Dr Greg Bahnsen - Critical Thinking: Deductive Proof Part 2 (Part 17)

Area number four. I was pointing out that line four denies what turns out to be the consequence in the conditional statement given in line one. The ONP is denied in line four. So by using modus colons, I can now deny the antecedent in line 1, which is M and M. So line 5 is justified. Line 6 is justified by Demorgan. Again, because when you have a negation outside of parentheses, where two things are disjoined, that can be rewritten as the conjoining of both of them being denied. So I have not M and not M, and then I can get not M easily by simplification. Follow that pattern. Okay? In Roman numeral 4, we're told for.

Each of the following, adding just three.

Statements to the premises will produce a formal proof of validity. Construct a formal proof of validity for each of the following arguments. All right, let's take a look at number two. At number two, and I'll work it out for you. He gives only one premise. The premise is not B or C and D. Now, it turns out that that premise can be rewritten by the law of distribution as not B or C and not B or or D. Everyone see that by distribution, I can rewrite it in the form of not B or C and not B or.

D.

Then the next step, which will be line three, is to assert not B or C by simplification of line two. And then line four is the conclusion B. And that follows by material implication, It's B if B then C. And that's just another way of writing not B or C. So line three is rewritten by material implications as if B then C. Okay, line. Excuse me, exercise number three.

Jay, would you do that one?

Okay. On line two, there's only one line of proof. On line two, by the rule of distribution from line one, I assert this and D or F and E or D. Very good.

And the next step.

The next step is to commute the 2.

Reverse the order of the blobs. Okay?

And then step so that you would have E or G and E or F. And then step one, by the rule of simplification, we would drop E or F and do F or E. Very good.

P. Try number four.

Number four. We have one line of proof. The second line we get.

We have one line of premise. Premise, one premise.

Okay, Second line, we get H and I and J from association.

Okay?

The H and I are in parentheses and the J is added.

Right?

And that just reverses the order mentioned by association.

Next.

Next, we commute what was in two. So we Get J and the blob of H and I.

Well, now you've done two steps and.

One there, haven't you? No, the next line should be I.

And H within parentheses and J.

No, that's my next step.

Yeah. Actually, you could take the next two.

Steps in either order.

You took it in one order, I took it in another. But go ahead and finish your way.

Then I concluded, or I finished with commuting line three so that it's J. And then I commuted H and I to I and H, so it's J and in parentheses I and H as our conclusion.

So we can make this clear, then the three steps are association, then commutation and commutation, whichever order you want to do the commutation step.

Right.

All right, let's look at the problem in Roman numeral 6 where we're asked to construct a formal proof of validity for each of the following arguments.

And.

Let'S do number three first of all. And I will try to do it to show you how it would work. I'm going to write the problem on the board and then we'll take each step.

We'll explain each step.

All right. In problem number three, we have two premises. The antecedent if G, then not H, therefore I. And the second premise is it's not the case G and H. The first step of proof that I've taken. Well, I should indicate that we're supposed to prove therefore I or not H. The first step to prove is to assert not G or not, which we're able to do by de Morgan's theorems. Applied to line two, we look at line two, we see that de Morgan.

Where you have a negation outside of.

A compound statement that has a conjunction, change the conjunction to a disjunction and negate both sides. In line four is a material implication operated on line three. Applied to line three in material implications, we have if P, then Q being the same as not P or Q. So if I take the not P part of this formula, which turns out to be not, notice how I'm treating the not G together as one blob. Again, now I'm going to negate that, and so that gives me G and that leads to not H. So that's material implication applied to line 3. Line 5 is the assertion of I by modus ponens from 1 and 4. Line 1 as a premise gave me if G, then not H as an antecedent to I. And I have proven if G then not H in line 4 and therefore by modus ponens. I can assert from lines 1 and 4 on line 5 a conclusion I. And having done that, of course, I could add anything I want to the I. And so line 6 is I or not H by addition.

To line 5.

Can everyone see how those steps were taken? Pete, would you do number seven?

Number seven?

Yeah. Why don't you just tell us in English as best you can what line 3, 4 and 5, etc will be and how you justify them.

Okay. Number 3 became not not X or not B from line 2. That's DeMorgan's theorem.

You have not not X.

Yes or not V. Why don't you just.

Oh, I see. I have. What I did is then your next step is going to be by double negation to get rid of the not not.

Okay.

So you have not not X or not V by De Morgan applied to line two. And then line three is going to be X or not V by double negation. Okay, go ahead.

Then line five will be not V or X by the commutation of line four.

Very good.

And then line six will be.

W.

Or W from lines one and five.

