Symbolic Logic

1.  Write two arguments in English, one in the form of modus ponens and one in the form of modus tollens. Then, write the arguments in symbols using sentence letters and truth-functional connectives. (If your computer does not have all the symbols needed, use some other symbol you do have access to and explain what its meaning is.) What advantages does being able to symbolize our arguments provide? Are there disadvantages to using this technique to make the structure of our arguments more explicit and clear?

 

150 words

6

Elementary Symbolic Logic

Heinz-Dieter Falkenstein

Against logic there is no armor
like ignorance.

—Laurence Peter

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CHAPTER 6Section

6.1 The Logic of Sentences

Now that we have looked at inductive arguments, deductive arguments, and the com-ponents—premises and conclusions—that make them up, we can turn to making our
understanding of deductive reasoning more precise. We do so by introducing a few sym-
bols for sentences and for the kinds of terms—such as and and or—that connect sentences,
and we use these symbols to look at the structure of both sentences and arguments. For
the most part, we look at material we have already examined; here we just use symbols to
make the various structures involved a bit more explicit. Arguments often get sidetracked
because of the information presented, and although that information is important, we can
avoid being sidetracked in this way by focusing on the structure of the arguments. Sym-
bols are helpful in keeping the focus on such structures.

What We Will Be Exploring

• We will look at how symbols can be applied to sentences, and then to arguments.
• We will examine the notion of a truth function and the kinds of specific logical properties sentences

possess.
• We will see how truth tables can be used to evaluate certain kinds of sentences, as well as testing

deductive arguments for validity.
• We will use basic symbolic logic to examine some earlier material and see how it can be made more

precise.

6.1 The Logic of Sentences

We begin by seeing how to apply symbols—sentence letters—to sentences, first using basic sentences and then using sentences constructed out of these basic sentences.

Assertoric Sentences and Sentence Letters

Earlier we looked at various strings of words: some were questions, some were commands,
and some were assertions. Here are some examples of the kinds of sentences we examined
earlier, as well some new, compound sentences, which assert more than one claim:

1. Cheddar cheese is better than American cheese.
2. The window is broken.
3. Turn left at the next light.
4. Art is a great dancer.
5. Art is very popular.
6. Art is very popular and he is a great dancer.
7. Art is very popular because he is a great dancer.
8. John and Mary got married.
9. John and Mary had a baby.
10. John and Mary got married and had a baby.
11. John and Mary had a baby and got married.
12. Can you hear that music?

Can you determine which of these make assertions or state some kind of claim? You’ll
find that most of the sentences on this list do. Sentence 3, however, is an imperative, and

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CHAPTER 6Section 6.1 The Logic of Sentences

sentence 12 is a question; we are not able to evalu-
ate those sentences, because they do not put forth
claims that can be evaluated as true or false; that
is, they are not assertoric sentences. In the type of
logic we discuss in this text, we are able to evalu-
ate only assertoric sentences. (Although we do
not look at them, other logics have been devel-
oped to consider such things as imperatives and
questions.)

A few of these sentences, however, need spe-
cial mention. Consider the sentences about Art,
his dancing skills, and his popularity. Sentences
4 and 5 are both assertoric sentences; they each
make claims that can be treated as either true or
false. When these sentences are put together in
sentence 6, we have a compound sentence that
asserts two claims, but because of the way they are
put together—with “and” connecting them—we
can continue to treat sentence 6 as an assertoric
sentence. But what about sentence 7?

Let’s imagine that we know it is, in fact, true that
Art is a great dancer and that Art is very popular.
If so, we know that the compound sentence—Art
is a great dancer and he is very popular—is also
true. But do we know that the reason Art is popu-
lar is his skill at dancing? Could it be the case that he is a fabulous dancer, very popular,
and yet be popular for some other reason than his dancing ability? That seems quite pos-
sible. So what we see is that we may know that both component sentences—sentences
4 and 5—are true, but still not know that sentence 7 is true. Again, this is due to the way
the sentences are put together. In sentence 7, they are put together using “because”;
since we don’t have enough information to know that the reason Art is popular is his
dancing skill, we can’t determine the truth of the sentence as a whole. We see some
more examples of this a bit later; at this point, however, we introduce the idea of a truth
function. The idea behind a truth function is simply that if we know the truth-values of
the components of a sentence, the truth-value of complex sentences put together out of
these components can be determined. For instance, if I know it is true that it is raining,
and it is true that there is thunder, I also know it is true that it is raining and that there
is thunder. For the reason we just saw, we cannot treat sentence 7 truth functionally: we
don’t know the truth-value of the entire sentence simply by knowing the truth-values
of its components, due to the term “because” that connects those two component sen-
tences. In contrast, sentence 6 can be treated truth functionally: we can determine its
truth-value by knowing the truth-values of its components, due to the term “and” that
connects the two component sentences. All of this is to make what is really one simple
point: “and” can be treated in a way distinct from “because,” and this difference means
that “and” can be treated as a truth-functional connective and “because” cannot.

Thinkstock

Logicians try to determine whether
they can treat a sentence truth func-
tionally by determining its truth-value.

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CHAPTER 6Section 6.1 The Logic of Sentences

Highlights: What Is a Truth Function?

Let’s assume it is true that Kirk likes hot dogs and that it is also true that Kirk likes hamburgers.
If so, we can determine that the more complex sentence Kirk likes hot dogs and hamburgers
is true.

If we can determine the truth-value of a complex sentence just by knowing the truth-value of
the components that make up the complex sentence, we can say that the complex sentence
can be treated truth functionally. A truth function is a function that permits one to go from
a set of truth-values to another truth-value. But it is probably easier to think of it this way: a
sentence can be treated truth functionally if by knowing the truth-value of the components
of a longer sentence, we can determine the truth-value of the longer sentence.

For instance, let’s assume Sacramento is the capital of California; so it is true that Sacramento
is the capital of California. But perhaps Paul isn’t very good at geography and believes that
Los Angeles is the capital of California. Using sentence letters, let “S” be “Sacramento is the
capital of California,” and “L” be “Los Angeles is the capital of California.” If we know that “S”
is true, and that “L” is false, we are not able to determine the truth-value of these sentences:

Paul believes that L.

Paul believes that S.

Since we cannot determine the truth-value of the longer sentence by knowing the truth-
value of its component sentence and the phrase “Paul believes that,” we can’t go from the
truth-value of the component sentences to the truth-value of the longer sentence. So this
isn’t a truth function, and this also shows (for some complicated reasons) that “Paul believes
that” cannot be treated truth functionally.

If we were to run into a sentence such as “Paul believes that L,” we would just symbolize it as
“P” and treat it as an entire sentence.

Logicians typically use the letters “P” and “Q” to represent sentences (if we need more
letters, we just continue along the alphabet, “R,” “S,” etc.). We can represent any assertoric
sentence—which we call simply “sentences” from now on—with a sentence letter, such as
“P.” So “Cheddar cheese is better than American cheese” can be represented as “P.” Simi-
larly, here are some of the other sentences from our list represented with sentence letters:

1. Cheddar cheese is better than American cheese.

P

2. The window is broken. P

3. Turn left at the next light. command, not a sentence

4. Art is a great dancer. P

5. Art is very popular.

Q

6. Art is very popular and he is a great dancer.

P and Q

7. Art is very popular because he is a great dancer. P

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CHAPTER 6Section 6.1 The Logic of Sentences

8. John and Mary got married. P

9. John and Mary had a baby. Q

10. John and Mary got married and had a baby. P and Q

11. John and Mary had a baby and got married. Q and P

12. Can you hear that music? question, not a sentence

13. Texas is larger than China. P

14. Wanda believes Texas is larger than China. P (have to treat as a single
sentence)

You will notice here that sentence 7 is represented by “P,” since, as we saw, we cannot treat
its components separately. But given what we saw about “and,” we are able to treat the
components of sentence 6 separately and show that structure by representing each com-
ponent sentence with individual sentence letters.

