Dr Greg Bahnsen - Critical Thinking: Quantified Logic Part 1 (Part 18)

Let's begin the prayer. We thank you, Father, for this day that you've given us. We thank you that you have given us life and above all, that you've given us to live again in our Savior, Jesus Christ. We do pray that his spirit would be with us as we study today. They would glorify your name, that we would think clearly and faithfully and that we would use our lives to show our love for you and service in your kingdom. We pray that you would help us to understand the lesson that is before us. Do pray that you would give us grateful hearts that we might use it to obey you. For we pray in Jesus name. Amen. All right, I'm going to do something a little bit bold and creative today. I've decided to put Kopi aside at the beginning of class anyway and try to explain chapter 10 to you. My own. As you can see here, I've written down my notes on two napkins here from the motel. And I got to thinking, why is it that a number of students seem to get intimidated, if not wilt, when they come to chapter 10 COPI on quantification theory? And as I was reading through the chapter again, I got to thinking he has a lot of explanations here, here, which are of interest to philosophers or those who want to do advanced work in logic. But I'm not sure that these explanations really advance the cause of understanding for the introductory student. And so it seemed to me that I might be able to put it in my own words and simplify everything that's there for you. And I'm going to at least give it a shot here. And if this works and we can do the exercises, then that would be great. And if not, then we'll go back to cofi, or I may go back to cofi just to pick up some of these interesting notions as well. The one thing we in case I don't get back to Kopfi, I want to say right now is chapter 10 makes it real clear that symbolic logic has a certain metaphysical outlook. That is to say, there are certain assumptions about the nature of reality which are built right into this use of logical notation. The relationship of individuals to predicates, for instance, which I'm not disputing that. I'm not saying that that should be controversial, but it is controversial. Not everyone agrees. We have the notion of a bare individual here, that if we have to think of something that has an identity apart from its attributes, and then you start building attributes for that thing, is that really the way things are in the world? Or is the identity of something the combination of its attributes? What holds these attributes? We can go on and on. But anyway, if I don't get back to Copia, I do want you to remember that much of this assertion assumes that you'll follow certain rules. Assume there's one individual. At one point Kofi says, okay, well, from a Christian standpoint we can do that. But how about from a non Christian standpoint? You have the right to assume that. How about you have to make sure you don't take one individual symbol, let's say a small letter A, and have that stand for two different individuals in the world? Well, again, that assumes a Christian view of the way things are divided up in the world. That's fine with us. But that does say something about the nature of reality, right? That predicates go with this individual and not with that, and you can't interchange and so forth. So the little rules that he gives you to follow out use existential instantiation before universal instantiation. All of that has behind it a certain view of the nature of reality. Okay, well, enough of that philosophical stuff. Let's just see if we can learn how to do the philosophy. Excuse me, the logic given to us in this chapter. What is quantification theory all about? Let me remind you of the outline of our course. We are now studying analysis. Correct. Analysis had to do with language, had to do with ordinary language fallacies. And then finally we got to formalize deductive logic, formal logic, deductive logic, whatever you want to call it. So you're spending some time this week on formal deductive logic. What goes into formal deductive logic? Well, so far we've seen there's two different kinds of argumentation that we've had to learn about. The first is categorical syllogisms. Categorical syllogisms is the logic of classes. Or, if you will, it teaches us how to use all and some. The words all and some can be translated into a notation in categorical syllogism. There's a bunch of stuff under that, of course. Then we came to the second part of formal deductive logic, and that's propositional logic, Where, if you will, it's the logic of connectives such as and, or if, then that's how far we've gotten in the course. We've learned how to formulate such arguments and how to test them for validity and invalidity, either categorical syllogisms or propositional arguments. That is to say, at one point in the class, we learned how to use all and some at another point in the class, we learned how to use and, or if. Then what we're going to do now, As we learn quantified logic, is to find a way to bring both together. In quantified logic, we're going to be able to speak of some and all as well as and, or if then propositions are connectors between propositions. So we'll look at propositions that are quantified, that is, propositions that have some indication of number, specifically some or all. You see how those are quantity remarks. If we can incorporate quantity indications, I.e. sUM and all, into our propositions that are related by nature and, or if then connected, then in a sense, what we've done is we've combined these two lessons in logic, that is, we've combined categorical and propositional logic. So today my goal is in this brash attempt to explain chapter 10 all on my own. My goal is just to show you all we have to do to actually add to what we've already learned so that we now can take some and all and sentences that contain them, or they can be translated into some or all as well as the connectives and, or if then. And then test arguments that are built up from them. Everyone with me so far. Our goal now is just to combine everything we've learned in symbolic are formalized logic. Okay, in order to accomplish this task of bringing together our classes, some and all, as well as our connectives and, or if then we're going to introduce a notation. I'm going to teach you a way of writing, or if you will, translating into a notation the indications of some and all. I'm just going to give you now a new way of writing down some and all. And the reason we're doing that is we already have a notation for testing propositions that have been formalized. We learned that in the previous two chapters. So we know how to write down a notation and tell arguments, but we have to add some new notation so we can incorporate the quantity remarks some and all. When I have an X in parentheses preceding something else, that's going to be the formalized proposition that is going to give you a quantity indication. And it will mean given any individual. Given any individual. Now, it's important for you to see that X has a special meaning. X is not like A B, C through W. A, B, C, D, E, F through W stand for particular individuals. So that if I wanted to refer to pat, I might use small letter P. Okay, but you can