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119
4Propositional Logic
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Learning Objectives
After reading this chapter, you should be able to:
1. Explain key words and concepts from propositional logic.
2. Describe the basic logical operators and how they function in
a statement.
3. Symbolize complex statements using logical operators.
4. Generate truth tables to evaluate the validity of truth-
functional arguments.
5. Evaluate common logical forms.
har85668_04_c04_119-164.indd 119 4/9/15 1:26 PM
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Section 4.1 Basic Concepts in Propositional Logic
Chapter 3 discussed categorical logic and touched on how
analyzing an argument’s logical

form helps determine its validity. The usefulness of form in
determining validity will become
even clearer in this chapter’s discussion of what is known as
propositional logic, another type
of deductive logic. Whereas categorical logic analyzes
arguments whose validity is based on
quantitative terms like all and some, propositional logic looks at
arguments whose validity is
based on the way they combine smaller sentences to make larger
ones, using connectives like
or, and, and not.
In this chapter, we will learn about the symbols and tools that
help us analyze arguments and
test for validity; we will also examine several common
deductive argument forms. Whereas
Chapter 3 introduced the idea of form—and thereby, formal
logic—this chapter will more
thoroughly consider the study of validity based on logical form.
We shall see that by adding a
couple more symbols to propositional logic, it is also possible
to represent the types of state-
ments represented in categorical logic, creating the robust and
highly applicable discipline
known today as predicate logic. (See A Closer Look:
Translating Categorical Logic for more on
predicate logic.)
4.1 Basic Concepts in Propositional Logic
Propositional logic aims to make the concept of validity formal
and precise. Remember
from Chapter 3 that an argument is valid when the truth of its
premises guarantees the
truth of its conclusion. Propositional logic demonstrates exactly
why certain types of prem-
ises guarantee the truth of certain types of conclusions. It does

this by breaking down the
forms of complex claims into their simple component parts. For
example, consider the fol-
lowing argument:
Either the maid or the butler did it.
The maid did not do it.
Therefore, the butler did it.
This argument is valid, but not because of anything about the
maid or butler. It is valid because
of the way that the sentences combine words like or and not to
make a logically valid form.
Formal logic is not concerned about the content of arguments
but with their form. Recall from
Chapter 3, Section 3.2, that an argument’s form is the way it
combines its component parts
to make an overall pattern of reasoning. In this argument, the
component parts are the small
sentences “the butler did it” and “the maid did it.” If we give
those parts the names P and Q,
then our argument has the form:
P or Q.
Not P.
Therefore, Q.
Note that the expression “not P” means “P is not true.” In this
case, since P is “the butler did
it,” it follows that “not P” means “the butler did not do it.” An
inspection of this form should
reveal it is logically valid reasoning.
As the name suggests, propositional logic deals with arguments
made up of propositions,
just as categorical logic deals with arguments made up of

categories (see Chapter 3). In
har85668_04_c04_119-164.indd 120 4/9/15 1:26 PM
philosophy, a proposition is the meaning of a claim about the
world; it is what that claim
asserts. We will refer to the subject of this chapter as
“propositional logic” because that is
the most common terminology in the field. However, it is
sometimes called “sentence logic.”
The principles are the same no matter which terminology we
use, and in the rest of the
chapter we will frequently talk about P and Q as representing
sentences (or “statements”)
as well.
The Value of Formal Logic
This process of making our reasoning more precise by focusing
on an argument’s form has
proved to be enormously useful. In fact, formal logic provides
the theoretical underpinnings
for computers. Computers operate on what are called “logic
circuits,” and computer programs
are based on propositional logic. Computers are able to
understand our commands and
always do exactly what they are programmed to do because they
use formal logic. In A Closer
Look: Alan Turing and How Formal Logic Won the War, you
will see how the practical applica-

tions of logic changed the course of history.
Another value of formal logic is that it adds efficiency,
precision, and clarity to our language.
Being able to examine the structure of people’s statements
allows us to clarify the meanings
of complex sentences. In doing so, it creates an exact,
structured way to assess reasoning and
to discern between formally valid and invalid arguments.
A Closer Look: Alan Turing and How Formal Logic Won the
War
The idea of a computing machine was conceived over the last
few centuries by great thinkers such as Gottfried Leibniz,
Blaise Pascal, and Charles Babbage. However, it was not until
the first half of the 20th century that philosophers, logicians,
mathematicians, and engineers were actually able to create
“thinking machines” or “electronic brains” (Davis, 2000).
One pioneer of the computer age was British mathematician,
philosopher, and logician Alan Turing. He came up with the
concept of a Turing machine, an electronic device that takes
input in the form of zeroes and ones, manipulates it according
to an algorithm, and creates a new output (BBC News, 1999).
Computers themselves were invented by creating electric cir-
cuits that do basic logical operations that you will learn about
in this chapter. These electric circuits are called “logic gates”
(see Figure 4.2 later in the chapter). By turning logic into cir-
cuits, basic “thinking” could be done with a series of fast elec-
trical impulses.
Using logical brilliance, Turing was able to design early com-
puters for use during World War II. The British used these early
computers to crack the Nazis’ very complex Enigma code. The
ability to know the German plans in advance gave the Allies a

huge advantage. Prime Minister Winston Churchill even said to
King George VI, “It was thanks
to Ultra [one of the computers used] that we won the war” (as
cited in Shaer, 2012).
Science and Society/SuperStock
An Enigma cipher machine,
which was widely used by
the Nazi Party to encipher
and decipher secret
military messages during
World War II.
har85668_04_c04_119-164.indd 121 4/9/15 1:26 PM
Statement Forms
As we have discussed, propositional
logic clarifies formal reasoning by
breaking down the forms of complex
claims into the simple parts of which
they are composed. It does this by using
symbols to represent the smaller parts
of complex sentences and showing how
the larger sentence results from com-
bining those parts in a certain way. By
doing so, formal logic clarifies the argu-
ment’s form, or the pattern of reason-
ing it uses.

Consider what this looks like in mathematics. If you have taken
a course in algebra, you will
remember statements such as the following:
x + y = y + x
This statement is true no matter what we put for x and for y.
That is why we call x and y
variables; they do not represent just one number but all
numbers. No matter what specific
numbers we put in, we will still get a true statement, like the
following:
5 + 3 = 3 + 5
7 + 2 = 2 + 7
1,235 + 943 = 943 + 1,235
By replacing the variables in the general equation with these
specific values, we get instances
(as discussed in Chapter 3) of that general truth. In other words,
5 + 3 = 3 + 5 is an instance
of the general statement x + y = y + x. One does not even need
to use a calculator to know that
the last statement of the three is true, for its truth is not based
on the specific numbers used
but on the general form of the equation. Formal logic works in
the exact same way.
Take the statement “If you have a dog, then you have a dog or
you have a cat.” This statement
is true, but its truth does not depend on anything about dogs or
cats; its truth is based on its
logical form—the way the sentence is structured. Here are two

other statements with the
same logical form: “If you are a miner, then you are a miner or
you are a trapper” and “If you
are a man, then you are a man or a woman.” These statements
are all true not because of their
content, but because of their shared logical form.
To help us see exactly what this form is, propositional logic
uses variables to represent the
different sentences within this form. Just as algebra uses letters
like x and y to represent
numbers, logicians use letters like P and Q to represent
sentences. These letters are therefore
called sentence variables.
Bill Long/Cartoonstock
Formal logic uses symbols and statement forms to
clarify an argument’s reasoning.
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Sentence: “If you have a dog, then you have a dog AND you
have a cat”
Form: If P then P Q
and
Section 4.2 Logical Operators
The chief difference between propositional and categorical logic

is that, in categorical logic
(Chapter 3), variables (like M and S) are used to represent
categories of things (like dogs and
mammals), whereas variables in propositional logic (like P and
Q) represent whole sentences
(or propositions).
In our current example, propositional logic enables us to take
the statement “If you have a
dog, then you have a dog or you have a cat” and replace the
simple sentences “You have a dog”
and “You have a cat,” with the variables P and Q, respectively
(see Figure 4.2). The result, “If P,
then P or Q,” is known as the general statement form. Our
specific sentence, “If you have a
dog, then you have a dog or you have a cat,” is an instance of
this general form. Our other
example statements—”If you are a miner, then you are a miner
or you are a trapper” and “If
you are a man, then you are a man or a woman”—are other
instances of that same statement
form, “If P, then P or Q.” We will talk about more specific
forms in the next section.
At first glance, propositional logic can seem intimidating
because it can be very mathemati-
cal in appearance, and some students have negative associations
with math. We encourage
you to take each section one step at a time and see the symbols
as tools you can use to your
advantage. Many students actually find that logic helps them
because it presents symbols in
a friendlier manner than in math, which can then help them
warm up to the use of symbols
in general.

