Lesson 2: An Introduction to Logic

Purpose

This lesson will familiarize you with some basic features of the primary tool of the philosopher, logic.

Learning Objectives

After completing this lesson, you should be able to accomplish the following:

  1. Define the terms deductive logic, inductive logic, and fallacy.
  2. Recognize the basic shape of formalized arguments.
  3. Construct a truth table to demonstrate the validity or invalidity of an argument in sentence logic.
  4. Explain the difference between validity and soundness.
  5. Recognize some common fallacies in reasoning.

Commentary

Logic

As noted in Lesson 1, philosophers do not simply "feel" their way around issues. Philosophers offer arguments for their positions. Unfortunately, those not familiar with logic can sometimes find these arguments confusing or even intimidating. In order to alleviate such apprehensions, we will begin our study of philosophy with an introduction to logic.

Arguments are sets of sentences. One of these sentences is the conclusion; the others are premises. In a good argument, the premises provide reasons for believing the conclusion. They do not simply provide motivations. The truth of the premises should increase the likelihood of the conclusion being true, not simply make belief in the conclusion seem more pleasant. In other words, arguments present evidence for a conclusion, not just motivation for accepting the conclusion. In a famous skit on Monty Python's Flying Circus entitled "The Argument Clinic," two people set out to have an argument, but only one understands the correct definition of the term argument.

Consider the following two arguments.

Premises (P) are listed above the line, the conclusion (C) beneath it.

P1. Either the maid did it or the butler did it.

P2. The maid didn't do it.


C. Therefore, the butler did it.

P1. Ninety-nine percent of all persons who drank the well water became sick.

P2. I drank the water.


C. Therefore, I will be sick.

There is an important difference between these two arguments. In the first, the argument is structured in such a way that it is impossible for the premises to be true and the conclusion to be false. The truth of the premises secures the truth of the conclusion. Another way to say this is that if the premises are true, the conclusion is guaranteed to be true. This means that the argument is deductive.

But this is not so with the second argument. The truth of the premises only seems to make the conclusion more likely. If the premises are true, the conclusion is very likely to be true, but it need not be so. We call these arguments inductive arguments.

We will be primarily interested in deductive logic in this course. If a deductive argument is set up correctly, the truth of the premises assure us of the truth of the conclusion. For instance, consider the following arguments concerning the singer Bono:

P1. If Bono is a penguin, then Bono can fly. (false premise)

P2. Bono is a penguin. (false premise)


C. Bono can fly. (false conclusion)


P1. If Bono is a rock star, Bono can make music. (true premise)

P2. Bono is a rock star. (true premise)


C. Bono can make music. (true conclusion)

Both of these arguments are valid. An argument is valid if the truth of the premises secures the truth of the conclusion. If a deductive argument is set up correctly, it will be valid.

Consider this argument:

P1. If Bono is a monkey, Bono is a mammal.

P2. Bono is a monkey.


C. Bono is a mammal.

Is this argument valid? The second premise is clearly false. And yet the argument is valid. For it is impossible for both premises to be true and the conclusion to be false. There are two valuable things to be learned from this. First, just because an argument is valid, it does not follow that its premises are true. Second, just because an argument has a true conclusion, it does not follow that it has true premises. The last example illustrates both of these points.

Does a true conclusion guarantee that an argument is valid? No. Consider this example:

P1. Bono is a human, a monkey, or a squirrel.

P2. Bono is not a monkey.


C. Bono is a human.

Clearly, the conclusion is true. But the truth of the premises does not guarantee this fact. So the argument is not valid. It would be valid only if we added a third premise: "Bono is not a squirrel."

In order to know the value of an argument, then, we must know more than whether or not it is valid. We must know whether or not it has true premises. If an argument has true premises and is valid, we call the argument sound. To put it another way:

Validity + True Premises = Soundness

Do sound arguments have true conclusions? Absolutely! If an argument is valid (the truth of the premises secures the truth of the conclusion) and the premises are in fact true, the conclusion must be true. There is nothing better one can say about an argument than to say that it is sound.

Most of the arguments that we will study in this course are deductive arguments. Arguments that are not deductive will be highlighted so that we can further analyze them. At this point, however, you should be aware that there are only two ways to attack a deductive argument. First, you can point out that the argument is invalid. Second, you can make the case that the argument has a false premise (and is therefore unsound). Any other attempt to attack a deductive argument will result in bad reasoning.

