 Hi there. I'm Professor Matt Brown and I'd like to welcome you back to fill 320 deductive logic. This is our last lecture for Unit 5. Today we're talking about QL with identity, right? So this is an extension to our language QL adding an additional logical symbol, right? And that is the identity symbol which we'll use the equal sign to represent. So the equal sign means that x is identical to y. It applies to terms. It applies to variables or constants. It is a two-place predicate, right? But it's a special one. It's one that has a special logical meaning. We will also show the negation of this predicate this way with a slash over the equal sign. And so when we add identity to QL, we call our formal language QL with identity, right? So we have QL by itself and we have QL with identity, right? So let's look at some examples of when we might use identity. First, we can look at some normal sentences that we don't need identity. Clark Kent is a man. We can just use a simple atomic expression where capital M is the predicate man and lowercase c stands for Clark Kent is a constant. Okay, Clark Kent is a superman. We could represent with two predicates, super and man. So Clark Kent is super and Clark Kent is a man. Or we could use one predicate to just refer to superman, a superman. That'd be kind of a weird way to do it, but you could do it that way. But with the expression Clark Kent is superman, it's different, right? Clark Kent is a proper name. Superman is also a proper name. And so we would use the expression c equals s, right? The identity predicate applied to c and s, right? So this is a different kind of expression. Similarly, Clark Kent is not Batman, right? We would express with the negation of the identity symbol in this way with lowercase b, being a constant for the proper name Batman. And if we wanted to say something like no one besides superman fights crime, right? Because superman is a proper name again, that would be difficult to do without the identity predicate. I would do it this way. I would say for all x, if x fights crime, capital F meaning fights crime, then x is superman, right? Use the identity predicate there. That says anyone who fights crime is superman, right? We could also use the equivalent expression, right? It is not the case that there exists someone x such that x fights crime and x is not superman. Another way to say no one fights crime besides superman, right? Now you might ask yourself, do we really need this identity predicate? It's just a special predicate, right? We could use a normal predicate like i, a normal sentence letter like i to mean x is identical to y in this way and not use a special symbol for it, right? And in a sense, you know, that's okay. That's ql without identity. That's what you would have to do. But we wouldn't be able to do anything special with it because it would be no different from any other predicate in our language. And we might want something a little different. We might want to treat it like a special logical symbol so that, for example, we could introduce special proof rules, identity introduction, identity elimination rules when we do proofs and qls, which we will do in unit seven, right? So it's worth having a long identity also allows us to translate expressions of quantity, expressions like there is at least one or there are at least two or more generally, there are at least some number of f's or there are at most some number of f's or there are exactly, right? So when we're trying to express specific quantities, the identity predicate really helps us out. Let's look at how these works. And FYI, if you look at page 151 at the back of the book, these are all kind of conveniently laid out on a reference sheet, right, that you can use to study. So if we say there's at least, there are at least blank number of f's, right? If it's one, that's just simple existential quantifier, there exists at least one, right? But if we want to say there are at least two, what we need to say is that there exists some x1 and there exists some x2 such that x1 is f and x2 is f and x1 and x2 are not identical, right? This tells us that there have to be at least two. And you can see how this goes as we increase our number to three, our expression gets a little more complicated. We need three quantifiers, three variables. We apply f three times and then we have to get the non-identity between all the different variables, right? So with four, it becomes even more complicated. And we can define an expression for there are at least nf's or any number n in this way. Now when it comes to there are at most blank f's, you know, one way to say there are at most n things are f is just to put a negation sign in front of one of the symbolizations and say it's not the case that at least n plus one. So we could just take, if we want to say there are at most one, we could just say it is not the case that there at least two, right? Just to negate the one more than what we want to say there are the most of. But perhaps a simpler expression that is equivalent is can be done using the universal quantifier in the way expressed here in the book, which we say, you know, to say there's a there's at most one, we say for all x1 and all x2, if x1 and x2 are both f, then x1 must equal x2, right? And similarly, we can do this for two and three and so on any number there at most, right? So when we're saying there are at most, first thing is we need one more variable than the number we're trying to limit ourselves to. And we need to generate this expression where we're saying are at least one of our variables is equal to the other, right? This tells us that there cannot be more than the number in question, right? That's what those expressions are getting at, right? And then if we want to say there are exactly one or two or three or even zero, right? One thing we could do is just conjoin the at least and at most expressions, but there is a simpler way to do it. And