only use A through W for individuals. That's just the rule of this notation. And now we're going to use some special notation. X means given any individual. And therefore this X in parentheses is like a place marker. It indicates given any individual. And then you would have to fill in. Now if we were to fill in the individuals, what would they? They would be A, B, C, D, et cetera, to W. For instance, A, B, C, D through W. The small letters will be used as individual constants. Or if you will, they stand for particulars. So if we're talking about people, and I'll try to keep the illustrations there, although we could be talking about horses, we could be talking about rocks, we could be talking about trees or birds or whatever you want for our purposes, the individuals in our illustrations will be people. A stands for one individual. Let's say it's Dr. Bogson. Now let's make B for Dr. Bogson. A stands for Melissa, C stands for P, D stands for J and so forth. That's how we use the little letters of the Alphabet. They stand for individuals. Now again, we could be talking about other. We could be talking about molecules for that matter, although it'd be hard to know how to predicate about them. But anyway, any particular thing. But when we use X and put it in parentheses before thing, we're talking about about any individual. Any individual. Not a particular one, but anyone. So X is a place marker. Now I'm going to give you another notation. We have this interesting backwards E and then X in parentheses. When we use the backwards capital E X that'll stand for there is at least one individual. Notice this makes an existence claim. Think of it as capital E, backwards E for existence. This makes an existence claim. There is one individual. Actually at least one individual. Because if I want to say that 100 people have beards, I would still put backwards capital E X because that would mean that there's one or more. There exist one or more individuals. Okay, now let's go back and remember what we're trying to do. We're trying to find a way to take our propositions that we can test in symbolic way logic and put quantifiers on them. Use the sum and all that we learned about in categorical syllogism. Now we have a notation for it. Given any individual would mean all individuals. There is at least one individual means some individual. Here's a new notation. A capital letter. I'm going to use H here, but it could be A, B, C, D, whatever. A capital letter will now stand not for a proposition by itself. In the past we've said like if H, then K, not K, therefore not H. Where they stood for particular propositions. Now, let's let the capital letter stand for any predicate, any predicate. Let's kind of review our English grammar. What's a predicate? Predicate says something of a subject, right? So the predicate. Predicate could be. Is old. The predicate could be runs fast. Anything you can predicate of an individual can then be symbolized with a capital letter in predicate. So verbs, adjectives, classes, anything that can be predicated of an individual. Alright, let's put it all together. If these small letters A, B, C, D, et cetera stand for individuals and the capital letters stand for predicates, then notice what happens. Now, this is just the style that has been developed and so accept it in this way. A lot of people say, why do you write it backwards? There's a reason. It's because this is controlled by the predication rather than by the subjects. The subjects are quantified. The predicates control the logical relations. Anyway, you write the predicate first. I'll write capital H and then a small B next to it. If you followed my lectures so far, you realize that that stands for a sentence now, or what we'll call a proposition. This. Now it doesn't look like a proposition. It's not written out in English. But that's the whole point. We're trying to get a formalized notation in our formula that stands for a proposition. And what proposition does it stand for? Bahnsen is a human. Well, if H stands for human, I didn't say that. You could also say humorous. Okay, so Bahnsen is humorous or Bahnsen is honorable, or whatever H stands for, or whatever B stands for. But if you know what they stand for, this symbolism up on the board, capital H, small B is a sentence, It's a proposition. Now, it might be helpful to remember that when logicians say that this stands for a proposition, they call it a propositional function. To say it's a propositional function is to say that it stands for it functions as a proposition. Now, if we understand that H B is a proposition or stands for one, and if we understand our new notation of x in parentheses and backwards E in parentheses, notice we can make a proposition using these as well. Now, we're not going to refer to particular individuals. Small A, B, C, D, so forth refers to particular individuals. These refer to quantities, all and some. And so we can make a sentence in a new way by saying, now what I've written on the board is Parenthesis x, capital H, small X. What did we say? Parentheses X would stand for given any individual. So now when we look at this, we can read this sentence. It says given any individual, that individual is H. So parentheses X, capital HX means given any individual, that individual is human, honorable, humorous, whatever H stands for, Okay? HB is a proposition about a particular. About Bonsen. Parentheses X. HX is also a proposition. Now, not about Bond or about Jay or Melissa. It's about all individuals. Given any individual at all, that individual is human. If I were to put in parentheses backwards ex, then hx, that would be another way to make a proposition or to symbolize a proposition. This one quantifying not in a universal way, but in a particular way. Now I'm saying there is at least one individual such that it is human. So I wrote on the board parentheses backwards ex and then capital H, X. And that reads, there is at least one individual such that. That individual is a human or is honorable is humorous. And with this notation that I've now taught you, we can now quantify our propositional logic. We can now take propositions that are connected with and, or if, then connectives, and we can add to them indications of quantity, either universal quantity, given nex or particular quantity. There is at least one individual. We can quantify our propositional logic by adding prefixes, prefixes to our prophecy propositions. You've already learned about how to relate propositions. Now, what we're going to do is take those propositions that are related to each other and put prefixes on them. So let me just give you an example. Previously we talked about. I'll write this down. If H, then K. If K then. And what conclusion can we draw from this? If H and L, by what rule? Hypothetical syllogism. Exactly right. So everyone's with me so far. Now, what if I were to. This is written up here on the board. Everyone understands it from the previous lessons. What if I were to have run in here, forget today's lesson thus far, and I had just written something in front of each one of these letters. I'm just going to scribble up here. We'll call this the blob. I was going to put blobs in front of each one of these letters. And then I. And then when you came into class today, I