Dr Greg Bahnsen - Critical Thinking: Quantified Logic Part 2 (Part 19)

And let's continue. We're on page 336. Let's do number five. And Pat, let's have you try that one. You don't use the Morgan theft. First of all, let's get rid of the negation in front of the quantification.

Well, we take the contradictory of that form.

Okay, You want to look on the right hand side of the right hand column for one that has a not given any x as the beginning. Why are you calling it an E form here? Let's take a look at it together. Page 329 on the right hand column for each of those biconditionals, do you see one that has the form not given any X?

Yes.

Okay. No, the first one is not not given any X. That's not. There is an X that the first line in the right hand column has not. Then backwards ex. That's not what you're looking for because the problem you have in number five begins with not parenthesis X.

Okay.

Yeah, I must have looked at the wrong one. Okay, well, let's work it through then. Using line two is our equivalence which has the not anything X. We're going to reformulate that as there is an X such that. Actually, we have to go down further. I may have misled you a little bit here. It's the fourth line down because the second line has two negations. Really, it has. And I want you to look back on page 329. The second line of equivalences in the right hand column has not X and then not phi X. You see that? And that's not really the form that we're working with. If you look further down the columns on page 329 at the bottom, you'll find the not X without another negation. So now let's use that equivalence pattern. We have not given any X becomes what? On the other side it reads there is an X such that not. Okay, so now back on page 336 and number five, the not given any X will become what there is an X subject.

Not. Good.

And then whatever's in the parentheses. Now you have the negation in front of the parentheses with an ex in front of it all. Now how are we going to get that negation inside the parentheses or how are we going to strip it off first in front of the parentheses? We're going to use Demorgan's theorem. And Demorgan says you can put the negation inside and apply it to both of the compounds and change the connective from or to and so when we apply Demorgan's, then we end up with there is an X such that not not ix. No, no. And you have to change the or to an and with the Morgan not jx. And then finally, by double negation, we can get rid of the not not ix. And the final answer will then be there is an X such that ix and not jx. Okay, Pete, would you do number seven?

Number seven. We are given not ex or backwards ex. And if we look at our table on 329, you can see that's the.

Third.

Column down on the right. To replace it, we end up with any X than the not or the negation sign.

Let's remember to say this right. For any x or given any x not X or not whatever's in print, not, whatever the properties.

And if we apply that, then we end up with given any X not, not. Then the rest in the parentheses. Using the double negation, we end up with given any X. Then what's left in the parentheses MX or nx?

Then you apply Demorgan to it.

No, it's just left it in given any X MX or mx.

Now, wasn't the first step the first step you took, that is giving the equivalence gives us given any x not the case bracket, not the case parentheses, MX or nx parentheses, bracket. All right, and so what you did is you did something intuitive, but it's not allowed by the rules. You took the negation outside the bracket and applied it to the negation inside the bracket. And in that way, well, I take it back. Maybe you are allowed to do that by copies rules. I get a little confused when I do the double negation. I expect it to be legitimate. When I catch you guys doing it, then you see how I jump on it right away. Now, that'll be fine. So the answer comes out for any X, it is the case MX or nx. Good work, number nine. Melissa.

Okay, I have the answer. I don't know if I can explain it. Right.

Well, let me take you through it step by step and you tell me the answers.

Okay.

This begins with a negation outside the quantifier. It says not. There is an X such that. And when we look at the equivalences, how should that be translated? So the negation's not in front?

Well, because negation. I look back on page 329 at the square again, and because the negation was in front, I have used the contradiction to be a principle and turns it into X in parentheses and then in parentheses, negative 2.

No, it should Be X in parentheses, not something. Correct.

No, well, that. When you. No, because when you get a prediction of that, when you do the equivalent of a negative ex, negative five or.

Ex, when you do the contradiction of an ex, then you end up with an X naught.

She's looking at line one.

Yeah.

So that.

I did the same thing she did. We had the not ex then, not the quantity. I took that to be what, my number one sets. And so I just dropped the negatives.

That.

As you applied double negation as your. In a sense, as your first step inside the brackets. So you have inside there now a QX or rx.

Negative QX or rx. Yeah. And then I did material application, and I got.