Many philosophy teachers (myself included) have heard their students complain, "These philosophers are just living in their own world. They play by their own rules." In other words, the philosophers are playing a rigged game meant to fool the unsuspecting into believing things that they ought not to. But understanding how arguments can and cannot be attacked helps convince students that philosophy is not a rigged game. Still, students are often confused by the arguments in the philosophical literature they read.

One further point of caution might be helpful here: never confuse an account of a person's position on an issue with an argument for that position. When students first begin to study logic, they very often will see arguments in everything they read and hear. In one respect this is fortunate, for it shows they are taking an interest. In other respects it is very unfortunate, for it is just as much a part of philosophy to give an orderly and informative account of one's position as it is to argue for it. Try to see the difference in philosophy, in your other studies, and in everyday life. To challenge a person's argument before they offer one is to rush off with Don Quixote to tilt at windmills.

Connectives

In this course, we will focus on what is called sentence logic. It is called that because the key element it focuses on is the sentence. We will want to know which sentences imply the truth of other sentences. To make things easier, we will use variables to represent sentences. For instance:

H = "I like ham."

T = "I like turkey."

R = "I like rice."

In choosing a variable, it is best to choose one that reminds you of the sentence it represents. Obviously I chose "H" because it reminds us best of "ham."

Now consider the following argument:

P1. If I like ham, then I like turkey.

P2. I like ham.


C. Therefore, I like turkey.

This argument is valid. Moreover, it looks similar to other arguments that we have considered in this lesson, for it has the following shape:

P1. If H, then T

P2. H


C. Therefore, T

Any argument that has this shape is valid no matter what the variables represent. For instance, let's look again at the following:

P1. If Bono is a monkey, then Bono is a mammal.

P2. Bono is a monkey.


C. Therefore Bono is a mammal.

Suppose that

M = "Bono is a monkey."

A = "Bono is a mammal."

(Notice that I cannot use M to represent the second sentence, because it has already been used. And we don't want to get confused by using the same variable to represent two different sentences. Similar problems would result in algebra class if we let one variable represent two different numbers.)

Then the argument above takes the following shape:

P1. If M, then A

P2. M


C. Therefore, A

So let's make a generalization—for any sentences P and Q, the following shape of an argument will always be valid:

P1. If P, then Q

P2. P


C. Therefore, Q

This is the most famous of all argument forms. It is called modus ponens. In the Latin, this means "a way of building up." If you want to "build up" or support Q, this is the best way to use P to do it.

There is another famous argument form called modus tollens. To understand it, consider the familiar question, "If you're so smart, how come you're not rich?" One might take this to be offering the following argument:

P1. If you were smart, then you would be rich.

P2. You are not rich.


C. Therefore, you are not smart.

Again, this is a valid argument (although I think premise 1 is dubious). What is its shape? Let the following variables represent the sentences:

S = "You are smart."

R = "You are rich."

So the argument seems to be

P1. If S, then R

P2. Not R


C. Therefore, not S

Arguments with this shape are usually meant to take a "toll" on one's opponent in a debate. In the Latin, modus tollens means "a way of tearing down."

Sometimes, premises in arguments go unstated under the assumption that they are taken for granted. For instance, "If Kathy was in the supermarket today, Jim would have seen her." We are left to fill in the fact that Jim didn't see Kathy. And then, by modus tollens, we are left to infer that Kathy was not in the market. If you become proficient at spotting these unspoken premises, you will be able to see the underlying logic in a great many of your day-to-day conversations.

Modus ponens and modus tollens are the two forms of argument most commonly used. There are more, but we need not study all of them. What we need to know is how to determine whether a given argument is valid. Consider the following argument:

P1. If God exists, then God is all-powerful, all-good, and all-knowing.

P2. If God were all-powerful, all-good, and all-knowing, then God would stop all evil.

P3. God has not stopped all evil.


C. Therefore, God does not exist.

Whatever one thinks about the soundness of this argument, it is certainly valid. But how can we show this to be the case? It is not as easy to see this as was the case when we studied modus ponens. So let's begin by trying to discover the shape of this argument.

Thus far, we have used symbols (variables, actually) to represent sentences. We will now use a new group of symbols to represent connectives. Connectives do just what their name implies: they connect sentences. For instance, the symbol "&" represents the connective and. Now let's return to the following three sentences:

H = "I like ham."

