Dr Greg Bahnsen - Critical Thinking: Categorized Syllogism Part 1 (Part 10)

Let's begin this morning with a word of prayer. Our Heavenly Father, please forgive our sins this day as we come to you in the name of Christ our Savior, you do receive us on his behalf for the sake of his righteousness. We do pray that his death would atone for all of our transgressions and that we would be forgiven according to your mercy. We do confess that we would have nothing and that we could not stand before you if it were not for your grace, for the provision that you have made by your spirit, that we might have life and enlightenment, that you have given us the gift of faith, that we could stand before you. We do confess that everything that we are shall ever be comes by your blessing. We do pray that you would be with us today to help us to become more like you, to think more clearly, to be drawn closer to the truth. We do pray that even through our lessons and logic, we might understand that we are coming to know you, to know your mind. And we do pray that you would help us to glorify you in all that we do and say in this class. For we pray in Jesus name. Amen. All right. Today we turn from our discussion of language and ordinary language fallacies to deductive or formalized logic. This is a big step that is still part of that aspect of our course that I'm calling analysis. The first part of the course dealt with argumentation, what that amounts to theology of argument. The second major part of our course, I'm designating analysis, an analysis of language and then of the fallacies that are found in ordinary discourse. And now, as part of our discussion of analysis, we turn to formal logic or deductive logic. And this brings us to chapter five in Kopi. In Kopi's section on deductive logic, the middle part of this book, he will discuss the logic of categorical statements, and then we'll study the logic of connectives like and, or if, then. And. I think if you keep those two segments in mind. To be clear what's going on in this part of the course, we are now going to begin that section that deals with categories or classes of things, often called categorical syllogism. The logic of categorical syllogism. Let's turn to page 161, chapter 5, and COPI, and work through this chapter and the exercises that were assigned to you. The first thing he has to do, of course, is to explain what a categorical proposition is in terms of this notion of classes or sets that we've been introduced to briefly up to this Point we're going to be at this juncture in our class, our interest is going to be in the logic of the words all some and not. So no matter how complicated this gets, you might just keep that in mind. All I'm doing here is learning to regiment the logic of how we use all some and not. And this requires that we think in terms of objects in this world being part of classes. 5.1 Categorical propositions in Classes Copie reminds us that a deductive argument is one whose premises are claimed to provide conclusive grounds for the truth of its conclusion. Okay, so to say that an argument is intended to be deductive is to say that it's supposed to offer conclusive grounds. Every deductive argument is either valid or invalid. You don't have strong or weak or varying degrees of them. You have valid or invalid. And when is it valid? It's valid if it's impossible for the premises to be true without the conclusion also being true. And everything else is invalid. Remember, it's impossible that you have true premises and a false conclusion. Impossible. So in a valid argument, the truth of the premises will mean that the conclusion is true as well. He tells us in the next paragraph that the study of syllogism, or what we call categorical syllogism, begins with the philosopher Aristotle. I think that's a good historical note for you. In the footnote, when Aristotle wrote on what we now call logic, the book in which he wrote was entitled Organon, or if you will, English translation, instrument, the organ. The instrument. And so Aristotle saw logic as the instrument of thinking, the organ by which, or the instrument by which we can draw correct conclusions. Page 162 categorical propositions. Categorical propositions are about classes, and they affirm or deny that a certain class that we'll call S is included in a class P, either in whole or in part. And although we're not going to be using the Venn diagrams, I will, for the purposes of reinforcing our lesson, do something akin to them. If I have a class, things what is called an S's, whatever they may be, they can be seagulls, they can be sailing ships, they can be anything at all that can be identified as a separate class. And I say that all of the members of this class are also members of another class, then what I've done is I've included them in a larger class. So here we have S and here we have P. You can visualize. Then, for those listening by tape, the S circle is completely inside the P. Circle. In that case, every S is a P as well. But now what if I have these circles intersect each other so there's this partial overlap. And then we have this S circle on the left and this P circle on the right. There's this center section that is S and P both. In this case, some of the S's are P's. Make sense if you've had advanced math, you know about set theory and inclusion and exclusion, I'm assuming, and that's all we're talking about here. A class is a collection of all objects that have some specified characteristic in common. And we're going to learn now to relate different classes, the classes of seagulls and sailing ships and soup cans, if you will, all the S's in the world. Of course, it doesn't have to begin with an S. But just to help you remember, a universal affirmative proposition is one that says all the S's are P's. It's universal because it talks about all the S's and it's affirmative because it is positive. It's saying yes, all of them are P's. On the other hand, a universal negative proposition says of all the S's that none of them are P's. So again, it's universal about the S's, but negative in what it's claiming. Not all S is P or actually all are not. The particular affirmative proposition is one. That is our second diagram on the board that has some overlap, but not complete overlap. And so sum of the S's are P's. And you notice copie tells us 2/3 of the way down to 163, the word sum will mean at least one. It does not mean two or more, it means at least one. And then finally we have particular negative propositions which do not talk about all of the S's, but some of them at least one. And it's a denial that some. It's a denial that some RP comes out reading, some S is not P, some S is not P. And you have some exercises then just introduce us to this notion of universal and negative. Universal in particular affirmative and negative propositions and how they are to be classified. So let's look at page 164. Identify the subject in predicate terms and name the form of each of the following propositions. All right, let's take number one. Some historians are extremely gifted writers whose works read like first rate novels. What is the subject term?

Historians.

What is the predicate term?

Extremely gifted writers whose works read like first rate novel.

Yeah, the fact that it's a long description doesn't change the fact that that's what the class is. The class of extremely gifted writers is worst read like first rate novels. Okay, and then is tell me what particular form this has. Particular.

Affirmative.

Particular affirmative. Okay then number two. No. Athletes who have ever accepted pay for participating in sports are amateurs. What is the subject?

Athletes who have ever accepted pay for participating in sports.

What's the predicate?

Amateurs.

