Dr Greg Bahnsen - Critical Thinking: Introduction to Propositional Logic (Part 13)

Okay. Which of the terms in the conclusion are distributed? Okay. Are they distributed in the premises?

Yes.

Where is S distributed in the major premise, Right. No, I is S. And where is K distributed in the minor premise? All K is I. Okay, so we don't have a distribution problem. Do we have a negative premise used anywhere?

Yes.

Then you have to have a negative conclusion, which you do, so that rule is not violated. So have we covered all the rules? Existential assumption is not violated. We've already said that, so it appears to be valid argument. The conclusion drawn is that there's no knowledge of substance. Now, if you don't like that conclusion, what are you going to have to do with it? You're going to be able to attack this logically? No, you're going to have to go after one of the premises as not being true. And that's not too difficult to do if you look at them, as a matter of fact. All right, the next exercise it will do. All right, number three. Everyone who smokes marijuana goes on to try heroin. Everyone who tries heroin becomes hopelessly addicted to it. Therefore, everyone who smokes marijuana becomes hopelessly addicted to it. Is there a problem with this argument?

The problem is. Well, I thought the problem at first is those two universal premises, but well.

Before we even translate it. Do you not see just in reading it, that there is. What do we call this? An amphibole? An oblique reference here. Notice the first premise is everyone who smokes marijuana goes on to try heroin. Everyone who tries heroin becomes hopelessly addicted to it. Becomes hopelessly addicted to what? To heroin. Everyone who tries heroin. Therefore, everyone who smokes marijuana becomes hopelessly addicted to it. To marijuana. Or was it supposed to mean everyone who smokes marijuana becomes hopelessly addicted to heroin? Is that the way you interpret it? If we do not interpret it that way, then it's going to be dismissed out of hand as permitting the fallacy of equivocation. But let's take it in that form.

Do you want to leave it in it then? Because if you change it to heroin, the conclusion is going to.

The conclusion is everyone who smokes marijuana becomes hopelessly addicted to heroin. Okay, we'll schematize that as All M is A, where A stands for people addicted to heroin. M is people who smoke marijuana. So all people who smoke marijuana are people who become addicted to heroin. That's the conclusion. All M is A. Now, if that's the case, what would be the major premise?

All people are going to try heroin are people who will become addicted to heroin.

Okay. So you have the class those who are addicted to heroin. And the other class is those who try heroin. Okay, so you have all H. H being the class of those who try heroin. All H is A those addicted to heroin. By the way, you notice when we translate it that way, so that you don't have temporal indicators that the sentence is strictly speaking false. All people who try heroin are people who are addicted to heroin. That doesn't follow. I mean, that isn't true at all. First time you try it, you can't be addicted to it. Okay, so A has got to be pushed will become addicted to heroin rather than presently R. Anyway, we'll put it as all H is A. Then the second premise would be what?

All people who smoke marijuana are people who are going on to find out.

All those in the M class who smoke marijuana are people who are in the H class will try heroin. So the first premise becomes all H is A. Second all M is H. Conclusion, all M is A. Are we able to test this?

Yes.

Okay, are any of the rules broken?

I thought the rule of you want to have two universals.

Well, if you have two universals, then your conclusion must be universal and it is. Is the middle term distributed?

Yes.

Okay, so we have no distribution problem. Is the what term is distributed in the conclusion? Is it distributed in any premise? Yes, in premise two. So the distribution rules are. Okay, we've already tested for existential assumption, so Rule 6 taken care of. So we just have the quality rules, right? Do we have two negatives? Do we have any negatives? So we can't have any quality problems. So what do we conclude?

It passes valid.