You may also want to notice sentences 10 and 11. These sentences help show that logic may
treat two sentences differently from how we might treat them in conversation. In ordinary
speech, we might think that sentences 10 and 11 are different, each indicating a different
order of events—sentence 10 indicating that the marriage took place before the baby was
born and sentence 11 indicating that the baby was
born before John and Mary got married. But logic
doesn’t make this distinction. In logic, the two sen-
tences really say the same thing: two things hap-
pened, with no indication of which occurred first.
We always want to be careful not to introduce
more information than the sentences themselves
allow us to consider; in these two sentences, the
structure does not indicate the sequence of events.

Truth Functions and Bivalence

In the previous section, we looked at truth functions; the basic idea is that we have a
truth function if we can determine the truth-value of a longer, compound sentence by
knowing the truth-values of the shorter, component sentences that make it up. We used
examples of longer sentences with “because” and “believes that” as instances where we
can’t determine the truth-value of the longer sentence just by knowing the truth-values
of its component sentences. This is the reason that “because” and “believes that” are not
truth-functional connectives.

This may sound complicated, but another example may make it clear. Let’s symbolize the
sentence “One of George Washington’s little brothers was named John” as P. Sentence P is,
in fact, true. But even if we happen to know that this sentence is true, would the following
sentence (which we call “Q”) be true?

(Q): It is well-known that one of George Washington’s little brothers was
named John.

Stop and Think: Assertoric Sentences

See if you can identify which of the following
are assertoric sentences and which are not.

• You must work harder.
• Paris is the capital of Hawaii.
• Paris is the capital of France.

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CHAPTER 6Section 6.1 The Logic of Sentences

It seems pretty unlikely that Q is true, even if P is
true. Q introduces a new issue—how well-known
this is—and we can’t be very confident that it is
well-known (or not). So here we know the truth-
value of the component P but still don’t know the
truth-value of the longer sentence Q. Since we
can’t move from knowing the truth-value of P to
knowing the truth-value of Q, this doesn’t give us
a truth function. As we saw earlier with “believes
that,” “it is well-known that” also doesn’t work as
a truth-functional connective.

We look more closely at these truth-functional con-
nectives as we go along. We have already seen that
“and” can be such a connective: we can determine
the truth-value of P and Q if we know the truth-
value of P and we know the truth-value of Q. The
connective “or” also can be a truth-functional con-
nective: consider the following sentence, which
we symbolize as R:

(R): I will take calculus, or I will
take accounting.

If we let “P” be “I will take calculus” and “Q” be
“I will take accounting,” we know that if P is true
and Q is true, R is true. That is, knowing the truth-
values of the component sentences—in this case

they are both true—we can determine the truth-value of the larger sentence R. Whatever
the truth-value of P is, and whatever the truth-value of Q is, once we know those truth-
values, we can determine the truth-value of the larger sentence R. So this gives us a truth
function, and “or” operates as a truth-functional connective.

So far in this discussion, we have assumed that there are only two truth-values—that
sentences are either true or false. We should probably make this assumption explicit, and
to do so we now introduce the term bivalence. Bivalence simply means “two values,” and
here we operate on an assumption of bivalence (pronounced buy-VAY-lence). That is, we
use only the two truth-values of true and false, and no other truth-value is ever used. This,
generally, is a pretty safe assumption, but it may be worth pointing out that there are rea-
sons not to assume bivalence. For instance, we may not be willing to say that the following
sentence is true or false:

It will rain in Cleveland, Ohio, on October 14, 2056.

Bivalence says that either this sentence is true, or it is not true (false). But given that we do
not know now, or have any way of knowing now, whether it is true or false, some logicians
and philosophers hesitate to say that this sentence is either true or false. Others also point
to sentences that use vague terms, such as “tall” or “bald,” to call bivalence into question.
Imagine Steve is six feet, two inches; do we want to say the sentence “Steve is tall” is either

Keith Brofsky/Thinkstock

Bivalence says that a sentence is either
true or false, but some logicians hesi-
tate, especially if sentences use vague
terms like “tall.”

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CHAPTER 6Section 6.1 The Logic of Sentences

true or false? Again, some philosophers and logicians worry about such vague or ambigu-
ous terms enough to cause them to hesitate in assuming that all sentences are either true
or false, or assume bivalence.

These are complicated—and quite interesting—issues that are mainly studied in the
philosophy of logic and the philosophy of language. But in this discussion, we assume
bivalence in order to avoid these kinds of complications. Therefore, every sentence we
consider has one of two possible truth-values—true or false—and no other truth-values
are possible. We saw in Chapter 2 that the kind of logic we are using here treats the past,
present, and future versions of sentences the same—technically, we ignore the tense of the
sentence—so we do not have to worry about problems with statements about the future.
Nor do we have to worry about vague terms; we consider “Frank is bald” to be either true
or false; in other words, we treat such sentences bivalently.

Truth-Functional Connectives

We have already seen two truth-functional connectives: “and” and “or.” We saw that we
can determine the truth-value of compound sentences that are built with these connec-
tives if we know the truth-value of the components of such sentences. We can now make
all of this more precise and introduce the remaining standard truth-functional connec-
tives. Then we can begin to symbolize sentences and arguments in order to look at their
logical structure.

Conjunction (&)
Since we already looked at “and” a bit, that is where we start our discussion. As a connec-
tive, “and” is symbolized by “&” and is called the conjunction. Let’s take a sentence and
translate it into symbols.

Charlotte likes to swim, and Charlotte likes to play basketball.

Let “P” be “Charlotte likes to swim,” and “Q” be “Charlotte likes to play basketball,”
which gives us

P and Q

Using the new symbol for conjunction to express “and,” we get

P & Q

We now need to figure out when this conjunction, P & Q, is true. We define such sentences
as true in only one situation: when all of its component sentences are true. If it is true that
Charlotte likes to swim, and it is true that she likes to play basketball, then it is true that
she likes to do both. So “P and Q” is true when P is true and Q is true, and only when

P

is true and Q is true. If we discover that it is false that Charlotte likes to swim, or false
that she likes to play basketball, or false that she likes to do both, then the conjunction as
a whole is false. So the conjunction is defined as true only when all of its components are

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CHAPTER 6Section 6.1 The Logic of Sentences

true. We can express this definition in what is known as a truth table, which lists all the
possible truth-values and allows us to give a complete display of all sentences that are
structured as conjunctions, or as “P & Q.” (We look more closely at truth tables later, to see
how they can demonstrate various properties of sentences and arguments.)

P Q P & Q
T T T
F T

F

T F F
F F F

Truth tables are a handy way of displaying all the possible truth-values that can be assigned
to sentences, and they allow us to define the various truth-functional connectives just by
using the truth-values of true and false. They can be used to demonstrate the properties
of such sentences as those that are necessarily true, or true for all possible truth-values of
the component sentences (known as “tautologies”), and those that are necessarily false, or
false for all possible truth-values of the component sentences (contradictions). They can
also demonstrate other things, including that certain kinds of arguments are valid or not
valid. Once you see how they work, truth tables can be extremely useful, although we will
have to introduce a bit more material to see all the things truth tables offer.

Disjunction (v)
The other connectives are defined in similar ways. Previously, we looked at “or” a bit, and
now we can explain it more explicitly by symbolizing it and then giving a truth table that
defines it.