Now, you don't have to say W.

Or W.

Oh, I, I.

Okay.

Because that's a psychology.

I just.

Again, I, I compress another step.

I understand what you're doing. You've done it, right?

You have W or W by constructive dilemma.

Constructive dilemma, yeah. And then line seven is W from the tautology of line six.

Very good.

Okay, Pat, at what point did just become unclear? Okay, let me see if I can. Let me schematize it on the board for you. Okay. I've written the first two premises which are given to us on the board. The second one says it is not the case that not X and B. Now, Demorgan's rule of appreciation or Demorgan's theorem, allows us to rewrite something when there's a negation outside the parentheses conjunction inside as the negation of both those things. Now, disjoin. You understand Demorgan's theory. We're going to rewrite that now as line three. And so since what we have is A not X in the first position, then we get not not X or not V. That by Demorgan to be very mechanical. And then we need to get rid of this double negation. So 4 becomes X or not B. And that by double is the rule of double negation. See? Double negation and not not. So there's a rule there where you can drop the two that becomes X or not V.

Okay.

And then by commutation I can rewrite line 4 as not v or X. And then you'll notice that not V is the first antecedent. Look at line 1. In the first conditional, not v is the antecedent. In the second conditional, x the antecedent. And so, by the rule of constructive dilemma, I can disjoin the consequences of each of those conjoined conditionals. I know the language gets confusing, but look at it this way. If not B leads to W and I've got that, then I've got W, and I have to add that I have X, which leads to W. So I have one or the other of.

Those, not B or X, in which case I have one or the other.

Of the consequences, W or W. So line 6 becomes W or W by constructive dilemma. Does that make sense? And then line 7 believes you by the law of tautology, W or W.

Max W.

And that's what we were supposed to prove. Good job, Keith. JJ, tie number nine. What is your third line going to be?

My third line will be taken from line one.

By exportation.

I get the opposite of not D and not E, then not F.

Okay, we've obviously approached this differently. That's not my line three. But I do use exportation later on. How many steps will it take you to get to the conclusion?

Eight.

The conclusion is step eight.

Three.

You?

Yes. The conclusion of step seven for me.

Yeah.

But then again, I've been known to cheat, so. I mean, some of these things, like topology and double negation, I just fly over. So maybe that's what I did.

No, it's just the way I did it. I'm pretty sure step two for me, line four is by commutation. On line three, instead of having not D and not E, therefore not S, I have not E and not D there. Then not S, I just plop the two in the blob.

Okay.

And then from there I see where I got. See where I got where I got.

Now.

I don't know if I can do that.

Okay, well, let's go back and let me propose a way of doing this.

Well, what I did is I dropped.

The D out of the bracket.

Now that I look at it, I'm not sure I can do that.

No, you can't.

So let's go back and try it this way. Let's make step D3. I'll write this on the board for you. All right. I've written the two premises that are given to us on the board. I would suggest that the first step of proof we'll Call line three the assertion of if D, then this compound or blob fn, not D. And we're able to do this by looking at line two and using the rule of transportation. Line two, the rule of transportation. Transportation is a kind of modus pollen, remember? So in line two you notice that I have not G as the consequent. If I deny that I get G, and then I have to deny the P which went before it and that then drops off that negation and I get F and not d. So line 3 by transportation from line 2.

Reads.

If G, then in parentheses fn, not.

G. Line 4 is everyone with me.

So far that step napalg. Then line four will be if not D, then a compounding parenthesis F then.

E.

If not D, then in parentheses if F then E. And that line is justified by transposition. But the transposition is applied only to a portion of line one, the component of line one that amounts to the consequence of the implication that is in parentheses. We have not E then not F. And if we apply transposition to that, we get double negations. So we have if F then E. Reverse the order and get rid of the double negation. Now line 5 in my proof.

Will.

Be compound statement as the antecedent not D and F, not D and F then E. So we have if it is the case not dnf, then E. And that is justified by exportation on line four. Exportation allows us to move the parentheses when we have two implications like up in this way. And that's all I've done is I've moved it so that not D and the F are now conjoined and they become then the antecedent to E. Line.

6.

Becomes the compound as an antecedent F and not D. And the consequence is E. So to read it again, if it is the case F and not D, then E. And this line is justified by commutation of line four. I should say line five. Pardon me, commutation of line five. I've reversed the order of mention inside the parentheses here. So I have F and D rather than. I mean F and not D, rather than not D and F. And now that I've done this, line 7 is the conclusion which was to be proven if G then. And we can get if G then E by hypothetical syllogism from lines three and six, from lines three and six. So up here on line three I have if G, then a blob.