T = "I like turkey."

R = "I like rice."

How would we symbolize the sentence "I like ham and turkey"? We simply put our connective in between the variable H and the variable T, like so:

H & T

Connectives allow us to take simple sentences such as H and T and create compound sentences. This particular connective is called a conjunction. The important thing to bear in mind is that a conjunction can be true only if all of the items that it connects are true. So how do we find out if the compound sentence H & T is true? Here is the rule for any conjunction:

P
Q
P & Q
Row 1 True True True
Row 2 True False False
Row 3 False True False
Row 4 False False False

This table tells us that "P & Q" can be true if and only if P is true and Q is true. In all other cases, rows 2–4, the compound sentence "P & Q" is false.

Another connective that we often use is the word or. For instance, we might say, "I like ham or rice." Hence, "H or R." In logic, we use a "wedge" to stand for the word or.

H v R

Compound sentences using "&" are called conjuncts. When we use v, we call the resulting compound sentence a disjunct. Under what conditions do sentences like this count as true? The most lenient approach suggests that at least one of the simple sentences, either H or R, must be true—perhaps both, but at least one. This is the approach logicians generally prefer. The following table gives us the rule for knowing when a disjunct is true:

P
Q
P v Q
Row 1 True True True
Row 2 True False True
Row 3 False True True
Row 4 False False False

The simplest of all the connectives to understand is the negation. The negation of "I like rice" can be expressed in many ways:

  • I don't like rice.
  • The sentence "I like rice" is false.
  • Rice is not something I like.
  • It is not the case that I like rice.

Logicians prefer the last of these. Why? Because it is simple. If you wish to negate any sentence, simply place the words "It is not the case that…" in front of it. When working with symbols and variables, your job is even easier: you negate the sentence R by placing the symbol called a tilde in front of it: ~R. This tells us "It is not the case that I like rice." The rule for the negation is:

P
~P
Row 1 True False
Row 2 False True

This table tells us two things of importance. The first is that no sentence (whether we represent it with the letter P or any other variable) can be both true and false at once. Logicians refer to this as the law of noncontradiction. A sentence and its negation cannot both be true.

Secondly, every sentence must be either true or false. This second principle is often referred to as the law of excluded middle. In other words, there is no middle ground. Some clarification might be needed on this point. Consider an egg. Is it spherical or cubical in shape? Someone might be tempted to say that the former is closer to the truth. So maybe there is some middle ground between truth and falsity? Not so. The person who says that the egg is spherical might be closer to apprehending the truth, but his sentence is nonetheless simply false. It is not the case that eggs are spherical. They are ovals (or something akin to an oval).

So far we have studied three types of connectives: the conjunction, the disjunction, and the negation. We call the last of these a connective even though, strictly speaking, it doesn't really connect anything. There are only two other kinds of connectives: the conditional and the biconditional. As you probably expect, these are somewhat similar, so we will deal with them together.

Conditional sentences are sentences that take the form "If…, then…." The part of the sentence that follows the word if is called an antecedent. The part following the word then is called a consequent. So in the sentence "If I like turkey, then I like rice," the part that we have chosen to designate as T is the antecedent, and the part we have called R is the consequent.

You should be familiar with these sentences from our discussion of modus ponens and modus tollens. Both of these argument forms use conditional sentences. Under what circumstances is a conditional sentence true? This is a more difficult question than was raised with regard to any of our previous compound sentences.

Logicians prefer the following rule: any conditional sentence is true unless the antecedent is true and the consequent is false.

In other words, if we symbolize the conditional sentence with an arrow (or "greater than" symbol), we get the following rule:

P
Q
P > Q
Row 1 True True True
Row 2 True False False
Row 3 False True True
Row 4 False False True

This table has some interesting implications. For instance, the sentence "If Bono is a monkey, then Bono is God" turns out to be true. The antecedent is false and the consequent is false, so according to row 4 above, the whole compound sentence is true. Also, the sentence "If Bono is God, then Bono is a rock star" turns out to be true according to row 3.

