Dr Greg Bahnsen - Critical Thinking: Propositional Symbolic Logic Part 1 (Part 14)

Okay, let's begin our class this morning with a word of prayer. Lord, we ask that you would sustain us today, both physically and mentally as we take on this new chapter in our book. Would you pray that we would not lose sight of the fact that we studied these things for your glory? We pray that you would forgive us for the ways in which we fall short of that glory, the ways in.

Which we don't show that we love.

You with all of our mind, that you would forgive us for the sake of Christ our Savior, indeed encourage us and give us grateful hearts that you have shown your grace and mercy to us and that we would, for the sake of Christ this day, live in.

A way that would please you and.

Be more obedient to your word. We do ask for the strength of your spirit to that end and also for the blessing of your hand and all of our daily needs. In Jesus name, amen. Well, now we get into the algebraic part of logic. It probably would be good for encouragement, if nothing else, that I remind you where we are in the course, this course on critical thinking. First of all, explain to you what an argument was and dealt with some of the theology of argumentation. Can't do critical thinking if you don't know argument, analysis, premises, conclusions, so forth. Secondly, we began a consideration which we are still in, of what analysis is analysis of language, understanding its multiple functions, how it works, analysis of fallacies in ordinary language. And now we're in the midst of a discussion of formal reasoning. And in formal reasoning or deductive logic, we consider first of all the relationship of classes to each other. So I'm just going to put on the board this morning class relationship logic in your book called categorical syllogism. All some not. Secondly, we consider proposition relation logic.

And.