4.2 Logical Operators
In the prior section, we learned about what constitutes a
statement form in propositional
logic: a complex sentence structure with propositional variables
like P and Q. In addition to
the variables, however, there are other words that we used in
representing forms, words like
and and or. These terms, which connect the variables together,
are called logical operators,
also known as connectives or logical terms.
Logicians like to express formal precision by replacing English
words with symbols that rep-
resent them. Therefore, in a statement form, logical operators
are represented by symbols.
The resulting symbolic statement forms are precise, brief, and
clear. Expressing sentences in
terms of such forms allows logic students more easily to
determine the validity of arguments
that include them. This section will analyze some of the most
common symbols used for logi-
cal operators.
Figure 4.1: Finding the form
In this instance of the statement form, you can see that P and Q
relate to the prepositions “you have a
dog” and “you have a cat,” respectively.
Sentence: “If you have a dog, then you have a dog AND you
have a cat”
Form: If P then P Q
and
har85668_04_c04_119-164.indd 123 4/9/15 1:26 PM

Conjunction
Those of you who have heard the Schoolhouse Rock! song
“Conjunction Junction” (what’s your
function?)—or recall past English grammar lessons—will
recognize that a conjunction is a
word used to connect, or conjoin, sentences or concepts. By that
definition, it refers to words
like and, but, and or. Logic, however, uses the word conjunction
to refer only to and sentences.
Accordingly, a conjunction is a compound statement in which
the smaller component state-
ments are joined by and.
For example, the conjunction of “roses are red” and “violets are
blue” is the sentence “roses
are red and violets are blue.” In logic, the symbol for and is an
ampersand (&). Thus, the gen-
eral form of a conjunction is P & Q. To get a specific instance
of a conjunction, all you have to
do is replace the P and the Q with any specific sentences. Here
are some examples:
P Q P & Q
Joe is nice. Joe is tall. Joe is nice, and Joe is tall.
Mike is sad. Mike is lonely. Mike is sad, and Mike is lonely.
Winston is gone. Winston is not forgotten. Winston is gone and

not forgotten.
Notice that the last sentence in the example does not repeat
“Winston is” before “forgotten.”
That is because people tend to abbreviate things. Thus, if we
say “Jim and Mike are on the
team,” this is actually an abbreviation for “Jim is on the team,
and Mike is on the team.”
The use of the word and has an effect on the truth of the
sentence. If we say that P & Q is true,
it means that both P and Q are true. For example, suppose we
say, “Joe is nice and Joe is tall.”
This means that he is both nice and tall. If he is not tall, then
the statement is false. If he is not
nice, then the statement is false as well. He has to be both for
the conjunction to be true. The
truth of a complex statement thus depends on the truth of its
parts. Whether a proposition is
true or false is known as its truth value: The truth value of a
true sentence is simply the word
true, while the truth value of a false sentence is the word false.
To examine how the truth of a statement’s parts affects the truth
of the whole statement, we can
use a truth table. In a truth table, each variable (in this case, P
and Q) has its own column, in
which all possible truth values for those variables are listed. On
the right side of the truth table
is a column for the complex sentence(s) (in this case the
conjunction P & Q) whose truth we
want to test. This last column shows the truth value of the
statement in question based on the
assigned truth values listed for the variables on the left. In other
words, each row of the truth
table shows that if the letters (like P and Q) on the left have

these assigned truth values, then the
complex statements on the right will have these resulting truth
values (in the complex column).
Here is the truth table for conjunction:
P Q P & Q
T (Joe is nice.) T (Joe is tall.) T (Joe is nice, and Joe is tall.)
T (Joe is nice.) F (Joe is not tall.) F (It is not true that Joe is
nice and tall.)
F (Joe is not nice.) T (Joe is tall.) F (It is not true that Joe is
nice and tall.)
F (Joe is not nice.) F (Joe is not tall.) F (It is not true that Joe is
nice and tall.)
har85668_04_c04_119-164.indd 124 4/9/15 1:26 PM
What the first row means is that if the statements P and Q are
both true, then the conjunction
P & Q is true as well. The second row means that if P is true
and Q is false, then P & Q is false
(because P & Q means that both statements are true). The third
row means that if P is false
and Q is true, then P & Q is false. The final row means that if
both statements are false, then
P & Q is false as well.

A shorter method for representing this truth table, in which T
stands for “true” and F stands
for “false,” is as follows:
P Q P & Q
T T T
T F F
F T F
F F F
The P and Q columns represent all of the possible truth
combinations, and the P & Q column
represents the resulting truth value of the conjunction. Again,
within each row, on the left
we simply assume a set of truth values (for example, in the
second row we assume that P is
true and Q is false), then we determine what the truth value of P
& Q should be to the right.
Therefore, each row is like a formal “if–then” statement: If P is
true and Q is false, then P & Q
will be false.
Truth tables highlight why propositional logic is also called
truth-functional logic. It is truth-
functional because, as truth tables demonstrate, the truth of the
complex statement (on the
right) is a function of the truth values of its component
statements (on the left).
Everyday Logic: The Meaning of But
Like the word and, the word but is also a conjunction. If we say,
“Mike is rich, but he’s mean,”
this seems to mean three things: (1) Mike is rich, (2) Mike is
mean, and (3) these things are in

contrast with each other. This third part, however, cannot be
measured with simple truth values.
Therefore, in terms of logic, we simply ignore such
conversational elements (like point 3) and
focus only on the truth conditions of the sentence. Therefore,
strange as it may seem, in proposi-
tional logic the word but is taken to be a synonym for and.
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Disjunction
Disjunction is just like conjunction except that it involves
statements connected with an or
(see Figure 4.2 for a helpful visualization of the difference).
Thus, a statement like “You can
either walk or ride the bus” is the disjunction of the statements
“You can walk” and “you can
ride the bus.” In other words, a disjunction is an or statement: P
or Q. In logic the symbol for
or is ∨ . An or statement, therefore, has the form P ∨ Q.
Here are some examples:
P Q P ∨ Q
Mike is tall. Doug is rich. Mike is tall, or Doug is rich.
You can complain. You can change things. You can complain,
or you

can change things.
The maid did it. The butler did it. Either the maid or the
butler did it.
Notice that, as in the conjunction example, the last example
abbreviates one of the clauses (in
this case the first clause, “the maid did it”). It is common in
natural (nonformal) languages to
abbreviate sentences in such ways; the compound sentence
actually has two complete com-
ponent sentences, even if they are not stated completely. The
nonabbreviated version would
be “Either the maid did it, or the butler did it.”
The truth table for disjunction is as follows:
P Q P ∨ Q
T T T
T F T
F T T
F F F
Note that or statements are true whenever at least one of the
component sentences (the “dis-
juncts”) is true. The only time an or statement is false is when P
and Q are both false.
har85668_04_c04_119-164.indd 126 4/9/15 1:26 PM

Basic AND Gate
Basic OR Gate
Battery
Battery
P gate: Closed if P is true
Q gate: Closed if Q is true
The light
goes on only
if both P and
Q are true.
The light
goes if
either P or
Q is true.
Figure 4.2: Simple logic circuits
These diagrams of simple logic circuits (recall the reference to
these circuits in A Closer Look: Alan
Turing and How Formal Logic Won the War) help us visualize
how the rules for conjunctions (AND
gate) and disjunctions (OR gate) work. With the AND gate,
there is only one path that will turn on the

light, but with the OR gate, there are two paths to illumination.
Basic AND Gate
Basic OR Gate
Battery
Battery
The light
goes on only
if both P and
Q are true.
The light
goes if
either P or
Q is true.
Everyday Logic: Inclusive Versus Exclusive Or
The top line of the truth table for disjunctions may seem strange
to some. Some think that the
word or is intended to allow only one of the two sentences to be
true. They therefore argue for
an interpretation of disjunction called exclusive or. An
exclusive or is just like the or in the truth

table, except that it makes the top row (the one in which P and
Q are both true) false.
One example given to justify this view is that of a waiter
asking, “Do you want soup or salad?”
If you want both, the answer should not be “yes.” Some
therefore suggest that the English or
should be understood in the exclusive sense.
However, this example can be misleading. The waiter is not
asking “Is the statement ‘do you want
soup or salad’ true?” The waiter is asking you to choose
between the two options. When we ask
for the truth value of a sentence of the form P or Q, on the other
hand, we are asking whether
the sentence is true. Consider it this way: If you wanted both
soup and salad, the answer to the
waiter’s question would not be “no,” but it would be if you were
using an exclusive or.
(continued)
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Negation
The simplest logical symbol we use on sentences simply negates
a claim. Negation is the act
of asserting that a claim is false. For every statement P, the
negation of P states that P is false.

It is symbolized ~P and pronounced “not P.” Here are some
examples:
P ~P
Snow is white. Snow is not white.
I am happy. I am not happy.
Either John or Mike got the job. Neither John nor Mike got the
job.
Since ~P states that P is not true, its truth value is the opposite
of P’s truth value. In other
words, if P is true, then ~P is false; if P is false then ~P is true.
Here, then, is the truth table:
P ~P
T F
F T
Everyday Logic: The Word Not
Sometimes just putting the word not in front of the verb does
not quite capture the meaning of
negation. Take the statement “Jack and Jill went up the hill.”
We could change it to “Jack and Jill
did not go up the hill.” This, however, seems to mean that
neither Jack nor Jill went up the hill,
but the meaning of negation only requires that at least one did
not go up the hill. The simplest
way to correctly express the negation would be to write “It is
not true that Jack and Jill went up
the hill” or “It is not the case that Jack and Jill went up the
hill.”