said, could you please work this argument for me? And then you said, well, Dr. Johnson, what are those little blobs up there, those scribbles in front of there? I was, oh, well, just ignore that. Just do the Logical calculus that you've already learned. You say, okay, well, I can ignore that. If H then K, if K then L, then I can conclude if H then L and say, that's fine. Now I'm going to give you a different kind of prefix to put in front of them. Now, I'm not going to use little blobs in the place of those little blobs that I told you to ignore for the sake of doing your logic, doing the drawing the conclusion, or testing for validity and so forth. I'm going to use a prefix, not a blob prefix, but I'm going to use a quantity prefix. So what quantity prefixes do I have available? There's only two, right? So I'm going to have either X in parentheses or backwards ex in parentheses, which are used as prefixes in front of these letters. However. However. Now, this is a little dangerous using an illustration like this because in a sense, I something that was false to try to get you to think this way. These letters no longer stand for propositions. The H, the K, the L no longer stand for propositions. They stand for what? Predicates. They stand for predicates. That's the difference between a proposition and a predicate. A proposition is a subject and a predicate a predicate. It doesn't. So if we're going to use the capital letters for predicates, then we have to make them into propositions somehow. And the way in which we make them into propositions is by quantifying them. And what are the ways in which we can do that? We can either use a universal quantifier, a particular quantifier, or we can assign an individual to it. So those are the three ways to turn predicates into propositions. Here I have the predicate human in the first line. If I put parentheses X in front of it and X following. Now I. Now I have a proposition. Now, it says for any X, X is human. And that is, of course, the class of all humans. Here I have a K. What will K stand for? Cool. And I put backwards ex in front of it. And I have to put X here to finish the notation. Now I say there is an X such that X is cool. So this is a proposition. And this is a proposition, by the way, not a very convincing one, that for any individual given any individual, if that individual is human, then there is an individual such that that individual is cool. I don't expect it to make a lot of sense. I'm just making sure you know how to read across here. But that's not the only way I could have turned these into propositions. I've used this and this, that is the X and the backwards ex. I could have also used individual constants. So in this case, I take capital H and put a small B after it. I take capital K and put a small B after it. And now the line reads, if Bonsen is a human, then Bonsen is cool. Again, it may be a nonsensical statement, but you understand how to read it. Okay, I just want to make sure everyone's with me. So far, we're building this up slowly this morning, but I think it's not too difficult. We take the capital letters, they stand for predicates, and we have three ways in which we can make the predicate into a proposition. We can quantify it universally or particularly, or we can make it apply to a single individual. Okay, I'm going to add one more piece of notation, and then we're going to be up and running. So far in my explanation, I've been using capital letters for a predicate. H. Capital H standing for is human, is humorous, is humongous, whatever. Now, if we want to generalize about what we've been saying in terms of this quantified notation, let's have something stand for any predicate like that. And for this we use Greek letters, particularly phi. And. We could have others as well. But where a Greek letter, particularly phi psi appears, then you'll know that's a place marker for another kind of place marker. Get it? This is a generalization. Any place marker like the capital letter H. Okay, so that you're not confused, we're going to change to Greek letters like B and p to take the place of any capital letter that's a predicate. Now we have a generalized way to write down some things. Let's reintroduce the square of opposition with only the contradictories in it and write out what it would be like to say all humans are mortal, or all bugs are vicious, or all days are happy. Things like that. No days are happy. There are some days that are not happy. There are days that are happy. Notice what we have here are propositions that are quantified, referring to all or some. So I'm going to now go back to our notion of the square of opposition, and I'm just going to use the new notation. This is just a new way of writing it. That's all I've given you today is a new notation. If I want to say all of something has a certain predicate such as all humans are mortal, then are you My universal quantifier parenthesis X. And I'll say that quantifier governs everything in the following parentheses. So for any X, for any X, which is to say all X's are this way, for any X, phi will stand for what? Pat. No, no, no, no, no, no, no. Phi stands for what? A predicate, not individuals. A predicate. So fee would stand for mortal because X X will take place. Actually, fee must stand for capital M. Ultimately, this is just a way of putting any predicate in there. Okay, so I'm going to put phi X. So far I have for any X, X is phi, where phi stands for mortal. Now if I want to say that all mortal. No human. I said mortal and I should have said human. If I want to say all all humans are mortals, then if X is a human, then I have to use a different letter. Now for mortal, I'll use the C. Then X is the C. For any x, if x is phi, then x is c. That is a way of saying that everything that is phi is also a C, or all X's of this sort are of that sort as well. So it's a universal quantification. Now, that is contradictory to the idea that there is some human that is not mortal. So let's see if we can schematize saying there is is some that isn't. There is an individual that is human but doesn't have this quality of mortality. So now I'm going to make my diagonal line for the square of opposition for the contradictory. I'm going to go from the A position up here down to the O position. If you remember, that's an A proposition. This is an O proposition. There is at least 1x. So I have in parentheses the quantifier backwards ex. I quantify there is at least 1x. And now how do I want to describe this one X? Well, this X is phi. So I put phi X. That is, this X is human. And please notice, very importantly, I do not use the if. Then I'm not saying if this individual is human, he is not mortal. I'm saying he is human and is not mortal. It's the and that is being asserted here because this is an existential claim. Notice I'm claiming that there is someone that has both of these attributes, humanity and not mortality. So I say phi X and I put the dot, not the C X. There is an individual such that X is phi. And it is the case that X is not perceived or There is an individual such that X is human. And it's the case that X is not mortal. So that would be the very opposite. That would be contradictory to all humans are mortal, which is given any X, if X is phi, then X is C. Now let's do