The truth-functional connective “or” is written “v”—sometimes called a wedge—and the
connective is called the disjunction. So “P v Q” is read as “P or Q.” Using our earlier exam-
ple, we now symbolize it and then show its truth function using a truth table.

I will take calculus, or I will take accounting.

Letting “P” be “I will take calculus” and “Q” be “I
will take accounting,” we get

P or Q

Using the new symbol for disjunction, we get

P v Q

The disjunction is somewhat different than con-
junction, for it is true when either one of its com-
ponents is true, or when both its components
are true. So this sentence would be true if you
take calculus, or if you take accounting, or if you
take both calculus and accounting. It is false only
when all of its components are false; that is, if

Kablonk! Kablonk!/Photolibrary

With a disjunction connective, you
would paint a room peacock blue OR
golden wheat, not peacock blue AND
golden wheat.

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CHAPTER 6Section 6.1 The Logic of Sentences

you took neither calculus nor accounting, P v Q would be false. This is what its truth
table looks like:

P Q P v Q
T T T
F T T
T F T
F F F

At first, this may look similar to the previous truth table we saw for conjunction (“&”). But
notice the middle two rows; here, with disjunction (“v”), each of these sentences is true
because only one of its component sentences is true.

Disjunction, however, brings with it one ambiguity, or complication. Sometimes people
use disjunction in a way that means one thing or the other, but not both. Perhaps the
server at a restaurant tells you

With your dinner, you get soup or salad.

Generally, we assume this means one gets either soup or salad, but not both. This is what
is called “exclusive” disjunction, because it excludes the possibility of both things in the
disjunction. This can occasionally cause confusion. Imagine the annoying six-year-old
next door is bothering you, as he often does. You tell him, “If you go away for the rest of
the day, I will take you to play video games or to get ice cream.” He agrees, and the next
day you take him to play video games. A couple of days later, he returns for you to take
him for ice cream. He understood “I will take you to play video games or to get ice cream”
inclusively—video games, ice cream, or both—while you intended it exclusively—either
video games or ice cream, but not both.

Generally, the context in which such disjunctions are used makes it clear whether exclu-
sive disjunction or inclusive disjunction was the intended meaning. But we can eliminate
any such worries, because we define the “v” symbol to mean inclusive disjunction. So the
first row of its truth table, as we saw, indicates that “P v Q” is true when “P” is true and
“Q” is true. Whenever we see the truth-functional connective “v,” then we know the sen-
tence is true when P is true, Q is true, or both are true. In other words, “v” is inclusive
disjunction.

Negation (~)
When we negate something, we deny it; for us, that simply means we change the truth-
value, from true to false or from false to true. Negation is symbolized using “~” and is
often read “not.” The more explicit way to state the negation is “it is not the case that,” but
generally “not” does the trick.

Here are a few sentences and their negations:

It is snowing It is not the case that it is snowing
I like football I don’t like football
She doesn’t watch too much TV It is not the case that she doesn’t watch too much TV

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CHAPTER 6Section 6.1 The Logic of Sentences

As the last sentence indicates, some sentences have negations within them; we negate
such a sentence by denying what it says. And for all of these sentences, if we represent
the sentence in the left-hand column as “P,” the sentences in the right-hand column are
all written as “~ P.” As should be clear, if P is true, then ~ P is false; if P is false, ~ P is true.
This gives us a very concise truth table:

P

~ P

T F
F T

What we also see from this is that two negations—(~ (~ P))—cancel each other out.

In this kind of logic, then, “not-not-P” is the same as “P” (they have the same truth-
value). In fact, any even number (2, 4, etc.) of negations cancel each other out. Although it
is unusual and can be confusing to see a large number of negations, they are easily dealt
with. So, for example, if someone says, “It is not the case that I do not dislike football,”
we can represent “I like football” as “P.” This sentence then becomes “~ (~ (~ P))),” we
see we have an odd number of negations (here, 3). All but one cancel out, meaning that
this sentence is equivalent to “~ P.” This kind of example also shows that sometimes
dealing with sentences in their logical structure is easier than dealing with them in natu-
ral language.

Conditional (→)
Whereas negation is probably the easiest truth-functional connective to understand, the
conditional can be a little confusing at first. Let’s first look at some sentences that could
come up in conversation:

If it is too hot today, then she will turn on the air conditioning.

If I win the lottery, then I will buy a new car.

If you eat too many chicken wings at night, then you won’t feel very
good the next morning.

These are called conditional sentences, for the “if”
part of the sentence introduces a condition, and
the “then” part of the sentence indicates what fol-
lows from that condition. The condition, which is
introduced by “if,” is called the antecedent; what
follows, indicated by “then,” is called the conse-
quent. If we let the antecedent be symbolized by
“P,” and the consequent be symbolized by “Q,”
then we can represent the three preceding sen-
tences by writing them in this way:

P → Q

read as “if P then Q.”

Thinkstock

“If I win the lottery, then I will drive
around in a limo and shop all the
time” is an example of a conditional
statement.

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CHAPTER 6Section 6.1 The Logic of Sentences

Under what circumstances are conditional sentences said to be true (or false)? Philoso-
phers have spent a good deal of time investigating this question, for it is not entirely clear,
at least at first, what to do when the antecedent of a conditional is false.

Imagine you are in an English class, and your grade is based on three exams. We might
describe the situation with this conditional:

If I get A’s on all my exams, I will get an A in the course.

It seems clear that, if in fact you do get A’s on all your exams and you do get an A in the
course, we could say that the conditional as a whole is true. It seems equally clear that, if
you get A’s on all your exams and you do not get an A in the course, we should say that
the conditional states something that is false. But what happens if you do not get A’s on
all your exams? In this case, the antecedent is false, and you have not met the condition.
Perhaps you do not get A’s on all the exams and still get an A in the course. Or perhaps
you do not get A’s on all the exams and you do not get an A in the course. How are these
kinds of conditionals treated, then, when the condition (the “if” part of the conditional, or
the antecedent) is false?

Various responses have been made to this question; indeed, entire books have been writ-
ten about how one handles such a situation. One might think that the conditional as a
whole is neither true nor false if the antecedent is false; but in the kind of logic we are
considering here, that would be to introduce a third truth-value, which conflicts with
our basic assumption of bivalence—that all sentences are either true or false. On the
other hand, we might think that the conditional as a whole should be false; but this
seems to conflict with common sense. (For more consideration of this particular concern,
a nice discussion is offered by the philosopher Peter Suber here: http://www.earlham
.edu/~peters/courses/log/mat-imp.htm.) We have three options in giving a truth-value
to a conditional statement whose antecedent is false. We can give the conditional a third
truth-value, we can give it the value “false,” or we can give it the value “true.” Our
assumption of bivalence eliminates the possibility of a third truth-value; more complex
reasons than we can consider here eliminate the possibility of giving it the value of false.
So we are left with the only other option—that conditionals with false antecedents are
given the truth-value of true. This, then, gives us the following truth table, which defines
this connective:

P Q P → Q
T T T
F T T
T F F
F F T

In other words, a conditional is true except when the “if” part of the sentence (the anteced-
ent) is true and the “then” part of the sentence (the conditional) is false.

Here’s another way of thinking of the conditional. Caryn asks her teacher what she needs
to do to get an A in her class. The teacher says, “If you get A’s on all the tests, you will get
an A in the class.” Caryn asks, “So it couldn’t be true that I get A’s on all the tests and not
get an A in the class?” The teacher says that is right.