Similar problems affect the negation of claims such as “John
likes you.” If John does not know
you, then this statement is not true. However, if we put the word
not in front of the verb, we
get “John does not like you.” This seems to imply that John
dislikes you, which is not what the
negation means (especially if he does not know you). Therefore,
logicians will instead write
something like, “It is not the case that John likes you.”
When we see the connective or used in English, it is generally
being used in the inclusive sense
(so called because it includes cases in which both disjuncts are
true). Suppose that your tax form
states, “If you made more than $20,000, or you are self-
employed, then fill out form 201-Z.” Sup-
pose that you made more than $20,000, and you are self-
employed—would you fill out that form?
You should, because the standard or that we use in English and
in logic is the inclusive version.
Therefore, in logic we understand the word or in its inclusive
sense, as seen in the truth table.
Everyday Logic: Inclusive Versus Exclusive Or (continued)
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Conditional

A conditional is an “if–then” statement. An example is “If it is
raining, then the street is wet.”
The general form is “If P, then Q,” where P and Q represent any
two claims. Within a condi-
tional, P—the part that comes between if and then—is called the
antecedent; Q—the part
after then—is called the consequent. A conditional statement is
symbolized P → Q and pro-
nounced “if P, then Q.”
P Q P → Q
You are rich. You can buy a boat. If you are rich, then you
can buy a boat.
You are not satisfied. You can return the product. If you are not
satisfied, then
you can return the product.
You need bread or milk. You should go to the market. If you
need bread or milk, then
you should go to the market.
Formulating the truth table for conditional statements is
somewhat tricky. What does it take
for a conditional statement to be true? This is actually a
controversial issue within philosophy.
It is actually easier to think of it as: What does it mean for a
conditional statement to be false?
Suppose Mike promises, “If you give me $5, then I will wash
your car.” What would it take
for this statement to be false? Under what conditions, for
example, could you accuse Mike of

breaking his promise?
It seems that the only way for Mike to break his promise is if
you give him the $5, but he does
not wash the car. If you give him the money and he washes the
car, then he kept his word. If
Everyday Logic: Other Instances of Conditionals
Sometimes conditionals are expressed in other ways. For
example, sometimes people leave out the then. They say
things like, “If you are hungry, you should eat.” In many
of these cases, we have to be clever in determining what
P and Q are.
Sometimes people even put the consequent first: for
example, “You should eat if you are hungry.” This state-
ment means the same thing as “If you are hungry, then
you should eat”; it is just ordered differently. In both
cases the antecedent is what comes after the if in the
English sentence (and prior to the → in the logical form).
Thus, “If P then Q” is translated “P → Q,” and “P if Q” is
translated “Q → P.”
Monkey Business/Thinkstock
People use conditionals frequently
in real life. Think of all the times
someone has said, “Get some rest if
you are tired” or “You don’t have to
do something if you don’t want to.”
har85668_04_c04_119-164.indd 129 4/9/15 1:26 PM

you did not give him the money, then his word was simply not
tested (with no payment on
your part, he is under no obligation). If you do not pay him, he
may choose to wash the car
anyway (as a gift), or he may not; neither would make him a
liar. His promise is only broken
in the case in which you give him the money but he does not
wash it. Therefore, in general,
we call conditional statements false only in the case in which
the antecedent is true and the
consequent is false (in this case, if you give him the money, but
he still does not wash the car).
This results in the following truth table:
P Q P → Q
T T T
T F F
F T T
F F T
Some people question the bottom two lines. Some feel that the
truth value of those rows
should depend on whether he would have washed the car if you
had paid him. However, this
sophisticated hypothetical is beyond the power of truth-
functional logic. The truth table is as
close as we can get to the meaning of “if . . . then . . .” with a
simple truth table; in other words,
it is best we can do with the tool at hand.

Finally, some feel that the third row should be false. That,
however, would mean that Mike
choosing to wash the car of a person who had no money to give
him would mean that he broke
his promise. That does not appear, however, to be a broken
promise, only an act of generosity
on his part. It therefore does not appear that his initial statement
“If you give me $5, then I will
wash your car” commits to washing the car only if you give him
$5. This is instead a variation
on the conditional theme known as “only if.”
Only If
So what does it mean to say “P only if Q”? Let us take a look at
another example: “You can get
into Harvard only if you have a high GPA.” This means that a
high GPA is a requirement for
getting in. Note, however, that that is not the same as saying,
“You can get into Harvard if you
have a high GPA,” for there might be other requirements as
well, like having high test scores,
good letters of recommendation, and a good essay.
Thus, the statement “You can get into Harvard only if you have
a high GPA” means:
You can get into Harvard → You have a high GPA
However, this does not mean the same thing as “You have a
high GPA → You can get into Harvard.”
In general, “P only if Q” is translated P → Q. Notice that this is
the same as the translation
of “If P, then Q.” However, it is not the same as “P if Q,” which
is translated Q → P. Here is a
summary of the rules for these translations:

P only if Q is translated: P → Q
P if Q is translated: Q → P
har85668_04_c04_119-164.indd 130 4/9/15 1:26 PM
Thus, “P if Q” and “P only if Q” are the converse of each other.
Recall the discussion of conver-
sion in Chapter 3; the converse is what you get when you switch
the order of the elements
within a conditional or categorical statement.
To say that P → Q is true is to assert that the truth of Q is
necessary for the truth of P. In other
words, Q must be true for P to be true. To say that P → Q is
true is also to say that the truth of
P is sufficient for the truth of Q. In other words, knowing that P
is true is enough information
to conclude that Q is also true.
In our earlier example, we saw that having a high GPA is
necessary but not sufficient for get-
ting into Harvard, because one must also have high test scores
and good letters of recommen-
dation. Further discussion of the concepts of necessary and
sufficient conditions will occur
in Chapter 5.

In some cases P is both a necessary and a sufficient condition
for Q. This is called a biconditional.
Biconditional
A biconditional asserts an “if and only if ” statement. It states
that if P is true, then Q is true,
and if Q is true, then P is true. For example, if I say, “I will go
to the party if you will,” this
means that if you go, then I will too (P → Q), but it does not
rule out the possibility that I will
go without you. To rule out that possibility, I could state “I will
go to the party only if you will”
(Q → P). If we want to assert both conditionals, I could say, “I
will go to the party if and only if
you will.” This is a biconditional.
The statement “P if and only if Q” literally means “P if Q and P
only if Q.” Using the translation
methods for if and only if, this is translated “(Q → P) & (P →
Q).” Because the biconditional
makes the arrow between P and Q go both ways, it is
symbolized: P ↔ Q.
P Q P ↔ Q
You can go to the party. You are invited. You can go to the
party if and
only if you are invited.
You will get an A. You get above a 92%. You will get an A if
and only if
you get above a 92%.
You should propose. You are ready to marry her. You should

propose if and only
if you are ready to marry her.
There are other phrases that people sometimes use instead of “if
and only if.” Some people
say “just in case” or something else like it. Mathematicians and
philosophers even use the
abbreviation iff to stand for “if and only if.” Sometimes people
even simply say “if ” when they
really mean “if and only if.” One must be clever to understand
what people really mean when
they speak in sloppy, everyday language. When it comes to
precision, logic is perfect; English
is fuzzy!
Here is how we do the truth table: For the biconditional P ↔ Q
to be true, it must be the case
that if P is true then Q is true and vice versa. Therefore, one
cannot be true when the other
one is false. In other words, they must both have the same truth
value. That means the truth
table looks as follows:
har85668_04_c04_119-164.indd 131 4/9/15 1:26 PM
P Q P ↔ Q
T T T
T F F

F T F
F F T
The biconditional is true in exactly those cases in which P and
Q have the same truth value.
Practice Problems 4.1
Complete the following identifications.
1. “I am tired and hungry.” This statement is a __________.
a. conjunction
b. disjunction
c. conditional
d. biconditional
2. “If we learn logic, then we will be able to evaluate
arguments.” This statement is a
__________.
a. conjunction
b. disjunction
c. conditional
d. biconditional
3. “We can learn logic if and only if we commit ourselves to
intense study.” This state-
ment is a __________.
a. conjunction
b. disjunction
c. conditional
d. biconditional
4. “We either attack now, or we will lose the war.” This
statement is a __________.
a. conjunction
b. disjunction

c. conditional
d. biconditional
5. “The tide will rise only if the moon’s gravitational pull acts
on the ocean.” This state-
ment is a __________.
a. conjunction
b. disjunction
c. conditional
d. biconditional
6. “If I am sick or tired, then I will not go to the interpretive
dance competition.” This
a. conjunction
b. disjunction
c. conditional
d. biconditional
(continued)
har85668_04_c04_119-164.indd 132 4/9/15 1:26 PM
Section 4.3 Symbolizing Complex Statements
4.3 Symbolizing Complex Statements
We have learned the basic logical operators and their
corresponding symbols and truth
tables. However, these basic symbols also allow us analyze
much more complicated state-
ments. Within the statement form P → Q, what if either P or Q

itself is a complex statement?
For example:
P Q P → Q
You are hungry or thirsty. We should go to the diner. If you are
hungry or thirsty, then we
should go to the diner.
In this example, the antecedent, P, states, “You are hungry or
thirsty,” which can be symbolized
H ∨ T, using the letter H for “You are hungry” and T for “You
are thirsty.” If we use the letter
D for “We should go to the diner,” then the whole statement can
be symbolized (H ∨ T) → D.
Notice the use of parentheses. Parentheses help specify the
order of operations, just like in
arithmetic. For example, how would you evaluate the quantity 3
+ (2 × 5)? You would execute
the mathematical operation within the parentheses first. In this
case you would first multiply
2 and 5 and then add 3, getting 13. You would not add the 3 and
the 2 first and then multiply
by 5 to get 25. This is because you know to evaluate what is
within the parentheses first.
7. “One can surf monster waves if and only if one has
experience surfing smaller
waves.” This statement is a __________.
a. conjunction
b. disjunction
c. conditional
d. biconditional
8. “The economy is recovering, and people are starting to make