And what's the form of the proposition? Universal negative. Correct. Universal negative. Okay, number five. Some members of families that are rich and famous are not persons of either wealth or distinction. What is the subject?

Members of families that are rich and famous.

What's the predicate?

Persons of either wealth or distinction.

And the form?

Particular negative.

Particular negative. That's correct. And then number six. Some paintings produced by artists who are universally recognized as masters are not works of genuine merit that either are or deserve to be preserved in museums and made of available to the public. What is the subject? Subject to be?

Paintings produced by artists who are universally.

Recognized as masters and the predicate? Works of genuine merit that either are or deserve to be preserved in museums and made available to the public. And what's the form?

Some S is not key or particular negative.

Particular negative. Very good. Okay, let's see if we can get a little fancier about this. Now, 5.2 in copie deals with quality, quantity and distribution. He says the quality of a proposition is affirmative or negative according to whether class inclusion is affirmed or denied. So when you affirm you've got the quality which is positive or affirmative, and when you deny, then you have a negative quality. As to quantity, he says in the next paragraph, the quantity of a proposition is universal or particular according to whether the proposition refers to all members or only to some members of the class designated by the subject term. And in this regard, the quantifiers are the terms all know or none and some all know some and do. Pick up this note at the end of the paragraph. The quantifier no serves to indicate the negative quality of the e proposition as well as the quantity. Right? The quantity is universally negative. Well, the quality is negative and the quantity is universal. You all understand that you get two for the price of one with the word no. And then the copula is the verb to be or some version thereof, depending on the form of the sentence, has been, will be, is so forth. And then on the top of page 166 he says the schema for expressing a categorical proposition is to quantifier quantifier is going to be what, all, some, none, or no. Then the subject term, then the copula is, will be, has been, and then the predicate term. Halfway down the page, he refers to all S is P as an A proposition and says that it refers to all members of the class designated by the subject term. S. That should be real obvious to us. All the S's in the world or in the universe or in reality are referred to in an all S's P sentence, but it does not refer to all members of the class designated by the predicate P. Okay, let's just take all. All sailors are drunkards. Now, does that talk about every sailor there is. Is it universal about sailors? All sailors are drunkards? Yes. Does it tell us anything about all drunkards? No. We know that some drunkards are sailors, but we don't know about all drunkards. And so here we have a case where all the members are designated in the subject term, but not all the members of the of the predicate term are referred to. A proposition distributes a term if it refers to all members of the class designated by the term. So when I say all sailors are drunkards, my sentence distributes the term sailor, because it's universal, does not distribute the term drunkard because it doesn't tell us about every member of the class of drunkards. Okay, let's go from A to the E proposition. The E proposition is the universal negative. E propositions distribute both their subject and predicate terms. That should be easy enough to see if I can put it on the board. Here we have two circles without any overlap or touch between them. So we're not including the two classes in each other, nor are we saying this partial inclusion between the two. Here we have S and P as separate circles. No S is P. Then there's no overlap at all. Now if no S is P, I'm telling you about all the S's and about all the P's, aren't I? So in a universal negative, both the terms are distributed. When we turn to The I proposition, that is the particular affirmative. Some soldiers are cowards. Page 167. Notice that neither class is said to be either wholly included or wholly excluded from the other. So in that case, both the subject and predicate terms are undistributed. A particular affirmative proposition doesn't distribute any term at all. Then either the subject term or the predicate term. We don't learn about all of them. And then the O proposition, which is the particular negative, speaks of only a portion of the Subject class, some portions. Now it talks about some horses not being thoroughbreds, to use the example in Kopi. So of the predicate term, we know that none of them are going to be part of the subject term, even though the subject term is limited. So the RI Kofi, some horses are not thoroughbreds, says of this part of the class of all horses that it is excluded from the class of all thoroughbreds. That is, it's excluded from the whole class of thoroughbreds. So of the class of horses, some of them, which is the subject term, some horses are excluded from the entire class of thoroughbreds. So distributors distribution has to do with speaking of the entire class. When we say some horses are not thoroughbreds, are we talking about all horses? No, we're only talking about some. So that's not distributed. Are we talking about all thoroughbreds with respect to those sums? Yes. And so the predicate term is distributed. At the bottom of page 167 you have a very convenient chart given to you. Following the the AEIO chart. The A proposition being universal affirmative, all S is P is on the top line. And then to the right is the E proposition. No, S is P which is the universal negative. Then below the A you have the I form of proposition, which is the particular affirmative C. Some S is P. And then right next to it, on the same line underneath the E proposition, the particular negative sum S is not P. And if you ever have to, you remember that A, E, I, O configuration and then think to yourself about distribution. Now if you know what distribution means, you can figure out each one. But if you want to remember the chart, you notice that if you start at the top you have the subject term distributed. Move to the right of the chart you have the predicate term distributed. So we have subject distributed, predicate distributed. Now we're going to go subject and predicate again, bottom and then up to the left. So we make a full circle here, the bottom, the subject term is undistributed. And on the left hand side you have the predicate term undistributed. Now where do we get the subject term distributed and the predicate term distributed? Melissa?

In the E proposition.

Okay, And Jay, where do we get the subject term undistributed and the predicate term undistributed?

That would be the ICROP proposition.

Okay, so you have the full distribution in E, the full undistribution in I. And then you have the mixture at A and O. Okay, we have some exercises on page 168. Copie asks you to name the quality and quantity of each of the following propositions and state whether their subject and predicate terms are distributed or undistributed. Number one, some presidential candidates will be sadly disappointed people. What is the subject term here?

Some presidential candidates.

And what's the predicate term?

Sadly disappointed people.

What's the copula will be? Okay, and what kind of proposition do we have here? Particular affirmative. Particular affirmative. And is the subject term distributed? No. Is the predicate term distributed?

No.

Okay, so in the case of the I, we have no distribution at all. Very good. Number two, all persons arrested in the Watergate burglary were dupes of the Nixon law and order administration. What is the subject here?

First interested in the Watergate burglary.

What's the copula word? What's the predicate dupes of the Nixon.

