rational discourse. Name calling doesn™t count.

There are many devices in ordinary language for indicating premises and

conclusions of arguments. Words like hence, thus, so, and consequently are

identifying premises

and conclusions used to indicate that what follows is the conclusion of an argument. The words

because, since, after all, and the like are generally used to indicate premises.

Here are a couple of examples of arguments:

All men are mortal. Socrates is a man. So, Socrates is mortal.

Lucretius is a man. After all, Lucretius is mortal and all men are

mortal.

One di¬erence between these two arguments is the placement of the con-

clusion. In the ¬rst argument, the conclusion comes at the end, while in the

second, it comes at the start. This is indicated by the words so and after all,

respectively. A more important di¬erence is that the ¬rst argument is good,

while the second is bad. We will say that the ¬rst argument is logically valid,

or that its conclusion is a logical consequence of its premises. The reason we

logical consequence

say this is that it is impossible for this conclusion to be false if the premises are

true. In contrast, our second conclusion might be false (suppose Lucretius is

my pet gold¬sh), even though the premises are true (gold¬sh are notoriously

mortal). The second conclusion is not a logical consequence of its premises.

Roughly speaking, an argument is logically valid if and only if the conclu-

logically valid

arguments sion must be true on the assumption that the premises are true. Notice that

this does not mean that an argument™s premises have to be true in order for it

to be valid. When we give arguments, we naturally intend the premises to be

true, but sometimes we™re wrong about that. We™ll say more about this possi-

bility in a minute. In the meantime, note that our ¬rst example above would

be a valid argument even if it turned out that we were mistaken about one

of the premises, say if Socrates turned out to be a robot rather than a man.

It would still be impossible for the premises to be true and the conclusion

false. In that eventuality, we would still say that the argument was logically

valid, but since it had a false premise, we would not be guaranteed that the

conclusion was true. It would be a valid argument with a false premise.

Here is another example of a valid argument, this time one expressed in

the blocks language. Suppose we are told that Cube(c) and that c = b. Then it

certainly follows that Cube(b). Why? Because there is no possible way for the

premises to be true”for c to be a cube and for c to be the very same object

as b”without the conclusion being true as well. Note that we can recognize

that the last statement is a consequence of the ¬rst two without knowing that

Chapter 2

Valid and sound arguments / 43

the premises are actually, as a matter of fact, true. For the crucial observation

is that if the premises are true, then the conclusion must also be true.

A valid argument is one that guarantees the truth of its conclusion on

the assumption that the premises are true. Now, as we said before, when we

actually present arguments, we want them to be more than just valid: we also

want the premises to be true. If an argument is valid and the premises are also

true, then the argument is said to be sound. Thus a sound argument insures sound arguments

the truth of its conclusion. The argument about Socrates given above was not

only valid, it was sound, since its premises were true. (He was not, contrary

to rumors, a robot.) But here is an example of a valid argument that is not

sound:

All rich actors are good actors. Brad Pitt is a rich actor. So he must

be a good actor.

The reason this argument is unsound is that its ¬rst premise is false.

Because of this, although the argument is indeed valid, we are not assured

that the conclusion is true. It may be, but then again it may not. We in fact

think that Brad Pitt is a good actor, but the present argument does not show

this.

Logic focuses, for the most part, on the validity of arguments, rather than

their soundness. There is a simple reason for this. The truth of an argument™s

premises is generally an issue that is none of the logician™s business: the truth

of “Socrates is a man” is something historians had to ascertain; the falsity of

“All rich actors are good actors” is something a movie critic might weigh in

about. What logicians can tell you is how to reason correctly, given what you

know or believe to be true. Making sure that the premises of your arguments

are true is something that, by and large, we leave up to you.

In this book, we often use a special format to display arguments, which we

call “Fitch format” after the logician Frederic Fitch. The format makes clear Fitch format

which sentences are premises and which is the conclusion. In Fitch format, we

would display the above, unsound argument like this:

All rich actors are good actors.

Brad Pitt is a rich actor.

Brad Pitt is a good actor.

Here, the sentences above the short, horizontal line are the premises, and

the sentence below the line is the conclusion. We call the horizontal line the

Fitch bar. Notice that we have omitted the words “So . . . must be . . .” in the Fitch bar

conclusion, because they were in the original only to make clear which sen-

tence was supposed to be the conclusion of the argument. In our conventional

Section 2.1

44 / The Logic of Atomic Sentences

format, the Fitch bar gives us this information, and so these words are no

longer needed.

Remember

1. An argument is a series of statements in which one, called the conclu-

sion, is meant to be a consequence of the others, called the premises.

2. An argument is valid if the conclusion must be true in any circum-

stance in which the premises are true. We say that the conclusion of

a logically valid argument is a logical consequence of its premises.

3. An argument is sound if it is valid and the premises are all true.

Exercises

2.1 (Classifying arguments) Open the ¬le Socrates™ Sentences. This ¬le contains eight arguments

‚| separated by dashed lines, with the premises and conclusion of each labeled.

1. In the ¬rst column of the following table, classify each of these arguments as valid or

invalid. In making these assessments, you may presuppose any general features of the

worlds that can be built in Tarski™s World (for example, that two blocks cannot occupy

the same square on the grid).

Sound in Sound in

Argument Valid? Socrates™ World? Wittgenstein™s World?

1.

2.

3.

4.

5.

6.

7.

8.

2. Now open Socrates™ World and evaluate each sentence. Use the results of your evaluation

to enter sound or unsound in each row of the second column in the table, depending on

whether the argument is sound or unsound in this world. (Remember that only valid

arguments can be sound; invalid arguments are automatically unsound.)

