Dr Greg Bahnsen - Critical Thinking: Propositional Symbolic Logic Part 2 (Part 15)

All right, so the two rows, or it might be three or four or five depending on the number of variables.

The two rows to the far left.

Are just the variables. Now we have a set of rows that will represent the premises in the argument.

Let's say that I have a premise.

That is P or Q. That's one premise. I have another premise. And we'll say this other premise is not Q, not Q. And then the conclusion of my argument.

Is.

P. And over here I'm going to put my conclusion. This will be the third section of the truth table. My conclusion here is. Okay, now what I have to do is go back and look at these various rows and calculate the truth value of each one of the compounds or simple state variables. And that's going to be done on the basis of the combinations laid out over here on the left hand side of the truth table. And we just spent some time learning how to do this. This is not hard, by the way. In the case of the conclusion line, in the case of the conclusion line, since I don't have a compound, I just have one of the variables, then I can go through and fill in for each one of these lines what it is on the basis of what's already been assigned. So for line one, it's going to be true. For line two, true. For line three, false. For line four, false. Now, if I had a compound statement here, I would have to do, you know, I'd have to look at this and evaluate. It wouldn't be as automatic as that. For instance, here's a compound thing. Let's go through and do this premise P or Q. Now, given that we have the assignment of truth values true and Q, is.

The compound P or Q true?

Yes. All you need is one to be true, and so that whole compound is true. Now, the Q is true. So look at this next row. If Q is true, is not Q true? No. Then it's false. Let's go down to the next line and consider this possibility of truth assignment. True and false. If I have P or Q that is false or true, is the whole compound true? Yes, but there is at least one true component divided in two. Now. Now, in the third line of this particular table, Q has been assigned a value of true. So what is not Q's value going to be false? Going to be false. That's right. And now we only have one more line to consider. The assignment of false and false to the variables P and Q. Now, if P is false and Q is false, then what's the compound P? Or Q going to. When you false on this particular line.

In the chart, Q is said to.

Be false, what is not Q going to be true? True. Now we have finished the chart, it's completely done. We look at every possibility. I'm going to back up here. We look at the specific form of the argument. We realize that the truth value of the compound, the compounds, we only have one attribute, can be determined by the parts. And now what we have to remember is an invalid argument is one that has true premises and a false conclusion. So the first thing we have to do is to go through that segment of the truth table that lists our premises to see where we have all true premises. Any line in a truth table that has a false premise can be excluded automatically. Because the only thing we're testing is to see if we have true premises and a false conclusion. So we have to have all true premises of these lines on the true table, where do we find all true premises? Which line? I'm sorry, the second blind alone has all true premises and the rest have at least one false premise. So that will help us determine validity or invalidity. In some truth tables, depending on what your premises are, you might have many more than one line. So don't think that in a truth table, you're just looking for one line. You got to check for every line that has all true premises. We only happen to have one there. Now, in this particular case, when they're all true premises, is the conclusion also true? And therefore it is not an invalid argument? Now if we had on a different illustration, put this together in such a way that a false was found in the conclusion row on the line where you have all truths for premises, then you would have had true premises and a false conclusion as a logical possibility. And thus you would have to rule that whole form out, that specific form out as an invalid argument. And does that help everyone understand how.

We use truth tables?

When you ever have it, is it.

Possible to have two true premises and a true conclusion, then further down have.

Two true premises and a false conclusion? Yes, because what you're looking at are different possibilities. So you might have on a truth table, going to the record in the premise rows, let's say we have three possibilities.

You might have a false, a true.