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CHAPTER 6Section 6.1 The Logic of Sentences

This gives us another way of seeing how to think of the conditional. We can use a truth
table to show that since Caryn’s version of the teacher’s statement says the same thing as
the “if . . . then” statement, then they will have the same truth functional definition. Let
“P” represent “I get A’s on all the tests,” and “Q” represent “I get an A in the class.” Here
are our sentences, then, in symbols:

If I get A’s on all my tests, I get an A in class: P → Q

It is not the case that I get A’s on all my tests and not get an A in class: ~ (P & ~ Q)

And here is a truth table showing their truth functions. Since they say the same thing, they
have the same truth function:

P Q P → Q ~ P ~ Q (P & ~ Q) ~ (P & ~ Q)
T T T F F F T
F T T T F F T
T F F F T T F
F F T T T F T

Biconditional (↔)
The last truth-functional connective we look at is
called the biconditional. In contrast to the sym-
bol for the conditional, the arrow for the bicon-
ditional (↔) points in both directions, because it
establishes a connection between the component
sentences that is stronger than that established by
the conditional. The biconditional is read as “if
and only if.” We would read “P ↔ Q,” then, as “P
if and only if Q.” This establishes the kind of con-
nection between component sentences that might
be seen in a definition. Here’s an example:

A two-dimensional object with
three sides is a triangle if and only
if a triangle is a two-dimensional
object with three sides [this is just
the definition of a triangle]

Here, we can symbolize the first part (“a two-dimensional object with three sides is a
triangle”) with P, and symbolize the second part (“a triangle is a two-dimensional object
with three sides”) with Q. That gives us:

P ↔ Q

which is read, again, as “P if and only if Q.” Here is another way of stating this: if it is
a two-dimensional object with three sides, it is a triangle; and if it is a triangle, it is a
two-dimensional object with three sides. As you might have already realized, this is the
kind of truth-functional connective one sees with definitions, as one might run across in
mathematics.

Stop and Think: Conditional Sentences

Here are some conditional sentences in
English. Identify which part is the anteced-
ent and which part is the consequent and
symbolize them with the truth-functional
connective “→.”

• If we buy popcorn at the movies,
then we will spend too much money.

• Emma looks tired if she doesn’t get a
good night’s sleep.

• If they make us wait any longer, we
should go somewhere else.

• If I have to do any more homework,
I’m going to scream.

• We should download a different song
if they don’t have the one we want.

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CHAPTER 6Section 6.1 The Logic of Sentences

We can return to our earlier example to see the difference between the conditional and the
biconditional. Remember our earlier conditional stated

If I get A’s on all my exams, I will get an A in the course.

or, as symbolized

P → Q.

At this point, it is helpful to use the concept of necessary and sufficient conditions, which
we discussed in Chapter 3, to explain the conditional, the biconditional, and the difference
between them. In this example, P—the antecedent—is the sufficient condition for Q. That
is, if P takes place (if you get all A’s on your exams), that is enough to know that Q takes
place (that you get an A in the course).

Highlights:

Necessary and Sufficient Conditions

In a conditional sentence, such as “if P then Q” (P → Q), the antecedent is said to be the
sufficient condition for Q, and Q is the necessary condition for P. These are very helpful
terms and can be very useful in understanding conditional sentences. Here are examples of
a sufficient condition, a necessary condition, and a necessary and sufficient condition. It is a
good idea to get comfortable with this language.

Sufficient Condition
If I live in Chicago, then I live in Illinois.

(Here we see that if we know a person lives in Chicago, that is enough to know that that
person lives in Illinois; to live in Chicago is a sufficient condition for living in Illinois.)

Necessary Condition
If a person is to be elected the president of the United States, he or she must be 35 years old
(or older).

Here we see that one must be at least 35 years old to be elected president, although obviously
that isn’t enough to get elected. It is then a necessary condition to become president, to be
35 years old or older.

Necessary and Sufficient Condition
If Amy is a female parent, then she is a mother; if Amy is a mother, then she is a female
parent. (Or Amy is a female parent if and only if she is a mother.)

It is required both, in order to qualify as a mother, to be female and to be a parent; if you
are a female and a parent, that’s enough to know you are a mother. So, here, being a female
parent is both necessary and sufficient for being a mother (and being a mother is both
necessary and sufficient for being a female parent).

A website that goes into more detail on these relationships can be found here:

http://plato.stanford.edu/entries/necessary-sufficient/

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CHAPTER 6Section 6.1 The Logic of Sentences

Let’s imagine the course requirements are much stricter than the ones indicated so far.
Perhaps the teacher says the only way to get an A in the course is to get A’s on all the
exams. The teacher might put it this way: If you get A’s on all your exams, you will get
an A in the course, and if you get an A in the course, you will have gotten A’s on all your
exams. Symbolized with our truth-functional connectives, we can write it this way:

(P → Q) & (Q → P)

In other words, P is a sufficient condition for Q, and Q is a sufficient condition for P. We
know that if you get A’s on all your exams, you get an A in the course, and it is the only
way to get an A in the course. We know one gets an A in the course if and only if one gets
A’s on all the exams. This, then, is the biconditional, and as we see, stating the relation-
ship with a biconditional can be expressed as we just did: using a conjunction (“&”) of
two conditionals, expressing the idea that each antecedent is a sufficient condition for the
consequent. Combining the two conditionals, then, gives us the biconditional, which we
can define using the following truth table:

P Q P ↔ Q
T T T
F T F
T F F
F F T

Perhaps the easiest way to remember this truth-functional connective is by seeing that it is
true when the component sentences have the same truth-value. Where both “P” and “Q”
are true, “P ↔ Q” is true; where both “P” and “Q” are false, “P ↔ Q” is true. The bicondi-
tional is false only when the two sentences have different truth-values.

The Truth-Functional Connectives
• “~” not/it is not the case that [negation]
• “v” or [disjunction]
• “&” and [conjunction]
• “→“ if . . . then [conditional]
• “↔“ if and only if [biconditional]

P Q ~ P P V Q P &Q P → Q P ↔ Q
T T F T T T T
F T T T F T F
T F F T F F F
F F T F F T T

Symbolizing Sentences

Now that we’ve discussed sentence letters—P, Q, and, if we need them, R, S, etc.—and the
truth-functional connectives, we can use these to symbolize sentences. On occasion, we
also use parentheses to group parts of sentences; this helps make the structure clearer and
helps avoid any ambiguity that might otherwise arise.

1. Cynthia can play the guitar and the piano.

2. Either I’m crazy or everyone loves logic.

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CHAPTER 6Section 6.1 The Logic of Sentences

3. If we don’t pay the electric bill, our power will get cut off.

4. Henry likes online education because he can go to class in his pajamas.

5. I will go to the game only if I can get a ticket.

Let’s take each sentence, show how to symbol-
ize its components (if any), and then use the
truth-functional connectives to give a complete
symbolization.

1. Cynthia can play the guitar and the
piano.

Let P represent “Cynthia can play the guitar” and
Q represent “Cynthia can play the piano.”

P and Q
P & Q
2. Either I’m crazy or everyone loves logic.

Let P represent “I’m crazy” and Q represent
“everyone loves logic.”

P or Q
P v Q

Table 6.1 Sentences

In summary, we use two kinds of letters when we symbolize sentences and arguments: sentence letters
and truth-functional connectives.

Sentence Letters
• Also called variables
• The symbols that indicate propositions, sentences
• p, q, r, s, t, and so on as needed

Truth-Functional Connectives
• The symbols that indicate the relationship between propositions
• &, v, ~, →, ↔

Connective Meaning Variables & Connectives Example

& Conjunction
(and)

p & q I will take accounting and I
will take calculus

v Disjunction
(or)

p v q I will take accounting or I
will take calculus

~ Negation
(not)

~ p I will not take accounting
It is not the case that I will
take accounting

→ Conditional
(if-then)

p → q If I take accounting then I
will take calculus

↔ Biconditional
(if and only if)

P ↔ q I will take accounting if and
only if I take calculus

Thinkstock

Joey goes to school, plays soccer, and
plays guitar. How would you symbol-
ize that sentence?