The blob is F and not D.

On line six I have that Blob, F and not D leading to E. So to put it all down together, if G, then the blob, and if the blob, then E. Therefore, if G then E. So by hypothetical syllogism, we get our conclusion. Everyone followed that. Pretty nifty. Well, let's try a few more of these now in a natural language that has to be translated into symbolism in Roman numeral seven. Let's try number three. Let me ask you this. Did you find these remarkably easy?

Yes, you should.

And that, I think, should encourage you that even though some can get real complicated, sometimes you come back to it. And this is more like what you're going to run into in real life, obviously. And when you get these, hey, now, it's pretty easy to do it. All right, let's see. Jay, why don't you walk us through number three.

We are to use the letters A.

D and I for these premises. I'll read it. And then you tell me how you would write them down and how you would finally do the proof. If a political leader who sees her former opinions to be wrong does not alter her course, she is guilty of deceit. How would you write that one down?

If not A then B.

Correct.

And then another premise.

And if she does not? All for her course, she is open to a charge of inconsistency. Line 2.

If A, then I.

If A then I. The next parameter. She either alters her course or she doesn't.

Very good.

And then the conclusion.

Therefore, she is guilty of deceit.

Or else she is open to the truth of inconsistency. And that's what you're supposed to prove.

That was B or I.

Very good. Now what?

What are the lines in your proof? Line four. Okay.

Line four would be by conjunction, not A then B and not D A than I.

Right.

In essence, what you've done is you've combined lines one and two.

Correct.

By conjunction.

Okay.

Then line five. I would by commutation, take line two and make it not A then A. I mean not A or A. A Commut. By commutation on line three, I reversed the order of A or not A and made it not A or A.

Okay.

So line five becomes not A or a. And then line six sets you up.

So line six is the conclusion. B or I by constructive dilemma from.

Lines three and five.

Very good, Pete. Walk us through number four. We only have to use F and A. It is not the case that she either forgot or wasn't able to finish. How would you schematize that I said.

The negation of F or not A.

Okay.

Now you got to get the punctuation in there.

You have negation and then parentheses. Yes.

S or not A. Yeah.

Within the parentheses you have F or not A. It is not the case that she forgot or not able to finish the conclusion. Therefore she was able to finish is what A A.

Okay.

Now how are you going to get to A from not F or not A?

Okay. Line two.

Yes.

I used to theorem correct. Came up with not F and not not A. Okay. There's a double negation for line three, so it becomes not F and A. Okay. I commuted those in line four and come up with A and not F. And then in line five took a simplification of line four. Very good.

And I appreciate the fact that you did all of the steps on the.

You know, the double negation and things.

Like that, because what I did is.

I just jumped ahead real fast.

My line 2 was not F and A, not F and a by de Morgan, rather than doing the not not A. And so since I had not F.

And A, then I just simplified.

But then again, I didn't put it in the order where the first one is what you drop.

I'm still not sure you have to.

Have it in any particular order. So you may have had a little bit longer proof than you needed, but you did pursue it in the right way. Okay, enough of these proofs.

This is what many people think of.

When, when they remembered their class in.

Logic trying to construct the proofs for deductive syllogism. And that's fine.

It's something of a game. It's a lot of fun. It's very powerful, it's very valuable. I want you to remember, however, that constructing proofs of validity for deductive propositional compound propositional arguments is only a very small part of the total thing that we would consider critical thinking.

This is just a subsection of logic, which is itself a subsection of critical thinking. So it may be the most mathematical and challenging and therefore the most memorable.

But it's only a part of what it's all about. Now I'm going to take another step back and evaluate what we've been doing. We have these very powerful rules of inference and replacement to use for proving the validity of a deductive argument. But Copy himself tells us something about them on pages 305 to 307. And so let's get some commentary from the expert here. He himself admits, first of all, these 19 rules of inference are somewhat redundant.

If someone wanted to say I want.

To reduce all the rules to a.

Bare minimum, so I just have as.

Few as possible to do all the.

Testing that is necessary.

Copie says, well, these rules are not a bare minimum. I have done a few of the patterns that can be proven so that.

You have a few more tools at your disposal.

So some of these rules will prove to be redundant.

A good example of that would be transposition.

Transposition gives you what you should know from modus.

Whole ones already.

The second thing he says about his.

Rules on page 306 is that the.

List of 19 rules has a certain sort of deficiency. That is to say, there are certain other rules which are intuitively valid that he could have added, but he has not done so. And so some he has added to make it helpful and therefore has become redundant. Some that he could have added and been redundant, he chose not to. And so the 19 rules are not all of the rules, however, and this is important 3/4 of the way down the page. He says, this set of rules is complete. The present list of 19 rules constitutes a complete system in the sense that it permits the construction of a formal proof of validity for any valid truth functional argument.