the other one. Let's try an E form, the universal negative. We would say for any X, if X is phi, then it follows that X is not a C. So what would be a universal negative that we could fit into here? No, no. Reptile is a mammal. Is that true? No, is it? Yeah, that's true. I hope so. Given any X, if X is a reptile, then X is not a mammal. C stands for reptile, the C stands for mammal. That's a way of stating a universal negative. So up on the board in the E position of our square of opposition, we have pretty parentheses X, then parentheses, cx, horseshoe curl or not a C, X, end of parentheses. Now, to make that universal negative claim is the direct contradiction of saying that there is a reptile that is a mammal. Don't worry about your knowledge of natural science here. Just for logical purposes, if we were going to assert the contradiction of this E form proposition, that is an I form proposition, we would say. There is an X. So I use X in parentheses backwards ex. There is an X such that X is phi. And now what is the connective going to be here? What? And. And that's right. Since I'm asserting the existence of this X, then I'm not saying if it is phi, it is PC. I'm saying it is both phi and faci. And so here we have our. Particular affirmative proposition, the I form proposition. There is an X such that every X is for B and X is for C. So now we have our square of opposition, our contradictories up on the board. Now, because these are contradictories, I want you to remember what that amounts to. If they are contradictories, then notice this, I'm now underlining the A form quantified proposition that is the opposite. That is logically equivalent to the negation of the O form, isn't it? So if I were to put a negation in front of all of this, now we're going to see in a minute. We don't want negations in that position. But just for the understanding, if I were to negate all of this right here, that would be logically equivalent to the contradictory. So negating the O form proposition is logically equivalent to an A form proposition. Likewise, negating an A form proposition is logically equivalent to the O form proposition works the other way as well, since we have two other contradictories. If I negate an E form proposition, that's logically equivalent to the I form. Likewise, if I were to negate the I form, that would be logically equivalent to the E form. Without learning anything else except notation, we can now combine our categorical syllogism and our propositional syllogisms using connectives and or if, then that's what we've been able to accomplish. I want to give you one little rule of thumb for procedure. Let me erase this off the board first. The one little rule that I want to add to this explanation is that the negation sign, the negation sign should be used only before predicates. Only before predicates. Now, what that means is if you have a negation sign outside of a quantification. So I've written up on the board, not parenthesis x, then parenthesis and some proposition is in there, whatever that happens to be. If you have a negation before a quantification, you want to write that differently. So the negation no longer is there. And how would you do that? By remembering the square of opposition that I just gave you. A zero. Well, depending on whether it's an A or an E form, proposition can be translated into its logical equivalence by rewriting it in either the O or the I form. If this is an A form proposition, then rewrite that without the negation by putting it in the form of its O counterpart. If it's an E form property, then rewrite it without the negation out here in the form of its I counterpart. Likewise, if the negation is in front of an existential quantifier negation parenthesis ex. If it is an I form proposition, then rewrite it in terms of of its E form counterpart. And if it's an O form proposition, rewrite it in terms of its A form counterpart. And then you can strip off the negation in front of the quantifier. That's if the negation appears before the quantifier. Now, there's another way. A negation can appear and not be directly connected to a predicate. I'm going to erase these negations off the quantifiers. Another way a negation can appear is in front of a parenthesis that governs a proposition. So now let's say I have x in parentheses, meaning given any X, then I have not. And now a parenthesis that gives me a problem. Or I could have a backwards Ex. In parenthesis, there is an individual such that and, then not in another parenthesis, giving me a proposition. And so here we have the negation appearing not in front of a predicate but in front of a parenthesis. And the way in which we can get rid of those materials is by using Demorgan's theorems. Because the connective or the compound inside of the parentheses is either going to be an and, or, an or, and, or, and. If you look at Demorgan's theorems at the back of your book, you'll notice that a negation in front of a common compound using and, or a negation in front of a compound using or can be translated into another notation, a logically equivalent notation where we've removed the negation from in front of the parentheses and put it inside the parentheses, and when it moves inside, then it's attached directly to a predicate. Okay, so this little note at the end of my explanation has only been a To say, don't have negations outside of parentheses. If it's outside the quantifier, use your square of opposition to rewrite it without the negation in front of the quantifier. If it's outside of the parentheses governing a compound proposition, then use Demorgan's theorems to move the negations inside the parentheses and get them out from the front of the parenthesis. Yeah, the parenthesis. Now I said inside, you're going to have either and, or. Or somebody might have said, well, Dr. Bonson, you could also have if. Then what? If it's a proposition that has the horseshoe or the arrow. For if, then. Well, then you use material implication and translate that into not something or something. And once you do that, then you've gotten rid of the horseshoe and you can use the Morgan again. Now we are ready to go back and show that we can. Now we can write propositions that have quantity in them as well as connective in them. And having done that, having written those propositions, we can now prove validity and invalidity. I gotta teach you one little trick about it, though. Let's go back to that point in my discussion where I said, what if you came in here today and I had written some blobby kind of thing in front of those letters that are connected? Now you know how to work the logic on those letters that are connected, but you have those blobby things in front. Let's go back to that. I had H, then K, K, then L. And I said, imagine I had some blobby thing here and here Here and here as a prefix. And I said, ignore the blobs and just do the logic as you've been taught. Just follow the rules of logic that you've been taught that you were able to do that. Now, if the propositions, remember these are propositions. If these propositions had prefix to them, not blobs, which you can easily ignore as just blotches on the blackboard, yet they had prefixed to them quantification indications, or if they had following them individual constants. And