Law and order administration?

Okay, what's the quality and quantity of this proposition?

The all.

What is the quality and quantity? Let's use the terms the technical given us. Oh, the A of the particular.

Universal Affirmative.

That's right. All S is P. So you have universal Affirmative. Is the subject term distributed? Yes, because it refers to all the persons arrested in the water heat burglary. Is the predicate term distributed? No. No, because there may be dupes of the Nixon law and order administration that are not being discussed here, but only those arrested. Okay, that's good. Let's look at number five. No. Leader of the feminist movement is a major business executive. What is the subject leader of the feminist movement? What is the quality?

The quality is. Negative.

What's the quantity?

Universal.

What's the copula is and what is the predicate?

Predicate is a major business executive.

Okay. Is the subject term distributed? Yes, it is. Because every one of the subjects, subject, class is referred to with the. No. Is the predicate term distributed?

Yes.

The predicate term is distributed because we have here a universal negative. Yes. Okay, so you have full distribution. Remember up in that corner the A, E, I, O chart, the E section. You have full distribution. That's the easiest one to remember. Okay, and then number six, all hard line advocates of law and order at any cost are people who will be remembered, if at all, only for having failed to understand the major social pressures of the late 20th century. What is the predicate term here?

People who will be remembered, if at all, only for having failed to understand.

The major social predicament. It's a long description, but you're right. That is the predicate term. Or the class that is named. What is the copula? R. R. Okay. And then what is the subject term?

Hardline advocates of law and order at any cost.

What is the quantity of this proposition?

It's all, it's inside.

Universal is universal and quality is affirmative. Right. I just want to make sure we distinguish those two. The all is the quantity, and the affirmative is that it's actually positive about this. All hardline advocates are people who will be remembered. Okay, let's talk about distribution here. Is the subject term distributed? Yes. Is the precedent term distributed?

No, it is not.

Okay, so we all understand distribution. Is that clear? And then we're going to move on to the square of opposition in section 5.3. When Pope returns to the traditional square of opposition, he does so so that we can learn about immediate inferences that can be made from certain statements. If you want to jump to the point of this section, you turn to page 172 and it gives you a chart of the aeio propositions. First of all, if they are true, and then the second four if they are false. He tells you what you can eat immediately infer. And we're going to see why you can immediately infer certain things in the course of this section, Copie tells us on page 168 certain important truth relations are correlated with the various kinds of opposition between A, E, I and O categorical propositions. First of all, he talks about contradictories to two propositions are contradictory if one is the denial or negation of the other, in which case they cannot both be true and they cannot both be false. All judges are lawyers. Some judges are not lawyers. The A and O propositions cannot both be true and cannot both be false. So they are contradictory. Everyone understand what a contradictory statement is. You can't have both true and both false. Now, the reason why I'm making sure you have that in your mind is because a lot of people in ordinary discourse talk about contraries and contradictories in a real kind of imprecise fashion. And so now, if we're going to use the technical jargon of logic on page 169, he says two propositions are said to be contraries, not contradictors, contraries. Yet they cannot both be true, although they might both be false. They cannot both be true, but they might both be false. And he gives us as an example of contraries, A and E propositions. All poets are idlers. No, poets are idlers. You can't have it both ways, can you? They cannot both be true. But is it possible that both would be false? Yeah, it's possible that Both of those are false. Can someone explain to me why.

Some.