Okay. Well, technically speaking, the form here, once you see that bracket, that closes off the stuff, you don't look at that as a not ex. Not. You look at that as a not ex because. Well, when the bracket's there, I mean, that's just. That's the way we play the game. But now what I'm suggesting is what you ended up doing is that inside the bracket, you saw that as a. Oh, you just dropped the brackets themselves. I see. I think he wanted you to take another step in there. Let's. Let's find out what you're.

No, that's not.

I thought that's how he's explaining it. Did anyone else get that?

That's how I did it. Okay.

I mean, I thought basically how he was saying.

When I say another step, what I mean is you. You just dropped the brackets. Why did he give us the brackets if we weren't supposed to work it through with another step?

You can go another step after that one. After the line is kids.

Okay, well, let's find out what your final answer was. What is the way it should be translated?

Well, then I do that material indication, X in parentheses, and then in parentheses, qx. If qx, then gx.

The final answer read across the line slowly. For any X, if Q, just read the symbols along the line. For any X parentheses.

If ux, then dx, rx. Excuse me. I don't know where I got dx. Excuse me.

Well, some kind of mistake. It should come out for any X, not X cubed or rx. It's the same thing.

That's the same thing.

By material implication.

Yeah, by material implication. But you see, the reason I ended up that way is that I treated the bracket as a full unit. I didn't break the bracket down, so I had to apply to Morgan's and so forth. So There are different ways of doing this, but the point is we end up with the same answers or logically equivalent answers. Let's move ahead to the stuff that's a little more challenging. Go now to page 341. Following section 10.4, Roman numeral 1 says, construct a formal proof of validity for each of the following arguments. And let me run through the first couple for you, and you check your work against this. Lines 1 and 2 in the formal proof have been given to me as premises. They're the first two lines written out in the problem there. I won't bother to read them across for you. But now line three is going to be CA and aa. And here we're just going to have to take it for granted. You know the difference between the capital and the small letters when I read it CA and aa. And I did that by existential instantiation EI on line two. Now why did I do EI before I do the UI in the next step? Because we have a rule that says you have to apply EI first, or else you're going to get your universe confused. Okay, Line four then is aa, then not ba, and that is proven by UI universal instantiation on line one. Everyone see that? So far all I've done is I've stripped off the quantifiers on premises 2, then 1. Now then, I can proceed with my proof. Line 5 will be AA, which is a simplification of line 3. Line 6 will be not BA, which I proved by modus ponens from 4 and 5. Line 7 will be CA, which is a simplification of Line 3. Line 8 will be CA and not BA, which is the conjunction of lines 6 and 7. And then number 9 will be the conclusion. There is an act such that Cx and not Bx, which is the existential generalization of line 8. You notice how what I did is I stripped off the quantifiers and then I put the quantifier back in. At the end I did an existential generalization so that I'm not dealing with a little letter A one particular individual. Now I'm going to generalize about that and say there is an X that CX and not bx. Does anyone have any questions about number one? If not, let me rush ahead to number two. Number two gives me two premises and a conclusion that is supposed to follow from them. I'll just take it for granted. You know those premises. You're looking at them on page 341. Line three will become DA, then not EA, DA, then not EA. This is a universal instantiation of line one. I've just taken line one and stripped off the quantifier and put an individual instance in there DA, then not EA. Line 4 will be FA, then EA, which is a universal instantiation from line 2. If you look at line 2, I strip off the quantifier and put my instance A in there. Line 5 is EA, then not DA. And I know this by transposition of line 3. Transposition says you can deny the consequent, put it in the place of the antecedent, and then make the new consequent the denial of the first antecedent. Is that real clear to you? Look at transposition in the rules. Anyway, I've reversed the order of mention and applied negation. So now I have ea, then not da. Line six will be fa then not da. Fa, then not da, which is hypothetical syllogism applied to lines four and five. Of course, as F, FA leads to ea. Five says EA leads to not da. So I put them together and get FA leads to not da. And line seven is my conclusion. I do a universal generalization on line 6 to apply UG to line 6. I say for all x or for any x, then fx, then not dx. A is taken out and X is putted back in its place with the quantifier prefixed. Everyone follow that. Let's go to Roman numeral 2. Construct a formal proof of validity for each of the following arguments in each case using the suggested notations. Let's look at number one. We have ax, BX and C. What does C stand for here? Path AX is given to us. BX is given. What does C stand for? Remember my lecture. What does a small letter refer to?