That'S what we're in the midst of studying right now. Now, this proposition relation logic begins with our ability to use and, or if then types of arguments in ordinary language, which is what we saw at the end of chapter 7 in disjunctive syllogism, modus ponens, modus tollens. Okay, so we see this in ordinary language. And I said that's the most important part of this section of the class is remembering those particular arguments and how they are used in ordinary language. However we want to become, we want a calculus by which we can do even more than test arguments in the simple modus ponens, modus tollens and disjunctive syllogism forms. And so our proposition relation logic, the logic of and, or, or if then our proposition, relation logic will become much stronger if we use a formal notation or what I'll call an algebra for it being able to use variables. So we strip away all the material content and just look at the form or relationship of things. That's what chapter eight is all about, symbolic logic. But please remember, as difficult as chapter 8, 9 and 10 will be in comparison, because we're not used to a formal language and we have to learn the rules of the game, as it were, as difficult as 8, 9 and 10 will be, they're just part of the whole process of critical thinking. One of the things you have to look at when you're doing critical thinking is formal validity in a person's reasoning. Sometimes you can test that in an ordinary language when you have such simple things as modus tollens or disjunctive syllogism. Other times you need to translate into a kind of notation and apply a calculus to that notation. That will help you figure out validity or invalidity. And it's good training for the mind. And it's sometimes necessary later on when you get involved in critical thinking, argumentation, analysis for you to do this. But you're not going to do a lot of it. My guess is not many people are going to do a lot of symbolic logic. However, when the time comes to do it, you have to know what it requires. So that's why we're going to go through this. It's just part of the course. We still have to get to explanation. That's Roman numeral 3, causal analysis, presuppositional analysis, and finally debate. Okay, now back into the forest. Here let's look at the trees. More particularly, symbolic logic. Page 253. In Kopi, he speaks of the value of special symbols. First of all, he says arguments presented in any natural language are often difficult to appraise because of the vague and equivocal nature of the words used, the ambiguity of their construction, misleading idioms. They may contain, their potentially confusing metaphorical style and the distraction due to whatever emotive significance they may express. Now, you've seen all that. We've been looking at this in the course. Natural languages have those kinds of problems when you're trying to evaluate formality validity because you have to understand what the person is saying. There's so many pitfalls along the way. So what the author says is we'll find it convenient to set up an artificial symbolic language that doesn't have the metaphors and the idioms and the ambiguity and other sorts of things. Or the emotive punch that sometimes gets in the way of considering an argument. The next paragraph reminds us this is not an altogether modern thing. Aristotle and others have had a notation, a calculus to use, but in the modern world it has become much more powerful. The way in which we write our logical notation and the way in which we can test syllogisms or arguments. He says in page 254. Another value to these special symbols is the aid they give in actual use and manipulation of statements and arguments. And he uses a good analogy here. You know how difficult it would be to learn the times tables or to do multiplication if you use Roman numerals? You can hardly imagine doing so because you've learned it in terms of a certain kind of notice Arabic numerals rather than Roman numerals. So in the same way, the notation that we use today is a great improvement and will help draw inferences and appraise arguments. And here's a paragraph for the sake of understanding the nature of logic that I want you to pay attention to right at the end of 8.1 in the middle of page 254. He says, from this point of view, paradoxically enough, logic is not concerned with developing our powers of thought, but with developing techniques that permit us to accomplish some tasks without having to think so much. Now, you may be chuckling because you said, well, I'm thinking awfully hard trying to do, you know, these homework problems. That's true, but that's because you're. It's like learning German. Do you think children who grew up in a German home in Germany, speaking German, think of German as difficult? No, it's natural. But those who, from outside learning the notation, learning the grammar, learning the rules, we may find it difficult. You're finding it difficult learning the logical stuff. But the point is, if you knew all this already, you could turn off your thinking process, because it becomes almost a mathematical process of going through the various possibilities to prove validity or invalidity. And so he says, it's ironic that this turns out to be something that does not develop our powers of thinking, but encourages us to not have to think so hard. The same way that a computer helps you, right computer will make it easier for you to do certain tasks, but will also take away some of the training that comes in doing the math or even spell checking or whatever it may be. Okay, so now let's start learning the language 8.2. The symbols for conjunction, negation and disjunction. He tells us on the top of page 255. A simple statement is one that doesn't contain any other statement as a component. A compound statement does contain another statement as a component. Okay, so when you connect two propositions, two statements, they become compound, or the resulting linguistic unit is a compound. And then in the next paragraph, he gives you a point about English, that you might accidentally have a statement in English by taking a broken part of a sentence. The example that he gives is, the man who shot Lincoln was an actor. Well, in terms of English statement development, the end of that Lincoln was an actor is a statement. It just accidentally turns out to be the case that could be a statement in the English language. And so he gives us a rule so we can eliminate such things. This is one of those things. You probably wouldn't have worried about this. You would have said, well, anybody understands that. But he's trying to give a mechanical calculus to determine what is a statement and what is not a statement. So that's all that paragraph is, to eliminate incidental statements. He says the second part of his rule that any replacement must be meaningful. If you say that's a statement, then you could put another statement in its place. You'd still have a grammatically meaningful sentence. Okay, forget all that. Let's get then into the. The notation itself. We have to learn conjunction, negation, disjunction, and punctuation before we get to the real fun one. And that is material implication. I must tell you, material implication is explained much better in later editions of COFI than when I first took logic in college with it. I mean, it comes out much nicer. And it's easier for me to explain now, too. But we're not even into that. Let's just begin with a few. We're going to work our way up casually or step by step. First of all, conjunction. When two statements are combined with an and then they are conjuncts, they become a conjunction. Remember, this is a conjunction compound statement because there are two propositions or two statements that are part of it. And he uses the notation of a dot for and a and B, where A is a statement and B is a statement. Charlie loves tuna and Mary despises cheese. A and B. No problem. You might want to know that in some logic books, the and is the amber sounds rather than a dot. But Kofi adopts the use of the dot. Now, page 256 tells us that every statement has a truth value, And the truth value of the compound statement is determined wholly by the truth value of its components. And then this is going to be called a Truth functional compound statement. Up on the board we have a compound statement, A and B. Now, what we want to know is what's the truth value of A and B? A compound, is that true or false? The compound. Well, can we determine the truth or falsity of it completely by looking at the component parts? If we can, it's considered a truth functional compound. That is, you don't need to know what A or B are to know whether the compound of A and B is true or false. As long as you know the truth value of A and the truth value of B, you can figure out the truth value of A and B. You don't have to. You don't have to know what they stand for. Okay? This is purely mechanical and formal. And for a compound, for a conjunction to be true, both of the compounds must be true. So if we arbitrarily assign the truth value of true to A and arbitrarily assign the truth value of true to B, then we know that the compound itself is true. Those both are true. You're saying, you're asserting A and B? Well, A and B can only be true if A is true and B is true. However, if we were to arbitrarily assume assign the value of false to B, we would be able to determine the truth value of A and B together, wouldn't we? What's the truth value of A and B now? It's now false because you have a true statement asserted and a false statement asserted. So the whole thing is not true. Well, the point is, we can always tell what the truth value of the compound is simply by looking at the truth value of the component parts. That's what makes it truth functional. We don't have to know what it's talking about. It could be about nuclear physics or hot fudge sundaes. Wouldn't make any difference. If we know the truth value, we can figure out. Now, why are we worrying about truth value? Because in the end, a valid argument, as the elaborate valid argument, true premises and a false conclusion, regardless of what the premises are about, then if we just assign truth values to them, we should be able to figure out the conclusion or whether it is validly followed. Copie points out on page 257 that the word and also has the sense in English, the sense of and subsequently, that is, it has the notion of temporal succession to it. But as we use it in logic, there is no temporal succession. It only is a truth functional term, meaning that the value of the component, excuse me, of the compound statement can be determined simply by looking at the values of the component parts. Now, there are other English words, when we find this in ordinary usage that have the same kind of force as the word and words like but yet also, still, although, however, moreover, nevertheless, sometimes the common and semicolon can have the effect of an and. So this is another case where you have to be able to understand the use of language before you can translate into the logical notation. This is the hobby horse that I told you about that I ride. If you don't understand language, you can't interpret what the person's really trying to say. You can't give a correct notational version of what they were saying. You can't put it into the symbolic logical form necessary. Okay, so now we've learned one word in logic. And I don't want you to think this is a big deal, just taught you how to use the word. And in logic, when you use the word and in this truth functional way, a compound statement statement is true only if both its components are true. Now, negation, In English, the notion of negation is best understood as prefixing to a phrase. It is not the case that or it is false that when I negate something, what I'm saying is not this. It is not the case that. Betty likes hot fudge sundaes. And in this book the notation of a curl is used for negation. In other books you'll find a simple straight line. So he found this. In another book he you would know that means not a copreadops the curl, which is another way of doing it. So curl A means it is not the case that whatever a asserts is true. It is not the case. And he says simply, the negation of any true statement is false and the negation of any false statement is true. Now, no one has any problem with that, right? We're working here with a two value logic. True and false, there's only two values. So the negation of a true statement has got to be false and the negation of a false statement has got to be true. Now what that means is as a truth functional notion, if I have not a up here, can you tell me just by the assignment of a truth value to a whether the not a is true or not? Can you determine whether or not A is true simply by knowing the truth value of a? Yeah, very easily. If I say that A is true, then not a is false. I haven't even had to ask what a is about. If I say A is false, I know that not a is true and I haven't even had ask what A is about. So it's truth functional. You can determine the value of this combined symbol here, the tilde or the negation, the curl and the A, simply by being told what the truth value of A is. Next we have disjunction. And here the author explains that the word or can be taken in two different ways. In English, probably in most languages, we can mean in a weak sense A or B. We can mean A and or B. Thus it's called inclusive. The inclusive use of or means that both could be true, but at least one is. So I'll put that over here. At least one, maybe both. Please remember that because this is different from A or B taken in the exclusive sense. In the exclusive sense, we read at most one. Okay? So it's important to know that when we use the word or in logic, we're going to take either the inclusive or the exclusive use of the term. And for truth functional purposes, in logic, we don't need the exclusive or the strong sense of or. So I'm going to erase that off the board. See, first of all, he acknowledges, in an ordinary language you could use it either way. But we're not going to do that. We're going to eliminate the strong sense and use only the weak sense. For truth functional purposes, A or B means at least one is true, maybe both, but at least one. On page 260, you notice that the word unless can also be used to form a disjunction of two statements. A unless B might be translated in notation as A or B mentions a comment about the English word and how it would come over in the notation. Then fourthly, we have to know something about punctuation. If you don't know how to punctuate a long compound statement, then the meaning of the statement is ambiguous. And he uses a good analogy here when he talks about about math. I'm going to put a mathematical notation on the board, Okay? Now, this notation, which I will just state, without trying to give meaning to it, two times three plus four. That notation is, is ambiguous as a statement. In math, that cannot be determined. We don't know what that is. And why is it? Because there's different ways of taking it. That's what an ambiguity is. That could mean two times the combination of three plus four. It could also mean two times three as one operation plus four. Now, these brackets which show us how to group functions together are called punctuation marks. They're ways of keeping clear the definite meaning of a long symbolic expression and we use parentheses and brackets in logic in the same way to avoid ambiguity and to make the meaning, he says on page 260 in Symbolic Logic, parentheses, brackets and braces are used as punctuation marks. It's important to see on the top of page 260 that a negation symbol is understood as applying to the smallest style statement that the punctuation permits. Negation applies to the smallest unit to which it can be applied. So that if I have negation P and Q and then I write another notation negation parenthesis P and Q close parentheses. These are different statements in the top one, which again for those listening by tape, just reads negation P and Q. The negation applies only to the P. Q is not being negated in any way. Here the negation applies to the smallest unit that it can apply to. However, where I have bracketed or put in parentheses pnq and proceed that with a negation mark, then the negation applies to the whole compound. Okay, let's see if we can figure out the truth value of what I have left up here on the board. I've put up on the or negation parentheses P and Q parentheses. Can we figure out the truth value of this? Provided we know the truth value of P and Q. If we can, then it's a truth functional statement. All we're saying here is we know the truth or the falsity of this long statement statement, not parenthesis P and Q parentheses. We know whether that's true or false, provided we know if P and Q are true are false. Given their truth values, we can figure out the truth value of the rest. So let's see if we can get this. I'm going to assign the truth value of true to P and false to Q. Now let's work this out. We have to work from the parentheses to out, he says on page 262, 2/3 of the way down. In working with such compound statements, we always begin with their inmost components and work outward. Alright, so we have the inmost component here and within the parentheses. When you can join a true and a false statement is is the compound true or false pat.