Poets might be idlers, some poets might be idlers, and so no poets are idlers is false. And all poets are idlers is false. So here we have all, none, some. Remember I told you this is all about all, none and some, or all, some and not, however you want to put it. Very good. Now a qualification has to be made. Koppi says it's obvious that this claim that A and E propositions are contraries is not correct if either the A or the E proposition is a necessary truth. Another way of speaking of necessary truth is logical truth or mathematical truth. A necessary truth is explained in the next sentence. If a proposition is necessarily true that cannot possibly be false, then it cannot have a contrary, because propositions that are contraries can both be false. So let's back up. What is a necessary truth? One that it's impossible to be false. If it's necessarily true, it's impossible that it'd be false. Now in the case of contraries, both can be false. So if we're dealing with a necessary truth which cannot be false, then it cannot have a contrary, can it? Because a contrary means it could be false. By calling it a necessary truth, you're saying cannot be false. Have I confused you? I hope not. It should be rather clear that if something's to be a contrary, it could be false. And that excludes then any necessary truths. A proposition that is neither necessarily true nor necessarily false is said to be contingent. A proposition that's neither necessarily true or false, so it's not a necessary truth or a falsity, it's contingent. In the case of contingent propositions, A and E are contraries. So we say A and E are contraries, provided they are contingent truths, not necessary truths. Everyone follow that. We're just making a slight point qualification about saying A and E are contraries. They're contraries only if they are contingent truths. Now he discusses subcontraries at the bottom of page 169. Propositions are subcontraries that they cannot both be false, although they might both be true. In the case of contraries, they cannot both be true, but both can be false. In the case of several contracts, they cannot both be false, so they might both be true. So you have the opposite situation. Both can't be true up on the A and E line. Both cannot be false on the I and O line of our little four cornered chart of propositions. But then a qualification has to be made here too. On the page 170. It says it's obvious, this claim, that I and O propositions are subcontraries. To call them sub contraries is to say they cannot both be false. One of them's got to be true. That is not the case if either the I or the O proposition is necessarily false. For if a proposition is necessarily false, that is cannot possibly be true. It cannot have a subcontrary, because propositions that are subcontries can both be true. On the top line, to review for you, A and E can both be false. On the bottom line, I and O can both be true. So if you're dealing with something that's necessarily false, it can't be one of the things that could be true, in which case subcontrary doesn't apply to it. And again, the qualification that we make is that if I and O propositions are contingent, then they are sub contraries and both could be true, but both cannot be false. Any questions so far? I don't want to go too fast. This is really not difficult. But because it's a new jargon, we sometimes get messed up thinking we don't understand. Okay, then the top of page 171, we get a new type of implication here, that of subalternation. The super altern, which is going to be either A or E, the universal, affirmative or negative. The A or E, the super altern implies the sub altern. So that you can deduce an I, some S is P from an A, all S is P. If all S is P, then obviously some S is P. And if no S is P, then it is obvious that some S is not P. So. So that is subalternation. Subalternation. You cannot go the opposite direction, though clearly you can't move from some SSP to all ssp. So super alteration, moving from the particular to the universal is not valid. However, you can immediately and perhaps the sub alternate from the super alternate. And then all of this is summarized very nicely for you in the square of opposition found in the middle of page 171. 171. I think it would be helpful if we went through here and wrote down in the case of these terms what they are going to to mean to each other. I think if that were right there in the chart, it would be helpful to you. So let's begin with contradictory. Going Back to page 168, he says two propositions are contradictory if one is the denial or negation of the other. They cannot both be true and cannot both be false. So in the very middle of the square, where you have the lines of contradict contradiction, they cannot both be true. Cannot both be false. Then the contraries, which are the top line of the square of opposition, remember it's defined for us on page 169. Here he says they cannot both be true, although they might both be false, cannot both be true, but might both be false. And I don't know if there's going to be enough room or if you want to clutter up your diagram, but do remember the qualification, provided they are contingent truths. Provided they are contingent truths. Now, with respect to the subcontraries, the I O line on the bottom of the chart, we have that defined on page 169. They, they cannot both be false, although they might both be true. You can't have both of those false, but you might have both of them true. And again, the qualification has to be made that we're here dealing with contingent truths. And then finally we have the implication of subalternation, where we say that the, the implication follows from the super altern to the subaltern. So if you will, running down the chart from A to I and from E to O, we do have a direct implication. A implies I, O implies. Excuse me, E, E implies. Oh, okay. Now, two thirds down the page on 171, Copi explains where a conclusion is drawn from only one premise, there is no mediation. When we talk about syllogisms, in a little while we'll have two premises and a conclusion. Premise one, premise two conclusion. And it's thought that the first premise is mediated through the second to the conclusion. But where a conclusion is drawn from only one premise, there is no mediating premise, no premise two. The inference is said to be immediate. So all of this has just been a way of introducing us to the immediate inferences that can be drawn from the categorical universal, affirmative, universal negative, particular affirmative, in particular negative. If you look at the form of the proposition regardless of the content S's and P's, then you can draw certain immediate inferences which are now listed for us on page 172. A, E, I and O. If they are true, you can draw the immediate inferences that are listed. In the case of all S is P, you know that no S is P is false. Some S is P is true, some S is not P is false. And then the next line from the E proposition, no S is P. If that's true, then the all S is P must be false. The sum S is P is false. And the sum S is not P is true. Given the I proposition, which is universal. Excuse me, particular affirmative. Some S is P. If that's true, then the E proposition that no S is P must be false. The A proposition that all S is P and the O proposition that some is not cannot be determined. There is no immediate inference there. You have only the contradiction.

Okay.

And then with respect to O proposition affirmative. No, I'll give this right particular negative. Particular negative, some S is not P. If that's true, then A must be false because A says that all S is P. But the E proposition that no S is P and the I proposition that sum is P cannot be determined. Just knowing that sum is not.

Why can't you determine sum F is.

P. Sum F is P would be the I proposition. You have a. A particular affirmative. All we know is that from an O proposition that sum is not P, some S is not P. Now, if sum S is not P, does that prove that sum S is? Well, you don't know. Maybe all is not. But you couldn't. All you know. It's like if you've examined 70% of the people in Lynchburg and found out they had tuberculosis. You could say some people in Lynchburg have tuberculosis, but from that you could not automatically conclude that all have it.

But could you determine that some do not have it? Isn't that what it's saying?

No, because all you're talking about is the ones that you do know. So you don't know about the others. They're undetermined. Does that help explain that? And then we have. Now, what if A, E, I and O are false? The second group here, if all S is P is false, then you know that O is true. Because that's the contradiction. You know that E and I are left undetermined. So that's interesting. Notice that in the top four, if the A proposition is true, you can tell about E, I, and O, whether they are true or false. But if the A proposition is false, you can only tell about the O proposition that it's true and the other two are left undetermined. Now the next line. The E proposition, no S is P. If that's false, if it's not the case that no S is P, then we know that the I proposition is true. The contradictory some S is P must be true then. But the A and O are left undetermined. And again, notice the corresponding line above where it says at E being given as true, you can get the truth value of all three of the other possibilities. But if E is given as false, then we can only draw one inference and the other two are left undetermined. Now, I want you to compare lines three and four. In the top four, you notice how two are left undetermined. If you compare lines seven and eight, all are determinable. So you have the reverse of what we've just gone over. What is left undetermined has a lot to do with whether you are dealing with a true or a false proposition. Does that clear to everybody what I'm talking about? If you compare lines 3 and 4 to lines 7 and 8, you notice that in lines 3 and 4, two of the other corresponding of the square of opposition are left undetermined. But if you know that I and O are false, then you can determine every other corner whether it's true or false. Then the exercises on page 172 present some challenge to us. What can be inferred about the truth or falsehood of the remaining propositions in each of the following sets? One, if we assume the first to be true. Second, if we assume it to be false. Let's look at number one, and let's begin with the second part of the exercise. If we assume it to be false, that is, if the first proposition is assumed to be false, that is. A, all successful executives are intelligent people. I'm going to go slowly here. Now, this is what you take for granted. That's false. It is not the case that all successful executives are intelligent people. If we go slowly, tell me, first of all, what is the form of that proposition?

That's an A proposition.

That's an A proposition. And we're taking it for granted that it is false. Now let's go through the other three and see what we can conclude. B, no successful executives are intelligent people. Now, what is the form of that proposition? That no successful executives are intelligent people.

That's an E form.

That's an E form. Very good. Now, if the A form is false, what can we say about the E form?

It's undetermined.