The variable?

What's it stand for? A variable.

Of what particular?

Of some particular. An individual constant. In the. In this case, which constant is mentioned? The argument says no athletes are bookworms. Carol is a bookworm, therefore Carol is not an athlete. What do you think C stands for? What particular?

Carol.

Yeah, Carol is a particular. We have also athletes and bookworms mentioned here, but they aren't particulars, they're classes. Okay, so number one is going to be schematized for all x.ax then not bx. That's the premise that no athletes are bookworms for all X. If X is an athlete, then X is not a bookworm. Premise two will be. Listen to this.

Bc.

Bc. There is no quantifier on that because the next premise is particular. Carol is a bookworm. And I'm glad this exercise is here.

To remind us after we've learned all.

This prefix of X and prefix of backwards ex. We don't always use them. If you're talking about an individual constant, one particular person or thing, you don't need that. And here we have BC. Okay, line 3 of the proof will be AC then not BC. AC then not B. And how do I get that? From a universal instantiation of line 1. The instantiation strips off the quantifier all X and I can put in there any individual constant I want. Now, why did I choose C? Was I just in a mood for choosing C rather than D or R or something like that? No, I chose C because I knew the argument was about C was about Carol. And so if we're going to apply these class remarks to the particular individual, then I'm going to choose C in my argument. I'm just talking out what should be obvious to you. And then my Conclusion is not AC by modus tollens from 2 and 3, 3 says AC then not BC. Line 2 is the contradiction of that. And consequently I can conclude that the antecedent is false, not ac. Now, I may have combined a double negation in there again, but the rules of our class are the teacher's allowed to do it. Okay, number two for that one, because.

It'S A E proposition. Can we say all athletes are not bookworm, Or would we have to go bookworms if bookworms are not at weak or. No.

It says no athletes are bookworms. Doesn't that amount to the E form fee X then not to see X?

Yes.

What I wrote down is AX then not bx. What did you write down?

If BX then not ax.

Well, why did you reverse bookworms and athletes?

Because I translated all athletes are not bookworms. Right? No, I did.

This has no athletes or bookworms. That means that with all athletes they are not bookworms. You don't want to say if a person is a bookworm. I suppose you could do it either way.

Because if we have proposition and both are distributed. That's right.

And both are distributed, then that allows for conversion, no. Is that what it is? We're applying conversion to do that the way I did it makes it a real short argument.

Yeah, well, this makes it shorter.

So what were. Yeah, I completed it in four steps.

Oh, that's what I did, too.

Okay, so what we're saying is that could have been written either as BX then not ax or ax then not bx. Let's look at number two. Here we have dx, ex and fx. All dancers are exuberant. Some fencers are not exuberant. Therefore some fencers are not dancers. And premise 1 will be for all x dx then ex. If x is the dancer, x is exuberant. Premise 2. There is an x such that fx and not ex. Remember the and rather than the if, then X is an F and it's not the case that X is an E. Line 3 will be FA and not EA. Existential instantiation of line 2. I strip out the quantifier and put an individual constant in there. Line four will be da, then ea. That's a universal instantiation of line one. Notice again that I did the existential before the universal instantiation, because that's the rule. Line 5 is FA, a simplification of line 3. Line 6 is not EA, a simplification of line 3. Line 7 is not DA, which is modus polens applied to 4 and 6, 4 said DA, then EA 6 said not EA 4. Therefore, through 6 leads leads to not DA. At line 7, line 8 will be FA and not DA, which is a conjunction of line 5 and line 7. Line 5 is FA, line 7 is not DA. So in 8 I can combine them. Conjunction FA and not DA. And now I want to generalize that. Get rid of the individual constant which is A, and I make that there is an X such that X is F and not X is B. Or let me read it across. There is an x such that fx and not dx by existential generalization of line 8. Do you have any questions about these problems in the homework? Actually, the proof. Proof is rather easy because what you're learning here is just how to strip out the quantification or put in an individual constant when it's necessary. The next section, 10.5. Test our ability to prove invalidity. When we have the quantifiers in the argument on page 346, we're asked in Roman numeral one to prove the invalidity of the following. Now, in order to do this kind of work, we have to translate the argument into a notation that doesn't have the quantifiers in there. And the interesting thing is, we can translate both the universal and the existential quantifiers by simply replacing them with the one particular individual. So when we do that, if you look at number one, line one becomes translated as aa and ba, capital A small A and capital B small A. Line 2 becomes CA and BA. Again, I've stripped off the quantifier and I put an individual constant in the place of the X, and the conclusion, which is universal, nevertheless comes out ca, then not aa. Ca, then not aa. And again, the fact that it's universal doesn't change my test procedure because I'm testing it with respect to one individual.