I said, well, I'm going to give you a way that you can ignore those blobs. You can strip them off, and if necessary, and it isn't always necessary, but then if necessary, to reintroduce the quantification at the end of the argument. Okay, so think of it this way. Dr. Bronson has taught us how to write these propositions with quantifiers. Now, if we can get rid of those quantifiers, we can use the rules of logic that we've already learned. And so he's going to now teach me how to strip off the quantifiers and then put them back in if they should be necessary. The stripping off process is called instantiation. And let's just ask ourselves why, because that'll help understand what it's all about. To instantiate something is to give an instance of it. To give an instance of it. If I say all humans are mortal, I instantiate that by saying Pete is mortal. So now I'm not talking about all humans. I'm talking about one instance of a human, in this case Pete. Does that make sense? I'm going to instantiate it. I'm going to make an instance out of it. So I have two kinds of quantifiers that I have to instantiate. Give an instance for one is the universal quantifier X in parentheses. The other is the existential quantifier backwards ex in par parentheses. These are the prefixes. Remember, for quantification, X in parentheses. For the universal. For universal, we'll say U backwards ex in parentheses. For the existential quantification, I'll just put an E next to it. So I have universal and existential. If I'm going to instantiate them, let me abbreviate that with an I. So I'm going to instantiate a universal proposition. I'm going to instantiate an existential proposition. So these will then be rules for translating. Now what I actually have in the case of a universal proposition, it's not just an X in parentheses or with an existential proposition, just the backwards ex in parentheses, it's going to look more like this. For all X, P is X, X is phi. Pardon me. For all X, X is phi. So I have X in parentheses, and then in parentheses, phi, X. For the existential, I have a backwards ex in parentheses and then in par. Everyone with me so far. Of course, if I wrote this out fully, I would have something else, wouldn't I? I have a facie in there. So though you don't find it in this long form in the book because you don't need it, you can really abbreviate it and be much more economical because of the way I've explained things to you. I'm going to fill this out and I'm going to put X is phi, then X is C. And of course, this won't be that. We'll put an. And here for the exponential output, there is an X such that X is phi and X is PC. Now, right now, now, I'm not talking about any particular individual. I'm talking here about any given individual. That is, when I use the X in parentheses. Given any individual. If I instantiate this universal proposition, all I do is, is I strip off. I strip off the quantifier out here, the X in parentheses. Okay? Just drop that and take one instance of all of those X's that are fees and proceeds. And so instead of X, I now use an individual. I'm going to use B here for Bonson. Okay? So in the place of X, I now put an individual constant, a small letter, A, B, C, D, E, F through W standing for Bonsen or Pete or J or whoever it may be. Actually, it's anybody arbitrarily. So universal instantiation amounts to. To dropping the X in parentheses and replacing all the X's with an individual constant, an A or a B or a C standing for a particular individual. And the way in which Copie notes, any individual constant is by the Greek letter nu. Well, it began in English, but it looks like a V. Now, I think that's confusing because V is already used as a wedge, but you need to be able to orthographically see the difference between the V and the new. Yeah, I know there isn't a whole lot. Look for a little curvy, a little more curviness here on this side. This is straight. Anyway, Colby uses a Greek letter to stand for any individual constant in the same way that he uses a Greek letter to stand for any predicate. So when you read the rule of universal Instantiation. You see that nu there? And it kind of throws you. All you have to remember is nu. The N in Greek stands for any individual constant. A, B, C, D, E, F, G. The small letter A, B, C, D, E, F, G. So one more time, just want to make sure that we've all got this. If I'm doing a universal instantiation, I can drop the quantification, the X in parentheses, and I can replace all the X's with a small letter A or a small letter B or a small letter C. Now how do I strip off. Remember, all I'm teaching you is how to strip off the quantification. How do I strip off the quantification in the case of an existential. An existential proposition where I have in parentheses backwards ex and then some formula X is phi and X is facet. How can I strip that off? By using ei existential instantiation. In the case of existential instantiation, you do the very same thing. You strip off the quantification. Everyone see that how I erase the quantification, then go through here and drop all the X's and put in there an individual constant. Again, I'm going to use B. So now I have B is phi and B is C. And I don't have any quantifier out here. And I did that through existential instantiation. I've just given an instance of that existence claim. Okay? And there's just one little kicker. Please remember, you play by the rules. And the rule says you can't choose for your individual constant, one that's already been used. If somebody's come along and use that constant to stand for another individual, you can't pick it up. That wouldn't be fair. That'd be like one person going in to take another person's GRE exam. GRE exam is given massively to all these people. One of the biggest cheating scams, of course, is to get someone who knows more than you to go in. You pay them to sit down and take the exam under your name. Well, that's cheating, isn't it? And in the same way in logic, it's cheating if you let the B be put into place here if B has already been used somewhere else to identify a different individual. So if the individual constant has been used previously in the argument or in the the context, as Copie says, it cannot be assigned here, but you can. Other than that, you can choose any individual constant you want, provided you haven't used B in another line of your proof. So now we know how to strip off the quantifications. And once the quantifications are stripped off, we can go ahead and use the laws of inference or the rules of inference and replacement that we've already mastered previously. So here we if you wanted to be a cynic, this, I think will actually help you understand what we've done in the class. So I don't want you to be cynics. If you wanted to be a cynic, you could say, oh, so you took all this time to teach us how to write the logical notation with the quantifier, and now what you've done is you've shown us how to strip it off. I said, that's right. However, in some arguments you don't want to leave your argument dealing with an individual constant. Sometimes you want to go and reintroduce or add back in the quantification after you've done your proof of validity or whatever, or invalidity. Actually, you don't have to put it back in if you're proving