A true, a true, false and true, a true, true and true, a false, false and true, a true, true and true. Okay, now you're going to get different combinations depending on what the compounds are up here. But you might have one line that has but the conclusion of the argument turns out to be true here. But down here, on this combination of different things, the conclusion proves to be false. Now, when you have that because of the different values assigned to the variable, then you know you have an invalid argument. Some forms of the argument will lead to the truth in that conclusion, but not all forms. And therefore the specific form is not reliable. It's not reliable because there's at least one way in which you could have true premises and a false conclusion. The author tells us on page 283 that a given argument can be a substitution instance of several different argument forms. And there is no reason, he says in the middle of the page, why an invalid argument form cannot have a valid argument at the substitution instance, but the specific form of a valid argument must be a valid argument form. On the other hand, a valid argument form can have only valid arguments as substitution instances. Essentially, what he's telling us here is that in terms of argument forms, if you have an invalid one, you might have. This is, in a sense, what I was just explaining to Pete. You might have some arguments that are valid arguments, but in other cases they are not. However, a valid argument form has only valid substitution instances. So to prove that a given argument is invalid, we must prove that the specific form of that argument is invalid. That, by the way, explains why when I taught you the truth tables in the Dr. Bonson way, I began by saying, remember what invalid means. It will not be enough to test for validity in the sense that you only look at the argument possibilities where the conclusion is true. You must look for those where you have true premises and a false conclusion that will tell you whether it's invalid. To prove that a given argument is invalid, we must prove that the specific form of that argument is invalid. Let's do some exercises just to catch on to this on page 285. Roman numeral 2 copy says, use truth tables to prove the validity or invalidity of each of the argument forms. In group B above, we have to go back to Roman numeral 1 and look at the group B argument forms. He says, use truth tables to prove the validity or invalidity of these. And let's begin with number one. Okay, let me walk you through number one. Just as a way of refreshing what we just learned about truth tables. Number one has as a premise, if P, then Q, it has as a conclusion. Therefore, if not Q then not P. First thing we have to do with the truth table, we have to get our variables. In this case, we only have two of them, P and Q, and how many possibilities we're going to be two to the nth power. Two to the seventh power can be four. True and true, True and false, false and true, false and false. So those are my variables. Now, how many premises do I have in this article?

Just one.

And that premise says if P, then Q. And then the conclusion is this compound statement. If not Q, then not P. Now I just have to figure out how to fill in the chart for the various possibilities. If P then Q, where I have if true, then true is true. If P and Q when I have true and false is false. Everybody with me so far. That's what material implication is all about. Okay, If P, then Q when I have a false antecedent is true. When I have a false antecedent on the last line here, here is true. Okay, now let's fill in the truth value of the conclusion, which is a compound statement that there's not Q. If not Q then not Q. Okay, on the first line, Q is true, so not Q becomes false. We know that a false statement applies anything, so it's automatic because the background is true. On the second line, Q is false, so the not Q becomes true. If true, then not. Key. The P on this line is assigned true, not P becomes false. Therefore, looking at the whole thing, you have if true, then false, that's false. Now in line three, he was taken as true and not. Q therefore becomes false. And anything that is false implies anything. So we can automatically put that down as a true statement. And then in line 4, Q is false, not Q becomes true. T is assigned the value of false, not Q therefore becomes true. And so what you have here is a true implying a true. And that is true. Now that we've got the chart filled in, all we have to do is look for all those lines that have true premises. In this case, there's only one premise. So pretty easy. We go down to this column here and we look at this, this and this, lines 1, 3 and 4. Now, do we have any that have true promotion premises? Any lines, true premise, any false conclusion? Here, line one, the conclusion is true as well as the premise being true. In line three, the premise is true, the conclusion is true. In line 4, the conclusion is true and the premise is true. And so we don't ever have the combination of true premise and false conclusion. And thus we conclude this is a valid argument form. Everyone follow that. Okay then, Melissa, let's have you walk through the next one. Number three.

Okay? On the board we've written down the different truth combinations for the variables seen here.

So we have four lines on this truth table.

And so that it's including this C. Therefore it can just look at a satisfying. Write down the conclusion from that.

Yeah, each line with respect to the conclusion of interviews. Because that's just one of the variables itself. And of course, we have to write it down in the same order that's found over here, don't we? Okay.

And the primal is pnc. So if we know P is true, if you know P is true and Q is true, then first time is true. At least one. Could be at least one.

No, that is a disjunction. In the case of an and statement conjunction, both have to be true in order for the compound to be true.

Right. So in the second combination, what he was doing through the planet and the third where he was fall from two anti happy false because the happy two trees.

That's right. So now we have filled in the two table completely. And the next step is to look for the row or rows that have only two premises. The first row, that's the only one. Now, is the conclusion true or false on that row. So you can have two premises and a true conclusion. That's the only possibility. When you have true premises, therefore the argument is valid. Very good. I think we're getting the hang of this. And I don't want to spend a great deal of class time doing what should become fairly mechanical. So let's just try one more. All right, we'll do number 11 and we'll have Jay walk us through it.

Okay, we got three variables, so we need eight columns.

Very good.

And then I just went through and I figured out if P is in Q and I filled out the table for all eight columns, then I do the same thing for if P then R, the same thing for if Q, not if Q, Q or R. And in doing that, I found that on the very last column, very last line, you'll have two, three premises and a false conclusion.