Okay, so look at the conclusion. Everyone who smokes marijuana is someone who will become hopelessly addicted to heroin. That's a valid conclusion. Is it a true conclusion? So what are you going to go back and look for? If you're arguing this point, you're going to look for a false premise, aren't you? You might in this case find a couple of false premises. I think that's why the author's doing this, is he's forcing these conclusions on you logically that you know are not true. You have to realize that a sound argument has to have not only a valid form, but also true premises. Okay, I think we're going to move on. Section 7.4 and 7.5 in copie deal with syllogistic argument that is incomplete in one way or another. The first kind of incomplete argument is an enthymeme. He says on page 231, often only part of the argument is expressed, the rest being understood. In an argument that is stated incompletely, part being understood or only in the mind part, is called an enthyme. Why is it that we find enthematic arguments in ordinary language? He says the reason is easy to understand. A large body of propositions can be presumed to be common knowledge. People don't bother to repeat as a proposition things that everyone takes for granted. Moreover, rhetorically, the anthematic way of expressing an argument is very powerful and persuasive. So you state one proposition and then go to your conclusion. Because you take it for granted, your audience will be able to supply the missing proposition. And so he says on page 232, a cardinal principle in supplying suppressed premises is that the proposition must be one dispute can safely presume their hearers to accept as true. And then we have just a piece of nomenclature for you to pick up on in the next paragraph. It says enigmatic arguments are traditionally divided into different orders, depending on which part of the syllogism has been left unexpressed. If the first premise is left unexpressed, it's a first order enthymeme. If the second premise is left unexpressed, it's the second order enthymene. And if it's the conclusion that is not explicitly stated, it's a third order enthyme. At the bottom of the page, a debate point is made that we'll come back to when we get to that part of our course on critical thinking. But it says if one of the premises is missing, it may be that only the addition of an implausible proposition as premise will make the argument valid, while with any plausible proposition added, the argument is invalid. Pointing this out is a legitimate criticism of an infammatic argument. Of course, an even more crushing objection is to show that no additional premise, no matter how implausible, can turn the enthymeme into a valid categorical syllogism. He says the difference between enthymemes and syllogisms is not a logical matter, it's only a rhetorical matter. And so we're going to test enthymemes in the very same way that we've been testing syllogisms up to this point. It's just that we're going to have to be clever enough to provide the missing premise in the enthymatic argument. Any questions about enthymemes? This is just a matter of supplying what is missing, missing in a person's argument. Let's look at the next section before we do any exercises? 7.5 deals with Sorites. He says sometimes a single categorical syllogism will not suffice for drawing a desired conclusion from a group of premises. Somebody gives a couple of premises and draws a conclusion. They say, well, there's something missing here. Now they got premises and a conclusion. What is missing here is not just one premise, as in an emblematic argument, but one or more categorical syllogisms that form a bridge from the opening premises to the conclusion. And when you have that kind of hidden information, or if you will, implicit argumentation, it's called a sorite. And for these, as he says on page 236, a stepwise process of argumentation must be resorted to where each step is a separate categorical syllogism. So you end up with a chain of categorical syllogisms connected. How are they connected? The conclusion of the first becomes the premise of the second. And then you test each one of the lengths in the chain, remembering that a chain is no stronger than its weakest length. He says, where such an argument is expressed in thematically, with only the premises and the final conclusion stated, it's called a sorites. Sorites may have three, four or any number of premises. Not very encouraging is. It could be a lot of work to analyze all this. But if you're going to do critical thinking, sometimes what you have to do is take this man's premises and its conclusion and construct the categorical syllogisms that are necessary to move from those premises to the conclusion, and then test every link in the chain to see if it's logically valid. Any questions about sorites, then?

Are you supposed to do all this stuff when you're debating?

Well, a later part of our class will talk about what you're supposed to do when you're debating. I don't know how to take your question. I would say implicitly, if you're going to understand a man's argument, you must do that. Yes, but now, whether you would do that naturally, debating with him privately or publicly, that's another matter. That's a question of what is necessary to expose the structure of the argument and refute it. Sometimes if you took a lot of time laying out, well, this argument has implicit in it all this, you might lose your audience completely. But if that's the only way to show that what he's been saying is invalid, then you might have to do it. So you're asking a question about debate and strategy in terms of critical thinking. That's what you have to do. If people have three or four premises and a conclusion where it's not in standard form, you got a. Well, of course one thing I do with my students is I'll send it back to them and say you translate this into standard form. Don't make me do all that work for you. What are you taking for granted? But yeah, if you're analyzing Leibniz or Bertrand Russell or somebody you can't send home to do this homework over again, you have to do this. You have to try to figure out a way to make it work. Or maybe it's somebody you like, like Calvin. Somebody says, why don't buy this argument, it doesn't seem valid to me. And then you want to show that it is valid, you'd have to construct the intervening steps of categorical syllogism. Alright, now that we understand what enthymemes and psorites are, let's back up and do a couple. Section 7.4 dealt with enthymemes and we have exercises on page 233. Name the order and discuss the correctness of each of the following enthymemes. Write the arguments in standard form, adding a missing premise or conclusion to make the completed argument valid if possible, using parameters if necessary. Notice how every step along the way we're where again you might have to do something that you've just learned or previously learned. But now we know how to use parameters, we know how to translate. What we're learning here is how to provide a missing premise. And I'm not going to do everything. Let's just see if we can figure out what the missing premise is. Number two, the soul through all her being is immortal, for that which is ever in motion is immortal. So we have one premise here that says the soul is immortal. All things which are souls are things which are immortal. That's the conclusion. The premise is that which is ever in motion is immortal. All things which are in motion are immortal. Now how am I going to get from things in motion are immortal to the soul is immortal implies understanding that all things possessing souls are possessing mortal souls. Yeah, you can give up the language of possessing soul. I think you can just use all souls. The conclusion is all souls are things that are immortal. The only premise given to us is things which are in motion are always in. Pardon me. Things which are always in motion are things that are immortal. So now what premise is missing? Let me put it up on the board here so it becomes real obvious to us.

This is not difficult. You're looking for what will probably be a middle term that is missing or a connection to a middle term that is missing.

So it's not right. The missing term isn't that all souls are immortal.

Souls. Well, that's not missing because that stated the very first thing. The soul through all her being is immortal. That's the conclusion. All S is I. All souls are immortal. So now we have a premise here about things that are immortal. That which is ever in motion is immortal. So things which are always moving. I'll call that M. Things which are always in motion. He says that which is ever in motion is immortal. So all ever moving things, all M IM is I. So the premise says all M is I. The conclusion says all S is I. How are we going to get from here to here? What's missing? Okay, it should be real obvious. Don't look for something difficult here. All souls are immortal. We're told that all things in perpetual motion are immortal. So then.