again, it's represented here in the book, right? Zero is a special case. There are exactly zero means that there are none. So we use the same expression for saying that no x are f, right? But for one, two, three, right? What we're going to do is we're going to say there is one or there is two or there are three. And then we say it is not the case that there are more than, right? So we're doing it with this expression here. For one, there exists an x such that fx and it's not the case that there's a y such that fy and x is different from y, right? That's saying that there is one and it's not the case that there's any other one that's not the same. And similarly, two, three, the expressions get more complicated as you get more variables, but we can in principle represent any number, right? And what our first part of the expression is saying is for two, there's an x1 and an x2 such that x1 is f and x2 is f and x1 and x2 are not the same. And there's not some other thing, right, that is also f that is not the same as those, right? So the expressions are complicated and it may take you a little while to work through why they make sense, but you can always refer back to this page 151 reference sheet to help you learn these translations, right? We can also use the same kind of structure to specify how big our universe of discourse is, right? Just by taking the predicate f out of there, right? We can use that to specify the size of the universe of course. For example, to say there are at least two things in the universe of discourse is just to take that there are at least two f's, right? Just take the f part out and say there exists an x and there exists a y such that x is not equal to y. That can only be true if our universe of discourse has at least two things in it. Similarly, if we want to say there is at most one thing in the universe of discourse, we can say that for all x1 and all x2, x1 equals x2, that's just there's at most one f with the f part of the expression taken out. There are exactly two things in the universe of discourse. Again, similarly, we use the exactly two f's, but we take the f bits out. So we say there exists an x and there exists a y such that x is not equal to y. That gets us the at least two. And then there's not some z that is not equal to x and not equal to y gets us the restriction to at most two, right? So that gives us our universe of discourse's size, right? We can specify it in that way. I also want to point out some equivalencies that you can substitute in QL. This will become more useful later when we're doing proofs, but it's helpful to think about now, right? For the purposes of translation between QL and English in either direction, these expressions mean the same thing, right? They're logically equivalent, and we can prove that later. So negating the existential quantifier applied to conjunction, right? So this says something like it is not the case that some a's are b's, right? Or it might be that no a is b, right? Is equivalent to the for all x, if a x, then not b x, or b x, then not a x, right? All of those are equivalent ways of saying that there's nothing that is both a and b. This expression here, there does not exist an x and a y, such that a x and a y and x equals, x is not equal to y, is equivalent to this expression for all x and for all y, if a x and a y, then x equals y, right? So again, that's a substitution that's equivalent and are both equally valid ways of saying, right, that there's only one, there's only one of them, right? This expression here, there exists an x, a x, and for all x and all y, a x and a y implies that y equals x, right, is equivalent to these two expressions here. It might just be a more intuitive for you way to translate the same kind of thing, right? It became the same kind of expression of quantity. Also equivalent by the way to this, there exists an x and for all y, a y, if and only if x equals y, that's perhaps the simplest version of that expression. Also, the identity can go in either direction, x equals y, y equals x, they're equivalent, right? The last thing I want to talk about with respect to QL with identity is what we call definite descriptions. This actually played a pretty important role, the notion of a definite description played a pretty important role in the philosophy of language in the 20th century, and what it allows us to do is rather than using constants to refer to specific individuals, we can use definite descriptions to do so, right? So for example, a sentence like the present king of France is bald, we could use a constant to refer to the present king of France and translate the sentence that way, but there are reasons we might not want to, right? One of them being that there is no present king of France, right? France no longer has a king, right? Superman fights crime. Superman is a problematic term if our universe of discourse is actual people, right? Living people, right? Both of these are problematic as constants, right? We might want to say instead that the superpowered man from Krypton fights crime, right? Assuming there's only one of them, right? That definite description may be a better way to do it, right? Similarly, the present king of France is not bald. We'll talk about how to translate that in a second, but basically when you have expressions like the chimera we talked about in a previous lecture or Superman, if our universe of discourse is something like actual living creatures or actual existing creatures or people in the real world, right? We can't use a constant to refer to those things because those things don't exist in our universe of discourse, and so having a description, a way of referring to a specific person using predicates may be better, a specific object or thing. So let's look at the first one. The present king of France is bald. We can translate this in this way, right? You'll notice that part of this expression looks just like the exactly one expression of