Okay, let's test that and see if you're right. Let's pretend that I didn't stay up late at night and work out all of those eight rows. But I just kind of sat there and I played it for a minute and I said, well, now what if K2 and R are. Were all false? Let's just take that as the assignment. And that's what you have in your last line. Now, if P then Q, if false, then false would turn out to be a true compound statement. You have that on your line there. Now the next premise is if P then R, that Is if a false, then a false. Well, that's true as a compound statement. So both of the premises are true. Now let's look at the conclusion Q or R, where both are false. Is it false or false? At least one has to be true so that compound statement becomes false. Consequently, we have a line on the chart where every premise is true but the conclusion is false. And that is sufficient to prove that invalidity. Any argument of this form that we find in number 11 will prove to be an invalid argument. Follow that. Very well. All right, we're going to move ahead and learn some more about symbolic Logic. In Section 8.5 of COPI, beginning on page 286, Section 1 talks about statement forms and statements, but I'm not going to bother with it because, as he says, we now make explicit a notion tacitly assumed in the preceding section. That can be a little confusing. You think? Well, I'm supposed to be learning something new. I mean, what's this all about? Well, in a sense, you already understood this, that you did what you were required to do and understand in the previous section. So now let's go on to the notion of topology. Contradictory and contingent statement form. Section 2 on page 287. In the middle of the page, Copie introduces a distinction between statements whose truth can be discovered only by observation, that is, only by empirical study, and statements whose truth can be known simply by their logical necessity. If I say Washington was born in Spain, George Washington was born in Spain, there's nothing you can do to tell if that's true or false except empirical study of history, the facts of history. But if I say it's either the case that Washington was born in Spain or he was not born in Spain, doesn't make any difference what you know or don't know about history, that's logical true, one or the other of those is going to be true. Now then, he gives a definition of tautology that you can understand in light of what we have just learned. He says a statement form that has only true substitution instances is called a tautologist statement form or a tautology. A tautology is a statement form where all the substitution instances are true instances, and it gives you a little truth table there to show you in the case of P or not P, no matter how you assign the truth values to P and not P, the combination P or not P is always going to be true. So where you have a truth table in that line has only truth, or that row, I should say that row has only true in it, then you know you're dealing with a tautology. Okay, now on the top of page 288 it says a statement form that has only false substitution instances. That's the opposite of what we just learned. One where you have only Fs in that row is said to be self contradictory or a contradiction and is logically false. So if you have all truths, you have a tautology. If you have all falses, you have a self contradiction. Then the next paragraph says statement forms that have both true and false statements among their substitution instances are called contingent statements. They might be true and they might be false. They're all false, they're self contradictory, they're all true, they're tautological. And if they're true and false, then they're contingent. That shouldn't be all that difficult. Any problem? Then what do we mean when we say, when we use the expression material equivalence? In the middle of page 288, Kofi says two statements are said to be materially equivalent. Equivalent, or I think it's easier to remember this expression equivalent in truth value. Two statements are said to be materially equivalent when they are either both true or both false. When they are true together or false together, they are materially. Okay, so he gives a a truth table here for P and Q. And he said look at the lines where P is true and Q is true. And you notice that P and we're using the triple line equation, material equivalent P material equivalent Q is true as well. Where both P and Q are false. P material equivalent Q is true. And the reason for that is the material equivalent sign simply means you have the same value on both sides of it. And if you have the same value, then the compound is itself true. Have I lost anybody? This asserts an equivalence. And so they either are equivalent or they aren't. If you have true and a false, they're not equivalent. So then the total, the compound P material equivalent Q would not be true. Whenever two statements are materially equivalent, they materially imply each other. And so we can read the material equivalence sign, the three line equation there as if and only if and in logic that's called a biconditional. Both sides imply the other if and only if amounts to if P then Q, and if Q then P. So now we know what a topology is, a self contradiction and a contingent statement. We also know what material equivalence between two statements amounts to. How is that different from logical equivalence? The opposite, at the very bottom of the page. Two statements are logically equivalent when the statement of their material equivalence is a Tautology. This is a recursive definition. You've already learned what material equivalence is. You just learned what psychology is. And so when the statement of a material equivalence is a tautology, then you have a logical truth. So look at the truth table down below. He has P and not P and not not P. And then he puts down the statement of the material equivalence of P and not not P. That's the last row in this truth table. Now is the statement of that material equivalence. The material equivalence is the three line equivalent sign. Is the statement of that material equivalence a tautology? Well, now you say, well, Dr. Bonson, a tautology is a statement where all of the values in the truth table for that row are. And you look at this and say, well, all of them are for this one. So that material equivalence, that statement of material equivalence is a tautology. So we now have a new term for this P material equivalent not, not P. We call that a logical truth. A logical truth. A logical truth is one where the statement of a material equivalence is pathological. The difference between logical equivalence on the one hand and material equivalence on the other hand is very great and very important. He says logically equivalent