All things emotion are.

Souls.

No, you don't want to say all things in motion are souls. He wouldn't say that. He'd say all souls are things. Okay, now if we begin here, let's just see if we can do this by a chain of reasoning. All souls are things perpetually moving. All things perpetually moving are immortal. So you see, S goes to M. M covers this end. So you have S and I covered. So all S is I. All souls are things which are immortal. Okay, is this a valid argument? I mean, after a while you see some that are in a certain form, you know, So I mean, I know.

That it is, but if you apply the rules, you'll see that none of.

The rules are broken by that argument.

May not be a good argument for.

The substance of it, but the form is fine.

Okay, let's do another one of these. Number 19. Any reader who completes this argument is a good student, for it is difficult. Now, in this, what is the conclusion that the author is drawing?

The conclusion is any reader who completes this argument is a good student.

Okay, let's get a notation for that. How would we put that in standard form?

All readers who complete this argument are good students.

Shall we say, all will use C for completing the argument. All who complete this argument, all C is G. G for good students. Okay. The conclusion is all C is G. Now, what premises do we have for it is difficult. Now that, of course has to be expanded. What is the author saying when he says it is difficult? What's the it?

This argument. Now this. The it is the argument.

The problem.

The problem Right. Okay.

Would it be the argument?

No, he's saying any reader who completes this argument for it is difficult. The completion of this argument is difficult. Okay, so the premise is the completion of this argument is difficult. How are we going to put that into standard notation? The premise that is offered is for it is difficult amounts to saying that completing this argument is difficult. And we can expand this using parameters to say, all who complete this argument are people who do difficult things. So now our premise is all C is D. Our conclusion is all C is G. And to supply the missing premise, we have to get DNA together in that second premise. And so can you now supply the missing premise at long last?

Yeah, it would be all people able to complete difficult arguments are good students.

All people able to complete difficult arguments.

Which would be D. All D is.

Is G. People who are good students. Okay, so now putting it together, all C is D, all D is G. Therefore all C is G. And of course, I think you can see just by looking at it by transition that that argument is a valid argument. But does everyone understand how we went back and rewrote the argument to supply the missing premise? If you don't understand in a more intuitive way, most of these I think you can see intuitively if you just look at the content. But if you don't, by writing out your premises in notation and remembering what each notation refers to. In this case, we saw that we had a premise, all C is G. All who complete this argument do things which are difficult and a conclusion, all who complete this argument are good students. So we have C and D, C and G. And somehow we've got to bring those together. And the only way to do so is by using D and G as connectives. And so we have to have some kind of premise that mentions those who do difficult things and those who are good students. And that's exactly what's been supplied. So that's a good answer. Let's just do one of the Sorites on page 237. Translate each of the following store rights into standard form and test its validity and we'll choose as our as our exercise number two. Do you have any difficulty with this, Melissa?

Yes, I have difficulty in changing the sentence, such as Jenkins develops blundering into a proper categorical form.

Right. That's why this textbook is set up in a regular recursive fashion of teaching. In order to do this exercise, you need to go back and apply the things you've learned previously. So at one point he taught us how to translate non standard propositions into standard form. So you have to go back and it's like, do that step and then you take. Then you put it all together and do the step of testing it so forth. These are not.

It's not like.

Okay, now we have a new direct type of exercise to do. This is now an exercise that is made up of applying two or three other things that we've learned previously. So number two gives us the conclusion. Jenkins is inexperienced and argues with three premises. No experienced person is incompetent. 1. Secondly, Jenkins is always blundering. 2. And then lastly, as a premise, no competent person is always wondering. So what we need to do is to translate each one of these into standard form and then start looking for categorical syllogisms that will provide the missing steps from the premises to the conclusion. Can we begin with the conclusion? Do you have a standard form translation of the conclusion?

Yeah, that you could say all J is non E.

All J. All Jenkins is non E. Inexperienced.

Or you could go I, which is probably what you want to do. Go I for inexperienced.

Okay, now we do see the word experienced used in premise one. So if you use experienced and inexperienced, you're going to have two terms. Eventually you're going to have to be reduced. How are you going to do that?

Well, I just took the inexperienced and that's basically the same thing as not.

Yeah, exactly. You should leave it as non experienced, I would think. But then you said. Or you could take it as inexperienced, but then you have a different term.

Then what you do is you take the experience and make it non I or non inexperienced. You see what I'm saying?

I understand what you're saying. I'm not sure how that is going to help us here. Let's stick to the conclusion. All Jenkins are things which are non. You're going to say non experience. Okay, so here we have non I.

Non E.

Non E. Thank you. We won't erase that mistake from the tape. Premise one, no experienced person is incompetent. How do you schematize that?

No, E is non C.

Okay, now why did you say non C? Just explain for us.

Because on premise three, you have no competent person is always wondering. So I just wanted to keep my terms as complements.

Very good. No, E is non C. You have experience and competence mentioned in 1. You have experience mentioned in the conclusion. So we have to get from competence. Excuse me. Yeah, from competence to experience. How do you Translate premise number 2? All J is B. Jenkins is always blundering. So it's always the case that all J.