False.

Okay, so this much of it we know to be false. Now, what does negation mean? When you negate a statement? Just statement here in parentheses which is false, then what does the total thing become? True? True. Yeah, this one is very easy. There's no mystic difficult part to this or aspect to this. P and Q together is false. If we assign true and false to the component Parts and negation of something that is false is true. So now, that whole thing that we began with, which is right here, not parenthesis, P and Q parentheses, is on this assignment of truth values true. Are there other assignments of truth values possible? Sure. Let's go back and redo this. Then here's the assignment of truth values. I said P will be true, Q will be false. We don't know what P and Q are, but if one is true and the other is false, we know that the not PNQ is itself true. Now, what happens if we say assume Q is true and P is true, the statement in parentheses is it true. And then the negation of the statement in parentheses, what's its value? False. False. And so we have been able to determine the truth value of a compound statement simply by knowing the truth values of its component parts, the P and the Q, and learning how to play the game of negation, conjunction, disjunction, and knowing something about how to do the grammar or the grouping of these things, whether you're brackets. How to use the punctuation. As he says on page 262, we get to apply our elementary steps of learning the language of logic. Basically, this is just the test if you understand how to use the negation, conjunction and disjunction, as well as punctuation marks. And so Exercise Roman one says, which of the following statements are true? Let's go through and do a few of these. Number one, Rome is the capital of Italy. Wedge. Rome is the capital of Spain.

Do you want to get indicated of the whole.

Yeah, just go through the thing and tell us how you did it.

Rome is the capital of England. It's. And then V is or Rome is a capital of span, which is false. So the whole statement is true because it is at least one.

Okay, notice the operator here is the V or the disjunction sign. And what does that mean? At least one is true. So now we're going to test, is it the case that at least one is true? If you find that one one of them is true, then the whole compound is true. And of course, in this one the first part is true, and so the whole compound will be true. And that's just because of the way we defined or in our logical notation, which comes also from the use of the word, at least part of the use of the word in the English language. Okay, Number three, not London is the capital of England, and not Stockholm is the capital of Norway.

Okay, but not London is the capital of England is false. And Not Stockholm is the capital of Norway is true. So you have a false and a true statement, and therefore the whole statement is false.

Okay, Number five. Not Rome is the capital of Spain or not Paris is the capital of France.

Not Rome as the capital of Spain is true and not Paris as the capital of France is false, and therefore it is false.

Rethink here. What's or mean?

Then it's true.

It's one of those compounds is true. Because you have the negation of a false statement, which then makes the whole thing true. So that part of the compound is true. Number seven.

Stockholm is the capital of Norway and Stockholm is not the capital of Norway.

False.

The same thing, but true and false.

At the same time.

Okay, but now you're doing this semantically, you're doing it from the content, right?

Yeah, but.

Okay, first, stockholders. The capital of Norway is false. Is false.

Stockholders, not the capital of Norway.

It's true.

And a false and a true are false.

You understand? What I'm getting at is you are right, intuitively right. But I want you to do it in a truth functional fashion. Okay? Let's go to Roman numeral number two. If A, B and C are true statements X, Y and Z are false statements, which of the following are true? Number one. Not A or B.

That's true. Because it is one.

Number three. Not Y or C.

That's true.

Explain why.

Same reason. Neither one of them are or one of them is true.

Okay. Is not Y true?

So that's false.

C is true, not Y is true also. Okay, okay.

So not Y will be true and.

C will be true.

So both true and the statement will still be true.

That's right. Number five. Parentheses A and X into parentheses or parentheses B and Y.

Okay? In parentheses. A and X. A is true and X is false. Therefore the whole parenthesis is false. Then parenthesis B and Y. B is true and Y is false. Therefore that whole parenthesis is false and you have two falses connected by a wedge and therefore the whole statement is false.

Very good, very good. Number seven. Not parenthesis C and Y parenthesis or parenthesis A and Z. Parenthesis C is true, Y is false.