Yes.

Can you do that instantiation with AABA and caba? When you're doing that twice, don't you.

Have to use a different variable?

Or am I on the wrong page?

Now we're on page 346. No, the answer to your question is you do not have to use a different variable because as the author emphasizes in the chapter, we are not applying universal instantiation and existential instantiation. This is not a use of ei. He says this is rather a test case for one individual in your hypothetical universe. And for that reason you can strip off the existential and universal quantifications and just use the same individual throughout. Okay, now, once we've done this, it's fairly easy to figure it out. The conclusion has to be false if I'm going to show invalidity. And the only way I can make an if then false is if I have a true then false, that is if true, then false sequence. So CA has to be true, and then it leads to not aa, which has to turn out to be false. So AA would have to be true, which when negated becomes false. And so AA is true, CA is true, true. How about ba? Well, if you look at line one or line two for that matter, you notice that you have conjunctions with ba and the only way to make those lines true is to have everything conjoined through, and therefore BA would have to be true as well. So AA is true, BA is true, and CA is true. And in that universe, notice my friends, in that universe, A, whoever A happens to be, let's say it's Melissa in that universe. If it's true that Melissa is A and Melissa is B and Melissa is C, those predicates all apply to her, then this argument is invalid. You follow that it's a test of a hypothetical universe. Take any individual and let's look at number two. Number two can be tested to be shown to be invalid if we strip off the quantifiers and just take some individual. So let's reformulate what number two would say. It would have da, then not ea, EA then fa, therefore fa, then not da. We have to make that conclusion false if we're going to prove this invalid, since we have an fa then not da. The only way to do so is to have FA true and then not DA false. For not DA to be false, DA has to be true. So we've now assigned two of our three values. DA is true, FA is true. This leaves ea. Now, if you look at line 2, you have EA, then FA. Since FA is true and anything implies a true premise in propositional and truth functional logic, then we are not able to tell anything about EA from line two. But if you look at line one, which has to be true also in order to show invalidity, line one says da, then not ea. Well, DA is true. And if DA is true, then it must imply something else which is true. Or else the whole line turns out to be false, which doesn't show invalidity. Therefore, ea, not EA would have to.

Be true.

In which case EA would have to be false when negated becoming true. So we've proven the invalidity of numbers 1 and 2. Let's do the same thing with Roman numeral 2, where we have to do some translation work first. In number one, we have ax, bx, and cx. All anarchists are bearded. All communists are bearded, therefore all anarchists are communists. Pretty good argument, huh? Hopefully you can see through that almost intuitively. But let's schematize it. That's going to be written down when we translate it into our notation. Aa. Well, actually, I've stripped off the quantifiers already. I hope you can see that when you're proving invalidity, you don't have to worry about the quantifiers. You're just going to take one individual and test every proposition. So I'm not going to put it into a form that says given any X or there is an X such that. I'm just going to use one individual. Aa, then ba, ca, then ba. Conclusion aa, then ca. Now we want to make the conclusion false. So AA becomes true, CA becomes false. Now the premises have to be true. If you look at premise two, we have ca, then ba. If CA is false, then BA could be anything. But if you look at premise one, which has to also be true, AA is true, and that leads to ba.

And the only way that whole line.