This is the problem I had. When we translate Jenkins as a class, it's both all and some.

So I wasn't sure whether I should.

Be taking both or just one of them.

Well, you have to try it both ways. Sometimes according to the book. But Jenkins is always blundering. It's going to be universal. Jay will be.

I took that always blundering to be an always blundering person or person who blunders.

Yeah. So that's fine. But now in number two, Jenkins is always blundering says something about the class of Jenkins and about the class of blundering. Okay. And what it tells us is that. If it didn't say always blundering, if it said Jenkins sometimes blunders, then we would say what? Some J is B. No, because J stands for the class of Jenkins rather than the class of things Jenkins does. Whereas in the conclusion, Jenkins refers to the class of Jenkins himself as a person. Actually, the conclusion could be stated therefore some J is non E. Isn't that right?

It could be either way.

It could be either way. But because we have J already in use, then we have to be careful on two. Jenkins is always blundering. Let's just take it as a universal all J. Is V. What about always in there?

Doesn't that describe more of A? That's kind of an adjectival phrase.

Yeah, but it's an important point. It's a quantifier that we can't overlook. And again, remember, another quantifier could be used sometimes, in which case that always and sometimes should be schematized differently.

You didn't want to put it all the paints in the beginning.

But what would happen if you'd say all Jenkins? Are people always wondering?

But they're not always wondering.

That is kind of build the always into the description of the class. People who always blunder.

Yeah, that's just what I did.

Okay, I think that's probably. Probably is the way to go. Let's do that then. So the second one becomes all J is what is say B? People who always wonder. And the third premise, no competent person is always blundering becomes no competent person.

Is the person who always blunders.

No, C is B is. Okay, now, for the sake of those listening by tape, then our translation of the three premises and the conclusion is, number one, no E is non C. Number two, all J is B. Number three, no C is B. And our conclusion, all J is non E. Okay, let's begin to reduce some terms or to put together Some categorical syllogisms that are implicit here. What's the first step?

First step was I took number one and number three for the premises and I took them together all, and I made no E is non C. And I outverted it and got all E.

Is C. Okay, Perform an obversion on one and you get all E is C. Next.

Then I took the third premise, no C is B. And from the conclusion of those two I got no B is E. All.

E is C. No C is B. And you brought as your first conclusion.

Then no B is E.

No B is E. All right, so we're now at our first sub conclusion, no B is E. What step do you want to take next?

From there I took the second statement, all J is B.

Okay?

And combining that with the sub conclusion, I got no J is E.

Let me just repeat this. Our first sub conclusion was no B is E. Our second premise, from the argument itself or from the problem itself, is all J is B. And when we combine these two, then you're drawing the conclusion that no J is E. Okay, so now we have a second sub conclusion, no J is.

I just took that sub conclusion and inverted that and got all J is.

Non E. So now we're doing not a syllogism, but what we call an immediate inference. So you're applying obversion. We know that all obversions are valid in Boolean interpretation. And to obvert means to change the quality. And therefore this negative proposition must become an affirmative proposition. Instead of no J, we're going to have all J and we have to change the predicate term to its complement. So instead of all J is E, we end up with all J is non e, which was to be proven. And so now we've gotten from the premises to the conclusion, supplying a number of either immediate inferences or syllogisms, each of which is valid. Everyone understand how to do soarites. I realize they are painful. You have to do translation, you have to do immediate inferences. Sometimes you have to reduce your terms. You have to test your syllogisms. But that's the process of critical thinking if you're looking at deductive logic dealing with classes. We're going to end that portion of our course then and move ahead now to copie chapter seven, subsection six. 7.6 on page 238. And I can't go right into it because this is one of those major changes in the course and because Kopi has put both of these things in one chapter, I'm afraid Many of his students, and sometimes my students, they kind of get confused. They say, well, what is all of these different procedures and techniques and so forth about?

It all breaks down into two kinds of formal logic. What I've been repeating over and over for you, when we get to that point, we're right now on analysis, that aspect of critical thinking which we call analysis. The first thing to do in analysis is linguistic analysis. You know, some understand language, how people use it, and so forth. The second thing we looked at were ordinary language fallacies, which we all had fun looking at. And they proved to be one of the most profitable parts of the courses. If you remember what the fallacies are.

That'S where most of the mistakes are made.

Most people don't think in formalized notation. Logic teachers may not like that, but that's just a fact of life. And so these fallacies are very important. And thirdly, we're moving away from ordinary language into formalized language. Okay, so now we're looking at formal deductive reasoning moving into languages evaluated with respect to their form rather than in their semantic content. And we're looking at these arguments in a formalized notation. So in both those senses, this is formal logic or formal deductive reasoning. Formal deductive reasoning, at least for beginning students. Looking at COPA here and so forth, breaks down into two types of analysis or reasoning. First of all, categorical reasoning or categorical syllogism. And categorical syllogism, as I have said repeatedly, only deals with three words, essentially all, some and not. So the this is the logic of all, some or not. Or to put it even better, it's the logic of class inclusion and exclusion. The logic of classes. How do different classes of things relate to each other? By inclusion, exclusion, all things, some things, not all, or some not so. It's just these three words, really. And what we have been studying after we got to the place that we learned how to test a categorical syllogism, what we've been studying is how to put the things we hear in ordinary language into standard form. That's basically what we've been doing this morning after we looked at the rules for testing categorical syllogism. But categorical syllogism is not the only kind of formalized logic that we deal with. There's a second that we're now turning to. And what I'm getting at is it concerns me a little bit as a teacher that Kofi is turning to a different kind of reasoning in the middle of this chapter, all under the rubric.