You should not be too disturbed by this. For one thing, there are plenty of conditional sentences described by rows 3 and 4 that do seem more plausible. For instance:

  • If Abraham Lincoln had been run over by a steamroller, then he would be dead today.
  • (false antecedent and true consequent)

  • If Superman were real, then Krypton would have been real.
  • (false antecedent and false consequent)

More important, there is no danger of being misled by an argument if we follow this rule. Consider an example I discovered in a "Peanuts" cartoon. In the cartoon, Lucy explains to Schroeder:

If we were to get married someday, then you would learn to appreciate me.

Of course, Schroeder does not think that the consequent of this conditional is true. He denies that he can comprehend it. He claims that it is like trying to imagine what lies beyond the boundaries of time and space. Why is he so stubborn about granting Lucy this premise? Clearly, if Lucy tries modus ponens, Schroeder will deny the second premise of her argument:

P1. If we were to get married, then you would appreciate me.

P2. We will get married.


C. You will learn to appreciate me.

And if she tries modus tollens, she will simply be proving Schroeder's point:

P1. If we were to get married, then you would appreciate me.

P2. You won't appreciate me.


C. We won't get married.

The only way Schroeder could get hijacked into a conclusion to which he does not feel obligated is if he accepts the conditional sentence as true and assumes that it is true by virtue of being the sort of sentence illustrated in row 1 of our table above. But our understanding of the conditional does not obligate us to this simply because we grant Lucy her conditional premise. So Schroeder would be more rational to reply, "Your sentence is true due only to the fact that it has a false antecedent."

This leaves us with the biconditional. The biconditional says that two sentences have the same truth value: that is, either both are true or both are false. This is sometimes expressed by the phrase if and only if. For instance:

  • My brother is a bachelor if and only if my brother is an unmarried male.
  • My sister is a bachelor if and only if my sister is an unmarried male.

Both of these compound sentences are true. That the first one is true seems pretty obvious. And on close inspection, the second one should strike you as true as well. Of course, the second biconditional is true because both simple sentences in it are bound to be false. In either case, the biconditional states that the truth of its component sentences stands or falls together. So the rule for the biconditional is:

P
Q
P < > Q
Row 1 True True True
Row 2 True False False
Row 3 False True False
Row 4 False False True

We use a "double arrow" to represent the biconditional. This is because the biconditional sentence "P < > Q" is just an abbreviated way of saying both that "P > Q" and "Q > P." The double arrow represents two conditional sentences.

Please take the time to memorize the expression if and only if. This denotes a biconditional. It is often abbreviated "iff." This abbreviation and expression will show up in many arguments made in these lessons.

At this point, you should know how to read any compound sentence consisting of two variables joined by one connective. Unfortunately, premises can be more difficult than this. Try to use the following variables and your symbols for connectives to formalize the following:

H = "I like ham."

T = "I like turkey."

R = "I like rice."

  • I like turkey, but I do not like ham.
  • I like both ham and turkey.
  • I like ham if and only if I like rice.

The following sentence will be more difficult:

If I like ham and turkey, then I like rice.

If we symbolize this as

H & T > R

we run into a difficulty, for this formula can be read in a second way. In addition to "If I like ham and turkey, then I like rice," it could mean:

I like ham and if I like turkey then I like rice.

If we wish to say the former and not the latter, we need to employ parentheses:

(H & T) > R

This lets us know that it is the conjunction (H & T) that is the antecedent of a conditional sentence that has R as the consequent. Therefore, we need to affirm both H and T in order to make the move to R.

Try to put parentheses in their correct places when formalizing the following sentences:

  • I like rice if and only if I like turkey and ham.
  • I like turkey and ham, or I like rice.
  • I like rice and ham and if I like rice, then I like turkey.
  • It is not the case that I like turkey or ham.

Truth Tables

Truth tables are a simple, effective means of demonstrating whether or not an argument is valid. Remember, an argument is valid if and only if the truth of the premises guarantees the truth of the conclusion. Let's consider modus ponens. Any argument with that shape will be valid. But how can we demonstrate this?

First, count the number of variables that represent sentences. In the case of modus ponens, you will find two: P and Q.

P > Q

P


Q

Second, make a column for each variable.

P
Q
   

Third, raise the number 2 to the power of the number of variables you have just counted. Since we counted only two variables, this means raising the number 2 to the power of 2, or 2 × 2. This gives us the number 4. This will be the number of rows in our truth table.

P
Q
1.
2.
3.
4.