It's undetermined. So? So we're not able to tell with respect to B if it's true or false. If it's not true that all successful executives are intelligent, we don't know whether it's the case that no successful executives are intelligent. Now let's go to C. Now we find the proposition. Some successful executives are intelligent people. Some S are. What's that form called? Some SRP it's an I proposition. That's an I proposition. And if we know that The A form is false. What can we say about the I form?

If the I form. Is undetermined.

Is undetermined. Right. If it's not true that all successful executives are intelligent, that leaves open that maybe some are. Now, because we have the content there. In some ways, you can almost intuit this. If somebody says it's not true that all are intelligent, you can say, well, okay, but then maybe some are. So the proposition that some are is left undetermined. That's a possibility. Okay, now D. Some successful executives are not intelligent people. If you know that it's not true. If you know that it's not true that all successful executives are intelligent, what can you say about the proposition that some successful executives are not intelligent?

It's true.

It's true. By what type of inference? This is not the exercise. How do you know that it's true?

Because it's an O combination.

What kind of inference is that? Going from all.

It's a sub alternation.

No, no. If it were a subalternation, you'd have some are here, you have, some are not. Oh, yeah, that's right. Exactly. Contradiction. Did you have that feeling of frustration? You said I learned all that for nothing. Anyway, what I want you to hold on to is remember the contradiction. You know, A and O, you know, contradict each other, and E and I contradict each other. So that's why I want to stress that one. Well, that was only the first run on these. Let's go back and do it again. Now, on this time through, we'll assume that what is listed as A here is true, that all successful executives are intelligent people. If it is the case that all successful executives are intelligent, all S is P, then what can we say about B that says no, S is P?

That would be false because it's contrary.

Right. And contraries, they cannot both be true. Good. How about C? Some successful executives are intelligent. If all successful executives are intelligent, that's true.

Because it's a subalternation.

That's right. And then let's look at D. Some successful executives are not intelligent people.

People.

If all successful executives are, is that A, is that implication that can be drawn? Is D true or false given. And why? By what? Immediate inference pattern by.

Subcontractor.

No, it's not a subcontrary. Subcontraries are on the same line. What you have here is an A and an O. Contradictory. Yeah, they're contradictory, which is, of course the main one. We want to remember that. If you know that all S is P, you know that it cannot be true. Then that sum is not. This is a lot of fun, isn't it? Let's try number two again. We have to go through here on two passes. First, on the assumption that number two is true. Well, I'm sorry that the first of the four listed propositions in number two is true. So 2A. No animals with horns are carnivores. No animals with horns are carnivores. Let me just first of all identify the form of that proposition. No, S is P. So we have an E proposition. Now, if an E proposition is known to be true, let's take the second line down and test it. Some animals with horns are carnivores. Here you have the particular affirmative. So an I proposition. If E is true, what can you say about I?

It's false.

By what inference pattern?

Contradictory.

Contradictory. Now, I know we're all trying to keep the form in mind, but let's just listen to this. If this was an ordinary discourse, you'd be able to figure it out. If I said it's true. No animals with horns eat meat are carnivores. No animals with horns are carnivores. Somebody else says, well, some animals with horns are carnivores. Haven't we got a direct contradiction? I mean, one of us has got to be right and the other has got to be wrong. Can't be both horns. So that's a contradictory. Let's look at C. Some animals with horns are not carnivores. Remember that we're assuming that no animals with horns are meat eaters. None of them with horns are. What can we say about the proposition that some with horns are not?

That would be true because.

Follows directly.

From the fact that none of them are.

If none of them are, then some of them are. Then some of them are not. C says some. S is not P. You have the particular negative, which is an O proposition. And we're assuming that the A proposition is true. That's an 80 proposal. Thank you for correcting me. We're assuming that the E form of that is true. And so if E is true, then O is. Oh, good. I thought you said false. It follows by subalternation. Exactly right. Now listen D. All animals with horns are carnivores. What can we say about that? If we assume that no animals with horns are carnivores. Horse.

It's false because it's a contrary.

That's right, it's contrary. They cannot both be true. Can they both be false? Very good. Now we have to do it Again, change the register here. Let's take the first proposition to be a false proposition. No animals with horns are carnivores. It's not true. So we deny that. We deny that no animals with horns are carnivores. That means that our E proposition is false. Let's take the next proposition. Some animals with horns are carnivores. Some S is P. Is that true or false?

That would be true.

Okay, and by what inference? Pattern? Subalternation? No, let's begin with the A. In our illustration, we know that it's false that no animals with horns are carnivores. Now, what kind of proposition is no S is P? No S is P. E is an E. And then the one we're testing is some animals with horns are carnivores.

Which is an I.

That's an I. Now, what do we know about E and I propositions? They are contrary.

They're contradictory.

They are contradictory. Yeah, we have to get that right. Now, if they're contradictory, if you know that one is false, the other must be. Must be true. Must be true. Okay, so B is going to be true. Given. I mean, I'm looking at the illustrations. B in the illustration must be true if A in the illustration is false. Let's look at number C. Now, some animals with horns are not carnivores. Here you have some. S is not. P is a particular negative, which is an O proposition. Now, if we know that the E proposition is false, what do we know about the O proposition?

It's undetermined. It's a subalternation.

Now, can we see this? Can we see this from just an intuitive grasp of it? Somebody says, no animals with horns eat meat. You say, well, that's false. Oh, well, then I guess some animals with horns eat meat. Well, we don't know that either. We can't determine that just from the denial of the universal negative. And then finally, the D in our illustrations, all animals with horns are carnivores. What can we say about that? If we assume that it's false that no animals with horns are carnivores, it.

Would be undetermined, because it likewise can.

Wait, if you say that all are not, doesn't it say that some. That all are.

They're contrary? So they can both be false.

They can both be false. Exactly. Because if somebody says all, somebody says none. There's also the possibility of some. And since some exists as a possibility, then we can't go from the negation of one of the alls to the other universal. If that makes sense to you. Now there are going to be further immediate inferences that can be drawn from our categorical propositions. Now things start getting exciting. We get into conversion and obversion and. What's the other one called?