statements have the same meaning and may be substituted for one another. Take the example above. He stated a logical equivalence between P and not not P. He says not, not P has the very same meaning as P. Therefore, any place you find P you could put in not not P. That's because they are logically equivalent. Material equivalence, however, is simply that they merely happen to have the same truth value. Things are materially equivalent if they have the same same truth value. That does not mean they can be substituted for one another. Go Back to section 3 on page 288 where material equivalence was introduced. Here we have the statement that P is materially equivalent to Q. Now, do they have the same truth value? Yes. In the lines where it's true they have the same truth value, does that mean you could put Q any place you find P? No. That should be rather intuitive to you. However, if P is materially equivalent to not not T, or is logically equivalent so that there's only true values in that row, then you could substitute them anywhere. But because in material equivalence you don't have all truths in the row, that is you don't have a logical truth or logical equivalence, you can't automatically substitute one for the other. Is that clear enough? So we have material equivalence. Well, we have topology, material equivalence and then logical equivalence. And now De Morgan's theorem two logical equal equivalences which are particularly important express the interrelations among conjunction, disjunction and negation. That is going to tell you something a little bit more about these elementary notions that we've been introduced to today. The interrelations between and, or and the gaseous not. And what De Morgan's theorems give us are tautological biconditionals. Tautological biconditionals. Now, since in a logical equivalence you can substitute one for the other anywhere because they have the same meaning, De Morgan's theorems are therefore helpful because they give us something that can be substituted, a form that can be substituted for another form because they are logically equivalent. The statement of Demorgan's theorems is found on the page of top of page 290. And what it says is the negation of the disjunction of two statements is logically equivalent to the conjunction of the negations of the two statements. So let's just go slowly here. So you see how this works. If you have the disjunction of two statements, that's going to be P or Q. We're dealing here with the negation of P or Q. He says the negation of the disjunction is logically equivalent. Notice I use the term three lines for logical equivalence. Well, it's a material equivalent that is tautological. The negation of the disjunction of two statements is logically equivalent to the conjunction of the negation of the two statements. So let's negate the two statements not T conjoined with not Q. Okay? Now because we understand what logical equivalence is, whenever you see something, two things on the sides of this triple line of equivalence, meaning logical equivalence, this right here, the not parenthesis P or Q can be substituted anywhere for the not P and not Q and vice versa. They are logically equivalent statements. And then he says, then there's another aspect of this, the negation of the conjunction of two statements. The negation of the conjunction of two statements. So here we have P and Q being negation is logically equivalent to the disjunction of the negation of the two statements. That is the same as saying not P or not Q. And then in section six, what we have is a couple of different ways to abbreviate if P then Q. We know the meaning of if P then Q. That if we know what material equivalence means. It turns out that if P then Q is the same as saying you don't have a case. That is, you negate the following P and not Q. If you Had P and not Q, in the case of if P, then Q, you'd have a true premise and a false conclusion. And so we don't want that. We negate then that combination. So then let me state this in notation for those who are listening by tape. Not parentheses, P and not Q. Parentheses is logically equivalent, the three line equivalent to if P and Q. I think everyone can see that, can't they? That is just a statement of the meaning of if P and Q. You cannot have a situation where P is Q true and Q is false. Now that can be simplified even further as you have either not P or Q logically equivalent to if P, then Q. Now, I've already taught you this because I gave you a little shortcut for testing. If P, then Q compounds for their truth value. I said if P is false, if P is false, you know anything follows from it. So the compound will be true. And that's what you have here. When you have the negation of P in the formula or in the compound, not P or Q, and there's one of the possibilities. If P is false, then you know that if P, then Q will be true. On the other hand, if P is true, then it will have to be the case that Q is true. So either P is false or Q will have to be true. So now what you have here are two different ways of noting this statement. If P then Q, you all understand logical equivalence. Then anywhere in an assignment or a problem where you find if P then Q, you may write in its place not P or Q, or you may write in its place not the conjunction of P and not Q. These are all equivalent expressions and therefore are some substitutable for each other. All right, let's look at some exercises to put this new information to test on page 291, Roman numeral 1. Copie says, for each statement in the left hand column, indicate which, if any, of the statement forms in the right hand column have the given statement as a substitution instance and indicate what you're venting is the specific form of the given statement. Now it will be helpful so that we don't get confused, which is easy for me to do here. If we remember that a variable that is one of those lowercase letters can stand for a compound statement. So if you have a statement like if A, then B, the sort of thing that might be bracketed as a unit, that compound can take a single variable as its representation. But you can't go the other direction. You cannot take a simple statement that is just An A or a B or something and have a compound of variables represent it, okay? So anything in your statement which can be bracketed together even though it's a compound can take a single variable, but you cannot move the other direction. You cannot take a simple statement and turn it into a compound of variables. Now, if you will follow that rule, I think we can make sense of these exercises and the way they're to be done. So let's begin with number one. We have the statement A or B of the variable or of the statement variables. We find in the right hand column which correspond to A or B.