more money.” This
a. conjunction
b. disjunction
c. conditional
d. biconditional
9. “If my computer crashes again, then I am going to buy a new
one.” This statement is
a __________.
a. conjunction
b. disjunction
c. conditional
d. biconditional
10. “You can post responses on 2 days or choose to write a two-
page paper.” This state-
ment is a __________.
a. conjunction
b. disjunction
c. conditional
d. biconditional
Practice Problems 4.1 (continued)
har85668_04_c04_119-164.indd 133 4/9/15 1:26 PM
It is the exact same way with logic. In the statement (H ∨ T) →
D, because of the parentheses,

we know that this statement is a conditional (not a disjunction).
It is of the form P → Q, where
P is replaced by H ∨ T and Q is replaced by D.
Here is another example:
N & S G (N & S) → G
He is nice and smart. You should get to know him. If he is nice
and smart, then you
should get to know him.
This example shows a complex way to make a sentence out of
three component sentences.
N is “He is nice,” S is “he is smart,” and G is “you should get to
know him.” Here is another:
R (S & C) R → (S & C)
You want to be rich. You should study hard and go to
college.
If you want to be rich, then you
should study hard and go to
college.
If R is “You want to be rich,” S is “You should study hard,” and
C is “You should go to college,”
then the whole statement in this final example, symbolized R →
(S & C), means “If you want
to be rich, then you should study hard and go to college.”
Complex statements can be created in this manner for every
form. Take the statement (~A &
B) ∨ (C → ~D). This statement has the general form of a

disjunction. It has the form P ∨ Q,
where P is replaced with ~A & B, and Q is replaced with C →
~D.
Everyday Logic: Complex Statements in Ordinary Language
It is not always easy to determine how to translate complex,
ordinary language statements
into logic; one sometimes has to pick up on clues within the
statement.
For instance, notice in general that neither P nor Q is translated
~(P ∨ Q). This is because P ∨ Q
means that either one is true, so ~(P ∨ Q) means that neither
one is true. It happens to be equiva-
lent to saying ~P & ~Q (we will talk about logical equivalence
later in this chapter).
Here are some more complex examples:
Statement Translation
If you don’t eat spinach, then you will neither
be big nor strong.
~S → ~(B ∨ S)
Either he is strong and brave, or he is both
reckless and foolish.
(S & B) ∨ (R & F)
Come late and wear wrinkled clothes only if
you don’t want the job.
(L & W) → ~J

He is strong and brave, and if he doesn’t like
you, he will let you know.
(S & B) & (~L → K)
har85668_04_c04_119-164.indd 134 4/9/15 1:26 PM
Truth Tables With Complex Statements
We have managed to symbolize complex statements by seeing
how they are systematically
constructed out of their parts. Here we use the same principle to
create truth tables that allow
us to find the truth values of complex statements based on the
truth values of their parts. It
will be helpful to start with a summary of the truth values of
sentences constructed with the
basic truth-functional operators:
P Q ~P P & Q P ∨ Q P → Q P ↔ Q
T T F T T T T
T F F F T F F
F T T F T T F
F F T F F T T
The truth values of more complex statements can be discovered
by applying these basic for-
mulas one at a time. Take a complex statement like (A ∨ B) →
(A & B). Do not be intimidated

by its seemingly complex form; simply take it one operator at a
time. First, notice the main
form of the statement: It is a conditional (we know this because
the other operators are within
parentheses). It therefore has the form P → Q, where P is “A ∨
B” and Q is “A & B.”
The antecedent of the conditional is A ∨ B; the consequent is A
& B. The way to find the truth
values of such statements is to start inside the parentheses and
find those truth values first,
and then work our way out to the main operator—in this case
→.
Here is the truth table for these components:
A B A ∨ B A & B
T T T T
T F T F
F T T F
F F F F
Now we take the truth tables for these components to create the
truth table for the overall
conditional:
A B A ∨ B A & B (A ∨ B) → (A & B)
T T T T T
T F T F F
F T T F F
F F F F T
In this way the truth values of very complex statements can be
determined from the values of
their parts. We may refer to these columns (in this case A ∨ B

and A & B) as helper columns,
because they are there just to assist us in determining the truth
values for the more complex
statement of which they are a part.
har85668_04_c04_119-164.indd 135 4/9/15 1:26 PM
Here is another one: (A & ~B) → ~(A ∨ B). This one is also a
conditional, where the anteced-
ent is A & ~B and the consequent is ~(A ∨ B). We do these
components first because they are
inside parentheses. However, to find the truth table for A & ~B,
we will have to fill out the
truth table for ~B first (as a helper column).
A B ~B A & ~B
T T F F
T F T T
F T F F
F F T F
We found ~B by simply negating B. We then found A & ~B by
applying the truth table for con-
junctions to the column for A and the column for ~B.
Now we can fill out the truth table for A ∨ B and then use that
to find the values of ~(A ∨ B):

A B A ∨ B ~(A ∨ B)
T T T F
T F T F
F T T F
F F F T
Finally, we can now put A & ~B and ~(A ∨ B) together with the
conditional to get our truth
table:
A B A & ~B ~(A ∨ B) (A & ~B) → ~(A ∨ B)
T T F F T
T F T F F
F T F F T
F F F T T
Although complicated, it is not hard when one realizes that one
has to apply only a series of
simple steps in order to get the end result.
Here is another one: (A → ~B) ∨ ~(A & B). First we will do the
truth table for the left part of
the disjunction (called the left disjunct), A → ~B:
A B ~B A → ~B
T T F F
T F T T
F T F T
F F T T
Of course, the last column is based on combining the first
column, A, with the third column,
~B, using the conditional. Now we can work on the right
disjunct, ~(A & B):

har85668_04_c04_119-164.indd 136 4/9/15 1:26 PM
A B A & B ~(A & B)
T T T F
T F F T
F T F T
F F F T
The final truth table, then, is:
A B A→~B ~(A & B) (A→~B) ∨ ~(A & B)
T T F F F
T F T T T
F T T T T
F F T T T
You may have noticed that three formulas in the truth table have
the exact same values on
every row. That means that the formulas are logically
equivalent. In propositional logic, two
formulas are logically equivalent if they have the same truth
values on every row of the
truth table. Logically equivalent formulas are therefore true in
the exact same circumstances.
Logicians consider this important because two formulas that are
logically equivalent, in the
logical sense, mean the same thing, even though they may look
quite different. The conditions

for their truth and falsity are identical.
The fact that the truth value of a complex statement follows
from the truth values of its compo-
nent parts is why these operators are called truth-functional.
The operators, &, ∨ , ~, →, and ↔,
are truth-functions, meaning that the truth of the whole sentence
is a function of the truth of the
parts.
Because the validity of argument forms within propositional
logic is based on the behavior of
the truth-functional operators, another name for propositional
logic is truth-functional logic.
Truth Tables With Three Letters
In each of the prior complex statement examples, there were
only two letters (variables like
P and Q or constants like A and B) in the top left of the truth
table. Each truth table had only
four rows because there are only four possible combinations of
truth values for two variables
(both are true, only the first is true, only the second is true, and
both are false).
It is also possible to do a truth table for sentences that contain
three or more variables (or
constants). Recall one of the earlier examples: “Come late and
wear wrinkled clothes only if
you don’t want the job,” which we represented as (L & W) →
~J. Now that there are three let-
ters, how many possible combinations of truth values are there
for these letters?
The answer is that a truth table with three variables (or
constants) will have eight lines. The

general rule is that whenever you add another letter to a truth
table, you double the number
of possible combinations of truth values. For each earlier
combination, there are now two:
one in which the new letter is true and one in which it is false.
Therefore, to make a truth table
har85668_04_c04_119-164.indd 137 4/9/15 1:26 PM
P
T
T
F
F
T
F
T
F
T
F
T
F

T
F
T
F
Q R
with three letters, imagine the truth table for two letters and
imagine each row splitting in
two, as follows:
The resulting truth table rows would look like this:
P Q R
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
The goal is to have a row for every possible truth value
combination. Generally, to fill in the
rows of any truth table, start with the last letter and simply
alternate T, F, T, F, and so on, as
in the R column. Then move one letter to the left and do twice
as many Ts followed by twice
as many Fs (two of each): T, T, F, F, and so on, as in the Q

column. Then move another letter to
the left and do twice as many of each again (four each), in this
case T, T, T, T, F, F, F, F, as in the
P column. If there are more letters, then we would repeat the
process, adding twice as many
Ts for each added letter to the left.
With three letters, there are eight rows; with four letters, there
are sixteen rows, and so on.
This chapter does not address statements with more than three
letters, so another way to
ensure you have enough rows is to memorize this pattern.
P
T
T
F
F
T
F
T
F
T
F
T
F

T
F
T
F
Q R
har85668_04_c04_119-164.indd 138 4/9/15 1:26 PM
The column with the forms is filled out the same way as when
there were two letters. The fact
that they now have three letters makes little difference, because
we work on only one operator,
and therefore at most two columns of letters, at a time. Let us
start with the example of P →
(Q & R). We begin by solving inside the parentheses by
determining the truth values for Q & R,
then we create the conditional between P and that result. The
table looks like this:
P Q R Q & R P → (Q & R)
T T T T T
T T F F F
T F T F F
T F F F F
F T T T T
F T F F T