Contraposition.

Yeah, contraposition. That one is a lot of fun. And the reason why you're all glaring at me this morning is that after you worked through this section, figured out how to do these silly things, you got to Existential Import and found out that obversion and conversion and contraposition all get chucked. Is that right? Not completely. Not completely. But you don't have, you know, a real easy time with immediate inferences anymore with them. Okay. You're still going to want to know this when on down the line, but I want to work through it for a few minutes this morning so we know at least what the procedure is, even if it doesn't become what you do every day of your life. Okay. As we come to further immediate inferences on page 172 or section 5.4, we will be introduced to conversion, obversion and contraposition. In the case of conversion, what is called a process of conversion in logic, you interchange the subject and predicate terms. Now, that's probably the easiest of the three. You just take the subject term and make it the predicate term, and the predicate term make it the subject term, And certain immediate inferences can be drawn. Then, when we use conversion as our process, these are listed for us at the bottom of page 173. Before we look at that list, however, in the middle of 173, we have to understand what it means to convert by limitation. This is jargon that comes from the development of the history of logic. And let's not worry about why it's called what it is. But Copie says by a combination of subalternation and conversion, you can take sum P is S as validly inferred from all S is P. And when you do this, both sub alternation and conversion, then you call that conversation conversion. By limitation, what you do is you interchange subject and predicate terms. That's conversion. And then you change the quantity. By changing the quantity, you're limiting. Okay, that's obvious. When we go from universal to particular, you're limiting. But limitation can go the other direction, too. You can go from particular to universal. Anyway, if we convert and then change the quantity, then we've converted by limitation. And so, notice at the bottom, page 173, taking all s is P, we can by limitation infer some P is S. You can almost tell that intuitively I think if some S is P then there have to be some P's that are s. Now E, no S is P by conversion. You just replace the S with the P and the P with the S and you get no P is S. So the converse is true. If no S is P and obviously no P is S either, that's what's still on the board. You have the two classes separated completely. The I proposition, sum S is P by conversion becomes sum P is S, which is fine. And now notice the ringer. Though not all conversions are valid, that will carry their truth value with them. If some P is not S, the converse is not valid. So make sure you note that you cannot take an O proposition and convert it.

Does that make it true or false?

I'm sorry?

Does that make it true or false or simply invalid?

It means that if you go from an O proposition to its conversion, you do not have a valid inference. And so that leaves it undetermined. Now, on page 174 we learn about obversion. This gets a little more complicated. To do an obversion, you have to understand what a complementary class is. And remember, a complementary class is not the same as a class contrary class. Heroes and cowards are contraries. They are not complementary classes. A complementary class, to get more exact, is if you have the class of heroes. The complementary class is the class of non heroes. Now whether all non heroes are counting cowards is another matter. Maybe they are, maybe they aren't. But you mustn't confuse the contrary with the complement. The complement is the straight out negation of the class or the negation of the class defining attribute. So if we speak of the class of birds, the complementary class is the class of all non birds. Y' all got that? Now that we understand what a complementary class is, the non P, if you will, if P is the predicate to obvert a proposition. This is the bottom of page 174. To obvert a proposition, we change its quality. Quality is to be affirmative or negative. We change its quality and replace the predicate term by its complement. Obversion deals with predicate terms. You have the SP pattern. When you obvert, the P is going to become non P, and you're now going to change the quality from affirmative to negative or from negative to affirmative. Oh, this is so much fun. And the list of how to do these things or the result of doing it is on the bottom of page 175. The inversions beginning with A, all S is P Now remember, what we're going to do is replace the predicate term with its complement and change the quality. The quality here is affirmative at least must now become negative. No S is non p. Yep, it worked. Just wanted to make sure we understood the definition. You see how that comes for us. No S is P. The E proposition. Change the quality from negative to affirmative. Replace the predicate with its complement. Now you have all S is non P. Some S is P is particular affirmative. We want to change its quality to particular negative and take the P and make it a non P. So now we have sums is not non P. By the way, is that intuitive that one to you? If some S is not non p, if you do the double negation, you have sum S is P. That's exactly what you had over on the other side, right? Sum S is P. Okay? And then at the O proposition we have sum S is not P. Change the quality from negative to affirmative and replace the predicate term P with not P. So now you are going to have sum S is non p. And again, that should be rather obvious as all we've done here is taken the not p and made it into a class of non P's. So obversion is kind of a tricky one. But most of the obversions become pretty clear, even by intuition. And now we're really going to have fun. Contraposition to form the contraposition. Do you ever wonder, do people sit around? They had nothing better to do in the Middle ages but to figure out all these different tricks you could play. Actually, they worked so long in the different kinds of inferences and they saw patterns. And this all gets summarized and boiled down for us here. To form the contra positive of a given proposition, we replace its subject term by the complement of its predicate term. So the subject term gets dropped, the predicate term's complement takes its place, and we replace the predicate term by the complement of the subject term. This is kind of like conversion, right? It's conversion. The S becomes P and the P becomes S. Except we're using complementary classes instead of the ones that are mentioned. Do you all see that? That it's just conversion using complementary classes. We replace the subject term by the complement of the predicate and replace the predicate term by the complement of the subject. So it's conversion using complement. Contraposition is what it's called. And the have to observe limitation when following the procedure of contraposition. And that takes place when you're contra positioning. When you're contrapositioning an E proposition, it will become true only by limitation. And what does it mean when we limit something? We change its quantity. So on the chart that you find two thirds of the way down page 177, look at E. In the case of E, you have no S is P. That's universal. Now we're going to want to change its quantity by limitation. So it's going to have to become sum. You notice over on the other side, the contrapositive sum P is found. So that's the limitation part of it. Let's go through each one of them. A, all S is P. Okay? S becomes non P. P becomes non S. So all non P is non S. I'm just going through the steps so you make sure you see it in your own mind. In the case of E, no S is P would become no. Non P is non S. Except we've learned from the reading of the chapter that that's only true by limitation. So the no becomes sum. Some non P is not some S. Excuse me, some non P is not some S. We have to change it from universal to particular. Then I. There is no contra positioning of an A proposition. Excuse me, of an I proposition that will be valid so that's left undetermined. And then O, some F S is not P. The S becomes non P. Some non P is not non s. And you notice this very nice paragraph following. It's such a relief. He says there are many other types of immediate inference. You're going, no, I don't want to learn more of these. These are crazy. He says there are many other types that have been classified and given special names. But since they involve no new principles, we shall not discuss them here. When he says they involve no new principles, every one of the other ones becomes some kind of recursive use of the three that you already have, have some combination of these new names given to it. Okay, so let's go to our exercises. Now, on page 178 in exercise Roman numeral one, we are to state the converses of the following propositions. Indicate which of them are equivalent to the given propositions, where the exercise asks for us to determine whether it is equivalent. He is referring to logical equivalence, and they are logically equivalent. The chart shows us all the logical equivalences except on those lines that say not valid. If the contra positive of the I proposition is not valid, then the contrapositive is not logically equivalent to the I proposition you began with. So that's just a different way of speaking. And we're going to take as Our exercise? No. People who are considerate of others are reckless drivers who pay no attention to traffic regulations. Let's do this slowly. What is the subject in this proposition?