We'll have C, which is the specific form, number one.

Yes, C does correspond to it, and it does so specifically, it is the specific form. Now, what if we were to make the mistake of being tempted to think, for instance, that F is also a statement form for number one? The reason we'd be tempted is we say, well, there the operative grammar or the thing that is prominent is the. Or you have if P, then Q or R and S. However, when you have P, Q, R and S, you have to remember you cannot have that many variables standing for only a simple statement or two. And therefore this does not prove to be an argument form of which A or B is a substitution. It's not a statement form of which A or B would be a substitution. The key here can be found perhaps on page 286, where when he defines a statement form, he says it's a sequence of symbols containing statement variables such that when statements are substituted for the statement variables, the result is a statement. But here's this qualifying expression. The same statement being substituted for the same statement variable throughout. The same statement substituted for the the same statement variable throughout. So back on page 291, if we look at F, here we have statement variables, but that doesn't correspond to A or B because you would not have the same statement substituted for the same variable throughout. You wouldn't have a one for one correspondence. Okay, let's try another one of these. Okay, let's try number two, where we have a statement C and not D. Go ahead.

That would correspond with letter D as the specific form.

B as in boy? No, D as in dog, D as in dog. That is the specific form.

And could A be the substitute substitution instance?

Yes, I believe that A could be a substitution instance where Q takes not D all the way through, and P would be the same as C. So it has the same form even though it's not the specific form, because the negation is not there. I'm not claiming that Cofie is altogether clear on this statement and statement form stuff, so I'll be a little tentative about it. Number three, not E, then parentheses, F and G.

Take the same. We can use B as well as.

C. How could we use C?

Because if we take E, if the opposite of E, then quantity of F and G and we use the definition of material implications. This was asking earlier whether we could do that.

Well, he's not. He's only asked, asking us to do the elementary statement and statement form.

Okay, then I would just use D, where P stands for the opposite of E and G stands for F and.

G. And it would be a substitution form. Okay, Now if we're not allowed to use the negation of a statement to put a variable for that negation, then there wouldn't be any of these forms that fit. And if we are allowed to do that, then the fact is we still don't have the specific form in the case of number three found anywhere in the column on the right hand side of the page. Okay, let's try one more. Number four. If H, then bracketed I and J, which of the statement variables on the right hand side corresponds to that?

Does that just substitute the instance or the specific form? Then maybe the specific form is if a compound variable can be represented by a single one.

All right, I follow your reasoning. Is there a specific form found? Okay, let's go to Roman numeral 2. I think this one in Roman numeral 1 is taking too much time. It's not all that important. I think what you do learn here is that a bracketed statement can have a variable stand for it, but you can't. You can't have a compound variable represent a simple statement. Number two, Use truth tables to characterize the following statement forms as topological tautologists, self contradictory or contingent. And let's just take a couple of them. We'll begin with number one. Where we have square bracket P, then parentheses, if P then Q, square brackets then Q. We have only two variables here, P and Q. And it shouldn't be all that difficult to write the truth table up on the board. So let me just turn to A. All right, we have our rows up on the board. Now for P and Q, we have the various combinations of truth values. Taking the innermost piece here, if P then Q, you have the various possibilities. Let's examine that. If true, then true is true. If true, then false becomes false. If false and true becomes true. And if false, then false is a true statement. Now we have to consider. If P, then this combination is P, then Q, So P taken as true and the combination is true would be a true implication. P taken is true, and the compound being false would be a false implication. So far, so good. P taken as false has to be true. P taken as false has to be true. Very good. Now we just have one more step. That's to test the total statement. And in the case of the total statement, we have a true implying a false. That is the long if P, then if P, then Q is true and Q is Q, which is false. So that means it is true. In the third line we have if P, then is. If P, then Q is true, implying Q which is true. So that's true. And then finally, in the last line, we have if P, then if P, then Q being a true statement, implying Q which is a false statement, so it is false. So you filled in the truth table correctly. Why did you call that a contingent statement? Because in the conclusion, we have both.