The next thing to do is to exhaust all the possible combinations of truth values for P and Q. Why is this necessary? If an argument is valid, then the conclusion should be true in every case that the premises are true. At this point, we are simply trying to find all the cases in which the premises might turn out true. And that means exploring all options.

In order to make sure that you do not skip any possible combination of truth values, I recommend the following: Take the number of rows you are dealing with (four, in this case) and divide by 2. This gives us the number 2. Therefore, place two ts, representing truth, immediately under the first variable.

P
Q
1. t
2. t
3.
4.

Then place two fs, representing falsehood, in the remaining rows under the first column.

P
Q
1. t
2. t
3. f
4. f

Because we went by twos—two trues and two falses in the first column—we will go by half that number in the second column.

P
Q
1. t t
2. t f
3. f t
4. f f

Although this probably seems overly meticulous, please take note that there is no combination of truth values not covered by our table. We have covered all of the possibilities. Once this has been practiced a few times, it will seem almost too easy.

The next thing to do is to make a column for our premises. We already have one, namely P, on our table. That is the second premise, and there is no need to make another column for it when we already have one. But we do need to add the following:

P
Q
P > Q
1. t
2. t
3. f
4. f

We know that conditional sentences are false only when the antecedent is true and the consequent is false. The only row in which that occurs is row 2.

P
Q
P > Q
1. t t t
2. t f f
3. f t t
4. f f t

Now, place an asterisk next to each row in which all of the premises are true.

P
Q
P > Q
1. t t t *
2. t f f
3. f t t
4. f f t

In this case, there is only one row in which all of the premises are true. Is the conclusion also true in that row? It is. This demonstrates that there is no possible case (or row) in which the premises are true and yet the conclusion is false. To put it another way, there is no way for the premises to be true and the conclusion false. We can thus be sure that the truth of the premises secures the truth of the conclusion: the argument is valid.

The rule is this: an argument is invalid if you find any row in which the premises are true and the conclusion is false. Otherwise, it is valid.

Let's try to construct a truth table for the difficult argument presented earlier in this lesson.

P1. If God exists, then God is all-powerful, all-good, and all-knowing.

P2. If God were all-powerful, all-good, and all-knowing, then God would stop all evil.

P3. God has not stopped all evil.


C. Therefore, God does not exist.

The first thing we need to do is take a look at the general shape of the argument. Let's assign variables to the simple sentences.

E = God exists.

A = God is all-powerful, all-good, and all-knowing.

S = God stops all evil.

So the argument is:

P1. E > A

P2. A > S

P3. ~S


C. ~E

Now we are dealing with three variables. So we first make a column for each. Please note that we are not counting E and ~E as two simple variables. ~E is a variable plus a connective. We must deal with it later.

E
A
S
     

In order to find out how many columns we need, we raise the number 2 to the power of 3 (the number of our variables).

2 × 2 × 2 = 8

So we will need eight rows in order to examine all possible combinations of truth values.

E
A
S
1.
2.
3.
4.
5.
6.
7.
8.

Now we must make sure not to miss any possibilities. I recommend dividing 8 by 2. That means we place four ts and four fs in the first column.

E
A
S
1.
t
2.
t
3.
t
4.
t
5.
f
6.
f
7.
f
8.
f

For the next column we go by half of 4: that is, two ts then two fs until we reach the bottom of the table.

E
A
S
1.
t t
2.
t t
3.
t f
4.
t f
5.
f t
6.
f t
7.
f f
8.
f f

And finally, we go by half of 2 down the table. That is, one true, one false, one true, one false….

E
A
S
1.
t t t
2.
t t f
3.
t f t
4.
t f f
5.
f t t
6.
f t f
7.
f f t
8.
f f f

Now we make a column for each premise and fill in the appropriate truth value by checking the truth values assigned to the simple sentences in the first three columns.

E
A
S
E > A
A > S
~S
1.
t t t t t f
2.
t t f t f t
3.
t f t f t f
4.
t f f f t t
5.
f t t t t f
6.
f t f t f t
7.
f f t t t f
8.
f f f t t t

We will also need a column for our conclusion. We did not do this in the previous example, for in that example, our conclusion P already had a column. Here, our conclusion does not.

E
A
S
E > A
A > S
~S
~E
1.
t t t t t f f
2.
t t f t f t f
3.
t f t f t f f
4.
t f f f t t f
5.
f t t t t f t
6.
f t f t f t t
7.
f f t t t f t
8.
f f f t t t t

Are there any rows that assign only ts under the premise columns? If so, place an asterisk by that row.