Of arguments in ordinary language.

So I want you at least to.

See the seam or to feel the bump.

We're going over.

We're now turning away from argument that.

Deal with all some and not.

Now, here's Copy's way of explaining that. On page 238, he says there are different kinds of syllogisms, taking their names from the kinds of propositions they contain. The next paragraph, we may think of categorical propositions as simple.

Notice up on the board I call this categorical syllogism. That's syllogism dealing with simple propositions. And the propositions have to do with classes and their inclusion or exclusion. And then he says we may think.

Of categorical propositions as simple in contrast to compound propositions that contain other propositions as components.

With that one sentence, copious makes a major turn of the truck here, and he starts looking at something different. Now we're turning from what he calls simple propositions to compound propositions and what we will call connective logic, or, if you will, propositional symbolic logic. Now, the first part of this, however, doesn't deal with symbolic logic per se. We're going to get more into that in the next chapter. But what I want you to note is he's now dealing with the relationship of propositions rather than the relationship of classes. So here we have categories related to.

Each other or classes.

Now we're going to talk about propositions and their relationship to each other. And now if I can perform my, you know, reduce this or humble this thing, it's going to look very complicated, believe me.

By the time we get to quantification.

You'Re going to say yowee. Why do we want to do this? We're only dealing with now learning how to use the terms and or if then this is going to be the logic of and, or if then not is also in there, but I mentioned that already, and or not if then. So do you all catch the significance then of the shift that's being made? We're moving away from the relation of classes of things, Where we have the class of drunkards, the class of men, the class of sailors and what their relationship might be? We're moving away from class relationships, category relationships. We're now moving to propositional relationships. How do propositions relate to each other? And for that we relate them with the words and or not if them.

I still find it strange that coping one sentence could do this. I would have started a new chapter.

Here and then explained we're going to talk about propositional Relations, first of all in ordinary language. And then we're going to devise a formal calculus for doing this, a system of formal notation that will help us test the validity of propositional architecture.

Anybody have any questions?

You could call this again connective logic. How propositions connect with each other and. Or if then the other is class logic, the relationship of classes.

Okay, in section 7.6, we're introduced to two types of syllogism. And I said to some of you off record, and I want to make.

Sure I say it in the class.

For those listening by tape, that if.

There was only one thing that you studied in deductive logic, in terms of.

What is going to be the most.

Common procedure used in your life in analyzing others and arguing yourself?

If there was only one, I would hope that you wouldn't leave it at one. But if there was only one, this is the the most important thing that you would learn about deductive logic, the.

Use of disjunctive and hypothetical syllogisms. And the reason why I say it's the most important is because it's the most common. And since we're engaged in trying to learn to think critically, this is the.

Sort of thing you're going to run.

Into out in the field, as we say. Now there's good news. Not only is it the most common, it's also the easiest to understand. Of all the stuff we've been doing in deductive logic, I think you'll find this easier than analyzing categorical syllogisms. And the reason for it is you're going to recognize that you use these kinds of arguments all the time as well. And I'm going to give you a real quick overview of these, and then we can look at the book to see if I didn't cover first kind of propositional syllogism or connective syllogism he.

Discusses is called the disjunctive syllogism.

Disjunctive syllogism first of all lays out two options, this or that. I'm just going to schematize it as A or B. My first premise, then, is a compound premise. Notice that there are two propositions referred to here, A and B, and they are related to each other in terms of the disjunct, or. We say the doctor The Dodgers won the World Series or the Giants won the World Series. Is that asserting that the Dodgers won the World Series? Is it asserting that the Giants won.

The World Series in terms of the.

Way we use the word or in logic, both might be true. When I say A or B, all I'm saying is at least one of these is true. That is, you do not interpret this as either A or B, which would be a strong form of disjunction. We will interpret that weakly. The weak inclusive interpretation says both could be true, but at least one is at least one, perhaps both. So if my first premise is an A or B, and then I add to that that it's not the case that A is true, now I can draw a conclusion that B must then be true, must be true, it is necessarily the case. It follows from the premises that B is true, A or B, not A. Therefore B, that's called a dictator. Disjunctive syllogism. And just to reinforce what I said about this being not only the most important but the easiest part of the course, my guess is if you stop and think about your ordinary discourse, you use disjunctive syllogism explicitly or implicitly all the time. Now, your premises might not always be true, but you use this way of thinking. I left my car keys either on the television or in my bedroom, but I'm looking at the television and they aren't here. So then you go to the bedroom to get your car keys. Okay, don't fault the illustration. My point is, in our ordinary affairs, we're always doing that. We're always saying, here are the options. I eliminate the assumption. As I've covered, one of these assets got to be true. I know this one's not, so I do the other one. Disjunctive syllogism. The other kind of syllogism that is used very commonly is known as hypothetical syllogism. And he gives it to you in two forms, pure hypothetical and mixed hypothetical.