True and false, true and false statements. If it were to be completely we.

Had all true answers, we would have A. If you. If you had all trues, it would be tautological. Tautological. And if you had all falses, you would have self contradiction. Self contradiction. All right, let's do another one here. Let's take number four. If P, then bracket, not P, then parenthesis Q or not Q.

Want this one on the board?

Yeah, let's put it on the board.

Want me to explain it? Okay, we only have two variables here, P and Q. So I wrote all my true values of P and Q. Then we also have not P and not Q. So in my third and fourth column.

I wrote down those two values and they are correct. I can verify that they're the opposite of those assigned values and P and Q rows.

Now, our fifth column is Q or not Q. And if we look at our second and fourth column, one is true, one is false, therefore that is true.

And.

If we follow that down, they all end up being true. And in our sixth column, we have negative P or if negative P, then Q or not Q, which if we look at our third and fifth column.

Let me just note that what you're testing here amounts to the bracketed portion of the statement that we're finally going to be considering. Okay, go ahead.

Then if we look at our third and fifth column, we see that.

One.

Is false and then you have a.

False implying a true. Right?

And that will allow you to be true.

Okay?

So the answer no, the consequent is always true.

So from there, yeah, since, since the consequent is always true, then anything will apply it, right? So everything in that world will be true.

And then that is the consequence of.

The total statement, the total statement which will always.

And the consequence will always be true.

In the total statement. Therefore everything will be true. So every row turns out to be true when we analyze the problem, because the consequence of the if then statement is always true, as we saw in the row just preceding it. Now, what do we say when every entry in the truth table is true? It's a tautologist. Tautologist. Say, what if he had had some truths and false?

Then it would have been a contingent statement.

And what if you would have had all falses?

It would have been contradictory.

That's right. Let's go to Roman numeral 3 and do a couple of these. Use truth tables to decide which of the following biconditionals are tautologies. Okay, let's remember that a biconditional is the same as material equivalence. And our definitions of a logical equivalence is when the material equivalence is a topology. So number three is asking us to determine which of these are logically equivalent. You all understand that? And we're going to do number two here, which is parentheses. If P then Q being logically, excuse me, materially equivalent to parentheses, not P then not Q. And would you please put that on the board? All right, number three, number two.

Okay, I took variables P and Q.

And you have all the combinations, truth values there, okay?

Four of them. And got the truth value of if P is in Q, then I took the combinations for not P and not.

Q, there's four of those.

And got the truth values for not P and not Q. And then I.

Now wait a minute. You got the truth values for not P then not Q. Okay, Correct.

Excuse me, getting ahead of myself. And then I took those two if P then Q, and it's not P than not Q. And I compared those two and I found that the middle two columns. Middle two columns, lines.

Yes, middle two lines were false.

Therefore, it's not a pathology. It would be a contingent statement.

Now, the way in which you test a statement of material equivalence is that it is true when you have an equivalent value. So what's before being stated as equivalent is the if P then Q which is in this second row here or column. Thank you. In this third column we have if P then Q. Now that's stated to be equivalent to the if not P then not Q, which is your sixth column. So every place where we have a corresponding truth value, the Statement of by condition, by conditionality or the statement of equivalence is true every place where we don't have corresponding values and it's not. So if we go through looking at them, you have true and true, which is going to then be as a biconditional statement true, you have false and true. And so that is not equivalent. False is not equal to true. And so you have to put false down here under the statement of equivalence. Do you follow that? The reason why this is false is because this asserts that there's an equivalence between these two things here. But when you compare those two things on the truth table, you find a line where one is false and one is true. So they are not equivalent. On that line you have to put down the statement of equivalence is false. The statement of equivalence is false on the third line for the same reason, whereas the statement of equivalence is true on the fourth line, because on that line the two parts of the biconditional are equivalent truth value. So this has been done correctly. That is not a logical truth. Then as a contingent statement, is it a self contradiction? No. Okay, let's do one more before we call it a day here. Number three, I've written up on the board the eight different combinations of truth values for three variables P, Q and R. Now one portion of the statement that we are testing here.