The rules may not be as minimal.

As they could be, they may not.

Be as maximal as they could be.

But they'll work for proving any argument to be valid if it is valid. The next thing he says about his rules is that in a sense, they're mechanical. The proof of validity can be decided, or the issue the question of validity. I should say the question of validity can be decided quite mechanically, in a finite number of steps. No thinking is required, he says on.

The bottom of page 306, either in.

The sense of thinking about what the sentence and the sequence mean, what the statements and the sequence mean, or in the sense of using logical intuition to.

Check any step to validity.

Mechanical, in that you don't have to know anything about what the argument's about in ordinary language, and you don't have to use any logical intuition. You can just mechanically apply these, and.

If you do so, it may be.

Real pedantic and so forth. Through the mechanical application of those rules, you can prove the validity of any valid argument. However, on page 307, 2/3 of the way down, he admits that constructing a formal proof by these rules is not an effective procedure.

And he means by this we have.

No effective or mechanical rules for the.

Construction of formal proofs.

Here we must think or figure out where to begin and how to proceed.

And of course, that's the challenge in.

What you were facing in doing your homework for this lesson.

Now, I want you to see the.

Connection between these two remarks. He says that the proofs are mechanical in that you don't have to know the meaning of the ordinary language they represent, and you don't have to have any ethical intuitions. They're mechanical. So the proofs are mechanical, but constructing.

Excuse me.

The rules are mechanical, but constructing a proof, doing something with those rules is not mechanical. That takes figuring out. It takes some ability to play the game, if you will, to see the.

Puzzle and find a solution to it.

So then we do have now again, what is ordinarily remembered from logic class. How to prove a deductive argument using propositional logic, symbolic propositional Logic. We have 19 rules by which we can construct these proofs.

These rules, however, are redundant. They are deficient and yet complete.

They are mechanical, but not completely effective.

In that you still got to do.

Some work creatively yourself.

And on page 308, we've already covered this. Qopi does try to give you some guidelines for constructing proofs when you're having a difficult time. And I would remind you of that.

We're going to move ahead to section 9.3 now. 9.3. In order to understand what we're doing, we're kind of making a minor change of direction. I'm going to put up on the board an outline of where we are in the course. When we came to the section of the course on formal deductive reasoning, formal deductive reasoning, we first of all looked at the categorical syllabus. And under the categorical syllogism, after we learned the basic concepts, class, quality, quantity, distribution, and so forth, we learned about immediate inferences. Immediate inferences such as contradictions, conflict, contrary, subcontrary, obversion, contraposition, so forth. Then we moved on to consider mediated inferences. And this is where we got into the syllogism. And once we learned what the mediated inferences were all about, the syllogisms, we had to have certain tests for those syllogisms. And remember, we had three tests. Logical analogies, Venn diagrams, which we didn't master, and the rules for fallacies.

Now.

When we left categorical diligence, we came to the second part of formal deductive reasoning. We came to propositional argument. Propositional argument began at the end of.

What was it?

Cofie, chapter seven. At the end of Kofi 7, he introduces disjunctive syllogism, hypothetical syllogism, you know, modus ponens, modus colons and so forth. So we began propositional argument, looking at it in ordinary language. And then we turn to propositional argument, the logic of connectives and compounds, if you will. Here we're learning how to use and or if. Then we turn to it in a formalized or symbolic logic. A formalized or symbolic logic. I'm not going to have room for the whole outline, so I'm going to impress it here in its formalized form.

Remember, the first sub point at our.

Two here would have been in ordinary language. Now we're going to talk about informalized language. We were introduced to the basic concepts and notations, the wedge, the dot, the curl, the horseshoe, and what material implication was all about. And then Copie taught us about tautology, contradiction, contingent statement, material and logical equivalence. And then we understood something about Demorgan's.

Tautologies and a restatement of material implication to get an idea of where some.

Of our rules of inference will come from. So we have all this introductory material, and then what we've come to now, and this is why I'm putting it on the board, is we've come to test. Notice how this corresponds to what we have under our study of categorical syllogism. After we learned about immediate inference and then mediated inferences, we came to certain tests of the categorical syllabus. In the same way, we've been introduced now to propositional argument, specifically the formalized argument, having to do with compound statements and connectives like and, or if, then. And we are at the place where we're going to learn how to test how to test propositional arguments that have.

These connectives in them.