invalidity. But anyway, if in a proof of validity, you have to strip it out before you can use the rules of inference, do your deduction. By the time you get to your conclusion, you have to add the quantifications back in so that your concluding proposition has everything that was in the premises, and the premises had quantifiers. And so now the should have quantifiers. So real quickly, how can we add back in the quantification? Well, remember, we have universal propositions, we have existential propositions, and at this point they have been instantiated. We're looking at one instance, but we've got to get back to the quantified, not quantified in instance. I don't want to say that we want to get back to the quantified statement rather than just a statement about an instance. And so this is called generalization. Now we're going to generalize from the instantiated propositions to a universal, or we're going to generalize from the instantiated propositions to an existential. And thus we have the rules ug generalizing universal, eg, generalizing existential. And this is just the opposite of stripping off with ug. Now I go back and I take the B's out that were the individual constants. I replace the X and I can put the X in parentheses back in front as a prefix again. Now I've got for all x, x is phi, then X is I took out the individual constants and put in the X again and replaced the quantifier out. Now, something has to be understood. Again, as just a little note here, in order to use universal generalization and not mess up our logical procedure, we have to remember that the individual constant that was found in the formula that is now being generalized must be an arbitrarily chosen individual. Must be arbitrarily chosen. If you're going to go from a particular instance to the generalization of all, then obviously you can't pick some special exception, an individual, because you can't move from some one to all unless the only thing you're looking at in terms of the one is the feature that's being generalized. Now, you should, in a sense know this already from our informal fallacies. Correct? Isn't that hasty generalization when I go from what might be a unique case to every case? Now, how do you know that universal generalization? What is universal generalization? It is taking an individual constant and then generalizing so that it's true of all individuals. So when I go from phi A to for any X, X is phi, have I not committed the fallacy of hasty generalization? And the only way to make sure that I haven't is to write in this qualification, the individual constant had to be an arbitrarily chosen individual. So in the examples in copi, he says we talk about a triangle given any triangle. Now we have to test that given any triangle, we look at a particular one, we look at another particular. The point is, the only thing these things have in common is their triangularity. And so in virtue of that class characteristic, I can universalize about triangles. So in the case of universal generalization, the rule is written down in the book with a Y, with a Y instead of an X. If you have a formula that uses X, you know what that means. That's any individual. If it is Y, it's any arbitrarily selected individual. Okay, just remember the arbitrary selection aspect of this. And that's what accounts for why we can use universal generalization. Okay, not only is there universal generalization where the individual constants B, c, D and so forth now turn into X, we replace them with the X and then put parentheses X as a prefix out in front to universalize, we also have existential generalization. This is much easier. It doesn't have the qualification built into it. If there is a B that is a phi and a B that is a PC, then you can erase the B, put the X in there, and now put your existential quantification out in front as a prefix. Eg therefore enables us to add back in the existential quantification and replace the individual constant with X. Now, what have I taught you? I've taught you how to rewrite a sentence in English that has not only connectives and quantification. I taught you how to rewrite that into a new notation. And then for testing the validity of arguments that are in that rewritten notation that now contains quantification, I taught you how to strip off what I just taught you. And then at the end of the argument, if you need to, if the argument is supported by to conclude something that has quantification in it rather than a conclusion about a particular individual, I've shown you how you can reintroduce the quantification prefix if that's necessary. And there's only one last rule to remember. Remember what I said just a couple of minutes ago about this arbitrarily selected individual and what I said earlier about how a constant cannot be one that is used previously? Well, because of the nature of those rules, it so happens that in any argument where you are going to be using both existential and universal instantiation, you must use existential first. Now, what I have here up on the board still is universal generalization, existential generalization. I am not talking about generalization. I'm not talking about how you put the quantifiers back in. I'm talking about if you're going to be taking them out of an argument, if you're going to be stripping the quantifiers out of an argument. So we're with dealing with instantiation. Solve it UI or EI in cases where both must be utilized. And what cases would those be? Those would be cases where the argument has propositions of both the universal and the particular variety. If you have an argument that has lines in it, some of the lines or one of the lines has a universal quantifier, that is has the X in parentheses and then the rest of the formula, and it has a backwards E in parentheses, X and then the formula. In those cases, not all of them do. But in those cases where both of these kinds of quantification appear, you might, you must use ei. You must get rid of the existential quantifiers first before you use ui, that is the universal instantiations. And the reason for that is because in the case of ei, existential instantiation, you have to make sure that that particular that you choose, small letter A, small letter B, small letter C has not been used previously. Now you know how to prove the validity of a quantified propositional. Argument. Piece of cake, huh? Once you've used the particular in particular variable in ei, okay, can you use the same particular variable again? Can you use it again? As long as you're not doing ei? Like if you could you use the same particular again for ui? Yes, if you've done EI first. Okay, but if you've done UI and let's say you chose B out of all of the individuals out there, this is universal. I'm just going to choose B. Now you come to an existential proposition later in the argument and you say, okay, now I've got to choose an individual. You can't just go and choose B now. So in order to avoid doing that, you always do EI first. And then you can, after you've chosen B for your EI procedure. Let me put this straight. After you've done an existential instantiation where you choose B as your individual constant. After that you can choose B as your constant in a universal statement as well. But you can't do it the other way around.