So therefore C and Y is false. You take the negative and you get true. You have A is true and Z is false. So A and Z together are false. So we have a true four false. Therefore it's true.

Correct. Number nine. Not parentheses, X and Z or parentheses B and C. X is false and.

Z is a false statement. We take the negative of that.

Now what's X and Z together.

False statement, false.

Okay, Then you negate that and you.

Get a true statement or, and then the reason for or, and then B the true statement, and C the true statement. So you have at least one true, so the whole statement is true.

Okay, let me ask you all. Do you. Do you see that by the time you figure out not X and Z to be true, if the operator is the wedge, you don't have to worry about the rest of it. You know that at least one is true. Everyone catch on to that? Okay, let's go to Roman numeral 3, which says if A and B are known to be true and X and Y are known to be false, but the truth values of P and Q are not known, of which of the following statements can you determine the truth value? The reason we're doing this is that a truth functional calculus or operation or statement is one where the truth value can be determined simply from the component parts and the truth values of those component parts. So, number one, A or P, can we know if that's true or false or is it indeterminate?

Well, and we can know because a we know is a true statement. And regardless of what P is, A or P would be a true statement.

Very good. And that's what I was just getting at a moment ago in that exercise, number nine, when I said, by the time you figure out that one part of it is true, the rest you don't even have to worry about. Okay, number three, Q or not X.

We know that X is false. The negation of X is true, false is true. So therefore the whole statement would be true because it's connected by a wedge.

Very good. P or not P. Okay, we know.

That with a wedge, only one needs to be true, and you have P and not P. So one of those.

Is the opposite of the other.

So one of them will be true, therefore the statement is true.

Very good. You might have been misled because he tells you the truth value of P is not known, but he's covered both the possibilities and since it's connected by a wedge, one's going to be true. Number seven.

The statement's false because you can't have. It's being ordered because Q is true or false. And either way. Either way.

If Q is true, not Q is false.

Right?

And if Q is false, not Q is true. So no matter how you assign truth values, one way or another, you're going to have a conjunction of a true and a false here. That's what you were trying to say. Yes, that's what I thought. 9. Not parentheses. P and Q into parentheses, or P.

And that can't be determined because we don't know what either Q or P is. If P were to be a true statement, we'd be able to tell, but we can't do it. We're not given enough information.

What's your thought on this?

I thought it was false. And the reason I said it was false is because if you write out a truth table on this, you're always going to end up with opposites on either side.

Truth table on this.

Excuse me.

True. I thought it was true.

Because if you say you assign P is true and Q is true, you're going to end up with a true and a true. If you say P is false and Q is true, you're going to still end up with a true and a false. And if you say that P is true and Q is false, you're going to end up with a true and a true. So any way you work it out, you end up with a true on one side of the equation. So therefore it would be a true statement.

Let's do this another way. If P is true, then we know that the whole thing is true just from the last part of the statement where it says or P. So if P is true, then we know the whole thing's true. So all we have to look at is the possibility that P is false. Now, if P is false, looking at the P and Q that's in parentheses there and noting that the whole thing's going to be negated if P is false, it makes no difference what Q is, because the whole thing then is going to be false. The P and Q conjoined together will be false if P is false, and if P is false and you negate it the compound, then the compound becomes true. So as a piece turns out, no matter what P is, the whole thing that we have is our problem. Not P and Q or P will turn out to be true. Did everyone understand that?

Now that he explained it, yes. So I had it on the term.

What we have here is just two possibilities. P is going to be true or false. False. We know that if it's true, then because P stands as one of the disjuncts. The whole disjunctive statement is true if P is false because of the way it's combined with something else. In a conjunctive statement, we know the conjunctive statement is false, but that conjunctive statement is negated, so it becomes then true. So even if P is false, the whole thing turns out to be true. Okay, very good. Let's move on to 8.3. This is where things get a little dicier, because now we're going to talk about if then. But in order to understand the hypothetical or conditional statement if P, then Q, we need to stop and consider how that is used in logic. The author says on the top page 266, first full paragraph, a conditional statement asserts that in any case in which its antecedent is true, its consequent is true. Also, it does not assert that its antecedent is true. It does not assert that its consequent is true. It simply asserts that if the antecedent is true, the consequent is true. If the P is true, then the Q is true as well. This leads us, then to talk about implication. And the author notes for us, on the bottom of page 266 and the top of 267, there are really different kinds of implication. He refers to logical implication. For instance, A in your illustrations, if all humans are mortal and Socrates is a human, then Socrates is mortal. We know that from our rules of categorical syllogism. That would be a logical implication. Number two, though, is a definitional implication. If Leslie is a bachelor, then Leslie is unmarried. That's true by definition. You can imply the implication comes just by the lexical definition of the term. But there are other kinds of implication too. For instance, causal implication. If this piece of blue litmus paper is placed in acid, then this piece of blue litmus paper will turn red. And that's because acid turns litmus paper red. It causes it to turn red. That kind of implication is scientific or causal. And then he has something that might be called decisional or committal implication. Where you say, if something happens, then I will do the following. As I say, if you pay me $3,000, then I will rebuild the engine in your car. That kind of implication is an implication made because of a decision that has been asserted and so on. 267, he says each of these asserts a different type of implication, but they are not completely different. There is an identifiable common meaning. And all we want to find is what is the common meaning to all four of those kinds of implications. In regular English usage, there is. They have a lot of difference between them. Logical, definitional, causal, and decisional implications are completely different from each other, or very much different from each other. But you know, in every case, this much is true. If the antecedent is true, the consequent is true as well. So in the middle of page 268, the author explains to us what we mean by material implication. This is a fifth kind of implication then, which states the common logical core to the other four. He says any conditional statement, if P, then Q is known to be false in case the conjunction pn, not Q is known to be true, that is, in case its antecedent is true and its consequent is false. Do you all follow that if we know that the antecedent is true, we assert P and we know that the consequent is false, we deny Q. We put those two together, P and not Q, then we know the if P, then Q is false. So another way of stating it, this. This amounts to saying not parentheses, P and not Q end of parentheses. And this all comes not from some kind of technical logical insight. It all comes from what it means to say if P then Q. What is the common core to all those implications? What this means is you cannot have. You cannot have a true and a false assigned as the values P cannot be true and Q false. Every other combination is, okay, false and false, true and true, false and true. But this we do know, that if P is true, Q must be true, then false combination. So then, as the author tells you, it amounts to saying we cannot have. What combination can we not have? We're going to deny the combination of P being true.