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CHAPTER 6Section 6.1 The Logic of Sentences
3. If we don’t pay the electric bill, our power will get cut off.

Let P represent “we pay the electric bill” and Q represent “our power will get cut off.”

If not P then Q

~ P → Q

Here we could have used P to represent “we don’t pay the electric bill,” thus including
the negation within the symbolization. We would then symbolize the sentence as “P → Q.”

4. Henry likes online education because he can go to class in his pajamas.

Let P represent the entire sentence; here we cannot break the sentence up into smaller
components. As we saw, “because” doesn’t function truth functionally.

5. I will go to the game only if I can get a ticket.

Let P represent “I will go to the game” and Q represent “I can get a ticket.”

P only if Q

One might be tempted to symbolize this sentence “Q → P.” But notice what the sentence
says: it says I will go to the game only if I get a ticket. It does not say that if I do get a ticket, I
will go to the game. So “only if” here indicates that what follows it is a necessary condition
(see the earlier Highlights section on necessary and sufficient conditions); the necessary con-
dition then becomes the consequent of the conditional, and so we symbolize this sentence as

P → Q

Practicing symbolizing sentences is a good way to become comfortable with how one goes
about making the structure of sentences clear and how truth-functional connectives work.
Here is another set of sentences, slightly more complicated:

6. Robyn will be promoted if and only if she is deployed to Iraq.

7. If we go to the beach or the mountains, we will have fun.

8. If I win the lottery, I will not quit my job, but I will buy a new car.

9. Suki likes rice, but her husband does not.

10. They went to the movies, then went to dinner, then went dancing.

Let’s go through these sentences as we did the previous ones, noting some important
details as we go along.

6. Robyn will be promoted if and only if she is deployed to Iraq.

Let P represent “Robyn will be promoted” and Q represent “Robyn is deployed to Iraq”

P if and only if Q

P ↔ Q

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CHAPTER 6Section 6.1 The Logic of Sentences

Here we see that “she” in the second part of the sentence obviously refers to Robyn.

7. If we go to the beach or the mountains, we will have fun.

Let P represent “we go to the beach,” Q represent “we go to the mountains,” and R repre-
sent “we will have fun.”

If either P or Q, then R

(P v Q) → R

Here, for the first time, we need a third sentence letter, R; the antecedent of this con-
ditional is itself a disjunction. We also use parentheses to structure the sentence in the
appropriate way.

8. If I win the lottery, I will not quit my job, but I will buy a new car.

Let P represent “I win the lottery,” Q represent “I will quit my job,” and “R” represent “I
will buy a new car.”

If P then not Q and R

P → (~ Q & R)

Here we make the negation explicit with the symbols, although we could have just as
easily symbolized “I will not quit my job” as P. We also see in this example that “but”
is treated as a conjunction, which is the standard way logicians treat “but.” We also use
parentheses here to make clear that the two results—relative to the job and the car—are
both conditional on winning the lottery.

9. Suki likes rice, but her husband does not.

Let P represent “Suki likes rice” and Q represent “Suki’s husband does not like rice.”

P and Q
P & Q

Here we include the negation in symbolizing “Suki’s husband does not like rice”; we
could have made it explicit by using Q to represent “Suki’s husband likes rice,” and sym-
bolizing it as ~ Q. We again see that “but” is treated as a conjunction. We also see that the
second part of this sentence, “Suki’s husband does not like rice,” is a bit more formal than
the way the sentence is actually stated, because “her” clearly refers to Suki and “does not”
clearly means “does not like rice.”

10. They went to the movies, then went to dinner, then went dancing.

Let P represent “they went to the movies,” Q represent “they went to dinner,” and R rep-
resent “they went dancing.”

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CHAPTER 6Section 6.1 The Logic of Sentences

P and Q and R

P & (Q & R) or equivalently (P & Q) & R

Here we can simply state that the three things
occurred; as we saw previously in this chapter, the
logic we are using can’t indicate what the sequence
or order of events is. Because the only connective
used here is the conjunction “&”, it doesn’t matter
how we group the sentences using parentheses.
If different combinations of truth-functional con-
nectives are used, however, you need to be sure to
group the sentences in the way indicated by the
original sentences. But with more than two sen-
tences, as we have here, we need to group them
using parentheses. Here, where the connectives
are all conjunctions (“&”), how we group them
doesn’t matter. But in more complex sentences,
and when the connectives are different, that
grouping can be very important.

Symbolizing Arguments

Now that we have seen how to represent sen-
tences with sentence letters and construct com-
plex sentences out of simpler components using
truth-functional connectives, we can begin sym-
bolizing arguments. Symbolizing arguments has

one big advantage: it allows us to see clearly the structure of the argument, and often that
allows us to determine very quickly how good the argument is. Particularly in the case of
deductive arguments, focusing on the structure of the argument not only reveals better
how the argument works—or fails to work—but can also prevent us from being distracted

by the content of the sentences. Consider the fol-
lowing two arguments:

1. If we want effective crime prevention, we
should use the death penalty more.

We want effective crime prevention.

Therefore,

we should use the death penalty
more.

2. If we want effective crime prevention, we
should abolish the death penalty.

We want effective crime prevention.
Therefore,
we should abolish the death penalty.

Stop and Think: Symbolizing Sentences

Here are some sentences for you to symbol-
ize. As we have done with the previous sen-
tences, indicate what you use each sentence
letter to represent, and then replace the
English connectives (such as “and,” “but,”
and “if . . . then”) with our truth-functional
logical connectives.

• Mia likes the Dodgers, but Steve likes
the Padres.

• If they don’t see that movie, they will
be sorry.

• I sent my kids to bed early, and then
they were upset.

• If you watch too much television,
then you won’t get your work done.

• I will buy you some ice cream only if
you wash the car.

• Dan will move to San Francisco if and
only if he gets a good job there.

• My boss is helpful, friendly, and
generous.

• If we don’t go to church Friday night,
then we will go Sunday morning.

• Bob likes bluegrass music, but
Carolyn likes jazz and rap.

• The dress didn’t fit, so she took it
back and got a refund.

Jupiterimages, Brand X Pictures/Thinkstock

Just like in math, we see that using
symbols can help us express a complex
equation or argument.

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CHAPTER 6Section 6.1 The Logic of Sentences

If we focus all of our attention on the details of whether the first premise is true, we may
get distracted by arguments over the effectiveness of the death penalty. But logic shows us
that the first step is to determine whether the argument—here a deductive argument—is
valid, and we can do that by symbolizing it with sentence letters and truth-functional
connectives:

P → Q

P

Q

Once we determine that the argument is valid, then we turn to the question of whether it
is sound. As we saw earlier, a sound argument is a valid deductive argument with prem-
ises that are actually true. For arguments 1 and 2, we see that both arguments are valid
because of their logical structure. Once we have established that, we can then go on to
investigate which premises are true. In this way, we can focus on where the disagreement
really lies. But what about this argument?

If prayer is not effective, then

I do not get what I pray for.

I do not get what I pray for.
Therefore,

Prayer is not effective.

This may seem to be a rather controversial argument, but the logician need not worry
about the effectiveness of prayer at all. The logician can simply point out that the argu-
ment is not valid because it commits a fallacy: specifically, the fallacy of affirming the
consequent (Q). We can see this most easily by symbolizing the argument (here including
the negations within our symbolization):

P → Q

Q

P

We know this argument form is not valid, no matter what the premises say. If the argument
is structured in this way, it is never a good argument. We can save a lot of time examining
this argument, for we can reject it simply on the basis of its logical structure, which reveals
that it commits the fallacy of affirming the consequent.