And if you. I don't know if you can, if this will ever happen. But if you had to use the existent twice, you would have to assign a different value on the second set of existential, Correct?

Yes, because you cannot use an individual constant that's been used previously.

Because I'm looking at page 341 where they, where they were showing that argument, where they showed it was wrong because they had the A listed twice. So they would have to said like bbcb.

That's right. See, that's the whole point. You can't use an individual constant that has previously been used. Once it's been assigned, you can't show up for the GRE exam and use that name. So if we're examining the validity of the argument and we use ei, if we've already assigned small letter A, we can no longer use it.

But we can still do existential.

Exactly. It's just you have to choose now B, C, D or E. After A has been assigned, you know how to prove the validity of arguments. How can we prove the invalidity of arguments that have quantifiers in them at every step along the way? I tried to remind you, and I've written this on the board. We have tests for validity. We also have have tests for invalidity. Okay, so what is the test for invalidity? Well, first of all, what is the test for validity of a quantified propositional argument? It's the very same test you've learned previously for propositional arguments, as long as you strip off the quantifier, do the argument, then put it back in if you have to. If the conclusion is about a particular individual, you don't have to do that. So you can strip off the quantity quantifier and just do what you've been taught. Likewise, in fact, in an easier way, you can prove the invalidity of a quantified argument in the way that we have learned already, and that is assigning truth values such that the premises are true and the conclusion is false. We were doing that yesterday. We remember how to do it. It's a little bit of a game. It's a lot of fun. If you can assign truth values to each of the components such that the conclusion would be false but the premise is true, then you have proven the invalidity of the argument. But now the problem is, Dr. Vaughn, there are all these blobs in front of what we've learned previously. We've got all these quantifier signs out there. Don't they affect the way we do this? And the answer is, if you understand the nature of quantification and its notation, no, it doesn't affect it at all. All you have to do, if you don't understand why you don't understand intuitively the inner reason why this is, then just accept it from those who know logic, don't worry about this. You can translate the universal and the existential propositions into instances and then test the validity of the argument. Or try to make the argument invalid. That is to say, if you have an argument that has X in parentheses governing, let's say, px, then a cx, and then the next proposition or the next line is the existential quantifier backwards, ex in parentheses and then ex and not the cx. And then I draw a conclusion from this. I don't know what the conclusion is going to be. Let's just say X in parentheses, not the cx and not bx. Now, I have no idea what this all amounts to. I've just made it up so that you have statements in our notation. But now we want to test this for invalidity. And the way we do that is by assignment truth values. Problem is, Dr. Bonson, don't you see these blobs out here in front? Again, you've got these quantifier signs and we're dealing with any individual. So how do we do it? Well, the answer is take an instance. So translate, strip off. Again, strip off the quantifiers and just make all of these formulas Rewrite them in the case of one individual. So now we're going to have PB then cb. Okay, I just put I've taken an individual constant, not Y, not nu, not X. I'm replacing the X with any individual constant. So I put that in the place of all the X's in this argument, and it makes no difference whether they are universal or existential. The reason why it doesn't make any difference is explained in the textbook. But you can take it on faith too. In the nature of the case, you can test an argument given any model. If there is a unit of such that one individual put into all of these propositions shows it invalid, then you've shown it invalid. You only need one test case then. So that's what we're doing here. We're just taking the B and putting it in the place of X and all these formulas. And so now we have in the second line, Phoebe and not the cb. And I just want to really drive this home. You notice that all I. I just took what's in parentheses and rewrote it with B in the place of X. And it made no difference that in this case it was an existential quantification. And then in the first instance it was a universal quantification. Makes no difference. And then after I've rewritten it, now I test it by trying to make PB true and so forth. And if I get a combination where every line is in the premises is true and the concluding line is false, then I have proven it invalid. So in the case of proving invalidity, just take an instance and get rid of the quantifiers, rewrite it without the quantifiers and put in any individual. Because you're going to test this against a model or a hypothetical universe where there's an individual that fits in to every one of those premises, And that explains quantification. That's all there is to it. The Last section in Copie 10.6 has the same function as the discussion in Copi. After we learned categorical syllogism, after we learned how to do categorical syllogism, then he had to admit, but in English there are some tricky expressions. You have to be careful that you translate them properly. And then test the categorical syllogism. That is, getting from English to the notation or to the categorical syllogism forms might be a little trickier than you think. So that's what he told us previously. Now all I'm saying is this last discussion in Chapter 10 is analogous to that, he wants to warn us that with respect to certain English expressions, with respect to certain English expressions, you have to really understand what they are saying before you can properly give their translation in symbolic notation. And of course, this just re emphasizes what Dr. Bronson has called his hobby horse throughout this course. If you don't understand language and how it's used, formal logic does you no good at all. It may look intimidating, it may seem very powerful, but if you don't know how to understand logic, language, big deal. Somebody comes along and says, well, I've schematized your argument in formal notation and I've shown that it's invalid. You can always say, well, maybe you haven't schematized my argument properly because you haven't understood what I was saying. You haven't understood my premises. That is, you haven't rewritten them in your formal notation in a way that captures the truth of what I'm saying. But then, on the other hand, it's possible that he's, you know, hung you as well. It's just that if you don't understand language in the first place, then you can't use formal logic. Or all you can use formal logic for is playing math games. It won't do you any good in terms of doing argumentation. So that's all 10.6 is emphasizing. And he gives three examples. Watch out for the word or. It's not always being used as a disjunction in the logical sense. Secondly, he says, watch out for the word and. Sometimes the word and in English, when you rewrite it in logical notation is an or rather than an and. Some English. What this amounts to is some English sentences using the word and have the logical force of or instead of and. When he says oysters are delicious. Excuse me. When he takes as example oysters and clams are delicious, that and has the force of an or rather than the force of an and. And that's why when you write it in logical notation, the first line on the bottom of page 348 at the very bottom there gives it for all X. X is an oyster or X is a clam, then X is delicious. You obviously don't want to write it using and. Because notice the next line says for all X. If X is an oyster and X is a clam, then X is delicious. Think about it. That doesn't make any sense. Do you know anything that is both an oyster and a clam? The sentence is not trying to say these oyster clam hybrids are really delicious. It's Trying to say there are two things that are delicious and so to write it you have to say if it is this or that, then it is delicious. So that's one of those tricky cases where and comes into logical notation as or. And then the other case that he gives in English we find exceptive propositions. Page 349. All except previous winners are eligible. All but previous winners are eligible. Previous winners alone are not eligible. These accepted propositions have to be translated as a conjunction of two general propositions. An accepted proposition. Remember, this will always have to in symbolic notation, when you put it over into the logical symbols, an acceptive proposition is asserting two things. And so it's going to take two propositions to capture it. Or in some cases he says you can use the biconditional is logically equivalent to or if and only if Z the way it would be translated. So he has there in the middle of the page, page 349 for all x, x is an E if but only if X is not a P. So the three line equivalence is really, if you will, the horseshoe going both ways. That's why I like using the arrow. It's like the arrow that has two points on it. It's if this, then that, if that, then this, if and only if. Once we remember we have those difficulties of translation and pitfalls to watch out for, you should be able to take any proposition then in the elementary logical calculus that we have and be able to test at any argument and test it for its validity or prove its invalidity. Would you like to ask any questions about my explanation of quantification? All right, let's take a break. Alright, let's turn to Cofie and do some of the homework exercises and apply what we have learned about quantification. The first section of Exercises follows section 10.3 in the chapter and is found on page 335. He says, Translate each of the following into logical notation of propositional functions and quantifiers. This exercise says, let's see if you can do the translation into our new notation. That's what I taught you at the beginning of today's lecture, the new notation. He says, give the translation, but do so in a way that you do not have a negation symbol as the first symbol in your statement. So you have to remember to get the negation symbol out from in front and put in front of a predicate. Okay? Pat, would you do number one for us? How would you schematize Bats are mammals? It's an A form Proposition.