People who are considerate of others.

What is the predicate?

Reckless drivers who pay no attention to traffic regulations.

Okay, and then what's the form of this proposition? It's A, E, which is universal negative.

Right.

Okay. Now, conversion involves, just by review here, interchanging subject and predicate terms. So let's see if you can interchange the subject and predicate terms now.

Okay, no reckless drivers who pay no attention to traffic regulations or people who are considerate.

Very good. That's the conversion. And is that logically equivalent? Yes. Yes, it is. Good. Let's go to the next set of exercises which calls for us to do an obversion. In the case of obversion, just to review the chapter, we change the quality and then replace the predicate term by its complement. The predicate term is affected only and the quality. So, taking number one in the section of Roman numeral two, some college athletes are professionals. We want to change the quality and replace the predicate term with its complement. What would that give us then?

Some college athletes are not non professional.

Sounds right to me. Everybody get the same thing. Change the quality from some are to some are not. Change the predicate term to its complement. Non professionals. So some college athletes are not non professionals. And is that logically equivalent?

Yes.

Correct. It is logically. It'd be easier just to remember which two are not and which two have limitations on them. But for right now, we'll just use the charts. And then number three, state the contrapositive of the following propositions. We're only going to do number one. All journalists are pessimists. To get the contrapositive, we replace subject and predicate terms. That's like conversion. And we replace them with their complements. So let's do that.

All non pessimists are non journalists.

And is that logically equivalent? Yes, that's right. And then I also wanted you to to do number six, Roman numeral six, Beginning with the assumption that some saints were martyrs, beginning with the assumption that that is true. What may be inferred about the truth or falsehood of the following propositions, and we're going to look at a few of those in order to determine the truth or falsehood. Then we need to remember our immediate inferences of contradiction, contrariness, subalternation, conversion, obversion and contraposition. In order to tackle number six. Let's just see what Immediate inferences could be drawn from the proposition. Sums is M. We use S and M for saint and martyr. Some S is M. And the immediate inferences that we're familiar with are contradiction, contrary, subalternation. Here there is no subalternation because you're already in the particular. You're on the bottom line. And we know super alteration is not alternation. It's not allowed. So you have two there contradictory. What's the contradictory of some S is M. That's going to be all are. No S is M. Okay, so let's make a list of true and false inferences. If some S is M, then we know that it's false, that no S is m. Does everyone follow me? What we're attempting to do here is to make a list of inferences, the true and the false inferences that can be immediately drawn from our test proposition Some S is M. The first one was the contradictory. No, S is M. Now we have, because this is a particular proposition, we have to deal with sub contrary, not contrary. And in the case of sub contraries, we know that perhaps both can be true, perhaps both can be, but they cannot both be false. We have a proposition that is true, but in terms of sub contrary, the other might be true as well. Therefore we cannot determine truth or falsehood. And thus the O form of this is going to be in the undetermined category. Some S is not M is undetermined. As we have already said, subalternation doesn't apply here. So we've covered contradiction, subcontrary subalternation. Let's do conversion. Now, if some S is M, the conversion of that is going to be sum m is S. Is that a legitimate, is that logically equivalent? Okay, so we can put that down in the true column. Sum M is S. Obviously, some saints are martyrs. Some martyrs are saints. Now the obversion, where the predicate term gets replaced by its complement. So we're going to have non M's, non martyrs, and going to change the quality from affirmative to negative. So the obversion would become some saints are not non martyrs. We know that that is true. So we put it down as some S is not non m. And then finally, contraposition is not going to help us here because we know that the contraposition of an I proposition some S is m will not be valid. So we get no help from that.

What about taking the obverse of some M as s?

Well, now you're going to start a new thing. I'm only trying to do the immediate inferences. On some S is m. Some saints are martyrs. And I'm just taking it step by step. What we know thus far without applying our procedures twice, we just make one run through all the different immediate inferences. We can make an immediate inference about contradiction, sub, contrariness, left undetermined. Subalternation doesn't apply. Conversion helps obversion. We can get something. We can't get anything from contrapositive or contraposition. And now we have to go through and look at the test cases. If they don't fall into any of the ones we've looked at, then we have to start testing some other ones to see if we can make them logically equivalent. Okay, let's look at number one. What can we say about all saints were martyrs? If what we know is that some saints were martyrs, okay, all saints were.

Martyrs would be the super alternation, and you can't do that. So that would be undetermined.