A hypothetical syllogism is one where you have a premise that relates two propositions in an if then relationship. An if then relationship. If this, then that.

I'll say if P then Q.

That's in hypothetical form.

In the proposition if P, then Q is P being asserted as true. Pat, if I say if P, then Q is P being asserted to be true. No, no, because it says if P is Q being asserted to be true. No, no, because it's contingent on P being true. So if P, then Q does not assert either one of them to be true, but it does assert this, that if P is true, then Q must be true. It's asserting this whole thing. This is true. This relationship between P and Q holds.

Alright, now your author says a pure hypothetical sil syllogism is One that contains only conditional propositions.

So this is a conditional proposition, hypothetical or conditional proposition. Now, if I put another one down. If Q then R. The conclusion I'm going to draw is if P, then R. This should be real simple to follow. The reason why it's so simple to follow is that you've all intuitively and habitually learned to think this way. Geometry, sometimes called rules of transition. But you do this in ordinary life. You say, well, if this is true, then that's true. And if that's true, then this is true. Then going back, if the original thing is true, then the last thing is true. If P, then Q. If Q, then R. Those are both accurate statements. Then if P, then R. Now, let me ask you this. In the conclusion, P, Is P being assertive? Is R being assertive? No, it's still hypothetical, but it's telling us, here's the relationship of these propositions to each other. P relates to Q in such a way that if P is true, Q is true, Q to R. Likewise, and when P put those two together, we now know the relationship of P and R. That if one is true, if.

P is true, then R must be true. That's a pure hypothetical syllogism. A mixed hypothetical syllogism, on the other hand, has one conditional premise and one categorical premise.

A categorical premise is not compound, is it? It's simple. So if you can look up here on the board, I want to show you what this amounts to, just so you can visualize it. A mixed hypothetical. I have two hypothetical premises. One, if P, then Q. I have a second one. If Q, then R. Now, I'm going to give you a different kind of syllogism. I'm going to erase the hypothetical part of this, and I'm just going to put one proposition there. Now, it's not a relationship between propositions. It's just a simple categorical statement of P. And when I do that, all along I've asked you, is P being asserted as Q being asserted? Now, is P being asserted? Yes, that's the assertion of P. Now, if I assert P, what am I going to assert in my conclusion?

Q?

Q again, this is not advanced stuff. And it's not advanced because it's part of everyday life. We think hypothetically. If P, then Q. Well, I know that P is true, therefore Q must be true. What I put up here on the board is known as modus ponens. Is the technical term modus ponens, meaning.

The method of affirmation.

If P, then Q then I affirm what was the antecedent in the hypothetical proposition. And I conclude by affirming the consequent in the hypothetical proposition. Now I want to look at a corresponding fallacy. In fact, it's probably. In my estimation, this is anecdotal, not systematic. But in my estimation, this is the most common fallacy made in ordinary discourse people make. What I have up here on the board is a legitimate form of argument. If P, then Q, P, therefore Q. How about this, though? What if we say if P then Q and assert Q instead of P, and then conclude? Well, then P must be true. If P is true, then Q is true. Well, I know Q is true, therefore P must be true. It happens all the time. It's called the fallacy of affirming the consequent. The fallacy of affirming the consequent. If you affirm the consequent, you have no right to affirm the antecedent. If Shakespeare wrote Paradise Lost, go back our example. P will be Shakespeare wrote Paradise Lost. Q will be Shakespeare is a great author. Let me ask you this. Is it true that if Shakespeare wrote Paradise Lost, then he's a great author? Is this if then relationship true? Sure it is. If Shakespeare wrote Paradise Lost, he is a great author. No doubt about that. Because whoever wrote Paradise Lost is a great author. But now what am I going to do? I'm going to say, well, Shakespeare is a great author. I know Q to be true. But the thing is, he may be a great author for some other reason altogether has nothing to do with P that he wrote Paradise Lost. And so when I say if P then Q and assert Q and conclude P, I have committed the fallacy of affirming the consequent. Now, there's a related, appropriate way of arguing. If I say if P then Q and I deny Q, I say not Q, then I may conclude not P. And this is called modus tollens. The method of denial. The method of denial. If P then Q, Well, I deny I deny the consequent Q. And in denying the consequent, I must then deny the antecedent. Because if P then Q, if that connective is true, it is true that if P then Q, the denial of what is said to follow means you have to deny what led to it. So modus tollens. Denying the consequent means you must deny the antecedent. And then, just to round this out, let's notice the fallacy that goes with modus tollens. What if I deny the antecedent instead of the consequent, I.e. i deny P first premise. If P, then Q second premise, not P. And then I conclude, okay, well, then not Q. But you see, Q may be true for other reasons other than P. And so the denial of P does not logically entail the denial of Q. This is known as the fallacy of affirming the antecedent. So you've learned two fallacies. One, the denial of the consequent, the other the affirming of the antecedent. Once you see this relationship, you have, I think, a very powerful tool of reasoning hypothetical syllogism. Modus ponens. If P, then Q. P therefore Q. Modus tollens method of denial. If P, then Q, not Q, therefore not P. And they have related fallacies. If someone says if P, then Q, Q therefore P. That's the fallacy of affirming the consequent. And in the case of modus tollens, you have if P, then Q. And what you deny is the antecedent. And then deny the consequent. You've committed the fallacy of affirming the. Excuse me. Denying the antecedent.