And.

Q being false. Okay, so I'm just, I'm doing this in steps just to see you how, show you how easy you get this. Since the meaning of if then rules out if true, then false, then we can say it rules out. That's the negate, the combination of asserting P and denying Q. And he says in the next paragraph, no matter what type of implication is asserted by a conditional statement, this much is common to all of them. Decisional, definitional, logical and causal connections all have this much in common. And so, on the top of 269 down a few lines, the symbol horseshoe is completely unambiguous. What P horseshoe Q abbreviates is what I've written on the board, not the combination of P and not Q. This is what we call material implication. And it is not to be confused with other more usual types of implication. When we're talking in ordinary English with other people and we say, well, if this, then that, we hardly ever mean material implication. We usually mean causal implication, definitional implication, decisional implications, something like that. So the author tells you at the bottom of the page, no real connection in the sense that we understand connection and ordinary usage, no real connection between antecedent and consequent is suggested by a material implication. All it asserts is that as a matter of fact, it is not the case that the antecedent is true when the consequent is false. I want to stress two things here. Material implication is not what we ordinarily mean in the English language when we say if then. But material implication is what is common to all the different uses of if then in the English language. And all we're looking at here now is not causal information about the world, lexical information about the world, volitional information about an individual who made a decision if this, then that we don't look at any of that. We're only looking at the truth value connection. And in all of those different if then statements in English, the truth value connections are the same. And that's what we call material implication. When you say if P, then Q, you are denying that you can have P be true and Q be false. Page 270, almost at the bottom of the page. So our proposal amounts to suggesting that we ignore or throw away part of the meaning of a conditional statement when we translate it into our symbolic language. When we move from the real world of natural discourse and we start translating into the symbolic notation, we're not pretending that we're capturing all the meaning of an if then statement. We're only capturing the truth value concept connection in those statements. We call that a material implication. I think this would be the point at which I would refer you to the end of the chapter. 8.6 the paradoxes of Material Implication. I'm not sure why he shuffled this discussion down to the end. I can see some reason, but I still think I might have picked it up right at this point. If you understand what we have been saying about material implication, that all it rules out is an if true then false combination. Since what is ruled out and the only thing that is ruled out by material implication is the if true then false combination of truth value assignments, then what is ruled in is any of the other three possibilities you could have. If the antecedents is false, then of course anything could follow from it. It wouldn't make any difference whatsoever, because then you don't have the if true then false combination. If the consequent is true, then this is going to be ruled in by material implication, because you're never going to have the if true then false combination. And of course, if you have a false leading to a false if false then false doesn't violate the rule. The rule only rules out the if true, then false combination. So that's what we mean by material implication. And so the author says on page 292, if a statement is true, then for truth functional purposes, if a statement is true, it's implied by any statement whatever. Not implied causally, lexically, logically, it's simply implied because of the relationship of truth values. Moreover, if a statement is false, then it implies any statement whatever. A statement that is false implies any statement whatever. It's not ruled out by our definition of material implication. He says these seem paradoxical because we believe that the two things he's talking about in the illustration are utterly irrelevant to each other. And we believe that no statement true or false can really imply any other statement, false or true, to which it's utterly, utterly irrelevant. This paradox is easily resolved, however, when we acknowledge the ambiguity of the word implies. Then he repeats for you, there are different kinds of implication logical, definitional, causal, and decisional. And then this is crucial. But subject matter or meaning is strictly irrelevant to material implication, which is a truth function. Only truth and falsehood are relevant here. We're only talking about the relationship of statements in terms of their truth value, not the relationship in terms of empirical or scientific relations, or lexical or decisional, or what have you. Okay, so let's go back then to his discussion of material implication and pick it up again. Middle of page 271, the author tells us that the word if can be replaced by such phrases as in case or provided that, or given that on the condition that without changing the meaning. This is just to help us pick up in the English language what we're going to translate into our notation as an if, then, And then on page 272, he talks about necessary and sufficient conditions. If we write if P then Q, that is to say, Q is a necessary condition for P. Necessary, because if P is true, Q has to be true. It becomes a necessary condition condition. And that's the same as saying P only if Q. You can't have P without Q being true as well. So P only if Q. At the end of the next paragraph, he says P is a sufficient condition for Q, as symbolized by if P then Q. So Q becomes a necessary condition for P, and P becomes a sufficient, sufficient condition for Q. Whenever we write if P then Q, or that's what we are stating when we write if P then Q, If P is a sufficient condition for Q, then Q must be a necessary condition for P. On the other hand, if P is a necessary condition for Q, then of course, we have it written in the opposite direction. Then we say if Q, then P and Q becomes a sufficient condition for P. Hence, if P is necessary and sufficient for Q, Q is sufficient and necessary for P. Have we lost anybody here? This is just a way of showing how in a notation, necessary condition and sufficient condition are understood. I'll put it up here on the board. P then Q. In a situation like this, what we are saying is that P is a sufficient condition. So the antecedent is a sufficient condition. The consequence, since it must follow from P being true, the consequent, is a necessary condition for the antecedent. Q becomes a necessary condition for P. So this is just a way of translating the P horseshoe Q Q notation. P horseshoe Q means P is a sufficient condition for Q and Q is a necessary condition for P. But now what if we add to this the claim that Q is a sufficient condition for P, if Q is a sufficient condition for P. So let me put the whole opposite direction. We have if Q, then P, then P becomes a necessary condition for Q when it's written with the opposite direction. And one more thing that we Note on page 272, the author says. Says not every statement in English containing the word if is a conditional statement as we understand it in logic. We have to understand what the meaning or intention of a person is when they use the word if. He gives some nice examples. There's food in the refrigerator, if you want some. Now, if you read that in the logical way, if you want some. If you want some, then there's food in the refrigerator, meaning there's if you want some. When that's true, then there's food in the refrigerator must be true. Is that what that statement is trying to assert? That if you want food, then there must be some in the refrigerator? No. So here the word if is not being used in the logical sense at all. I see a humorous illustration of. It's humorous to me every once in a while when you have there are these bumper stickers out in yuppie land anyway, where a young lady might have a bumper circuit on her car that says, if you're rich, I'm single. And I look at that, and I always say, that's not logically true now, is it? Because whether I'm rich or the driver next to me is rich or not, you're still single. It has no connection between those two. And so the author is right. The word if doesn't always have this conditional force to it. And then he Repeats again. This is Dr. Bronson's hobby horse. You have caught on by now. He says the presence or absence of particular words is never decisive. In every case, one must understand what a given sentence means and then restate that meaning in a symbolic formula. So it comes right back down to this. All of this formal logic that we're learning, which is powerful and helpful, is of no use whatsoever if you don't understand language to begin with, if you don't know what people are really trying to say. So you read this bumper stick and you say, oh, that's an interesting thing. She is single or not single, depending on whether I'm rich or not rich. What a bizarre kind of thing that would be utterly ignorant to not catch on to how the if is being used in a sentence like that. So the symbolic logic is good, but it will not help you until you know how to translate from ordinary language into the formula. Okay, on page 273 we have an opportunity to test our understanding of this new step. And this step here again is just trying to explain how the notation works. We're not into any big time stuff. It's just like this is the rule of the game. When you use logical language, the if then means material implication. And that rules out only the if true, then false combination number one or Roman numeral one. If A, B and C are true statements, and X, Y and Z are false statements, determine which of the following are true. Okay, let's take the first one. If A, then B, that is a.