E
A
S
E > A
A > S
~S
~E
1.
t t t t t f f
2.
t t f t f t f
3.
t f t f t f f
4.
t f f f t t f
5.
f t t t t f t
6.
f t f t f t t
7.
f f t t t f t
8.
f f f t t t t *

There is only one way for the premises to turn out true, the way described in row 8. Is the conclusion also true in that row? It is. So the truth of the premises secures the truth of the conclusion. The argument is valid.

Exercises

Check your understanding of truth tables with the following exercises. Test each for validity and then check your answers against those provided. The progress evaluation and exam questions will not exceed these exercises in difficulty. Pay special attention to the last three questions. These will test your comprehension of the concept of validity.

Note: You will not submit your answers for grading.

  1. (A & B) > C

    ~C


    ~A

  2.  

    A
    B
    C
    ~C
    A & B
    (A & B) > C
    ~A
    t t t f t t f
    t t f t t f f
    t f t f f t f
    t f f t f t f *
    f t t f f t t
    f t f t f t t *
    f f t f f t t
    f f f t f t t *

    The premise columns are in bold. The asterisks to the right mark the rows in which both premises are true. Is the conclusion true in all of these rows? No. The fourth row demonstrates that it is possible for the premises to be true and the conclusion false. The argument is invalid. The premises will be true and the conclusion false whenever A is true and both B and C are false. When setting up your columns, did you remember not to deal with more than one connective at a time?

  3. (A v B) > C

    ~C


    ~A

  4.  

    A
    B
    C
    ~C
    A v B
    (A v B) > C
    ~A
    t t t f t t f
    t t f t t f f
    t f t f t t f
    t f f t t f f
    f t t f t t t
    f t f t t f t
    f f t f f t t
    f f f t f t t *

    If this truth table gave you trouble, it was no doubt due to the extra twist that the column for the consequent of the conditional (premise 1) is to the left of the column for the antecedent of the conditional. Remember the rule for a material conditional: always true unless true antecedent and false consequent.

    There is only one line in which the premises are true and in this line the conclusion is true as well. The argument is valid.

  5. P > (Q & ~Q)


    ~P

  6.  

    P
    Q
    ~P
    ~Q
    (Q & ~Q)
    P > (Q & ~Q)
    t t f f f f
    t f f t f f
    f t t f f t *
    f f t t f t *

    In the two lines in which the premises are true, so is the conclusion. The argument is valid. Can you see why it took merely a single premise to get to the conclusion? Notice that the consequent in the premise cannot possibly be true. It violates the law of noncontradiction.

    Again, did you remember to handle no more than one connective at a time? Did you attempt to handle the (Q & ~Q) column before the ~Q column? Or the P > (Q & ~Q) before the (Q & ~Q) column? That would be making things harder than they need to be.

  7. X > (Z <> Y)

    ~Y


    ~X

  8.  

    X
    Y
    Z
    ~Y
    Z <> Y
    X > (Z <> Y)
    ~X
    t t t f t t f
    t t f f f f f
    t f t t f f f
    t f f t t t f *
    f t t f t t t
    f t f f f t t
    f f t t f t t *
    f f f t t t t *

    The argument is invalid, as demonstrated by line four.

  9. ~(B & A)


    ~B v ~A

  10.  

    A
    B
    ~A
    ~B
    (B & A)
    ~(B & A)
    ~B v ~A
    t t f f t f f
    t f f t f t t *
    f t t f f t t *
    f f t t f t t *

    The truth of the premise secures the truth of the conclusion. The argument is valid.

  11. P > (Q v R)

    Q > S

    R > S


    P > S

  12.  

    P
    Q
    R
    S
    (Q v R)
    P > (Q v R)
    Q > S
    R > S
    P > S
    t t t t t t t t t *
    t t t f t t f f f
    t t f t t t t t t *
    t t f f t t f t f
    t f t t t t t t t *
    t f t f t t t f f
    t f f t f f t t t
    t f f f f f t t f
    f t t t t t t t t *
    f t t f t t f f t
    f t f t t t t t t *
    f t f f t t f t t
    f f t t t t t t t
    f f t f t t t f t
    f f f t f t t t t *
    f f f f f t t t t *

    This one is a monster. But if you go slowly and remember to take baby steps, it is no harder than any other truth table. There will be no sixteen-row truth tables on any exam.