And I'm going to put three underneath here. We have three kinds of tests we're going to be looking at, or have looked at actually. The first kind of test is the test by truth table. That's what we were working on yesterday. We learned how to prove an argument by truth table. But then what did we say? The truth table sometimes become unwieldy. So then we turn to, let's try to have chains of argument where every link in the chain can be justified by a rule. And so now we have just today learned about rules of deduction. You can test an argument by the rules of deduction, trying specifically to prove validity. We prove validity by the rules of deduction. So the first test was the truth table. The second test was proving validity by the rules of deduction. Now we are turning to a different kind of test. Sometimes we don't have to bother to contribute a truth Table or a test.

I mean, or a chain of steps.

Showing the validity of an argument. Sometimes we can settle the issue by showing that it's invalid. And so now what we're turning to is the proof of invalidity. So this would be the third type of test that we get under propositional argument. Under categorical syllogism, we had tests, three kinds of tests, and under propositional argument.

We have certain kinds of tests. So happens also three in number.

They're a little bit different, though. The first two are tests for validity. Well, the first one is a test for validity or invalidity.

The second one, by rules of deduction.

Proves the validity of an argument. And then the last one, which we're.

Looking at right now, this is what.

I'm introducing, is the proof of invalidity.

All right?

In COPI on page 315, we come to section 9.3. The proof of invalidity copy says if we fail to discover a formal proof.

Of validity for a given argument, this.

Failure does not prove that the argument is invalid and that no such proof can be constructed. It may mean only that we've not tried hard enough. So let's say you have an argument.

In front of you and you say.

I've worked and worked and worked on this thing, and I can't prove it to be valid, therefore it's invalid.

But then maybe somebody who's better in.

Logic or has more endurance or more creativity or is not feeling as tired.

As you or whatever it may be.

Can come up with an argument. You never know that an argument for validity might not be constructed by somebody else. So what would constitute, he says, what would constitute showing that a given proof is invalid? We know how to prove validity, and if we do prove it, then that settles it. But if we don't prove validity, does that mean we've proven invalidity?

Okay.

And he suggests that the method of.

Proving invalidity is related to the truth.

Table method, although it's a great deal shorter. You remember how we on the truth table method show an argument to be an argument form to be invalid. We put down all the various possibilities, and then we go through, after we filled in the case table, and we check for a line where there are true premises and the conclusion is false. If that's one of the possibilities, when all the combinations are considered, then you know you have an invalid argument form.

So if we were to make a big truth table for some of these.

Arguments, we could find out if they were valid or invalid. But what if we want to prove invalidity without doing the Whole table.

Well.

A little bit of creativity and working a puzzle might enable you to construct one line from the table whereby assigning the truth values in a certain way, all the premises come out true and the conclusion comes out false. And so if you do that, you.

Just find that one line playing around.

With different possibilities of assigning truth values, then you will have proven invalidity taken, if you will, a shortcut. And so Kopfie tells us in the.

Middle of page 316, first we ask.

What assignment of truth values is required to make the conclusion false?

Please don't forget that.

Don't start working out the truth values just by saying, okay, arbitrarily.

I'll make this a T, that an F, this a T, that an F.

And then try it.

And they'll say, okay, that didn't work. I'll go back.

No, help yourself out. Start with the conclusion, because you know that if you're trying to prove invalidity.

The only way to do that is.

To have a false conclusion with true premises. Everyone got that?

You start with the conclusion, assign the.

Truth values in such a way that it will be formed false, and then you work backwards to see if you can then make all the premises true. And copy ends the section by reminding.

Us a certain amount of trial and.

Error may be necessary. And I know there are people with different personalities and their minds work different ways, and so I don't mean to universalize this, but for myself I find this the easiest part, because it's just a matter of seeing if you can find out sequences. It's either off on true or false.

On each one of these.

And you know the rules and you know what you have to come out.

With for each line, either truth or false.

And so it's not nearly as difficult as constructing the formal proof of validity on some tough ones.

And of course, it's much easier than.

Having to go through writing out a long truth table. Okay, let's do some exercises and practice proving invalidity. Please remember this is the opposite of what we have been doing. We have been proving validity. But let's say we can't prove that an argument is valid. How would we prove that it's invalid? Our failure to prove it valid does not mean it's proven invalid. The way the proof it invalid is to assign truth values in such a way that all the premises are true.

And the conclusion turns out to be false.

Let me see if I can take a few of these exercises and show.

You how to go step by step in doing them. Let's begin with number two on page.

317, prove the invalidity of each of.

The following by the method of assigning truth values.