Yeah, Very good procedure. You notice the relationship is that of super alternation. And we've already learned that super alternation may not be logically valid. And again, notice the intuitive connection here. If somebody says some saints were martyrs and another person says, oh, then I guess all were martyrs, they'd say, wait a minute, that doesn't follow. Well, you can't tell if all were just because some were. So in that one, you should be able just to almost think about the relationship as you could understand it from your knowledge of English usage. Okay, number two, some non martyrs were not non saints.

Number two is undetermined because if you take the contrapositive of some non martyrs were not non saints, you would end up with some saints were not martyrs.

Yeah, let's go slow here for everyone, including the instructor, to follow here. Okay, you're going to take the contrapositive of number two, right? Okay. Now, why are you working from the test proposition rather than from the one you know to be true?

Because I find that that's easier. I don't know why I did it.

For testing, you found that to be easier. Well, let's see if it works out. Now you're telling me test take the contrapositive of the test proposition. The test proposition has the form of an O proposition. Some something is not. So when we take the contrapositive of it, we can get a logical equivalence. And what is the logical equivalence? The contrapositive.

Some saints, because we're not martyrs, would.

Be the contrapositive because all we're doing here is changing subject and predicate and making Using complement terms. And so you have some saints were not martyrs.

And if you look on your little chart that we made before, that is equal to an undetermined.

Very good. I'm proud of you. I hadn't thought of working backwards like that. My guess is working backwards may be a real pain sometimes. But since you were able to get to something on our chart, that's a good method. Okay, we'll move then to number five. Okay.

For number five, I worked backwards from some martyrs were not non saints. I took the aversion of that, which would be some martyrs were saints. And we find that in our chart under true. So that's a true statement.

Yeah. The chart that we worked out of, the immediate inferences to be drawn from our test proposition are from the one we took to be true. The exercise begins with some martyrs. Excuse me, I'll get it right. Some saints were martyrs. And then we found an immediate inference that means some martyrs are saints. And now if you do an obversion on that, you have to put the negative in there. It's going to become some martyrs are not. And you have to use the come complementary term, non saints. And that's what the exercise calls for. So we know that 5 is true first by applying conversion, then by applying obversion. And Pete was able to figure that out. As you see, take a shortcut by working backwards from what he was testing and then seeing if doing one of the immediate inferences on the test statement gave him agreement with one of the propositions that we already had inferred from the one we took to be true. I don't want to confuse you here. By working backwards, he found agreement with what was on our initial chart of immediate inferences. Okay. The next one we want to look at is number six. No martyrs were non saints.

What I did is I took the number six, no martyrs were non saints. And the logical equivalent through obversion is all martyrs were saints.

Okay. The first step is you performed an obversion on the test premise and came up with all martyrs were saints. Correct. So far, so good.

Now, if we look over in our true column, we see the phrase some martyrs were saints. And we can look at that and see that all martyrs were saints is the super alternation of what we know to be true, and therefore it would be undetermined.

Okay, Just so it's clear to those listening by tape, our baseline premise, some saints were martyrs is logically equivalent by conversion to some martyrs were saints. And so now the question is, can we get from some martyrs were saints through one of our inference patterns to what number six states no martyrs were non saints. Now, number six is logically equivalent through obversion to the statement that all martyrs were saints. So let's go back here. Now we're asking, can we get from our baseline statements equivalents some martyrs were saints by some inference pattern to our test propositions equivalents of Paul martyrs were saints, saints. And can we. Can you go from some to all? Well, that's what we call super alternation. And therefore we cannot draw that conclusion. And we have to then say that number six in our exercises is undetermined.

All mars were non saints. I found to be false because I did an odd version on the test premise.

Okay, so the test premise is all martyrs were non saints. When you do an odd version on it, what do you get?

I get all martyrs were non saints. Wait, I get no martyrs were saints.

That's right. The test premise is all martyrs were non saints. The odd version of it is no martyrs were saints. Okay, what's next?

And that's an E proposition. And when you convert the original proposition, it is contrary to not contrary. It's contradictory. Contradictory the proposition.

Okay, so let me summarize for those who are at home listening. If you do a conversion of our baseline proposition, the baseline, the one we take for granted is some saints were martyrs. The conversion gives you some martyrs. Were saying. Now the Test proposition number 10 in the book is all martyrs were non saints. And the obversion of that gives you no martyrs were saints. So it turns out that the obversion of the test premise is contradictory to the conversion of the baseline premise. And therefore we know the test proposition is false. Number 11 has a its test, the test premise, some non saints were not martyrs. And just to remind us our baseline, what we take for granted is some saints were martyrs. Okay, go ahead.

So I worked from the baseline and I averted it and got some saints were not non martyrs.

Okay? So if we do an odd version of our baseline statement, we get some state some saints. I'm going to write this down. Some s were not non martyrs. Correct. Okay, next step.

Then from there I did a contraposition and I got some martyrs were not non saints.

Now you're doing a contra position of the odd version of the obversion. So when we perform the contrapositive on that obversion, we're now to the point that we have some martyrs were not non saints.

Okay? And what we have on number 11 is some non saints were not martyrs. But you cannot take the contraposition we just gave you and convert it. That's an invalid procedures, therefore I said it was indeterminate.

The test premise would be the conversion would be the conversion of the procedure we've just gone through. Okay, so if you see that relationship, then you can rule it out. That's a very good procedure. Let's just make sure that we've explained this for everyone listening by tape. The first step was to use our baseline premise, the one we know to be true, and perform an obversion. The second step was to perform a contraposition on that obversion. And that gave us some martyrs were not non saints. The test premise is some non saints were not martyrs. And therefore the only way we could get from the test statement some non saints were not martyrs to the logical equivalent of our baseline premise that some martyrs were not non saints would be by conversion. And yet we have learned that the conversion of a o form statement is not logically equivalent to the statement itself. And so since the only way to get from the test statement to what is logically equivalent to the baseline statement is through an invalid procedure, then we have to rule this out as is it false or undetermined?

I put undetermined.

Yeah, it would be undetermined. Yeah. Since it would be an invalid inference, we cannot say that it's that what we're inferring to is necessarily true.