Can be true is if BA then is true. So you have AA is true, BA is true, CA is false. Exercise number two. We have dx, EX and fx. No diplomats are extremists. Some fanatics are extremists. Therefore some diplomats are not fanatics. This is an argument that is invalid. Let's see if we can prove it to be so. Let's translate our premises. And remember, we don't have to put the quantifiers in there. So DA then not EA is number one. No. Diplomats are extremists. If A is a diplomat, then A is not an extremist FA&EA. There is a fanatic that is an extremist FA&EA. Therefore the conclusion says DA and not FA, that is, therefore there is a diplomat who is not a fanatic. Okay, in the conclusion, which needs to be false to prove invalidity, we have a conjunction, in which case at least one side of that conjoined compound must be false. But if you look at line two, ea. Excuse me, FA and ea, for that to be true, both of those would have to be true. So we can automatically assign EA as true and FA as true. Now, if we go back to our conclusion, if FA is true, not FA is false, and so that satisfies the need of the conclusion to be false. In our first premise, we have DA then not ea. We know that EA is true and yet it is negated in the first premise, in which case not EA would be false. And the only way that you can have a false in the second position in a hypothetical statement and have the whole line be true is if what implies it is false also, and therefore DA has to be false. Everyone follow the logic here. And thus we have proven the invalidity of that argument using the truth table and assigning of the shortened truth table method of assigning truth values.

Then the last thing that you were.

Asked to do is in section 10.6, understand how the English language is sometimes tough to try translate into symbolic notation. And Kopi discusses that for three pages. And then we get some exercises to try out on page 350. And let me take number one in exercise, Roman numeral one, and show the trickiness of translation. Here we read, apples and oranges are delicious and nutritious.

And although this is mistaken, I'd like.

To put up here on the board what that might appear to be, translated as it might seem that what we have here is the statement that for any X.

We have a universal statement being made here.

You notice there's no restriction here.

It's all apples, all oranges.

So for any X, if X is an apple. And X is an orange, Then X is delicious and nutritious. What I've written on the board here is for any X in parentheses, ax and ox leads to parentheses, or is we have an if then in parentheses DX and nx.

So you see this in English and.

You say, well, the Words seem right. X is an apple and X is an orange because it says apples and oranges. Then X is delicious and X is nutritious. You see what the trick here is, though? I've written this down wrong.

Just. Just to show you in English you.

Have apples and oranges are delicious and nutritious.

The one use of and is not problematic. At the end of the sentence it.

Is asserting that two different things apply, delicious and nutritious. But the and at the beginning of.

The sentence is not talking about something.

That is both an apple and an orange.

And yet my notation here is talking.

About something that is both an apple and an orange. I'm dealing with an X, a particular X. I say ax and ox because there's an and. Here you see why that mistake would be made.

But that is of course not the.

Meaning in English at all.

The meaning is rather with respect to.

Any X.

If it is an apple or an orange, then it will be delicious and nutritious.

And so in order to express this.

Sentence correctly, we have to be very.

Careful to use an or instead of an am.

And he discussed that in the chapter. Let's look at one more here.

Number two. Some foods are edible only if they are cooked. Now we might be tempted to write that one in the following way. There is an X.

Notice we're dealing with some foods, not all foods.

So we begin our quantification with the particular. There is an X such that.

X is include.

And.

It is the case.

That X is edible if and only if X is correct. I put the biconditional.

I have not used the three lines.

Pardon me, I just used the two sided arrow because I think it's easier to see here we might think because it says only if they are cooked. And the author has told us that accepted propositions can be expressed in terms.

Of a biconditional, that it's easy to.

Use this particular biconditional. There is an X such that X is F, X is food and X is edible if and only if X is cooked. Okay, so let me remind you, what I've written on the board is the following backwards ex quantifying. There is an X such that bracket FX and now parentheses, ex, triple lines or double implication cx end of parentheses, end of bracket. Given what we have written up here on the board, is this biconditional a.

Statement of what the English sentence actually says? Let's think about it. Just try to be intuitive about it. This is saying that there's a food.

That has this characteristic about it if it is edible.

Then it is cooked.

Is that what they're saying? Some foods are edible only if they are cooked. The biconditional states of this food that if it is edible, then it is cooked. That's not what the sentence is talking about at all.

The sentence is not asserting that there.

Are some uncooked edible foods in he's.

Saying there are some foods that are.

Edible only if they are cooked. Okay, so it's more of that there.

Is a food such that if it.

Is cooked, then it is edible. So what we might have here is that instead of FX and ex biconditional cx, you might have FX and CX than ex.

Which would now read that.

X is a food, and if X is cooked, then X is edible.

Now the sentence is asserting that there.

Are such foods, like that some foods are edible only if they are cooked. So we've got that much right. But it hasn't expressed the whole meaning of the sentence, because the sentence is.

Not just telling us that there are.