Okay, if we look at number two, our goal, assign the truth values in such a way that all the premises will be true, but the conclusion will be false. So let's begin with the conclusion. Actually, before we begin with the conclusion, you might find it convenient to make a list for yourself. Just write across one line the letters.

That appear in the argument.

In this case we have the E, F, G and H. So I'll write E, F, G and H in a line across the top of my paper here. And what I'm going to do is.

I'm going to start writing underneath the.

Truth value that I assign to each one.

And if I have to change it, I'll scratch it out, and so forth.

So where do I begin? The first step in proving invalidity is to make sure that the conclusion turns out false. Here are the conclusion is G. Well, that makes it real easy. So I know that G has to.

Be assigned the value of false.

Everybody with me so far? Since the goal is to make the premises true and the conclusion false, and I see that my conclusion is just the letter G, can I assign G false? We're halfway there, or so it seems. Now then, what can I say about the other letters in this argument such that all the premises will turn out true? Let's work backwards. Premise number three says if H then G. Now, what's my goal? My goal is to make this line true.

I want the whole line H then G true if the G is false, because we know that if true, then.

False is a false.

If we make the implication from a truth to a falsity, then the whole implication is false.

Therefore, if I have an F in.

The second position of the implication, then I have to make sure I have an F in the first position. That is, H would have to be false as well. So now, on my little list keeping track, I can write H as an F automatically. By the way, we know these are invalid, and that's why we can be pretty definite. I know that G is false because of the conclusion having to be false.

And I now know that H has.

To be false in order to keep line 3 true. Now then, if we look at the second line, you notice we have G and H together in the consequence of that long implication we have not E and not F. Therefore, G and H. Now, G and H have already been assigned falses. And that means that the consequent of this implication itself is false. Because G and H are both false, putting them together in a conjunction means that conjunction is false. Now, if that conjunction is false, the only way to keep the whole line true is to have what implies it also be a false. Everyone with me, G and H have to be false according to the assignment already given a truth values. And since that G and H is the consequent of an implication statement, what implies it must itself be false. So that means we have to make what's in parentheses, not E and not f, false. Now, in a compound statement, where you have two propositions conjoined, if you want.

To make the compound or the conjunction.

False, then at least one of them must be false.

Maybe both of them are false.

In the case before us, the letters E and F, however, are negated. So at least one must be false. Maybe both are false means at least one must be true and thus negated.

Or both could be true and negated. But now if we which one shall we choose?

E or F? If we look above in the first line, we have the negation of E and F. We need to make this whole line true.

To make the whole line true, what.

Is in parentheses must be false because it is negated. To make what is in parentheses false, at least one of those compounds must be false. But perhaps both could be false. And so when it comes to E and F, we can assign the value of false to one of them and true to the other one. And the job has been accomplished. And what was that job? To make every premise false and yet the conclusion true. And thus we have proved the invalidity of number two. Let's look at number four. Try to be a little faster with this one. Our job is to make the conclusion false. The conclusion is M. Now again, let's write down what our various letters are that have to be assigned. We have M, N, O, P, Q, and R in this argument. So what assignment of truth values will show this argument to be invalid? The conclusion has to be false. If the argument is invalid and the conclusion is not M, to make not M false, M itself would have to be true. So now we can write down that we have one of the values established. M is true. Now we wish to have all the premises come out to be true as well. And all we know so far is that M is true. But if M is true. Notice line one. If M is true, we cannot have what it implies be false. So there we have M. Then this.

Compound N or O, N or O.

As a compound itself must be true. And if it's going to be true, then one of those two, N or O, @ least must be true. If we should make N false. Just go with me with this. If we should make n false, notice line two becomes no problem because anything is implied by a false statement. So making N false and O true satisfies what we have to do with line one would also satisfy what we need to do with line two. Now, how about Q, R and P? If we look at line four, Not R or P, that is not.

The compound of R or P, in.

Order to make that line true, what stands inside the parentheses must be false. And if what's inside that parentheses is false, then both of them must be false, because we have a disjunction. If one of them were true, it would not be a false disjunction. So in order to make the disjunction of R and P false and thus negated, so the whole line becomes true, R and P both have to be assigned the values of.

False. And when we do.

That, the last thing we have to decide from line three is what to make Q. Since R is known to be false, we cannot end up with Q being true, or else we'd have a true implying a false, and the line would prove to be false. But our goal is to make that line true. Consequently, our only option is to make Q false. So we're done. M is true, N is false, O is true, P is false, Q is false, R is.

False. Want to try another.