Chapter 2

Valid and sound arguments / 45

3. Open Wittgenstein™s World and ¬ll in the third column of the table.

4. For each argument that you have marked invalid in the table, construct a world in

which the argument™s premises are all true but the conclusion is false. Submit the

world as World 2.1.x, where x is the number of the argument. (If you have trouble

doing this, you may want to rethink your assessment of the argument™s validity.) Turn

in your completed table to your instructor.

This problem makes a very important point, one that students of logic sometimes forget. The

point is that the validity of an argument depends only on the argument, not on facts about

the speci¬c world the statements are about. The soundness of an argument, on the other hand,

depends on both the argument and the world.

2.2 (Classifying arguments) For each of the arguments below, identify the premises and conclusion

by putting the argument into Fitch format. Then say whether the argument is valid. For the

¬rst ¬ve arguments, also give your opinion about whether they are sound. (Remember that

only valid arguments can be sound.) If your assessment of an argument depends on particular

interpretations of the predicates, explain these dependencies.

1. Anyone who wins an academy award is famous. Meryl Streep won an academy award.

Hence, Meryl Streep is famous.

2. Harrison Ford is not famous. After all, actors who win academy awards are famous,

and he has never won one.

3. The right to bear arms is the most important freedom. Charlton Heston said so, and

he™s never wrong.

4. Al Gore must be dishonest. After all, he™s a politician and hardly any politicians are

honest.

5. Mark Twain lived in Hannibal, Missouri, since Sam Clemens was born there, and Mark

Twain is Sam Clemens.

6. No one under 21 bought beer here last night, o¬cer. Geez, we were closed, so no one

bought anything last night.

7. Claire must live on the same street as Laura, since she lives on the same street as Max

and he and Laura live on the same street.

2.3 For each of the arguments below, identify the premises and conclusion by putting the argument

into Fitch format, and state whether the argument is valid. If your assessment of an argument

depends on particular interpretations of the predicates, explain these dependencies.

1. Many of the students in the ¬lm class attend ¬lm screenings. Consequently, there must

be many students in the ¬lm class.

2. There are few students in the ¬lm class, but many of them attend the ¬lm screenings.

So there are many students in the ¬lm class.

Section 2.1

46 / The Logic of Atomic Sentences

3. There are many students in the ¬lm class. After all, many students attend ¬lm screen-

ings and only students in the ¬lm class attend screenings.

4. There are thirty students in my logic class. Some of the students turned in their

homework on time. Most of the students went to the all-night party. So some student

who went to the party managed to turn in the homework on time.

5. There are thirty students in my logic class. Some student who went to the all-night

party must have turned in the homework on time. Some of the students turned in their

homework on time, and they all went to the party.

6. There are thirty students in my logic class. Most of the students turned in their home-

work on time. Most of the students went to the all-night party. Thus, some student

who went to the party turned in the homework on time.

2.4 (Validity and truth) Can a valid argument have false premises and a false conclusion? False

premises and a true conclusion? True premises and a false conclusion? True premises and a

true conclusion? If you answer yes to any of these, give an example of such an argument. If

your answer is no, explain why.

Section 2.2

Methods of proof

Our description of the logical consequence relation is ¬ne, as far as it goes.

But it doesn™t give us everything we would like. In particular, it does not tell

us how to show that a given conclusion S follows, or does not follow, from

some premises P, Q, R, . . . . In the examples we have looked at, this may not

seem very problematic, since the answers are fairly obvious. But when we are

dealing with more complicated sentences or more subtle reasoning, things are

sometimes far from simple.

In this course you will learn the fundamental methods of showing when

claims follow from other claims and when they do not. The main technique

for doing the latter, for showing that a given conclusion does not follow from

some premises, is to ¬nd a possible circumstance in which the premises are

true but the conclusion false. In fact we have already used this method to

show that the argument about Lucretius was invalid. We will use the method

repeatedly, and introduce more precise versions of it as we go on.

What methods are available to us for showing that a given claim is a

logical consequence of some premises? Here, the key notion is that of a proof.

A proof is a step-by-step demonstration that a conclusion (say S) follows

proof

from some premises (say P, Q, R). The way a proof works is by establishing a

series of intermediate conclusions, each of which is an obvious consequence of

Chapter 2

Methods of proof / 47

the original premises and the intermediate conclusions previously established.

The proof ends when we ¬nally establish S as an obvious consequence of the

original premises and the intermediate conclusions. For example, from P, Q, R

it might be obvious that S1 follows. And from all of these, including S1 , it

might be obvious that S2 follows. Finally, from all these together we might

be able to draw our desired conclusion S. If our individual steps are correct,

then the proof shows that S is indeed a consequence of P, Q, R. After all, if

the premises are all true, then our intermediate conclusions must be true as

well. And in that case, our ¬nal conclusion must be true, too.

Consider a simple, concrete example. Suppose we want to show that Socrates

sometimes worries about dying is a logical consequence of the four premises

Socrates is a man, All men are mortal, No mortal lives forever, and Everyone

who will eventually die sometimes worries about it. A proof of this conclusion

might pass through the following intermediate steps. First we note that from

the ¬rst two premises it follows that Socrates is mortal. From this intermedi-

ate conclusion and the third premise (that no mortal lives forever), it follows

that Socrates will eventually die. But this, along with the fourth premise,

gives us the desired conclusion, that Socrates sometimes worries about dying.

By the way, when we say that S is a logical consequence of premises

P, Q, . . . , we do not insist that each of the premises really play an essen-

tial role. So, for example, if S is a logical consequence of P then it is also a

logical consequence of P and Q. This follows immediately from the de¬nition

of logical consequence. But it has a corollary for our notion of proof: We do

not insist that each of the premises in a proof actually be used in the proof.

A proof that S follows from premises P1 , . . . , Pn may be quite long and