animals, similar structure, right? But we're using the universal quantifier and a variable in this case. So I hope you did well on those. Let me know if you have any questions. I wanna take a moment to talk about how we pick a universe of discourse. In our last set of examples, we might be tempted to say that our universe of discourse is animals like the earlier examples. But that won't work, I think, because of the way that we're using animals as a predicate, right? It would be hard to represent six without that predicate A, right? So we might instead say living things. That would be a pretty simple universe of discourse, but specific enough. Everything would be too broad. Sometimes you're gonna be sort of struggling to make a choice when you have to specify a universe of discourse. Am I gonna go with a broader and simpler specification? Or am I gonna go more specific? So I might say living things, or I might say animals and plants, right? Those are different universes of discourse. It's problematic to use everything because it's not clear what everything applies to. And you can get into some paradoxical situations that way. Okay, I wanna spend a moment thinking about how we use the concept of a singular term, right? I want us to think about whether these English language phrases could be symbolized just with a constant, with a lowercase letter A through W or not. Take a second and see if you can figure out all of 12 of these. All right, let's see how you did. So number one, Bob Barker, that's a proper name. It could totally be a singular term, very reasonable. Some Bs, right? That's more generic. That's not a proper name. That's just some number of Bs. We would not use a singular term for that. Those Bs is a little trickier. You might wanna use a singular term for this or you might use a combination of the predicates, Bs and over there, something like that. I would say probably the best choice is to say no, not a singular term. It's a little debatable. My dog, Feynman, on the other hand, that's a specific object or specific creature, that's a proper name. Singular term is there. Numbers is also a category and so it would not be a singular term. The person with the highest grade in this class, would we use a singular term to refer to that? That is even more debatable. If we are sure that there is a unique person who answers that description, we could use a singular term to refer to them. If we know for a fact that there is one person with the highest grade in this class, but it might be better to try to represent it in a different way. We'll talk in the third lecture in this unit about the better way to represent that. Walter White, again, a singular term would be best here for Walter White. Even though Walter White is a fictional person, we might have our universe of discourse be fictional people or people in the show breaking bad and we'd wanna use a singular term in that case. A herd of elephants, not a singular term, nor is a herd of Asian elephants. Even though it's more specific, it's still a category. Our textbook, it's a little debatable. If I said instead of our textbook, I said for all X by P.D. Magnus, that would be a proper name for our textbook, but our textbook seems more like a predicate to me. The chimera. Now, if our universe of discourse was fictional creatures, we might say the chimera is a singular term. It refers to a specific mythical creature, not a type of mythical creature, but a specific one, the chimera. But if our universe of discourse is living creatures generally, and we don't know whether chimeras are living creatures, whether there is a chimera, that would be problematic. If the chimera doesn't exist in our universe of discourse, we couldn't use a constant to refer to it. So that's debatable, that's questionable. Same deal with today. Today names a specific day, but when I use it tomorrow, it names a different day. It's an indexical. And there's some debate about whether we should use singular terms to represent indexicals. If on the other hand, I said March 17th, 2024, that we would represent with a singular term if our universe of discourse was times, right? Or dates. Another important thing to think about in QL is the scope of the quantifier. That's what we call the part of the expression that the quantifier applies to. Or as we say sometimes, that the quantifier quantifies over, it quantifies over a certain part of the expression. That's its scope, right? So take a look at these expressions and see if you can figure out what the scope of the quantifier is. Let's see how you did. So in the first sentence, our quantifier is for all x and its scope is the whole rest of the expression, the fx, the thing it is next to. In number two, our quantifier is the same for all x and its scope is actually only this part that it is right next to. It is right next to the fx. The conditional here breaks the scope of the quantifier and so it is not within the scope. That means that the gx is, in fact, the x in gx is a free variable. We'll talk about what that means in a future lecture. In number three here, for all x is our quantifier and now that we have the parentheses right next to it, the quantifier applies to everything within those parentheses. So the scope is this larger part of the expression, the rest of the expression really. Now four is a little more complicated because we've got multiple quantifiers here. We have this existential quantifier, its scope is just the expression it's next to, fx, and we have the second quantifier, this is the second existential quantifier, it's next to a parentheses. So its scope is everything inside the parentheses. Okay, number five, we have this existential quantifier here. It is next to the parentheses and so it applies to everything inside the parentheses. Now embedded in this expression in the scope of this first quantifier, we have a second quantifier with a scope just on the atomic expression it is next to. And then here with six, it's a similar situation. We've got the existential quantifier here, applies to everything within the parentheses that are next to it. We've got this second quantifier here, also next to parentheses. So it applies to this whole expression here. And then we have the third quantifier, it only applies to what's next to it, which is just the gz. All right, so that's the basics of QL and how we use it to do translations and some of the things we have to think about when we're analyzing expressions in QL. Please have a look at some of the early practice problems and see if you can do some of the translations. And then I'll see you in the second lecture where we talk about how well-formed formulae work in QL. All right, bye.