F F T F T
F F F F T
The rules for determining the truth values of Q & R and then of
P → (Q & R) are exactly the
same as the rules for & and → that we used in the two-letter
truth tables earlier; now we just
use them for more rows. It is a formal process that generates
truth values by the same strict
algorithms as in the two-letter tables.
Symbolize the following complex statements using the symbols
that you have learned
in this chapter.
1. One should be neither a borrower nor a lender.
2. Atomic bombs are dangerous and destructive.
3. If we go to the store, then I need to buy apples and lettuce.
4. Either Microsoft enhances its product and Dell’s sales
decrease, or Gateway will start
making computers again.
5. If Hondas have better gas mileage than Range Rovers and you
are looking for some-
thing that is easy to park, I recommend that you buy the Honda.
6. Global warming will decrease if and only if emissions
decrease in China and other
major polluters around the world.
7. One cannot be both happy and successful in our society, but

one can be happy or
successful.
8. I will pass this course if and only if I study hard and practice
regularly, if I have the
time and energy to do so.
9. God can only exist if evil does not exist, if it is true that God
is both all-powerful and
all-good.
10. The conflict in Israel will end only if the Palestinians feel
that they can live outside
the supervision of the Israelis and the two sides stop attacking
one another.
har85668_04_c04_119-164.indd 139 4/9/15 1:26 PM
Section 4.4 Using Truth Tables to Test for Validity
4.4 Using Truth Tables to Test for Validity
Truth tables serve many valuable purposes. One is to help us
better understand how the logi-
cal operators work. Another is to help us understand how truth
is determined within formally
structured sentences. One of the most valuable things truth
tables offer is the ability to test
argument forms for validity. As mentioned at the beginning of
this chapter, one of the main pur-
poses of formal logic is to make the concept of validity precise.
Truth tables help us do just that.

As mentioned in previous chapters, an argument is valid if and
only if the truth of its premises
guarantees the truth of its conclusion. This is equivalent to
saying that there is no way that the
premises can be true and the conclusion false.
Truth tables enable us to determine precisely if there is any way
for all of the premises to be
true and the conclusion false (and therefore whether the
argument is valid): We simply create
a truth table for the premises and conclusion and see if there is
any row on which all of the
premises are true and the conclusion is false. If there is, then
the argument is invalid, because
that row shows that it is possible for the premises to be true and
the conclusion false. If there
is no such line, then the argument is valid:
Since the rows of a truth table cover all possibilities, if there is
no row on which all of the
premises are true and the conclusion is false, then it is
impossible, so the argument is valid.
Let us start with a simple example—note that the ∴ symbol
means “therefore”:
P ∨ Q
~Q
∴ P
This argument form is valid; if there are only two options, P and
Q, and one of them is false,
then it follows that the other one must be true. However, how
can we formally demonstrate
its validity? One way is to create a truth table to find out if

there is any possible way to make
all of the premises true and the conclusion false.
Here is how to set up the truth table, with a column for each
premise (P1 and P2) and the
conclusion (C):
P1 P2 C
P Q P ∨ Q ~Q P
T T
T F
F T
F F
har85668_04_c04_119-164.indd 140 4/9/15 1:26 PM
We then fill in the columns, with the correct truth values:
P1 P2 C
P Q P ∨ Q ~Q P
T T T F T
T F T T T
F T T F F
F F F T F

We then check if there are any rows in which all of the premises
are true and the conclusion is
false. A brief scan shows that there are no such lines. The first
two rows have true conclusions,
and the remaining two rows each have at least one false
premise. Since the rows of a truth
table represent all possible combinations of truth values, this
truth table therefore demon-
strates that there is no possible way to make all of the premises
true and the conclusion false.
It follows, therefore, that the argument is logically valid.
To summarize, the steps for using the truth table method to
determine an argument’s validity
are as follows:
1. Set up the truth table by creating rows for each possible
combination of truth values
for the basic letters and a column for each premise and the
conclusion.
2. Fill out the truth table by filling out the truth values in each
column according to the
rules for the relevant operator (~, &, ∨ , →, ↔).
3. Use the table to evaluate the argument’s validity. If there is
even one row on which all
of the premises are true and the conclusion is false, then the
argument is invalid; if
there is no such row, then the argument is valid.
This truth table method works for all arguments in propositional
logic: Any valid proposi-
tional logic argument will have a truth table that shows it is
valid, and every invalid proposi-
tional logic argument will have a truth table that shows it is

invalid. Therefore, this is a perfect
test for validity: It works every time (as long as we use it
accurately).
Examples With Arguments With Two Letters
Let us do another example with only two letters. This argument
will be slightly more complex
but will still involve only two letters, A and B.
Example 1
A → B
~(A & B)
∴ ~(B ∨ A)
har85668_04_c04_119-164.indd 141 4/9/15 1:26 PM
To test this symbolized argument for validity, we first set up the
truth table by creating rows
with all of the possible truth values for the basic letters on the
left and then create a column
for each premise (P1 and P2) and conclusion (C), as follows:
P1 P2 C
A B A → B ~(A & B) ~(B ∨ A)
T T
T F
F T

F F
Second, we fill out the truth table using the rules created by the
basic truth tables for each
operator. Remember to use helper columns where necessary as
steps toward filling in the
columns of complex formulas. Here is the truth table with only
the helper columns filled in:
P1 P2 C
A B A → B A & B ~(A & B) B ∨ A ~(B ∨ A)
T T T T
T F F T
F T F T
F F F F
Here is the truth table with the rest of the columns filled in:
P1 P2 C
A B A → B A & B ~(A & B) B ∨ A ~(B ∨ A)
T T T T F T F
T F F F T T F
F T T F T T F
F F T F T F T
Finally, to evaluate the argument’s validity, all we have to do is
check to see if there are any
lines in which all of the premises are true and the conclusion is
false. Again, if there is such a
line, since we know it is possible for all of the premises to be
true and the conclusion false, the
argument is invalid. If there is no such line, then the argument

is valid.
It does not matter what other rows may exist in the table. There
may be rows in which all
of the premises are true and the conclusion is also true; there
also may be rows with one or
more false premises. Neither of those types of rows determine
the argument’s validity; our
only concern is whether there is any possible row on which all
of the premises are true and
the conclusion false. Is there such a line in our truth table?
(Remember: Ignore the helper
columns and just focus on the premises and conclusion.)
har85668_04_c04_119-164.indd 142 4/9/15 1:26 PM
The answer is yes, all of the premises are true and the
conclusion is false in the third row. This
row supplies a proof that this argument’s form is invalid. Here
is the line:
P1 P2 C
A B A → B ~(A & B) ~(B ∨ A)
F T T T F
Again, it does not matter what is on the other row. As long as
there is (at least) one row in

which all of the premises are true and the conclusion false, the
argument is invalid.
Example 2
A → (B & ~A)
A ∨ ~B
∴ ~(A ∨ B)
First we set up the truth table:
P1 P2 C
A B ~A B & ~A A → (B & ~A) ~B A ∨ ~B A ∨ B ~(A ∨ B)
T T
T F
F T
F F
Next we fill in the values, filling in the helper columns first:
P1 P2 C
T T F F F T
T F F F T T
F T T T F T
F F T F T F
Now that the helper columns are done, we can fill in the rest of
the table’s values:
P1 P2 C

T T F F F F T T F
T F F F F T T T F
F T T T T F F T F
F F T F T T T F T
Finally, we evaluate the table for validity. Here we see that
there are no lines in which all of the
premises are true and the conclusion is false. Therefore, there is
no possible way to make all
of the premises true and the conclusion false, so the argument is
valid.
har85668_04_c04_119-164.indd 143 4/9/15 1:26 PM
The earlier examples each had two premises. The following
example has three premises. The
steps of the truth table test are identical.
Example 3
~(M ∨ B)
M → ~B
B ∨ ~M
∴ ~M & B)
First we set up the truth table. This table already has the helper
columns filled in.
P1 P2 P3 C

M B M ∨ B ~(M ∨ B) ~B M → ~B ~M B ∨ ~M ~M & B
T T T F F
T F T T F
F T T F T
F F F T T
Now we fill in the rest of the columns, using the helper columns
to determine the truth values
of our premises and conclusion on each row:
P1 P2 P3 C
M B M ∨ B ~(M ∨ B) ~B M → ~B ~M B ∨ ~M ~M & B
T T T F F F F T F
T F T F T T F F F
F T T F F T T T T
F F F T T T T T F
Now we look for a line in which all of the premises are true and
the conclusion false. The final
row is just such a line. This demonstrates conclusively that the
Examples With Arguments With Three Letters
The last example had three premises, but only two letters. These
next examples will have
three letters. As explained earlier in the chapter, the presence of
the extra letter doubles the
number of rows in the truth table.
har85668_04_c04_119-164.indd 144 4/9/15 1:26 PM

Example 1
A → (B ∨ C)
~(C & B)
∴ ~(A & B)
First we set up the truth table. Note, as mentioned earlier, now
there are eight possible com-
binations on the left.
P1 P2 C
A B C B ∨ C A → (B ∨ C) C & B ~(C & B) A & B ~(A & B)
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
Then we fill the table out. Here it is with just the helper
columns:
P1 P2 C
A B C B ∨ C A → (B ∨ C) C & B ~(C & B) A & B ~(A & B)
T T T T T T