It says.

If P then Q. And so it's easy to determine the truth value of that compound by checking against our columns for P and Q. And so we have if true, then true is true. If true, then true will be a true statement. If true, then false is false. If true, then false is false. If false, then true is true. And all the rest having false antecedents would mean that the compound will be true. And now we have to test that compound as the antecedent in a further compound statement. If P then Q, then R. So now we're going to test what amounts to the fourth column and the third column. So if we have if true, then true. If true, then false is false. If false, then true. True is true. If false, then false is true. If true, then true is true. If true, then false is false. If true, then true is true. If true, then false is false. Going through the eight different rows now we have another part of our statement that's being analyzed that says if Q then P. So we have to go through and notice that inference pattern working backwards here. If true, then true, be true. If true, then true is true. If false, then true is true. If false, then true is true. If true, then false is False. If true, then false is false. And then we have the last two lines are false antecedents. And so we know that the compound will be true. Then one more part of this biconditional has to be put together. We have the if Q, then P as an antecedent to the consequent R. So we're looking at this row, the if Q then P row to get one set of truth values. And they are said to imply the set over here in column R that to the variable R. So we have if true, then true is true. If true, then false false. Line 3 if true, then true is true. Line 4 if true, then false. If compound becomes false. Line 5 if false and true is true. Line 6 if false and false is a true compound. If true, then true is a true. And then finally it's true than false, a false compound. Now the only thing that remains for us to do is to check the biconditional itself. Number three says if P, then Q as one compound. Therefore R is equivalent to if Q, then P, then R. So we'll take the the columns that have these two. The first one, if P, then Q, then R is right here, which is the fifth column, and then the last one, if Q, then P, then R is the seven. And since what we are testing is the statement of equivalence between 5 and 7 columns 5 and 7, we have to go through and check to see if they are equivalent. They are equivalent in line one, they are equivalent in line two, they are equivalent in line three, they are not equivalent in line four, they are equivalent in line five, they are not equivalent in line six. They, they are equivalent in line seven, And they are equivalent in line eight. So we do have two lines where this biconditional, this assertion of an equivalence is false. Consequently, we do not have a logical truth. Here we have a contingent statement. Have any questions about how we've done that? And let's finish this chapter by taking Note of section 8.7. We've already looked at 8.6. These two pages, pages 293 and 294, have some value for us in terms of the history of logic and also an analysis of what logic is all about. Sometimes you'll have people say, well, the most important thing about logic are the three laws of thought, the law of identity, the law of contradiction, and the law of excluded middle. In fact, every once in a while, when I go out and I talk about how the Christian worldview is necessary for laws of logic and have people who do not understand what that claim is all about, will try to get into, well, what laws of logic are you talking about? Then I'll say, well, any laws of logic for their desire be laws of logic. You need the Christian worldview. But then they'll say, okay, well, now the most important laws of logic are. And then they'll say identity, contradiction, and excluded middle. It's taken for granted by many people, but that's really what logic is all about. That's what it comes down to. These three most primitive laws of logic. Your author now addresses that he could have excluded these two pages and just gone on and finished his book. But they're always going to be people who say, well, what happens to those traditions? Why aren't they discussed in the book? He says those who have defined logic as the science of the laws of thought have often gone on to assert that there are exactly three fundamental or basic laws of thought necessary and sufficient for thinking to follow if it were to be correct. If the principle of identity says if any statement is true, then it is true. The principle of contradiction asserts that no statement can be both true and false. Principle of excluded middle asserts that any statement is either true or false. And then he gives the way in which that could be stated. In notation. The principle of identity comes down to if P, then P, the principle of contradiction is P and not P is false. Or if you will, the negation of P and not p. And then the principle of excluded middle is P or not, not pee. Now, in the history of thought, in the history of philosophy, objections have been made to these principles from time to time. But your author says he doesn't think these objections are all that strong. My personal feeling is he makes too quick a dismissal of these objections, especially with respect to excluded middle. But that's okay, we'll go on. There are debates of rage through the history of philosophy, even about these three most fundamental laws of thought. He says that you can rescue the laws of thought, however, if you make certain qualifications. For instance, challenges to the law of identity. A is a. That say, well, now wait a minute. Things are always changing, so you can't say A is A. Your office is both. Statements whose truth values change with time are elliptical, or, if you