True statement, because both A and B are true. And when they're connected together by the horseshoe, that means true.

Jumping ahead to what we learned about the paradoxes and material implication, let me just give you a shortcut. If you see that the antecedent in this example A, you see that the antecedent is true, then you've got to test to see if the consequence is false. But if the antecedent happened to be false, you wouldn't have to go any further. On the other hand, if you saw that the antecedent. Excuse me, if you saw that the consequent were true, you wouldn't have to go any further because anything implies a true statement. So let me say this again. If the antecedent is false, your work is done. You know the whole thing is true. If the consequent is true, you know that your work is done. I want to make sure everyone understands that, because the only thing that becomes false using the horseshoe, the only combination that's false, is if true, then false. Then the whole thing becomes false. Every other combination is acceptable. Okay, number three. If B, then why? Because. Why is false? Well, it's more than that. It can't ever be that one or the other is true or false.

True, true.

It's because Y is false and B is true.

Yeah. What you have is an antecedent that's true and a consequent that's false. And that's what material implication rules out. Number five in parentheses, if A, then B. That's taken then as a antecedent to horseshoe Z. If A, then B, then Z.

That's because A is true and B is true. The whole statement is true.

And because Z is true, the whole statement in parentheses. Right.

The whole statement in parentheses is true.

Okay.

And then because Z is false, we know that can't be true statement, therefore it's false.

Did you have an antecedent that's true and a consequent that's false? Very good, number seven.

If A, then B, they're both true statements, so that that part is true and the consequence C is true. Therefore they're both true and the whole statement is true.

Yeah. Let me ask you this. In the parentheses, if A, then B, could you have just looked at B and known that the implication is true? Now, let me show you a faster way to have done this. Since the key operator here is that horseshoe outside the parentheses, everything inside the parentheses stands as one unit. And then you have that unit implies C. Now, you were told that C is true. That's a given in the exercise. But in material implication, anything can imply a true statement. So if you just know that C is true, you didn't even have to consider the parentheses. Okay, does everyone catch on to that? As long as what is implied, that is the consequent is a true statement, you can have anything before it false or true. Okay, Number nine. Number nine. We have if A, then the compound. If B, then Z.

Okay, if we look at if B, then C in parenthesis or Z, then in parentheses, we see that Z is false. Therefore we have to look at B. If we look at B, we see that's true. Therefore the whole statement in parentheses, parentheses is false. And if that's false, then we have to look back at A. We see that A is true, but the whole statement is still false.

Very good.

Because the consequence is false.

Exactly the way to do it. Okay, let's look at Roman Numeral 2. If A& B are known to be true, X and Y known to be false, but the truth values of P and Q are not known. Which of the following statements can you determine the truth value? First of all, if P, then A.

That would be true, because A is true.

And we've learned that anything that implies a true statement is accepted under material implication. Good reasoning. Number three. We have in parentheses if Q, then A. And that is supposed to imply Z.

I have a question about this. One, it has Z and it's the only one in the whole section. Two that has Z and it doesn't explain in the directions what Z is supposed to mean.

Yeah, isn't that interesting?

I thought maybe they made a typo or something.

I would think it would have to be. It's crucial to our. Interesting. Did anyone else catch on to that?

Because that's. That's crucial to our understanding of it because the antecedent is true.

Well, let's start with it. The antecedent is if Q, then A. We don't know what Q is. We know that A is true. So is the if Q then A part of this true or false? Melissa? The part in parentheses. If Q, then A does that whole thing true.

That's true.

And how do you know that? Since we don't know what Q is?

Because we know what A is.

That's right. So knowing that the parentheses is true and that's the antecedent, if this parenthesis, then Z, we have to know whether Z is false or true to determine. And I do think what we have here is a typo. Probably he meant for Z to be in the false category. Notice how he has the early Alphabet true and the latter Alphabet false, and in between is indeterminate. So we'll just judge this to be Z is false. Therefore, is number three true or false?

Oh, if Z is false, then that's the statement's false.

Very good, number five. If P, then P is used as the antecedent for implying X. X is.

A consequence and X is given as a false. And so we look to the beginning. If P, then P were given no values to P, there's really no way.

Of telling, isn't there? Doesn't the statement imply itself?

Well, if P, then P was a false statement.