We can now look at some of the arguments we saw earlier and symbolize them. Perhaps
you remember this argument from our earlier discussion:

If we go out to eat tonight, we should go eat pizza.

If we stay in tonight, we should order pizza.

We should either go out for pizza or order pizza.

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CHAPTER 6Section 6.1 The Logic of Sentences

We can symbolize it this way:

P: We go out to eat tonight

Q: We should go eat pizza

R: We stay in tonight

S: We should order pizza

We can then symbolize this argument as follows:

P → Q

R → S

Q v S

This structure makes it fairly easy to see how the argument works. As stated, it seems that
it is not valid, until we look at P and R, which state that (P) we go out or (R) we stay in. If
we think just a bit, we see that those are probably our only two options, for we are either
going to stay home or not stay home. If we interpret “not staying home” as “going out,”
then it seems that we have an unstated premise that is pretty easy to accept, namely “We
will stay home or go out.” If we make this premise explicit, then the argument is valid,
and its logical structure makes that clear:

P → Q
R → S

P v R

Q v S

This argument, known technically as a complex constructive dilemma, really functions in
a way similar to the argument form of modus ponens. In modus ponens, we have a condi-
tional premise and its antecedent affirmed:

P → Q
P
Q

In the complex constructive dilemma, we simply have two conditional premises, and the
antecedent of each is affirmed. Looking at the structure of these logical forms should make
it easier to see that both forms fundamentally rely on the same kind of logical inference.

How might this argument be symbolized?

All cats have four legs.

All dogs have four legs.

All cats are dogs.

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CHAPTER 6Section 6.1 The Logic of Sentences

It is probably obvious that this argument is not valid but, again, looking at its structure
makes the mistake clearer.

The standard way of treating such claims as “All A are B” is to see that this sentence says,
“If something is an A, then that thing is a B.” So saying, “All trout are fish” is to say, “If
this is a trout, then it is a fish.”

P: this is a cat

Q: this is a dog

R: this has four legs

Symbolized, then, the argument looks like this:

P → R

Q → R

P → Q

Some arguments—perhaps this one is a good
example—don’t require us to symbolize them in
order to realize that they are not valid. Others, par-
ticularly more complex arguments, may not be so
clear. But a fundamental feature of logic is that no
matter how complex or confusing the argument,
all arguments are constructed and evaluated in
the same way. So knowing how to symbolize
basic argument forms is very useful in examining
those with a more complicated structure.

Let’s imagine a company, called NewBrandToys,
that confronts a rather serious business problem.
To stay in business, it must make a profit and it

Table 6.2 Advantages and Disadvantages of Truth Tables
Advantages

• Assuming they are done correctly, truth tables guarantee results when using them to test for the
validity of arguments.

• Truth tables can also determine what kind of truth function a sentence has: whether it is always
true (known as a tautology), always false (known as a contradiction), or sometimes true and
sometimes false.

• Truth tables can show that sentences are truth-functionally equivalent to each other; sometimes a
confusing sentence can be shown to be equivalent to a statement that is much clearer.

• Truth tables display all the possible truth-values involved in a complex sentence or in an argument.

Disadvantages

• Truth tables work only for deductive arguments.

• Truth tables can get very large if there are a lot of different sentence letters (their size increases
exponentially).

NASA/Getty Images/Thinkstock

A man on video, with feedback. Tau-
tology refers to needless, redundant
repetition.

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CHAPTER 6Section 6.1 The Logic of Sentences

must maintain market share. But to make a profit, it must raise its prices; to maintain mar-
ket share, it must not raise its prices. Evidently, this company is in trouble. This is what
the argument looks like:

If NewBrandToys is to stay in business, then it must raise its prices and not raise
its prices.

Therefore,

NewBrandToys is not going to stay in business.

We then assign different sentence letters, and, of course, we don’t have to use P and Q:

N: NewBrandToys is to stay in business

P: NewBrandToys must raise its prices

Assigning truth-functional connectives, we get the final version of the symbolized
argument:

N → (P & ~ P)

~ N

We know, of course, that a company cannot both raise its prices and not raise its prices. If it
is to stay in business, however, NewBrandToys has to do exactly this. So we know it is not
going to stay in business. As we see clearly from the symbolization, for NewBrandToys
to stay in business, “P & ~ P” would have to be true. But if P is true, ~ P is false, so this
sentence is false, and, of course, if P is false, this sentence is false. “P & ~ P” will always be
false—it is a contradiction—so NewBrandToys is in trouble, because the only way it can
stay in business is to make a sentence true that will always be false.

Sometimes arguments are rather straightforward, as the ones we have just seen generally
are. However, some arguments get much more complicated. It is still good practice to fol-
low our procedure in order to see what the argument’s structure looks like. So consider
this argument:

If Angela wants to go to college, then she will have to either borrow
money or get a scholarship. She gets a scholarship if she does well in
high school, but she won’t do well in high school if she hangs out with
the wrong friends. If she borrows money, she will have to pay the money
back. Angela hangs out with the wrong friends. Angela wants to go to
college. So Angela will have to pay back the money.

It may not be obvious whether this argument is valid or not, but it is good practice—and
a good first step—to symbolize it. As always, we assign sentence letters to the component
sentences and then symbolize the structure of the argument using truth-functional con-
nectives. This time we use somewhat different sentence letters, here using letters that refer
to the sentences in a natural way, such as “S,” picking up on Angela’s need for a scholar-
ship. It is also important to see that the conclusion here is indicated by the word “so.”

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CHAPTER 6Section 6.1 The Logic of Sentences

A: Angela wants to go to college.

B: Angela will have to borrow money.

S: Angela will get a scholarship.

H: Angela does well in high school.

F: Angela hangs out with the wrong friends.

P: Angela will have to pay back the money.

We can now state this argument using these sentence letters:

If A then (B or S)

If H then S

If F then not H

If B then P

F

A

therefore

P

We can now complete the symbolization by apply-
ing the appropriate truth-functional connectives:

A → (B v S)

H → S

F → ~ H

B → P

F

A

P

This helps us see the structure of argument bet-
ter, but it still may not be obvious whether this
argument is valid. Is there a way of determining
that the argument here is valid or not valid? For-
tunately there is, a method using truth tables. If
we symbolize the argument correctly, using the
truth table method guarantees that we can find out
whether or not this argument is valid. However,
in the case of Angela, we also run up against the
limitations of truth tables, as we soon see.

Stop and Think: Examining Sentences

Here are a few arguments. Assign sentence
letters to the component sentences, and
then, using the truth-functional connectives,
symbolize the argument.

• Either Mackenzie likes ice cream
or Zach likes ice cream. Mackenzie
doesn’t like ice cream, so Zack does.

• If you go to the beach, then you will
go swimming. If you go swimming,
then you will be stung by a jellyfish. If
you are stung by a jellyfish, then you
will go to the hospital. So if you go to
the beach, you will go to the hospital.

• Zelda plays basketball if and only if
Agnes plays football. Agnes does not
play football; consequently, Zelda
does not play basketball.

• If Shanghai is larger than Houston,
then Houston is larger than Iowa
City. Iowa City is not larger than
Houston. Therefore, Houston is not
larger than Shanghai.

Here are two sentences and two arguments.
Try constructing a truth table for each. What
do you discover about the sentences? Are
the arguments valid?