One? Okay, let's look at number six. Now we have material equivalence, and so it becomes a little trickier. You have to make sure you have the same value whether it's true or false, the same value on both sides of the stated equivalence. And if you do have the same same value, then the whole line is true. So then how can we begin number six as its conclusion gives B or C. Now, we could begin by arbitrarily using some signs, but here's a better way of doing it. Let's start with the 1, 2, 3 fourth premise, which is simply not A. We know that that premise must be true and the conclusion come out false if we're going to prove invalidity. So to make premise not a true, A has to be false. So we've got that step taken care of. Now, if we go up to the top line one, we know that A is false. So whatever is put on the other side of the material equivalence must itself be false in order for the whole line to be true. Because in a material equivalence, you have the assertion of equal truth. Value and what's equal to the false value is false. Therefore B or C must be false. And yet in a disjunction, if either one is true, the disjunction turns out to be true. Consequently, both the disjuncts must be false in order to make the whole disjunction false and thus make it equivalent to the false A, which we've already assigned. Consequently, we know that B and C are false, making the entire line true. So how are we doing? We have A, B and C all assigned, and that means we're done. Let's just check it out. A, B, and C are all false, it turns out so. On line two we have B, which is a false made equivalent to C, or A, which are both false, making the disjunction false. So you have a false equaling a false. And the line which says there's an equivalence here is true. There is an equivalence between two falses, and then line C, or line three, pardon me, has C, which is false, being equivalent to to a disjunction A and B, which we said are both false. So the disjunction is false. Thus we have a false equivalent to a false, and that is true. To say false is equal false. So we've made every line true and the conclusion false. We've proven it.

Invalid. We having fun.

Yet? Okay, one more. Number eight. This can be done. Let's use the procedure that I've lined out. Let's write down every one of the letters that we find there, which turns out to be K. Through R. We have K, L, M, N, O.

P, Q and R. And we'll begin.

With the conclusion that if we possibly can, and in this case it's very easy to do, because the conclusion is if K, then R. Our goal in proving invalidity is to have a false conclusion with true premises. So how do we make that conclusion? K, then R false? There's only one way. The K has to be true and the R has to be false. That's the only way in material equivalence to have it be false. So now we know that K is true and R is false. Now let's go back up to the top. The first premise says if K, then this compound of L and M, we've already said that K is true. But now if K is true, and our goal is to make that whole line true, then we must make sure that L and M are true are true as well, because you have if A true, then A, and here's the compound L and M that L and M must be true, or else we'd have a false material implication to make L and M both true. Well, if we didn't make them both true, we wouldn't be able to have the compound itself, the conjunction of those two be true. So now we can make L. We can assign in our answer sheet true to L and true to M. We've got K, L, and M, and R. Done. Now, the next line asserts a disjunction between if L then N or not k. One of these will have to be true if the whole line is going to be true. Since we already know that K is true, as we found out by looking at the conclusion, then we know that not K will be a false in line two. But if that is false, that leaves us only if L, then N to be made true in order that the whole disjunct, that whole line be true. We already know that L is true. And so if L, then N must be true. And we have L as a true, N has to be a true as well. And so now N becomes true. We have K, L, M, and N assigned as well as R. Now let's jump down to the fourth line. And the reason I do that is because you have a conjunction there. And a conjunction line can only be true if both conjuncts are true and you've got one letter standing out there with a negation in front of it. So that makes that one easy to assign. That has to be a true. So the negation of Q being true, Q would have to be false. I've got to make that something of the conjunction true. And since Q is negated, Q has to be false so that it comes out true when negated. Now that I know that Q is false, I can put false on the other side of the conjunction where I have not P or Q. And remember, both sides of this conjunction have to be true. And so the not P or Q has to be true. And yet I know Q is false. Therefore it's left to not P to be true. And if not P is true, P itself must be false. So if we have P be false, then not P is true true, or Q, where Q is false comes out true. And then on the other side of the conjunction, not Q means not false, so true. So you have true and true, so that all works. And the only letter we haven't assigned would be O, and we don't need to worry about what O is assigned. I don't think. Okay, well, let's test it then. In line three, we have if O, then This compound P or not N. So if the compound is going to. If we want this line to be true, if we make O false, then it won't make any difference what P or not N turn out to be. But now since we know what N is, N is true, so by being negated it becomes false. In that compound P we already know to be false. So in that compound we have a false or a false. In which case, if O were to be true, you'd have a true implying a false and the whole line would prove to be false. Therefore O must be false. Well, it takes a little while, but if you go step by step because you know the rules and you know what you're looking for, this bi valent system, two valued system, where you have either true or false can pretty much be figured out. Now, of course, if the argument were a valid argument, you never would have found an assignment of truth values to show it to be invalid. This leads then to one last complication, the problem of inconsistency. Section 9.4. There is one form of valid argument that is very disturbing to us, and that's because it has inconsistent premises. What I mean by that is that if in the premises of an argument there is an inconsistency, anything can be validly deduced from those premises. And I'm just going to put up here on the board what you find really in COPI, page 319 in the middle. This simple argument you want to remember this is very powerful for analysis and debate purposes. If you wonder why your opponent is able so easily to come up with an answer to anything you bring up, you might start asking, is he contradicting himself in his premises? Because if he has contradictory premises, he can do anything he wants. He can draw any conclusion he wants and do so validly. Of course, if you have contradictory premises, one of them must be false and so you never have a sound argument. But it can look like you've got a good argument, at least a valid argument. Because let me just use P and Q. Well, no, he doesn't like us to use B and Q. Let's use A and B. If one of my premises is A and my other premise is not A, I'm just going to be real minimal here. This is what I've got as my premises, or at least as part of my premises. You could have all sorts of other things there too. D, E, F, G, H and so forth. But if there's an A and A not A in there somewhere, then notice what I can.