T T F T F T
T F T T F F
T F F F F F
F T T T T F
F T F T F F
F F T T F F
F F F F F F
Here is the full truth table:
P1 P2 C
A B C B ∨ C A → (B ∨ C) C & B ~ (C & B) A & B ~(A & B)
T T T T T T F T F
T T F T T F T T F
T F T T T F T F T
T F F F F F T F T
F T T T T T F F T
F T F T T F T F T
F F T T T F T F T
F F F F T F T F T
har85668_04_c04_119-164.indd 145 4/9/15 1:26 PM
Finally, we evaluate; that is, we look for a line in which all of
the premises are true and the
conclusion false. This is the case with the second line. Once you
find such a line, you do not

need to look any further. The existence of even one line in
which all of the premises are true
and the conclusion is false is enough to declare the argument
invalid.
Let us do another one with three letters:
Example 2
A → ~B
B ∨ C
∴ A → C
We begin by setting up the table:
P1 P2 C
A B C ~B A → ~B B ∨ C A → C
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
Now we can fill in the rows, beginning with the helper columns:
P1 P2 C
A B C ~B A → ~B B ∨ C A → C
T T T F F T T
T T F F F T F
T F T T T T T

T F F T T F F
F T T F T T T
F T F F T T T
F F T T T T T
F F F T T F T
Here, when we look for a line in which all of the premises are
true and the conclusion false, we
do not find one. There is no such line; therefore the argument is
valid.
har85668_04_c04_119-164.indd 146 4/9/15 1:26 PM
Answer these questions about truth tables.
1. A truth table with two variables has how many lines?
a. 1
b. 2
c. 4
d. 8
2. A truth table with three variables has how many lines?
a. 1
b. 2
c. 4
d. 8

3. In order to prove that an argument is invalid using a truth
table, one must __________.
a. find a line in which all premises and the conclusion are false
b. find a line in which the premises are true and the conclusion
is false
c. find a line in which the premises are false and the conclusion
is true
d. find a line in which the premises and the conclusion are true
4. This is how one can tell if an argument is valid using a truth
table:
a. There is a line in which the premises and the conclusion are
true.
b. There is no line in which the premises are false.
c. There is no line in which the premises are true and the
conclusion is false.
d. All of the above
e. None of the above
5. When two statements have the same truth values in all
circumstances, they are said
to be __________.
a. logically contradictory
b. logically equivalent
c. logically cogent
d. logically valid
6. An if–then statement is called a __________.
a. conjunction
b. disjunction
c. conditional
d. biconditional
7. An if and only if statement is called a __________.
a. conjunction
b. disjunction

c. conditional
d. biconditional
8. An and statement is called a __________.
a. conjunction
b. disjunction
c. conditional
d. biconditional
(continued)
har85668_04_c04_119-164.indd 147 4/9/15 1:26 PM
Utilize truth tables to determine the validity of the following
arguments.
9. J → K
J
∴ K
10. H → G
G
∴ H
11. K→ K
∴ K
12. ~(H & Y)
Y ∨ ~H

∴ ~H
13. W → Q
~W
∴ ~Q
14. A → B
B → C
∴ A → C
15. ~(P ↔ U)
∴ ~(P → U)
16. ~S ∨ H
~S
∴ ~H
17. ~K → ~L
J → ~K
∴ J → ~L
18. Y & P
P
∴ ~Y
19. A → ~G
V → ~G
∴ A → V
20. B & K & I
∴ K
har85668_04_c04_119-164.indd 148 4/9/15 1:26 PM

Section 4.5 Some Famous Propositional Argument Forms
4.5 Some Famous Propositional Argument Forms
Using the truth table test for validity, we have seen that we can
determine the validity or inva-
lidity of all propositional argument forms. However, there are
some basic argument forms
that are so common that it is worthwhile simply to memorize
them and whether or not they
are valid. We will begin with five very famous valid argument
forms and then cover two of the
most famous invalid argument forms.
Common Valid Forms
It is helpful to know some of the most commonly used valid
argument forms. Those presented
in this section are used so regularly that, once you learn them,
you may notice people using
them all the time. They are also used in what are known as
deductive proofs (see A Closer
Look: Deductive Proofs).
A Closer Look: Deductive Proofs
A big part of formal logic is constructing proofs. Proofs in
logic are a lot like proofs in mathematics. We start with
certain premises and then use certain rules—called rules
of inference—in a step-by-step way to arrive at the con-
clusion. By using only valid rules of inference and apply-
ing them carefully, we make certain that every step of the
proof is valid. Therefore, if there is a logical proof of the
conclusion from the premises, then we can be certain

that the argument itself is valid.
The rules of inference used in deductive proofs are
actually just simple valid argument forms. In fact,
the valid argument forms covered here—including
modus ponens, hypothetical syllogisms, and disjunctive
syllogisms—are examples of argument forms that are used
as inference rules in logical proofs. Using these and other
formal rules, it is possible to give a logical proof for every
valid argument in propositional logic (Kennedy, 2012).
Logicians, mathematicians, philosophers, and computer
scientists use logical proofs to show that the validity of
certain inferences is absolutely certain and founded on
the most basic principles. Many of the inferences we make in
daily life are of limited certainty;
however, the validity of inferences that have been logically
proved is considered to be the most
certain and uncontroversial of all knowledge because it is
derivable from pure logic.
Covering how to do deductive proofs is beyond the scope of this
book, but readers are invited to
peruse a book or take a course on formal logic to learn more
about how deductive proofs work.
Mark Wragg/iStock/Thinkstock
Rather than base decisions on
chance, people use the information
around them to make deductive and
inductive inferences with varying
degrees of strength and validity.
Logicians use proofs to show the
validity of inferences.

har85668_04_c04_119-164.indd 149 4/9/15 1:26 PM
Modus Ponens
Perhaps the most famous propositional argument form of all is
known as modus ponens—
Latin for “the way of putting.” (You may recognize this form
from the earlier section on the
truth table method.) Modus ponens has the following form:
P → Q
P
∴ Q
You can see that the argument is valid just from the meaning of
the conditional. The first
premise states, “If P is true, then Q is true.” It would logically
follow that if P is true, as the
second premise states, then Q must be true. Here are some
examples:
If you want to get an A, you have to study.
You want to get an A.
Therefore, you have to study.
If it is raining, then the street is wet.
It is raining.
Therefore, the street is wet.

If it is wrong, then you shouldn’t do it.
It is wrong.
Therefore, you shouldn’t do it.
A truth table will verify its validity.
P1 P2 C
P Q P → Q P Q
T T T T T
T F F T F
F T T F T
F F T F F
There is no line in which all of the premises are true and the
conclusion false, verifying the
validity of this important logical form.
Modus Tollens
A closely related form has a closely related name. Modus
tollens—Latin for “the way of tak-
ing”—has the following form:
P → Q
~Q
∴ ~P
har85668_04_c04_119-164.indd 150 4/9/15 1:26 PM

A truth table can be used to verify the validity of this
form as well. However, we can also see its validity
by simply thinking it through. Suppose it is true that
“If P, then Q.” Then, if P were true, it would follow
that Q would be true as well. But, according to the
second premise, Q is not true. It follows, therefore,
that P must not be true; otherwise, Q would have
been true. Here are some examples of arguments
that fit this logical form:
In order to get an A, I must study.
I will not study.
Therefore, I will not get an A.
If it rained, then the street would be wet.
The street is not wet.
Therefore, it must not have rained.
If the ball hit the window, then I would hear
glass shattering.
I did not hear glass shattering.
Therefore, the ball must not have hit the
window.
For practice, construct a truth table to demonstrate
the validity of this form.
Disjunctive Syllogism
A disjunctive syllogism is a valid argument form in which one
premise states that you have
two options, and another premise allows you to rule one of them
out. From such premises, it
follows that the other option must be true. Here are two
versions of it formally (both are valid):

P ∨ Q
~P
∴ Q
P ∨ Q
~Q
∴ P
In other words, if you have “P or Q” and not Q, then you may
infer P. Here is another example:
“Either the butler or the maid did it. It could not have been the
butler. Therefore, it must have
been the maid.” This argument form is quite handy in real life.
It is frequently useful to con-
sider alternatives and to rule one out so that the options are
narrowed down to one.
Ruth Black/iStock/Thinkstock
Evaluate this argument form for
validity: If the cake is made with sugar,
then the cake is sweet. The cake is not
sweet. Therefore, the cake is not made
with sugar.
har85668_04_c04_119-164.indd 151 4/9/15 1:26 PM
Hypothetical Syllogism
One of the goals of a logically valid argument is for the

premises to link together so that the
conclusion follows smoothly, with each premise providing a
link in the chain. Hypothetical
syllogism provides a nice demonstration of just such premise
linking. Hypothetical syllogism
takes the following form:
P → Q
Q → R
∴ P → R
For example, “If you lose your job, then you will have no
income. If you have no income, then
you will starve. Therefore, if you lose your job, then you will
starve!”
Double Negation
Negating a sentence (putting a ~ in front of it) makes it say the
opposite of what it originally
said. However, if we negate it again, we end up with a sentence
that means the same thing as
our original sentence; this is called double negation.
Imagine that our friend Johnny was in a race, and you ask me,
“Did he win?” and I respond,
“He did not fail to win.” Did he win? It would appear so.
Though some languages allow double
negations to count as negative statements, in logic a double
negation is logically equivalent to
the original statement. Both of these forms, therefore, are valid:
P
∴ ~~P
~~P
∴ P

A truth table will verify that each of these forms is valid; both P
and ~~P have the same truth
values on every row of the truth table.
Common Invalid Forms
Both modus ponens and modus tollens are logically valid forms,
but not all famous logical
forms are valid. The last two forms we will discuss—denying
the antecedent and affirming
the consequent—are famous invalid forms that are the evil twins
of the previous two.
Denying the Antecedent
Take a look at the following argument:
If you give lots of money to charity, then you are nice.
You do not give lots of money to charity.
Therefore, you must not be nice.
har85668_04_c04_119-164.indd 152 4/9/15 1:26 PM
This might initially seem like a valid argument. However, it is
actually invalid in its form. To
see that this argument is logically invalid, take a look at the
following argument with the same
form:
If my cat is a dog, then it is a mammal.