Some foods or which, when you cook them, are edible. It's saying there are some foods which are edible only if you cook them. Only if you cook them. And so we have to rewrite it again, and here's the way it will come out. I believe when it's done correctly, there is an X such that bracket fx that is X is a food, and use the dot here, parentheses. If not cooked and is not cx, then not edible. There is an X such that if X is a food, then not. If there is an X such that, if X is a food and if X is not cooked, then X is not edible.

Does that not express the whole meaning of this statement? You notice how this first line up here did not.

It's not a biconditional. It's not like edibility and cooking imply each other for some particular food.

Nor is it the statement that there's.

A food which is cooked in.

It does say that, but it says much more. It says there are some which are.

Not edible if not cooked. So not edible if not cooked.

And the whole point of 10.6 is just to show you how tricky the translation from English into logical notation can be.

And with that, we're going to put.

An end to our discussion of quantification. And in putting an end to our.

Discussion of quantification, we're also putting an end to our discussion of formalized logic, formalized deductive logic.

So the last thing I'd like to.

Do today is to have you just look up here on the board and.

Let me retrace where we have been.

Thus far in the course, all the.

Mathematical part of the course, well, not all, but virtually all the mathematical part of the course is now finished. The first thing that we studied as we began is the whole subject of argumentation.

We had to learn what an argument.

Is, what propositions are.

All seems so easy now, right?

What premises are conclusions, inductive and deductive. And we learned something of a theology of argument and saw that as Christians, we should be concerned with thinking clearly.

And consistently and knowing what implications are all about. So a theology of argumentation was given.

Then the second step in critical analysis.

Excuse me, in critical thinking, is to.

Master the tools of analysis. So after we learned about argumentation, we turned to analysis.

And under analysis we learned three things. First of all, how to analyze language.

That is not only the first thing we did.

As I have been stressing over and.

Over again, it's the most important thing that we do in critical analysis. It, you do not know, engage in the analysis of argumentation in an effective way. Somebody might say, well, Dr. Bonson, logicians can strip out all those problems and turn everything into symbolic notation, and then.

There'S no more ambiguity.

But by now you know better than that. In fact, we just finished a unit that pointed out how difficult translation might be in some cases. And therefore symbolic logic is a real tool once you've mastered linguistic analysis. So the first thing we did under analysis was language. Secondly, we looked at ordinary language fallacies. By ordinary language, I mean those fallacies, that reasoning that takes place in a normal natural language like English, Japanese, French, whatever it may be. So I spent some time in Kopi, and then even more spent some time in Engel, studying the fallacies, which is the most common thing I think you'll run into. And that'll be the thing you can best use from the whole course, frankly, is catching people in hasty generalization, ambiguities.

And things of that nature.

Then thirdly, we turned to a different subject. No longer are we considering ordinary language, but now we're looking at formal deductive reasoning, That is reasoning that takes place in a formalized notation, reasoning that is deductive in character. So this is what we've studied in the course. Argumentation and then analysis of language, ordinary language fallacies, and then formal deductive reasoning. The formal deductive reasoning part of our course dealt with categorical reasoning or syllogisms, dealt with propositional reasoning.

And then thirdly.

Dealt with quantified propositional reasoning. And so that you can kind of package all this categorical reasoning regiments what three terms in the English language? Some, all, not some, all and not exactly. Propositional reasoning dealt with connectives. And so we learned how to use what three expressions?

If then, or.

And. Or if then. So the first one is the logic of classes in their inclusion or exclusion. The second is the logic of propositional relations and, or, if, then, and the third, which we study today, is quantified propositional logic, which combines the first two, so that now we can use the all and sum along with our connectives and, or if, then. Does that make sense to you how these two then come into and that's where we are in the course, we've dealt with argumentation, now we've dealt with analysis.

And although we're winding down, I don't.

Want you to miss this point.

This is crucial.