So you would, you would go to.

The rules in the A form proposition, which would put the quantifier x, all x, just one X in parentheses. And then we'll assign it bx as they've given you, therefore mx. Very good for any X bx, then mx. Is that the way it reads in your homework assignment? Very good. Actually, there should be a parenthesis around the bx, then mx, because that shows that the quantification governs everything. In the parentheses, following number three, reporters are present. Jay.

Okay, it's same thing. It is a. This is, this is the one. We weren't sure whether it was all reporters or some reporters. Isn't okay, I. I took it to be all reporters that are there. So therefore, as an A proposition, well.

It cannot be all reporters that are there because then, because then it would be saying all reports that are there are there. That would hardly be worth saying, would it? This is one of those cases where again, you take an English sentence, you have to have some feel for what the actual meaning of the sentence is. Now, I think you're probably mistaken. I think the sentence means that there are some reporters present. Someone runs inside, says reporters are present. They're not saying every reporter in the world, they're saying there are some. So now, how would you translate that into notation?

Then you have the quantifier, the existential X.

Okay? In parentheses, backwards E, backwards E, X.

And then in parentheses, rx and px.

Very good. In the parentheses, following the existential quantification, you have rxn px. There is an X such that X is a reporter and X is present. Number five, Diplomats are not always rich. Pete. In parentheses, backwards E, then X, then in parentheses again, DX and not rx. Very good. There is an X such that X is a diplomat and X is not rich. You notice again that when we have propositions of the I or of the o form that you use the and in your symbolic notation rather than the if, then that's what these two are stressing. Okay, Number seven, no boy scout ever cheats. That's that E proposition, and I wrote it X in parentheses and then in parentheses, if bx, then not cx. Okay? So reading across, it looks like X in parentheses, then parentheses. Bx, then not cx. Very good. For all X's, it is the TR case that. Or for any X, if X is a boy scout, then X does not cheat. Good. And number nine, I'll do number nine. Snake bites are sometimes fatal. There is an X such that. Now, in parentheses X is S and X is F. There are some snake bites that are fatal. Is that pretty clear to everyone? Let's go to Roman numeral 2. And in Roman numeral 2, we're to find a normal form formula logically equivalent to the given one. Now, I didn't use the expression normal form formula in my lecture. That just means write the formula without the negation in front of a parenthesis. Make sure the negation applies only to predicates. So the task here is to rewrite each of the lines given to us so that the negation is removed from in front of the parentheses. Number one gives us not for any X. Then X is A. Then X is B. What you can do here is follow the equivalences that are found from the square of opposition. What Colby did is write them out for you on page 329. If you look at 329 in the very middle, you have four equivalences stated. The first one on the left hand side is your A form proposition. You recognize that for all X X is phi. The second one down is your I form proposition. There is an X such that X is phi. The next one, number three is the the E form proposition for all X X is not phi. And then finally you have the I form proposition. Excuse me, the O form proposition. There is an X such that X is not phi. So there you have it stated without the negations in front. If you look at the right hand column for each one of those, you notice you have the contradictory for what was on the left hand side in each line. And the contradictory therefore has the negation in front of it, not ex, not phi X, for instance. So in this case we know that when the not is written out there like that, the logical equivalent is what's on the left hand side. So, going back to our homework problem, you will notice that when we have something that is of the not X variety, which is the fourth line in the equivalence, that amounts to an O form proposition. So I rewrote it in O form. There is an X such that it is not the case. Parentheses ax, then bx. So the first step in getting this right is to translate that and get the negation away from the quantification. Everyone with me so far, now that I have the negation in front of my second parentheses, that is the negation is in front of parentheses ax then bx. I can. Well, first of all, I'm going to change what's inside the par parentheses by material implication to read not ax or bx, not ax or bx. And then by Demorgan's theorem, I can change this using double negation, I broke the rules. I combined two steps into one again, but using Demorgan and double negation, the answer comes out. There is an X such that parentheses, ax and not bx. Everyone get that answer? Okay, let's try another one. Jay, would you do number three for us? Okay.

Starting with the knot or the existential quantifier and looking back to that page, 329, I think it is.

At the.

Biconditionals there I can drop the negative sign and make it a universal and then put a negative sign in front of the quantity ex and fx.

Not the quantity, the proposition.

Excuse me, the proposition.

That's okay, go ahead.

Still thinking in terms of algebra. And then by de Morgan's theorem, I can convert this that to an or proposition not ex or not fx, which by material implication converts to if ex then not fx. And so my final form is given.

X.

Then we have proposition fex, then not fx.

Okay, given any X. Given any X. Okay, not just given X. Yes. Now, after you had done Demorgan's on this, you could have stopped because then you would have written it in a form where the negations are only in front of predicates. What you did is going beyond the need of the assignment. Actually, you could have kept going, right? Probably hundreds of ways in which you could. Well, not hundreds, but plenty of other ways you could have applied rules of replacement and done things to would have been sufficient to say for any x parentheses, not ex or not fx. By the way, in doing these, where you have to use Demorgan so many times, have you caught on to what de Morgan's amount to? When you have a negation outside of a parenthesis, De Morgan's moves the negation inside the parentheses to both the compounds and changes and to or.