    The argument is valid.

  13. (X v Y) > (Z & Y)

    ~Z


    ~X

  14.  

    X
    Y
    Z
    ~X
    ~Z
    (X v Y) (Z & Y)
    (X v Y) > (Z & Y)
    t t t f f t t
    t t f f t t f
    t f t f f t f
    t f f f t t f
    f t t t f t t
    f t f t t t f
    f f t t f f t
    f f f t t f t *

    There is only one row in which both premises are true, and in that row the conclusion ~X is true. The argument is valid. Again, never try to handle more than one connective per column.

  15. P < > Q

    ~P

    Q


    R

  16.  

    P
    Q
    R
    ~P
    P <> Q
    t t t f t
    t t f f t
    t f t f f
    t f f f f
    f t t t f
    f t f t f
    f f t t t
    f f f t t

    There is no row where all the premises are true. Is the argument valid? Yes! The only way for an argument to be invalid is for there to be a row in which all the premises are true and the conclusion is false. No row here meets both of these criteria.

  17. B


    A v ~A

  18.  

    A
    B
    ~A
    A v ~A
    t t f t *
    t f f t
    f t t t *
    f f t t

    In both rows where the premise is true, the conclusion is true as well. The argument is valid.

    Of course, it is impossible for there to be a line in which the premise is true and the conclusion false. The conclusion can never be false. That would violate the law of the excluded middle.

  19. A & B

    ~A

    ~B


    C

  20.  

    A
    B
    C
    ~A
    ~B
    A & B
    t t t f f t
    t t f f f t
    t f t f t f
    t f f f t f
    f t t t f f
    f t f t f f
    f f t t t f
    f f f t t f

    There are no rows in which all the premises are true. Is the argument valid? Check question 8 for an answer. Could such an argument ever be sound?

Fallacies

Fallacies are errors in reasoning that occur when little or no evidence is provided for a conclusion but an argument nonetheless has the appearance of giving good reasons for the conclusion. We are prone to accept such bad evidence when we like a particular speaker or appreciate the conclusion that they want us to reach. Nonetheless, fallacies are to be avoided if one wishes to be a good philosopher.

Some fallacies are referred to as formal fallacies. These occur when the shape of an argument is incorrect. For instance, there is an incorrect way to do modus ponens. It is called affirming the consequent.

P1. P > Q

P2. Q


C. P

The truth of these premises does not secure the truth of the conclusion. For instance, consider the following argument:

P1. If Joe is a Missourian, then Joe is an American.

P2. Joe is an American.


C. Joe is a Missourian.

Certainly the first premise is true. Let's assume that the second is also true. Does this guarantee the conclusion? Not at all! The premises might be true even if Joe is from New Hampshire. Moreover, we can show with a truth table that affirming the consequent is an invalid form of arguing.

P
Q
P > Q
1.
t t t *
2.
t f f
3.
f t t *
4.
f f t

This might be a little difficult to read due to the fact that the conclusion is represented by the first column and the premises are in the following two columns (in reverse order). Sometimes it does not hurt to rewrite a column in a place where it is easier to read. I will repeat a couple of columns to make matters clearer.

P
Q
P > Q
P
Q
1.
t t t t t *
2.
t f f t f
3.
f t t f t *
4.
f f t f f

In rows 1 and 3, both the premises are true. But in row 3, the conclusion is false. So the truth of the premises cannot guarantee the truth of the conclusion.

Similarly, there is a wrong way to do modus tollens. It is called denying the antecedent. The following argument was actually used against a philosopher named Bill Dembski. Dembski believes that he has scientific evidence for God's existence. An opponent responded that he did not like Dembski's claim because it means that if you disprove Dembski's argument, then you have "disproven God." In other words:

P1. If Dembski's argument were sound, then God would exist (certainly true).

P2. It is not the case that Dembski's argument is sound (or so it is feared).


C. God does not exist.

This is a fallacious inference. The form of the argument is

P1. P > Q

P2. ~P


C. ~Q

One can construct a truth table to prove that this is invalid.

Not all fallacies are formal fallacies. Some fallacies are committed because an arguer or philosopher makes other mistakes. These other fallacies are called informal fallacies because they do not concern the "shape" of the argument. There are more types of fallacies than can be covered by any one course in logic. We will cover a few that are pertinent to this course.