We did some exercises on this, and.

I think we can go over them now.

This should be fairly easy for you. The exercise calls for you to identify the form and discuss the validity or invalidity of each of the following arguments. Number one, if a man could not have done otherwise than he in fact did, then he is not responsible for his action. But if determinism is true, it is true of every action that the agent could not have done otherwise. Therefore, if determinism is true, no one is ever responsible for what he does.

That's a pure hypothetical.

Yeah, we have a pure hypothetical syllogism here. Could you give us. Could you schematize it for us?

Well, because of the. Those are conditional statements. Because of the if, then I agree.

Give me a schema for it. Symbolize it for me. What's the first premise? If, let's say A, A will stand for a man could not have done otherwise than he did. Okay, so you're saying if that, then what?

Then B.

He'S not responsible.

Yes.

Okay. Now, if determinism is true, it is true of every action that the agent could not have done otherwise. So what form does that have?

If C. Okay, then D.

D, it is true of every action that the agent could not have done otherwise.

C.

But I think you can see that's not going to work, is it? Okay, why don't you say with respect to. You started with A as a man could not have done otherwise than he did. And B is he's not responsible. Okay, then two becomes. If determinism is true, call that C, then it's true of every action that the agent could not have done otherwise. That's the same as A, isn't it?

Yeah.

Just stated more broadly. So if A, then B, if C, then A. The conclusion, if determinism is true, no one is ever responsible for what he does. You have determinism is true. Is C. No one is ever responsible. Is B. Okay, now is that valid? Yes, sure it is. If C, then A. If A, then B. Therefore if C then B. Although I find it easier to schematize these using letters that will help me remember what's in them. So you kind of threw me there, but that's okay. Number two, I can't have anything more to do with the operation. If I did, I'd have to lie to the ambassador, and I can't do that. What kind of syllogism is this? Okay, and that he takes. Say if I. Let's see, you're calling it modus ponens now. Modus ponens. And the assertion will be if will make I did, I'd have to go lie to the ambassador. Let's start with something else. What's the. What's the conclusion of this argument? He can't have anything more to do with the operation. Okay, let's. Let's call that C. Okay, I can't have anything more to do with the operation. C is the conclusion. Now, what is the first premise? First premise is if I did, I'd.

Have to lie to the ambassador.

Tell you what, let's say that C is I can have something more to do with the operation. So we'll call the conclusion not C. Does that make sense?

Yeah.

Not C. Okay, now the first premise will be if I did, I'd have to lie to the ambassador. So if I did means if I had something further to do with the operation. So now you have if C, then what?

Then I'd.

If C, then not.

See? No, no, no. It says if I did, that becomes if I had anything further to do with the operation. Or if you will, if C, then you would lie. Then I'd have to lie to the ambassador. It'll say L. If C, then L. Okay, what's the next premise? You can't do that. Okay, so not L, not L, therefore not C, not C. Now, you called that modus ponens. That's my only regret here. What should you call this If C, then L, not L, therefore not C. That's modus tolus, the method of denial. When you deny the consequent and then conclude by denying the antecedent that modus tollens number three. JJ I replied, if it was any of your business, I would have invited you. It is not. And so I did not. First premise.

If it was any of your business. Okay, so if B.

If B.

Then I would have invited you.

Okay, then I, I. The second premise.

Second premises. It is not.

It is not what?

Your business.

Okay, so that's a denial of what?

Of B.

Of B. So you have not B. And the conclusion, therefore not I and not I. Very good. What kind of syllogism is that?

It's a mixed hypothetical syllogism.

Is it a valid one? No, no, it's not. Although you hear that in ordinary discourse all the time, that kind of thing. If it was any of your business, I would have invited you. It's not your business, so I didn't invite you. That's not logically sound or valid. If be, then I, not be. What fallacy is it?

It's a fallacy of denying the antecedent.

Denying the antecedent. Very good. Number four. Men, it is assumed, act in economic matters only in response to pecuniary compensation or to force. Force in the modern society is largely, although by no means completely obsolete. So only pecuniary compensation remains of importance.

This is disjunctive.

This is a disjunctive syllogism. Make it real quick for us. What are the two disjuncts?

Force and pecuniary compensation.

Put it in propositional.

That's where I had a little trouble. Okay, either. Either pecuniary, either P or C. Okay, P is.

Men act in response to pecuniary compensation. And what's the other?

Men act in response to force.

You said P or F. F. Okay, I think you said C. I meant F. That's right.

I can't get my letters.

P or F. What's the next premise? Not, not F. Not F. Because force is completely obsolete. Therefore.

Therefore P. Very good.

So it's a valid disjunctive syllogism. Now, I would quibble over some things here having to do with ambiguities and so forth. One talks about what causes men to act and the other is what is of importance. But that's okay.

And he doesn't deny either one, though he denies that. He says it's largely, although by no means completely obsolete.