My cat is not a dog.
Therefore, my cat is not a mammal.
This second example is clearly invalid since the premises are
true and the conclusion is false.
Therefore, there must be something wrong with the form. Here
is the form of the argument:
P → Q
~P
∴ ~Q
Because this argument form’s second premise rejects the
antecedent, P, of the conditional in
the first premise, this argument form is referred to as denying
the antecedent. We can con-
clusively demonstrate that the form is invalid using the truth
table method.
Here is the truth table:
P1 P2 C
P Q P → Q ~P ~Q
T T T F F
T F F F T
F T T T F
F F T T T
We see on the third line that it is possible to make both
premises true and the conclusion false,
so this argument form is definitely invalid. Despite its
invalidity, we see this form all the time
in real life. Here some examples:

If you are religious, then you believe in living morally.
Jim is not religious, so he must not believe in living morally.
Plenty of people who are not religious still believe in living
morally. Here is another one:
If you are training to be an athlete, then you should stay in
shape.
You are not training to be an athlete.
Thus, you should not stay in shape.
There are plenty of other good reasons to stay in shape.
If you are Republican, then you support small government.
Jack is not Republican, so he must not support small
government.
har85668_04_c04_119-164.indd 153 4/9/15 1:26 PM
Libertarians, for example, are not Republicans, yet they support
small government. These
examples abound; we can generate them on any topic.
Because this argument form is so common and yet so clearly
invalid, denying the antecedent
is a famous fallacy of formal logic.
Affirming the Consequent
Another famous formal logical fallacy also begins with a

conditional. However, the other two
lines are slightly different. Here is the form:
P → Q
Q
∴ P
Because the second premise states the consequent of the
conditional, this form is called
affirming the consequent. Here is an example:
If you get mono, you will be very tired.
You are very tired.
Therefore, you have mono.
The invalidity of this argument can be seen in the following
argument of the same form:
If my cat is a dog, then it is a mammal.
My cat is a mammal.
Therefore, my cat is a dog.
Clearly, this argument is invalid because it has true premises
and a false conclusion. There-
fore, this must be an invalid form. A truth table will further
demonstrate this fact:
P1 P2 C
P Q P → Q Q P
T T T T T
T F F F T
F T T T F
F F T F F

The third row again demonstrates the possibility of true
premises and a false conclusion, so
the argument form is invalid. Here are some examples of how
this argument form shows up
in real life:
har85668_04_c04_119-164.indd 154 4/9/15 1:26 PM
In order to get an A, I have to study.
I am going to study.
Therefore, I will get an A.
There might be other requirements to get an A, like showing up
for the test.
If it rained, then the street would be wet.
The street is wet.
Therefore, it must have rained.
Sprinklers may have done the job instead.
If he committed the murder, then he would have had to have
motive and
opportunity.
He had motive and opportunity.
Therefore, he committed the murder.
This argument gives some evidence for the conclusion, but it
does not give proof. It is possible

that someone else also had motive and opportunity.
The reader may have noticed that in some instances of affirming
the consequent, the prem-
ises do give us some reason to accept the conclusion. This is
because of the similarity of this
form to the inductive form known as inference to the best
explanation, which is covered in
more detail in Chapter 6. In such inferences we create an “if–
then” statement that expresses
something that would be the case if a certain assumption were
true. These things then act as
symptoms of the truth of the assumption. When those symptoms
are observed, we have some
evidence that the assumption is true. Here are some examples:
If you have measles, then you would present the following
symptoms. . . .
You have all of those symptoms.
Therefore, it looks like you have measles.
If he is a faithful Catholic, then he would go to Mass.
I saw him at Mass last Sunday.
Therefore, he is probably a faithful Catholic.
All of these seem to supply decent evidence for the conclusion;
however, the argument form is
not logically valid. It is logically possible that another medical
condition could have the same
symptoms or that a person could go to Mass out of curiosity. To
determine the (inductive)
inferential strength of an argument of that form, we need to
think about how likely Q is under
different assumptions.
har85668_04_c04_119-164.indd 155 4/9/15 1:26 PM

A Closer Look: Translating Categorical Logic
The chapter about categorical logic seems to cover a completely
different type of reasoning
than this chapter on propositional logic. However, logical
advancements made just over a
century ago by a man named Gottlob Frege showed that the two
types of logic can be com-
bined in what has come to be known as quantificational logic
(also known as predicate logic)
(Frege, 1879).
In addition to truth-functional logic, quantificational logic
allows us to talk about quantities
by including logical terms for all and some. The addition of
these terms dramatically increases
the power of our logical language and allows us to represent all
of categorical logic and much
more. Here is a brief overview of how the basic sentences of
categorical logic can be repre-
sented within quantificational logic.
The statement “All dogs are mammals” can be understood to
mean “If you are a dog, then you
are a mammal.” The word you in this sentence applies to any
individual. In other words, the
sentence states, “For all individuals, if that individual is a dog,
then it is a mammal.” In general,
statements of the form “All S is M” can be represented as “For

all things, if that thing is S, then
it is M.”
The statement “Some dogs are brown” means that there exist
dogs that are brown. In other
words, there exist things that are both dogs and brown.
Therefore, statements of the form
“Some S is M” can be represented as “There exists a thing that
is both S and M” (propositions of
the form “Some S are not M” can be represented by simply
adding a negation in front of the M).
Statements like “No dogs are reptiles” can be understood to
mean that all dogs are not reptiles.
In general, statements of the form “No S are M” can be
represented as “For all things, if that
thing is an S, then it is not M.”
Quantificational logic allows us to additionally represent the
meanings of statements that
go well beyond the AEIO propositions of categorical logic. For
example, complex statements
like “All dogs that are not brown are taller than some cats” can
also be represented with the
power of quantificational logic though they are well beyond the
capacity of categorical logic.
The additional power of quantificational logic enables us to
represent the meaning of vast
stretches of the English language as well as statements used in
formal disciplines like math-
ematics. More instruction in this interesting area can be found
in a course on formal logic.
har85668_04_c04_119-164.indd 156 4/9/15 1:26 PM

Each of the following arguments is a deductive form. Identify
the valid form under which
the example falls. If the example is not a valid form, select “not
a valid form.”
1. If we do not decrease poverty in society, then our society will
not be an equal one.
We are not going to decrease poverty in society. Therefore, our
society will not be an
equal one.
a. modus ponens
b. modus tollens
c. disjunctive syllogism
d. hypothetical syllogism
e. not a valid form
2. If we do not decrease poverty in society, then our society will
not be an equal one.
Our society will be an equal one. Therefore, we will decrease
poverty in society.
a. modus ponens
b. modus tollens
e. not a valid form
3. If the moon is full, then it is a good time for night fishing. If
it’s a good time for night

fishing, then we should go out tonight. Therefore, if the moon is
full, then we should
go out tonight.
a. modus ponens
b. modus tollens
e. not a valid form
4. Either the Bulls or the Knicks will lose tonight. The Bulls are
not going to lose.
Therefore, the Knicks will lose.
a. modus ponens
b. modus tollens
e. not a valid form
5. If the battery is dead, then the car won’t start. The car won’t
start. Therefore, the bat-
tery is dead.
a. modus ponens
b. modus tollens
e. not a valid form
6. If I take this new job, then we will have to move to Alaska. I
am not going to take the
new job. Therefore, we will not have to move to Alaska.
a. modus ponens
b. modus tollens
e. not a valid form