Sentences
P ↔ ~ P
(P & ~ P) → (Q & ~ Q)

Arguments
P → (Q v ~ Q)
~ (Q v ~ Q)
~ P
~ (Q & ~ Q)
Q v

~ Q

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CHAPTER 6Section 6.1 The Logic of Sentences

6.1 Quiz

1. True or false? Assertoric sentences are sentences that can be assessed in terms of
truth and falsehood.

2. True or false? Compound sentences are sentences that contain more than one
claim.

3. True or false? Imperatives and questions are two examples of assertoric sentences.

4. True or false? “Rene Descartes was a radical skeptic” is an example of an asser-
toric sentence.

5. Which of the following is not an assertoric sentence?

A. Eat your cereal!
B. Love is all you need.
C. Smoking causes cancer.
D. Pets lower blood pressure.

6. What do we mean by the “truth function” of a sentence?

A. That the sentence is, in fact, true.
B. That we cannot think of the sentence as true or false.
C. That if we know the truth-value of the components of a sentence, we know

the truth-value of a complex sentence made up of each.
D. None of the above.

7. True or false? Any assertoric sentence can be represented with a “P” or “Q”
variable.

8. What does it mean to say that a sentence is bivalent?

A. That it is a complex sentence made up of two component parts.
B. That it has only two possible values: true or false.
C. That it contains one false and true subcomponent.
D. All of the above.

9. The logical notation for the truth functional connective “and” is

A. v
B. ↔
C. ~
D. &

10. The logical notation for the truth functional connective “or” is

A. v
B. ↔
C. ~
D. &

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CHAPTER 6Section 6.1 The Logic of Sentences

11. The logical notation for negation is

A. v
B. ↔
C. ~
D. &

12. The logical notation for the truth functional connective “if and only if” is

A. v
B. ↔
C. ~
D. &

13. True or false? In a sentence containing a conjunction, if at least one of its compo-
nents is true, we know the whole sentence is also true.

14. True or false? In a sentence containing a disjunction, if at least one of its compo-
nents is true, we know the whole sentence is also true.

15. True or false? When you negate a negated sentence, both negations cancel each
other out.

16. True or false? A conditional or “if, then” sentence is represented by the sign “→.”

17. Which of the following is a necessary condition for being over six feet tall?

A. Being less than five feet tall.
B. Weighing over 125 pounds.
C. Being at least five feet tall.
D. Being at least seven feet tall.

18. Which of the following is a sufficient condition for being over six feet tall?

A. Being less than five feet tall.
B. Weighing over 125 pounds.
C. Being at least five feet tall.
D. Being at least seven feet tall.

19. What is the correct symbolization of the following sentence?

Either we go up or we go down.
(Let p = “we go up,” q = “we go down”)

A. p → q
B. p v q
C. p & q
D. p therefore q

20. What is the correct symbolization of the following sentence?

The movie was boring, and the movie was far too long.
(Let p = “the movie was boring,” q = “the movie was far too long”)

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CHAPTER 6Section

6.2 Truth Tables

A. p → q
B. p v q
C. p & q
D. p therefore q

21. What is the correct symbolization of the following sentence?

If you go to the movie, you will be bored.
(Let p = “if you go to the movie,” q = “you will be bored”)

A. p → q
B. p v q
C. p & q
D. p therefore q

Answers
1. True. This type of sentence is the central concern of most systems of formal logic.
2. True. “Cesar ate dessert and Rebecca had a cocktail” is an example of a compound sentence because it

contains two separate claims.
3. False. Neither of these can be thought of in terms of truth or falsehood, so they do not qualify.
4. True. Even though this sentence is in fact false, it qualifies as assertoric because it makes sense to assess

it in terms of “true” and “false.” It is important not to make the mistake that all assertoric sentences must
also be true.

5. A. “Eat your cereal” is an imperative and hence has no truth value.
6. C. “Truth function” is a technical term that refers to our ability to deduce truth-value of a whole from the

truth-value of the parts of the whole.
7. True. This is logical shorthand for sentences and enables logicians to more elegantly represent complex

logical statements.
8. B. Rejoinder: Bivalence is attributed to assertoric sentences and refers to the fact that they are either true

or false; there is no other possible value we can attribute to them.
9. D. For example, “P and Q” would be notated as “P & Q.”
10. A. For example, “P or Q” would be notated as “P v Q.”
11. C. For example, “not p” would be notated as “~ p.”
12. B. For example, “P, if and only if Q” would be notated as “P ↔ Q.”
13. False. Conjunctions are true only if both of their components are true.
14. True. So long as one half of an “either/or” statement is true, the entire sentence is true.
15. True. ~ (~ that water is H

2
O means “it is not the case that it is not the case that water is H

2
O,” which in

turn is equivalent to “water is H
2
O.”)

16. True. “If p, then q” translates to “p → q.”
17. C. No one who is over six feet tall is not at least five feet tall as well. Thus, being at least five feet tall is a

necessary, though not sufficient, condition for being at least six feet tall.
18. D. If we know something is over seven feet tall, then we can be assured that it is also over six feet tall.

Thus, being over seven feet tall is a sufficient condition for being over six feet tall.
19. B. The “v” symbol represents “or” in disjunctive sentences.
20. C. The “&” symbol represents “and” in conjunctive sentences.
21. A. The “→” symbol represents the “if, then” relation in conditional sentences.

6.2 Truth Tables

Truth tables are a well-known method for seeing the properties of sentences as well as whether arguments are valid. Here we focus on symbolizing arguments, then testing
them for validity using truth tables.

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CHAPTER 6Section 6.2 Truth Tables

Truth Tables and Validity

Truth tables can do a number of things, including
testing deductive arguments for validity (which is
our focus). They do so by assigning truth-values to
the various components of the sentences that make
up an argument and then methodically display-
ing the truth-values of the sentences that make up
the premises and the sentence that serves as the
argument’s conclusion. The fundamental idea to
remember is our original definition of validity: a
deductive argument is valid if accepting the prem-
ises as true requires the conclusion to be accepted
as true. What we do, then, to test an argument for
validity with a truth table is to apply truth-values
to determine the truth-value of the premises and to
determine the truth-value of the conclusion. If the conclusion is true on every row of the truth
table where the premises are also true—if there is no row where the premises are true and the
conclusion false—then the argument is valid. This may sound a bit abstract at first; the best
way to learn how to test an argument for validity using truth tables is to start testing them!

We can begin with a simple argument:

Dave likes either soccer or hockey.

Dave does not like hockey.

Therefore,

Dave likes soccer.

As usual, we assign sentence letters, and then symbolize the argument:

P: Dave likes soccer

Q: Dave likes hockey

P or Q

not Q

therefore
P
P v Q
~ Q
P

We begin constructing the truth table by listing the sentences that make up the argument
(here, just P and Q) in the first columns. We then give a column for each premise, and a
column for the conclusion:

P Q P v Q ~ Q P

Daniel Schoenen/Photolibrary

Truth tables help us test an argument
for validity.

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CHAPTER 6Section 6.2 Truth Tables

We then apply truth-values to our original sentence letters:

P Q
T T
F T
T F
F F

Notice that we give alternating truth-values in the first column, under P, and alternat-
ing pairs of truth-values under Q. This guarantees we list all the different possible com-
binations of truth-values. Remembering the definitions for the various truth-functional
connectives we saw earlier, we then apply those to fill out the remainder of the truth
table:

P Q P v Q ~ Q P
T T T F T
F T T F F
T F T T T
F F F T F

At this point, using our fundamental idea of validity, we look at the table we have con-
structed, focusing only on those rows where both premises (“P v Q” and “~ Q”) are true.
Is the conclusion (“P”) true on each one of those rows? We see that there is only one such
row on our truth table, the third row. Because the conclusion is true where the premises
are true—there is no row where the premises are true and the conclusion is false—then
this argument is valid.