Okay, but you're trying to get the. The general skeleton or force of the argument. And it's a disjunctive syllogism.

Okay, I just want to make sure that it wasn't.

I gotcha.

Number five.

If each man had a definite set of rules of conduct by which he regulated his life, he would be no better than a machine. But there are no such rules, so men cannot be machines.

I said it as a fallacy of denying the decedent.

Very good. That's if P, then Q, not P, therefore not Q argument. The reason why I want you to go over so many of these is you really need to. Once you catch on to this, it's an extremely powerful tool. People are always denying the antecedent or affirming the consequent number six. Smith is the fireman or Smith is the engineer. Smith is not the fireman, therefore Smith is the engineer destructive. Is it valid? Yes. Right.

Smith is a, fireman is A. Smith is the engineer, is B. Smith is not the fireman. Which. Which affirms B.

No, it denies.

Denies B.

Denies A. Then you affirm B. That's the problem. You guys go to these A's and B's when there's nothing like that. I would have said F or E, not F, therefore E. At least I can keep track of what I'm talking about in the argument without writing it down. Number seven. If the first native is a politician, then the first native denied being a politician. The first native denied being a politician, therefore the first native is a politician. I like this one. What's the form?

It's a disjunctive.

Disjunctive.

Wait, number seven.

Yeah. You have an if and A, then there you're right.

All right.

It's.

Mixed hypothetical.

Right. Mixed hypothetical.

What form?

If.

P, then Q.

If P, then Q, not P, therefore Q. Which would be a fallacy of denying the antecedent.

Very good. Denial. The fallacy of denying the antecedent. Number eight. If the first native denied being a politician, then the second native told the truth. If the second native told the truth, then the second native is not a politician. Therefore, if the first native denied being a politician, then the second native is not a politician.

That's me, isn't it?

What's the form?

It's purely hypothetical.

And is it valid?

And yes, I have it as valid.

Okay. Number nine. If Mr. Jones lived in Chicago, then Jones is the brakeman. Mr. Mr. Jones lives in Chicago, therefore Jones is the brakeman.

That's a mixed hypothetical modus ponens.

Right. And Modus ponens is valid. Correct. If P, then Q. P therefore Q. Okay, one more number. 10. If the second native told the truth, then the first native denied being a politician. If the third native told the truth, then the 1st native denied being a politician. Therefore, at the time same second native told the truth, then the third native told the truth. Alright, we'll have S stand for the second native told the truth and F will stand for the first native denied being a politician. So our first premise is that if S, then F for the proposition that the third native told the truth will use t. So our second premise is if the third native told the truth t, then 1, the first native denied being a politician, which we've already called F. Okay, just to review. The first premise then is if S, then F. The second premise is if T, then F. And the conclusion that is given to us in the problem is therefore if the second native told the truth, then the third native told the truth. The second native telling the truth is what we called S. Therefore the third native told the truth, which we have called T. So the conclusion is if S then T. If S, then S. If T, T then F. Therefore if S, then T. What kind of argument is this modus opponent? Well, let's start with what kind of syllogism is this pure cylinder? A what?

Pure.

A pure what? Syllogism. Pure hypothetical syllogism. Thank you. There are a lot of pure syllogisms that aren't hypothetical. Okay, a pure hypothetical syllogism. You have all hypothetical assertions relating propositions in an if then pattern. Is it a valid one? Yes, it is. First premise is if S then F. The next premise is if T then F. The conclusion is if S then T. The way your author tells you at this elementary level to evaluate hypothetical syllogisms on page 2, 4, 40, is to make sure that the conclusion or the consequent of the first hypothetical proposition is the antecedent of the second hypothetical proposition. So that if you go if S then F, F would have to be the antecedent in the next proposition, if F then T. And then you could conclude if S, then T. But since we don't have that configuration configuration on the simple rule that you've been given, you know that's invalid. Okay, I'm going to let you on your own study the discussion of dilemma. We'll come back to that later in the course when we talk about debate techniques. The dilemma, as he points out, is mainly a rhetorical tool. It's not so much A matter of learning the procedures of logic. But it's a very valuable discussion discussion that he has here, especially the three ways of defeating an argument that's formulated as a dilemma. You can escape between the horns, or secondly, grasp the dilemma by the horns, or thirdly, formulate a counter dilemma. And I hope when you read that, you found that entertaining. The counter dilemma. What we have been introduced to then is, is the new or the second part of deductive logic, compound propositions or connective logic. And we have first seen it in ordinary language. That's why it came up in this particular chapter. When we turn to chapter nine, excuse me, Chapter eight, we're going to be looking at this in terms of symbolic logic. We're going to go to a formalized notation. The first thing Chapter 8 does is show us how to use the symbols. And then chapter nine will give us a method for testing syllogisms that are in that symbolic form, the method of deduction. So what I'd like you to do for tomorrow. Is to do all of chapter eight, part of which you have already begun, and then finish chapter eight and begin chapter nine. So, technically, your assignment tomorrow is to go from 8.4 to 9.2, 8.4 to 9.2, and you've already done chapter eight, sections one, two and three, but we're going to cover those tomorrow. So just to make it clear, tomorrow is going to be all of chapter eight and the first two sections of chapter nine.