quantity, right? But then we add in the predicate for bald at the end, right? So let me read through this carefully, right? The present king of France is bald, okay? There exists an x, right? Such that fx, right? We'll say f stands for the present king of France, right? So there exists some x such that x is the king of France, is the present king of France, and there is not some other thing y such that y is a king of France at present, and x is not the same as y, right? This tells us there is one and only one king of France, present king of France, and that same person is bald, right? B is our predicate for bald. And so this entire expression is what we call a definite description, right? Uses a definite description to express something of a non-existent person. Similarly, where we might use cs to say Superman fights crime, if Superman exists in our universe of discourse, we can use a definite description to express it if Superman doesn't exist. So we replace Superman with a description, the superpowered man from Krypton, which we will express in this way. So there is some x such that x is superpowered, and x is a man, and x is from Krypton. And there is not some other y such that y is superpowered, y is a man, and y is from Krypton, and x is not equal to y, and that same person fights crime, right? That's a more complicated definite description, but it allows us to talk about Superman by his sort of defining description without committing to him actually existing, right? Now, this third sentence here, the present king of France is not bald, is tricky because of how we might interpret the negation. One way we might interpret it is this way. It is not the case that the present king of France is bald, right? That's just putting the negation out there. It's not the case that the present king of France is bald. However, another way to translate it is to say that the present king of France is not bald, right? Is non-bald, right? So the predicate bald does not apply to the present king of France. Why does it matter? Why does it matter that we can translate it both ways? Well, it matters in this case because the true values of these sentences come out differently in the case where the present king of France doesn't exist. If there is no present king of France, then there does not exist an x that answers to this description. So this whole sentence would be false. If this sentence is false, then the negation of the sentence is true. So when we say the present king of France is not bald, if we interpret it in this first way, it is true when there is no person who is the present king of France. If we interpret it in the second way, which we call this the wide version and this is the narrow version, if we interpret it in the narrow way, it comes out false when there is no present king of France. Here I have several examples of sentences that involve either expressions of quantity or definite descriptions. And I want you to think about how to translate these into QL. You've got your symbolization key up at the top. So pause the video and try to work out translations for all six of these expressions. Okay, let's see how you did. First one, there are at least two cashiers, right? We're going to use the expression for at least two. Cashier is our only predicate in that sentence, so it follows pretty straightforwardly from the definition on page 151 of the book. There are at least two cashiers working for Kate, right? That's a little bit more complicated, right? But not overly much so. K is Kate. That's a constant. K is a constant. WXY is a relation of works for, X works for Y, and we just put K in the second place there. And so we say there are at least two things that are cashiers and work for Kate, and they're not equal to each other. That's the expression for quantity again. There is at most one, right? Again, we're using the basic expression for at most one. And the only difference here is that we have not just one, but two predicates, accountant and lazy. So AX and LX and AY and LY implies that they are the same, right? There is at most one, right? Everything that applies, these two predicates apply to is the same, right? It's the same thing. And again, exactly one, upbeat cashier working for Kate. So we've got three, we've got three predicates there that matter, cashier, upbeat, and working for Kate. And we can use this expression instead of using FX, it's all three of these. And then, same in the second part, for there is exactly one, right? So there is some X such that it's a cashier and a beat and works for Kate. And there is not some other thing Y such that it is a cashier and upbeat and works for Kate and is not identical to X. There is exactly one upbeat cashier and he works for Kate, right? Is different, right? This says there's only one upbeat cashier working for Kate. This says exactly one. This says actually there's exactly one upbeat cashier, period, and it happens to be that he works for Kate. So instead of the WXK appear and appearing in the limiting expression here, we actually pull that out and put it at the end, as we do with a definite description. Number six, we see is a definite description and a pretty simple one, which we represent with the standard template for definite description. There exists an X such that AX, X is an accountant, and there is not some Y that is also an accountant, right? So that's how we go from an accountant to the accountant, right? The accountant basically says there is exactly one accountant and that person is not lazy. So that is how we do definite descriptions and expressions of quantity using the identity predicate, right? That's how we translate using QL with identity. That is our last lecture for this unit. So please try out the rest of the homework problems. Try out doing some translations with identity. Let me know if you have any questions and good luck with the exam for this unit. I will see you in unit six, where we do formal semantics. Bye.