Here’s another example:

P → Q
Q
P

We may remember the problem with this argument from our earlier discussion—it is not
valid—but now we can show it, using a truth table. (If you forget the truth-functional
definition of the conditional [“→“], you can always go back and review it.)

P Q P → Q Q P
T T T T T
F T T T F
T F F F T

Again, we look to see what rows of the truth table show all the premises to be true; in
this case, it is the first and second row. But in the second row the premises are both true
and the conclusion false, and we need only one row where the premises are true and the
conclusion false to show that the argument is not valid. We have thus shown that this
argument is not valid (and we may remember that it commits the fallacy of affirming the
consequent).

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CHAPTER 6Section 6.2 Truth Tables

Constructing Truth Tables

The best way to learn how truth tables work is to take sentences (and arguments) and
construct truth tables for them. Here are the steps one takes to construct a truth table.

1. Identify the sentence letters.
2. Make a column for each sentence letter, and assign truth-values to each. (The

easiest way is to alternate truth-values in the first column—the one farthest to the
left.)

3. Identify the component sentences.
4. Make a column for each component sentence.
5. Assign truth-values to the component sentences based on the original truth-

values you assigned to the sentence letters.

Here is a sentence for which one might give a truth table:

[P → (Q v ~ P)] ↔ (~ Q & ~ P)

The complete truth table would have separate columns for these components, from left to
right (notice how we go from the smallest parts to the entire sentence in the final column):

P
Q
~ P
~ Q

Q v ~ P

~ Q & ~ P

P → (Q v ~ P)

[P → (Q v ~ P)] ↔ (~ Q & ~ P)

Let’s look at one more argument we saw earlier, that concerning NewBrandToys and the
problems it faced: namely, to stay in business, it had to raise and not raise prices. The argu-
ment looked like this:

N → (P & ~ P)
~ N

We can then use a truth table to see if, in fact, this is a valid argument.

N P ~ P (P & ~ P) N → (P & ~ P) ~ N
T T F F F F
F T F F T T
T F T F F F
F F T F T T

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CHAPTER 6Section 6.2 Truth Tables

Here we see that we have an argument with a single premise (“N → (P & ~ P)”), so we
need only check to see where that premise is true. This premise is true on two rows (the
second and fourth), and there are no rows where the premise is true and the conclusion
false. So this argument is valid.

You may have noticed something in particular about this argument. Looking at the conse-
quent of the conditional, “N → (P & ~ P)” is “P & ~ P”, you can see from the fourth row of the
truth table, “P & ~ P” is false on all possible truth-values. A sentence that is false under all
possible truth-values is called a contradiction; it is a sentence that cannot possibly turn out to
be true. This is why NewBrandToys is in trouble, of course: if they want to stay in business,
at least according to this argument, they have to do something that cannot be done (raise
prices and not raise prices); they have to make a contradiction be true, which it never is. In
fact this argument form is very old and very well known—old enough to be referred to by
the Latin term for “reduce to absurdity,” reductio ad absurdum. In this case, it is absurd to
raise and not raise prices; showing that it must do so to stay in business is to show that the
hope of staying in business under such conditions makes no sense, or is absurd.

Once you see how they work, truth tables are generally easy to use and have one signifi-
cant advantage: they are guaranteed to tell you whether an argument is valid or not (as
long as they are set up correctly). But it was mentioned earlier that there are certain limita-
tions to truth tables, and the argument we saw with Angela on page 158 shows this clearly.

Table 6.3 Common Deductive Argument Forms
Common Deductive Argument Forms (valid)
Modus Ponens Modus Tollens

If p then q If p then q

p Not q

Therefore q Therefore not p

Hypothetical Syllogism Hypothetical Syllogism

If p then q p and q

If q then r not p

Therefore if p then r therefore q

Formal Fallacies (invalid)
Denying the Antecedent Affirming the Consequent

If p then q If p then q

Not p q

Therefore not q Therefore p

We saw with the truth-functional connection of negation (“~”) that the truth table had only
two rows, for we needed only one sentence letter to display all the possibilities (for a sen-
tence is either true or false). When we used two sentence letters to show the definitions of the
other connectives, the truth table had four rows. In short, if an argument has one sentence
letter in it, it requires two rows; if it has two sentence letters, it requires four rows. To display
all the possible different truth-values for three sentence letters, we would need eight rows.
You probably see the pattern (two, four, eight); truth tables grow very large, very fast. (If
you’re curious, the size of a truth table is determined by taking the number two, and raising

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CHAPTER 6Section 6.2 Truth Tables

it to the power of however many sentence letters make up the argument. So two raised to
the power of two is four, to the power of three is eight, to the power of four is sixteen.)

The argument we saw about Angela has six separate sentence letters; that would require
a truth table with sixty-four rows (26 = 64) to give all the possible combinations of truth-
values. That’s an awfully big truth table, and unless you’re a computer or have a great
deal of spare time on your hands, you might want to look for better ways to show argu-
ments are valid (or not valid). Thus we see that truth tables have their limitations, but if
you want to make absolutely sure whether an argument is valid, you can always use a
truth table as we have just done, and—assuming you set it up correctly—you are guar-
anteed to get your question answered.

6.2 Quiz

1. True or false? Truth tables are unable to determine whether or not an argument is
valid.

2. True or false? In a completed truth table, if there is no row in which all the prem-
ises are true and the conclusion false, the argument is valid.

3. Which of the following is not a step in the construction of a truth table?

A. Determine whether all the premises are actually true.
B. Label the sentences with letters.
C. Make a column for each letter and assign a truth-value (T and F) to each.
D. Identify the component sentences.

4. True or false? Truth tables may be used on inductive as well as deductive
arguments.

5. True or false? While truth tables will work on complex arguments, they can
become quite long and time-consuming.

Answers
1. False. Determining validity is the main function of truth tables.
2. True. Truth tables “look at” all possible combinations of truth-values for the premises and conclusion to

determine if in any instance all the premises can be true and the conclusion false. If not, then the argument
is valid.

3. A. Remembering that logic deals with “the big if,” we do not concern ourselves with the actual truth-
values of the sentences when making truth tables.

4. False. Truth tables are designed to determine validity, and inductive arguments are not assessed in terms
of “valid” and “invalid.”

5. True. As long as the argument contains assertoric sentences using standard truth functional connectives, a
truth table can be used to determine validity. However, when there are a large number of sentence letters,
they can be extremely large and cumbersome.

What Did We Find?

• We saw how symbols can be applied to sentences and then to arguments.
• We looked at the notion of a truth function and how some sentences can be

treated truth functionally, and how others cannot be.

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CHAPTER 6

Some Final Questions

• We examined how truth tables can be constructed to evaluate certain kinds of
sentences as well as to test deductive arguments for validity.

• We used basic symbolic logic to examine some earlier material and see how it can
be made more precise.

• We used truth tables to look more formally at arguments we had examined
informally.

• We saw that we can construct conditionals that correspond to arguments and
that the conditionals that correspond to valid arguments are always true, or are
tautologies.

Some Final Questions

• What are the advantages of using symbols to represent sentences and arguments?
• How can we use truth tables to determine whether arguments are valid or not

valid? What other properties of sentences can we use truth tables to establish?
• Why can’t truth tables show that an argument is sound?
• Why can’t we use truth tables to evaluate inductive arguments?

Web Links

Paradoxes of Material Implication

For more consideration of truth tables, a nice discussion is offered by the philosopher
Peter Suber here:

http://www.earlham.edu/~peters/courses/log/mat-imp.htm

Necessary and Sufficient Conditions

See the Stanford Encyclopedia of Philosophy for more on necessary and sufficient
conditions:

http://plato.stanford.edu/entries/necessary-sufficient/

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