Let's think about this. If P were a false statement, you'd have if false, then false. Is that ruled out by material implication? No, no it's not. And so even if P were false, the if P, then P compound would be true. Under our rules of operation for material implication, if P were true, you'd have if true, then true. And that's acceptable.

Then if that Were the case if P, then P, since that would be.

True.

Then the consequence of X, consequence of X is false and the whole statement is false.

Could you have an if true then false combination which is ruled out? Number seven? If X, then parenthesis? If Q then X.

And if Q then X.

We.

Can'T determine that one, so we would not be able to determine the whole statement.

Wrong.

Remember the shortcut I gave you? A false statement implies any other statement. And in number seven we have an antecedent that is false, X. It says if X which we know to be false, then that's true. We cannot determine the meaning or the truth value, pardon me, of the if Q then X. But since a false statement applies any other, then we know that number seven is true. Okay, number nine, easy.

Number nine is false because we have.

A false statement for Y.

And on the other side, if we take all the parentheses, if you start with the false for P, since P begins it, that would make the whole statement true. And if you have P be true, it won't matter, it'll still be true. So therefore on that side you're always going to get a true and a true, and a false is a false.

Okay, you lost me a little bit there. We do know that we have a false consequence. So now maybe here's a ripe possibility for not fitting into the material implication. It's not permissible if it implies a false and the antecedent is true. Now, can we determine in the square brackets if P, then if Q, then P, whether that's true or false? P and Q are both not known. Right?

But if we take P to be true, it's obvious that the whole statement will be true because it's Q then P on one side.

Very good. So let's work from the inside out. Just to use our method. The smallest unit inside is going to be that. If Q then P, assume P to.

Be true, then that whole statement would be true.

That would be true, because any statement implies a true statement on truth functional or truth value, material implication. So if P is true, then we know that the whole square bracket would have to be true because you would have a true consequent. And anything implies a true statement. But now if P is false, then looking at the if P, then the bracket Q, then P, we know that a false statement applies anything. So that's going to mean that the square bracket, which I'll read out as square bracket P, whole horseshoe parenthesis, Q, horseshoe p parenthesis, square bracket, that whole unit now becomes true. And now we have this true long unit implying Y which is false. And so the whole thing now must be false, because that's what's ruled out by material implication. Good thinking. Let's try one more. Number 11. We have two compound statements in an if then relationship. The first parenthesis says if P then Q. And that's said to imply if X then P. And we can eliminate this very quickly. I mean, we can get to the answer. We may not eliminate it, but it's true.

Because if you look at the second compound, if you make P true, we know the given is X is false. You know, if we make P true in the statement.