The first of these is the fallacy of begging the question. This is often called circular reasoning. We commit this fallacy when we assume for a premise what we intend to prove as a conclusion. For instance, consider this argument:

P1. Football players are stupid.


C. Football players are dumb.

Certainly one does not have much reason to believe the premise unless one has already bought into the conclusion. Here is a more subtle example:

Smith: How do I know that you are a reliable, honest man, Mr. Jones?

Jones: Well, just ask Clyde.

Smith: Can Clyde be trusted?

Jones: You have my word on it!

In order to buy into Jones's argument, Smith must assume precisely what is at issue. He must assume from the outset that Jones is honest. And that is precisely what Jones is obligated to prove. The philosopher David Hume claimed that any attempt to prove that the sun will rise tomorrow would beg the question. We will later examine his argument.

Another popular fallacy is the post hoc ergo propter hoc fallacy. It is often referred to as simply the post hoc fallacy, but the entire phrase is Latin for "after this, therefore because of this." We commit this fallacy when we determine simply on the basis of the temporal sequence of two events that one event must have caused the other. For instance, a baseball player might rub his lucky rabbit's foot and later win a game. If he decides that the game was won because of the rabbit's foot, he commits the post hoc fallacy. It takes more to prove a causal connection between two events than just establishing that one event came first in time.

Consider the following: "When the Republicans came into office, crime went down. So chalk one up for the Republicans!" An example such as this was used on an episode of The West Wing. Martin Sheen's character quickly spotted the informal fallacy involved: "Post hoc ergo propter hoc!" Unfortunately, few in his audience understand any Latin.

Hume, the philosopher mentioned above, purported to demonstrate that even the claim that a pistol shot caused Lincoln to die is based on no better reasoning than post hoc ergo propter hoc.

Yet another fallacy is equivocation. We commit this fallacy by using one word with two different meanings in the same context. Often this is very subtle. Here is an obvious example:

P1. If I am at the bank, I can get some money.

P2. I am at the (river) bank.


C. I can get some money.

The word bank is being used with two different meanings. Consequently, the second premise does not really affirm the antecedent of premise one.

Here are some other examples:

  • Mary is a good dancer. Therefore, Mary is a good person.
  • You believe in the miracles of science. So why do you not believe the miracle of the Bible?

In each case, equivocation is made upon a single word, which appears two times. Does good mean "proficient" or "morally upright"? Does miracles denote "surprising events" or "supernatural events?" If we insist upon consistency in word meaning, it is easy in each case to see that the first sentence does not support the second.

Philosophers such as George Berkeley and John Stuart Mill have been accused of committing this fallacy at vital points in their famous arguments. Berkeley was trying to prove that the physical world did not exist. Mill was trying to prove that pleasure was the only intrinsically good thing for a person. Of course, we will examine these issues as well.

The last two fallacies that we will study are of particular importance to those just beginning philosophy. This is because each represents an inappropriate means of getting out of a tough argument. Neither one really works, but when backed into a corner it is hard not to want to employ these fallacious arguments.

The fallacy of composition occurs when one jumps from the claim that parts of an item have a certain characteristic to the claim that the whole item has that characteristic. Here are some examples:

  • Each grain of sand is small. So the pile of them must be small.
  • Each member of the choir sings softly; therefore, the whole choir will sing softly.
  • Sodium and chloride are poisonous; therefore, salt is poisonous.
  • Each member of the band is fantastic; therefore, the band is fantastic.

The fallacy of composition can also be committed in reverse. That is, we can unjustifiably assume that the properties of the whole must be properties of the parts. This is the fallacy of division, which would lead to statements like the following:

  • The student body is 60 percent female; therefore, each member of it is 60 percent female.
  • The St. Louis Rams are a great team, so the center must be a good player.

You may be asked to spot some fallacies on your exam. If you feel that you have not got the hang of this, find an introductory logic textbook at your local library. Doubtless it will feature many more examples of these fallacies. If the subject of fallacies is of interest to you, I recommend looking up the short story "Love Is a Fallacy" on the Internet. It is a humorous account of a young man trying to win a lady's affection with his logical prowess only to have her turn his skills against him.

Review Exercise

The following interactive exercise will help you review what you learned in Lessons 1 and 2.

Note: You will not submit your answers for grading.

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