Analysis concerns meaning. When we do analysis, we're seeking the meaning of expressions and combinations of expressions, what we might call arguments, to understand the meaning of what is being said. To analyze something is to know how to analyze the language. That should be obvious when we call it meaning. Also, what does a person mean? What's going on when a fallacy is committed? And then thirdly, under formal deductive reasoning, when we see propositions, we want to ask, what are the logical relations between these statements with respect to and, or if, then some, all and not. By studying the inner structure, if you will, and the interrelationship between propositions in this formalized notation, we are getting clearer on the meaning, the conceptual condition, containments of our statements. What is conceptually contained, what is involved in saying, on the one hand, all men are mortal, and on the other hand, Socrates is a man. You see, logic helps show us that that also means Socrates is mortal. Now, the fact that a person has not asserted Socrates is mortal is a piece of biography. It may be psychologically interesting that a person who says all men are mortal, Socrates is a man, does not go on to say Socrates is mortal. But logically speaking, they have said that that is what it means to say these two propositions. Am I making sense to you, then, by studying logic? If two or three or four or five things imply another, that another is part of the meaning, the conceptual meaning of those propositions. So let's say that I've got five propositions up here, and then I draw a conclusion. What I am getting at by studying logic is that this conclusion is part of the meaning of these five together. So analysis, breaking things down. Analysis shows us what is contained in the things that we say. If I took a specimen of blood to the hospital lab to have analysis done of it. They would break down the blood to find out what are the things that are in it. And in the same way, logical analysis amounts to finding out what are the conceptual parts, what is contained conceptually in this specimen of language or argument. When people say these five things, what is conceptually involved is this conclusion, or vice versa. When I analyze this conclusion, I might find that conceptually contained in that conclusion are these five premises. Okay, so just standing back and looking at this again. Section 2 in our course on analysis has been a study of meaning and, if you will, conceptual containment. What is conceptually contained in the statements.

That we make are the arguments that we use. So I'm going to erase this detail underneath here and go back to Roman numeral 1 and 2.

We have studied argumentation, we have studied analysis, but we have not finished what is necessary for critical thinking. Most people do not get as far as number one. Fewer get as far as number two, much less all the way down to a quite quantified propositional symbolic logic.

So you've made great progress.

But if we were going to round out our course and understand what's involved in being a critical thinker, we'd have to know more than about argumentation and meaning, more than about argumentation and analysis. Another thing we would need to study would be explanation.

Now, I don't want to quibble.

Somebody could say, well, Dr. Bronson, under analysis, aren't we studying explanation? How to explain the use of language and the relationship of propositions in an argument?

You could put it that way.

But again, I've said that the analysis of meaning is a matter of conceptual containment. We're looking at what things mean. When we study exploration, we want to explain why things happen. So explanation is the study of causal relations rather than linguistic or logical relations. So I'm putting causation down here. Explanation, then, is the study of causation. Why do things happen the way they do? Not what does it mean to say something happens? What is conceptually contained in a piece of argumentation, but rather, how do we account for things? What causes things to happen or a.

Particular thing to happen?

So critical analysis requires that we get involved in scientific reasoning, and what are the procedures and the standards or paradigms of scientific reasoning? And then once we've mastered meaning and causation, and I'm building this piece by piece, you have to understand very few people get as far as what we're talking about. But even those who get through argumentation, analysis, and explanation will often totally forget the Influence of presuppositions, worldview commitments, that is your underlying, most fundamental convictions about the nature of reality and how we know what we know and how we should live our lives. And so the fourth part of critical analysis would be presuppositional or the study of presuppositions. And then the last step that would have to be taken if we were to round it all out is that if we understand our own argumentation, how to do analysis of meaning, explanation of causes, presuppositional assumptions, then the question is.

How do we apply all this?

And that would be the study of debate. And I don't mean by debate simply what happens in a public auditorium when two combatants get up and debate with each other, although that is part of it. Debate takes place when you talk to.

Your next door neighbor over the backyard.

Fence, you know, and you say, oh, you don't believe that this candidate is the best for office. And then you start going back and forth on this consideration of that. That's a debate. Also the exchange of arguments. Argument is. What is argument, contention. No, argument is inferring a conclusion from premises. Now, when, when you exchange arguments, your neighbor has one point of view and gives premises that he or she thinks leads to that conclusion. And then you give premises that you think lead to your conclusion. Now you're exchanging argumentation, and that's what we call debate. And what is involved in effective debate? What is involved in understanding and refuting.

That point of view?

Or what is involved in effectively presenting.

Your point of view?

And I don't mean here, just tricks of debate. That is, how do we show the weakness of your neighbor's thinking? How do you show the strengths of your own in terms of analysis, explanation and presupposition? That's what would take to finish out the course on critical thinking. What we have done thus far is argumentation and analysis, and hopefully we'll finish out the other three components at a later time.