Okay, I can tell you already, you're making it more difficult than you have to. You began by saying, saying if we take P to be true and then you're going to look at the other alternative. But you don't have to consider any alternative when you remember that X is given as false. And a false statement can imply any other statement. And therefore the if X then P part is going to be true. Moreover, you know that any statement implies a true statement. And so it makes no difference what is on the antecedent side of this major horseshoe. The if P then X can be anything whatsoever. Because we know that if X then P is true. And we know that if X then P is true because X is false and a false statement applies in the other statement. Do you all follow the reasoning there? Very good. Then I think we're ready to move on. Alright, we're going to move ahead to section 8.4, which introduces the notion of argument forms. Now you all know what arguments are. Now all we have to learn is what the form of an argument is. The title gives it all away. Bottom of the page he says the Validity and Invalidity. He says validity and invalidity are purely formal characteristics of arguments. Which is to say that any two arguments having exactly the same form are either both valid or both invalid, regardless of any differences in the subject matter with which they are concerned. Okay, so we're just looking at the skeleton or structure of an argument, its form. And if a particular form is valid, it's valid no matter what you fill into the place markers, A's, B's, C's and so forth. And it's invalid. If it's an invalid form, it's invalid regardless of the content that you put in there. At this point, footnote nine is worth paying attention to. The author says that when he makes this claim, he's assuming that the simple statements involved are neither logically true nor logically false. And he assumes that the only logical relations among the simple statements are those asserted or entailed by the premises. And that he has to remind you of, because we're only dealing with truth functional arguments at this point. There are more complex arguments that rely on considerations that go beyond simple truth functionality. Now, why am I bothering to tell you that or remind you of the footnote is just so we get the point across, we're only regimenting a few terms in the English language, and we're only regimenting them in a truth functional way. We're not even using if then in the broad common are multiple ways in which it's used in the English language. We're only using if then in terms of material implication. Now back to argument forms. If the form is valid, regardless of what you fill in the place markers like the algebraic signs, no matter what you put in as the values, the whole argument will be valid. If the form is invalid, regardless of what you put in there, the argument that is a substitution instance of that form will be invalid. The author says now on page 276, a given argument exhibits its form very clearly when the simple statements that appear in it are abbreviated by capital letters. You notice he uses capital letters that in some way capture the main point in each one of the compounds, each one of the components. I should say now though, if we are interested in discussing forms of arguments rather than particular arguments, I.e. arguments that have been given to us about Shakespeare or Washington or what have you, if we're interested in discussing forms of arguments rather than particular arguments having those forms, we need a method for symbolizing the argument form. So he introduces the notion of a variable. Variables. He's stipulating this. You all remember what a stipulative definition is. He is stipulating that a variable will be a lowercase letter from the middle part of the Alphabet and that when you find a lowercase letter from the middle part of the Alphabet, it will be a statement variable. It A statement variable, as we shall use the term, is simply a letter for which or in place of which a statement may be substituted. Where capital W might have stood for the statement Washington was assassinated, P now stands for that or any. Any other statement. W, A, G, R makes no difference. What he's saying here is that this is a very bland, open ended place marker for a statement. Not a particular statement like the W or the D or the A above, but any kind of statement. Well, any kind that's fair game under these set of rules. A statement variable is simply A letter for which or in place of which a statement may be substituted, and then argument form can be defined. We define an argument form as any array of symbols containing statement variables, but no statements. Let's just stop there. We're only going to be using P, Q, and R. We're not dealing with definite statements, but only statement variables, such that when statements are substituted for the statement variables, and here's a crucial qualification, the same statement being substituted for the same statement variable throughout P has got to stand for the same thing every time. The result is an argument for definiteness. We establish the convention that in any argument form, P shall be the first statement variable that occurs in it, Q shall be the second, R the third, and so on. So P, Q, R will be assigned as to what appears first in the argument first, second, third, and so forth. Then at the bottom of the page it says any argument that results from the substitution of statements for variables, that is, statements for statement variables in an argument form is called a substitution instance of that argument form. For any argument, there are usually several argument forms that have the given argument as a substitution instance. And now notice this, he uses an argument if B, then G, G, therefore B. He says that is a substitution instance, that is filling in the blanks, if you will, a substitution instance of each of the four argument forms that follows here. Now, the first one is rather clear. If P, then Q, Q, therefore P looks exactly like if B, then G, G, therefore B. And because it looks exactly like it, it's called the specific form of the argument. That is the specific form. In case an argument is produced by substituting a different simple statement for each different statement variable in an argument that argument forms. I mean, in an argument form, that argument form is the specific form of that argument. The key here is that you substitute a different simple statement for each different statement variable. Let's contrast that to what we have in the next case. Another argument form is if P, then Q are therefore P. Now, he says that that argument form fits or would take as a substitution instance the if B, then G, G, therefore B. Now, what's the difference? Notice this, in the top substitution instance, you only have two variables, B and G, right? Two statements. But in the form below, you have P, Q, and R. P, Q, and R. You have three variables, but there's only two statements. And so, as you'll notice in the explanation, what you do here is that you have to substitute. B for P, and G must be substituted for Q and R in this. And so you no longer have the specific form but you do have the same form. How do you know you have the same form? Because the form is a conditional statement, a categorical statement, and a conclusion drawn from it. Okay, y' all following me. The form is the same, although the assignment of values is much different. And you don't have the specific form because if you use number two in this line of argument forms, you have to make two variables mean the same thing. You don't assign one statement to this variable, another statement to that variable, and on and on. Where you have variables that take more than one statement as their meaning, you no longer have the specific form, even though in four ways terminology it corresponds. If this, then that, that, therefore this. And now at the bottom of the page, he says the technique of reputation by logical analogy can be described more precisely if the specific form. Notice specific. If the specific form given argument has any substitution instance whose premises are true and whose conclusion is false, then the given argument is invalid. We may define the term invalid as applied to argument forms as follows. An argument form is invalid if and only if it has at least one substitution instance with true premises and a false conclusion. And reputation by logical analogy is based on the fact that any argument whose specific form is an invalid argument form is itself an invalid argument. There's a lot of fancy terminology, maybe here, but I don't want you to be lost. All he's saying is arguments have a skeleton, they have a form. And there are different ways of putting variables in there for the form. Sometimes a variable stands for one statement, the next variable stands for a different one. But sometimes the variables can be assigned to two or more statements, but the form still is conditional, categorical, and then a conclusion. Now, a given argument is proved invalid if a refuting analogy for it can be found. But as we saw before, thinking up such refuting analogies may not always be easy. Sometimes we're not able to find an analogy real quickly that we can give. Happily, it is not necessary to do so, because there's a simpler, purely mechanical test based upon the same principle. Given any argument, we can test the specific form of that argument. To test an argument form, we examine all possible substitution instances of it to see if any of them have true premises and false conclusions. And this is what then leads to the examination of all possible different arrangements of truth values by means of a truth table. A truth table is absolutely mechanical. And what you do is you can test the validity of an argument by looking at the specific form of the argument, go through all the possible assignment of truth values to the component parts. And then see if you have any lines where the premise lines are true and the conclusion row is false. And you'll know whether you have a valid or an invalid argument. Now, I'm going to go to the board and state this again, If I were doing it, I think I would have explained it this way. Let's review for a second class. What do we mean by valid? What do we mean by invalid? Let's take invalid. An invalid argument is one which can have true premises and a false premise. So by invalid, we know that we have true premises, All of them are true, and the conclusion is false. Now, if we don't ever have that combination, true premises and a false conclusion, then the argument is valid. Okay, now that's step one. Step two is realizing that in truth functional logic, you can always figure out the truth value of a compound statement as long as you know the truth values of the component parts. Okay? Truth value of the. Compound can be determined. By the constituent parts if you know their truth value. The next thing that we have learned is that there's such a thing as a specific form for every argument. The specific form is entirely general. There's nothing to do with the content of the propositions in the argument. So then we can test the specific form for an argument looking at the truth value assignment to see if all the premises could ever be true and the conclusion false. So let's take the simplest case where we have only two variables in an argument, P and Q. If we only have two variables, then we're only going to have so many possible combinations. In fact, the author tells you on page 282, the number of combinations will always be 2 to the nth power, where N stands for the number of variables you have. So we have two variables. N becomes 2. What's 2 to the second power? So you know that there's only four combinations here. By the way, you could have known that apart from his telling you, you can look at this and say, well, you could have P be true and Q be true. You could have P be true and Q be false. You could have P be false and Q would be true. And you could have P be false and Q would be false. So you have true, true, true, false, false, true, false, false. Now, I'm going to keep the illustration simple, but you should understand that if you have three variables, then what you have is two to the third power. So you have two times two times two, which is eight. You left eight rows and you have to get all the possibilities. Now, the only reason I'M putting this down is you've got to know all the possible assignments of truth value, all the possible combinations, and once you know all the possible combinations, then you can go back and look to see if you have any where all the premises are true and the conclusion is false. If you have any where all the premises are true and the conclusion is false and you've got an invalid argument. Okay, now this much of the truth table is just dealing with the variables P, Q, R, S, T. Be horrible if you got four of them. It'd be 2 times 2 times 2 times 2 what? 64. No, 32.

8 times 4, 32.

No, 8 times 2, 16 times 2. That's four variables you'd have 2 times 2 is 4 times 2 is 8 times 2 is 16. 16. So if it five variables would be 32. But anyway, I want to keep it simple. The only reason we list all the P and Q combinations is because that's the number of variables we have. The next thing that we have in the truth table will be the premises of the argument. So up here on the board you see that I have one section of my truth table to the left.