Do. I can Say A or.

B. This is line one, line two. Line one is A. Line two is not A. They're the givens. Now line three is A or B. How do I justify A or B? Addition. Addition to line.

One.

Okay, my next step is to conclude B. And how can I conclude B? Disjunctive syllogism. Disjunctive syllogism applied to lines two and three. Here I have A or B in line two, I have the denial of A. So that leaves me B. You see how simple it is? From contradictory premises in two steps you can prove anything, because B was not part of my premises at all. I can, you know, the moon is made out of green cheese, as Kofi.

Says. I can put anything I want in.

There. And so when you find people, by the way, and you're doing analysis of the thinking of others or of yourself, when you find contradictions in their thinking or your thinking, you need to.

Realize, therefore they can go anywhere they.

Want. They may not realize that that's what gives them the strength to go anywhere they want. But those contradictory or inconsistent premises will allow for any. Will allow for anything to be validly deduced. But now will they prove or establish the truth of the conclusion? No, they will not. Because since they are contradictory, they cannot both be true. Therefore either A or not A will be. And if in the premises you have a false premise, you'll never have a sound argument. That is, the truth of the conclusion will never follow from your premises because your premises are not all themselves true. So as Kofi points out on page 318, 2/3 of the way down, no truth value assignment can make the premises true because they are inconsistent with each other. Their conjunction is self contradictory. He points out then that the truth table in a trivial way will always show an argument within consistent premises to be true. And the reason why it will always show it to be that way is you'll never be able to find a row or a line on the truth table where all the premises are true and the conclusion is false, because you'll never have a line where all the premises are true. But since the truth table method says that it's valid unless you have true premises and a false conclusion, then in a trivial way, an argument with inconsistent premises turns out to be valid. On page 319, he shows you how by addition and disjunctive syllogism, you can prove anything from a statement and its contradiction. Then 2/3 of the way down, page 319, here's the kicker. It cannot possibly be A sound argument. If the premises are inconsistent with each other, they cannot possibly all be true. No conclusion then can be established to be true by an argument with inconsistent premises because its premises cannot possibly be true themselves. He kind of repeats this on the top of page 320. He says we shouldn't call this a paradox, and you don't have to worry about that. He feels that paradox is a technical term and you shouldn't use it in the everyday, ordinary way. But whether you call it a paradox or not, it certainly is a strange situation that inconsistent.

Premises can be.

Used to validly deduce anything, but not prove just anything. If you mean by prove, show that it's true and that. He ends this chapter by saying that this explains why consistency is so highly prized. Inconsistent statements cannot both be true, and so that's why we strive to be consistent. And then secondly, he says any and every conclusion follows logically from inconsistent statements taken as premises. Inconsistent statements are not meaningless. Their trouble is just the.

Opposite. They mean.

Everything. You catch that if somebody says today is Thursday, it's not the case that today is Thursday. You don't say, oh, that's meaningless, it's meaningful. In fact, it means everything. Because from today is Thursday and it's not the case that today is Thursday, you can conclude anything in the way that we've just up on the board. By addition and disjunctive syllogism, you can bring in anything you want from contradictory or inconsistent premises. Alright, for tomorrow we're going to be tackling the worst chapter in the book. I'm going to tell you right up front so you don't get discouraged. Quantification theory becomes a lot more complicated. It's fairly difficult for most people. However, it's a very powerful thing to learn if you want to master symbolic logic. So we'll have you read chapter 10 in COPI and do some exercises and we'll see if we can master this when we come back.