will, incomplete formulations of propositions that do not change. Instead of saying A is A, you could fully carry this out by saying A at T1 is A at T1, where T stands for some temporal designation with respect to the principle of contradiction, which is, I think, the easiest to defend of the three. Nevertheless, Hegelians, general semanticist and Marxist have attacked it saying that there are conflicting forces in the real world. And what your author says here, well, we shouldn't call conflicting forces contradictions. But then, of course, within a certain person's worldview, they may choose to use the term contradiction and try to draw certain analogies to logic and all that. Your author simply says, and my system of semantics and logic, I have a way of dealing with that. I don't call those contradictions. Then finally, the principle of excluded middle, page 294, has been the object of more attacks than either of the other principles. It has been urged that its acceptance leads to a too valued orientation, and that certainly is true. It does lead to a two valued orientation. And much of what we talk about in life is not too valued. Is it raining or is it not raining? Well, sometimes it's misting. So which is it, rain or not rain? An example that is common that he gives is that something is white or it's black. He says, although the statement this is black cannot be jointly true along with the statement this is white, one is not the denial or contradictory of the other. In common parlance, you would think black and white are the contradictions of each other. But your author correctly says, in a sense, white is the opposite of not white, and not white could be any number of things. Black, green, red, and so forth. So he says, if we had completely unambiguous and perfectly precise terms, then you could use the principle of excluded metal, that it would be perfectly true. Of course, that all depends on having a way to have exactly precise terms that everyone will agree to. So your author starts off by saying, these three laws of thought have been criticized, but in every case I think they could be defended. I think we could find a way to rescue those three laws of thought. However, in the middle of page 294, although the three laws principles are true, it may be doubted that they have the privileged and fundamental status traditionally assigned them. As a matter of fact, he says, there are other laws of thought which are much more fruitful for purposes of deduction, and hence more important than the three laws of thought. In the last paragraph on the page, he says that the tautologies that he's going to be using to guide the method of deduction are actually more powerful, more efficient, more fruitful, however you want to put it, and therefore more important than the three laws of thought. In a certain sense, you could say, you know, with the three laws of thought, you can't go any further than that. You can't develop a logic from the three laws of Thought because you don't have any and, or if then combinations or definitions from the three laws of thought. And yet, as Copy points out, this is the last line of the paragraph before the last one on the page. He says the three laws of thought can be regarded as the basic principles governing the construction of truth tables. It is certainly the case that you couldn't construct a truth table like we have up here if T didn't stand for T throughout F. Stanford. That's the principle of identity. You couldn't figure out how to do the rows if you didn't assume excluded middle and the contradiction is not to be acceptable. So in a sense, he says the three laws of thought are valuable because they are used when we construct the truth table. But if you don't use the truth table and get definitions for and, or if they, then you're not going to have a very powerful logical calculus at all. Okay, as we conclude chapter eight, what we have found is that we can test the validity of argument and symbolic logic by the truth tables. That is one method of testing. Let's go back to categorical syllogism. Do you remember after we learned what categorical syllogisms were, we said there are various ways that testing them. One, by logical analogy, two by Venn diagrams, which we didn't learn, and three, by certain rules. Now we're dealing not with categorical syllogism, but with propositional logic and, or if then. And he says you now learned method one for testing them, the truth tables. But at the beginning of chapter nine, he says the truth tables in practice grow unwieldy as the number of component statements increases. You notice that even as we went through the very simple exercises, now we've got a whole board full, and this is not the most complicated you could have. You could have other brackets and compound statements. So in practice, the truth tables become very unwieldy. A more efficient method of establishing the validity of an extended argument is to deduce its conclusion from its premises by a sequence of elementary arguments, each of which is known to be valid. Back under categorical syllogisms, go back a couple of days. Here we found that we could test the validity of a categorical syllogism by a Venn diagram, although we didn't take the time to learn that. But the Venn diagrams become rather complicated. And so what we did is we learned certain rules for testing syllogisms in a similar way. Analogous to this, we're now going to learn certain rules of deduction by which we can test the validity of an argument without having to go through the unwieldy process of constructing a truth table. And that's what Chapter nine is all about then turning from truth table testing of validity to the method of deduction as a way of testing validity or in invalidity. And so I'd like you to be prepared for Chapter nine tomorrow as we meet again.