(continued)
har85668_04_c04_119-164.indd 157 4/9/15 1:26 PM
Summary and Resources
7. If human perception conditions reality, then humans cannot
know things in them-
selves. If humans cannot know things in themselves, then they
cannot know the
truth. Therefore, if human perceptions conditions reality, then
humans cannot know
the truth.
a. modus ponens
b. modus tollens
e. not a valid form
8. We either adopt the plan or we will be in danger of losing our
jobs. We are not going
to adopt the plan. Therefore, we will be in danger of losing our
jobs.
a. modus ponens
b. modus tollens
e. not a valid form
9. If media outlets are owned by corporations with advertising
interests, then it will

be difficult for them to be objective. Media outlets are owned
by corporations with
advertising interests. Therefore, it will be difficult for them to
be objective.
a. modus ponens
b. modus tollens
e. not a valid form
10. If you eat too much aspartame, you will get a headache. You
do not have a headache.
Therefore, you did not eat too much aspartame.
a. modus ponens
b. modus tollens
e. not a valid form
Chapter Summary
Propositional logic shows how the truth values of complex
statements can be systematically
derived from the truth values of their parts. Words like and, or,
not, and if . . . then . . . each have
truth tables that demonstrate the algorithms for determining
these truth values. Once we have
found the logical form of an argument, we can determine
whether it is logically valid by using
the truth table method. This method involves creating a truth
table that represents all possible
truth values of the component parts and the resulting values for
the premises and conclusion

har85668_04_c04_119-164.indd 158 4/9/15 1:26 PM
of the argument. If there is even one row of the truth table in
which all of the premises are true
and the conclusion is false, then the argument is invalid; if there
is no such row, then it is valid.
Knowledge of propositional logic has proved very valuable to
humankind: It allows us to
formally demonstrate the validity of different types of
reasoning; it helps us precisely under-
stand the meaning of certain types of terms in our language; it
enables us to determine the
truth conditions of formally complex statements; and it forms
the basis for computing.
Critical Thinking Questions
1. Symbolizing arguments makes them easier to visualize and
examine in the realm
of propositional logic. Do you find that the symbols make
things easier to visualize
or more confusing? If logicians use these methods to make
things easier, then what
does that mean if you think that using these symbols is
confusing?
2. In your own words, what is the difference between

categorical logic and proposi-
tional logic? How do they relate to one another? How do they
differ?
3. How does understanding how to symbolize statements and
complete truth tables
relate to your everyday life? What is the practical importance of
understanding how
to use these methods to determine validity?
4. If you were at work or with your friends and someone
presented an argument, do
you think you could evaluate it using the methods you have
learned thus far in this
book? Is it important to evaluate arguments, or is this just
something academics do
in their spare time? Why do you believe this is (or is not) the
case?
5. How would you now explain the concept of validity to
someone with whom you
interact on a daily basis who might not have an understanding
of logic? How would
you explain how validity differs from truth?
Web Resources
http://www.manyworldsof
logic.com/exercises/quizTruthFunctional.html
Test your understanding of propositional, or truth-functional,
logic by taking the quizzes
available at philosophy professor Paul Herrick’s Many Worlds
of Logic website.
https://www.youtube.com/watch?v=moHkk_89UZE
Watch a video that walks you through how to construct a truth
table.

https://www.youtube.com/watch?feature=player_embedded&v=8
3xPkTqoulE
Watch Ashford University professor Justin Harrison explain
how to construct a conjunction
truth table.
Key Terms
affirming the consequent An argument
with two premises, one of which is a condi-
tional and the other of which is the conse-
quent of that conditional. It has the form
P → Q, Q, therefore P. It is invalid.
antecedent The part of a conditional state-
ment that occurs after the if; it is the P in
P → Q.
biconditional A statement of the form
P ↔ Q (P if and only if Q).
har85668_04_c04_119-164.indd 159 4/9/15 1:26 PM
http://www.manyworldsoflogic.com/exercises/quizTruthFunctio
nal.html
https://www.youtube.com/watch?v=moHkk_89UZE
https://www.youtube.com/watch?feature=player_embedded&v=8
3xPkTqoulE

conditional An “if–then” statement. It is
symbolized P → Q.
conjunction A statement in which two sen-
tences are joined with an and. It is symbol-
ized P & Q. Also, an inference rule that allows
us to infer P & Q from premises P and Q.
connectives See operators.
consequent The part of a conditional state-
ment that occurs after the then; it is the Q in
P → Q.
converse The result of switching the order
of the terms within a conditional or cat-
egorical statement. The converse of P → Q
is Q → P. The converse of “All S are M” is “All
M are S.”
denying the antecedent An argument with
two premises, one of which is a conditional
and the other of which is the negation of the
antecedent of that conditional. It has the
form P → Q, ~P, therefore ~Q. It is invalid.
disjunction A sentence in which two
smaller sentences are joined with an or. It is
symbolized P ∨ Q.
disjunctive syllogism An inference rule that
allows us to infer one disjunct from the nega-
tion of the other disjunct. If you have “P or
Q” and you have not P, then you may infer Q.
If you have “P or Q” and not Q, then you may
infer P.

double negation The result of negating a
sentence that has already been negated (one
that already has a ~ in front of it). The result-
ing sentence means the same thing as the
original, non-negated sentence.
hypothetical syllogism An inference rule
that allows us to infer P → R from P → Q and
Q → R.
logically equivalent Two statements are
logically equivalent if they have the same
values on every row of a truth table. That
means they are true in the exact same
circumstances.
modus ponens An argument that affirms
the antecedent of its conditional premise. It
has the form P → Q, P, therefore Q.
modus tollens An argument that denies the
consequent of its conditional premise. It has
the form P → Q, ~Q, therefore ~P.
negation A statement that asserts that
another statement, P, is false. It is symbol-
ized ~P and pronounced “not P.”
operators Words (like and, or, not, and
if . . . then . . . ) used to make complex state-
ments whose truth values are functions of
the truth values of their parts. Also known as
connectives when they are used to link two
sentences.

proposition The meaning expressed by a
claim that asserts something is true or false.
propositional logic A way of clarifying
reasoning by breaking down the forms of
complex claims into the simple propositions
of which they are composed, connected with
truth-functional operators. Also known as
sentence logic, sentential logic, statement
logic, and truth-functional logic.
sentence variables Letters like P and Q
that are used in forms to represent any
sentence at all, just as a variable in algebra
represents any number.
statement form The result of replacing the
component statements in a sentence with
statement variables (like P and Q), con-
nected with logical operators.
har85668_04_c04_119-164.indd 160 4/9/15 1:26 PM
truth table A table in which columns to
the right show the truth values of complex
sentences based on each combination of
truth values of their component sentences
on the left.

truth value An indicator of whether a state-
ment is true on a given row of a truth table.
A statement’s truth value is true (abbrevi-
ated T) if the statement is true; it is false
(abbreviated F) if the statement is false.
Answers to Practice Problems
1. a
2. b
3. d
4. b
5. c
6. c
7. d
8. a
9. c
10. b
1. ~(B ∨ L)
2. D & S
3. G → (A & L)
4. (M & D) ∨ G
5. (H & L) → R
6. G ↔ (C & M)
7. ~(H & S) & (H ∨ S)
8. (T & E) → [P ↔ (H & P)]
9. (P & G) → (X → ~E)
10. C→ (P & T)

1. c
2. d
3. b
4. c
5. b
6. c
7. d
8. a
9. valid
J K J → K J K
T T T T T
T F F T F
F T T F T
F F T F F
10. invalid
H G H → G G H
T T T T T
T F F F T
F T T T F
F F T F F
har85668_04_c04_119-164.indd 161 4/9/15 1:26 PM

11. valid
K K → K K
T T T
F T F
12. valid
H Y H & Y ~(H & Y) Y ∨ ~H ~H
T T T F T F
T F F T T F
F T F T T T
F F F T T T
13. invalid
W Q W → Q ~W ~Q
T T T F F
T F F F T
F T T T F
F F T T T
14. valid
A B C A → B B → C A → C
T T T T T T
T T F T F T
T F T F T T
T F F F T F
F T T T T T
F T F T F T
F F T T T T
F F F T T T

15. invalid
P U P ↔ U ~(P ↔ U) P → U ~(P → U)
T T T F T F
T F F T F T
F T F T T F
F F T F T F
16. invalid
S H ~S ∨ H ~S H
T T T F T
T F F F F
F T T T T
F F T T F
har85668_04_c04_119-164.indd 162 4/9/15 1:26 PM
17. valid
J K L ~K → ~L J → ~K P → ~L
T T T T F F
T T F T F T
T F T F T F
T F F T T T
F T T T T T
F T F T T T
F F T F T T
F F F T T T

18. invalid
Y P Y & P P ~Y
T T T T F
T F F F F
F T F T T
F F F F T
19. invalid
A G V A → ~G V → ~G A → V
T T T F F T
T T F F T F
T F T T T T
T F F T T T
F T T T F T
F T F T T T
F F T T T T
F F F T T T
20. valid
B I K K & I B & (K & I) K
T T T T T T
T T F F F F
T F T F F T
T F F F F F
F T T T F T
F T F F F F
F F T F F T
F F F F F F
har85668_04_c04_119-164.indd 163 4/9/15 1:26 PM

1. a
2. b
3. d
4. d
5. e
6. e
7. d
8. c
9. a
10. b
har85668_04_c04_119-164.indd 164 4/9/15 1:26 PM
59
3Deductive Reasoning
moodboard/Thinkstock

Learning Objectives
1. Define basic key terms and concepts within deductive
reasoning.
2. Use variables to represent an argument’s logical form.
3. Use the counterexample method to evaluate an argument’s
validity.
4. Categorize different types of deductive arguments.
5. Analyze the various statements—and the relationships
between them—in categorical
arguments.
6. Evaluate categorical syllogisms using the rules of the
syllogism and Venn diagrams.
7. Differentiate between sorites and enthymemes.
har85668_03_c03_059-118.indd 59 4/22/15 2:04 PM
Section 3.1 Basic Concepts in Deductive Reasoning
By now you should be familiar with how the field of logic views
arguments: An argument is
just a collection of sentences, one of which is the conclusion
and the rest of which, the prem-
ises, provide support for the conclusion. You have also learned

that not every collection of
sentences is an argument. Stories, explanations, questions, and
debates are not arguments,
for example. The essential feature of an argument is that the
premises support, prove, or give
evidence for the conclusion. This relationship of support is what
makes a collection of sen-
tences an argument and is the special concern of logic. For the
next four chapters, we will be
taking a closer look at the ways in which premises might
support a conclusion. This chapter
discusses deductive reasoning, with a specific focus on
categorical logic.
3.1 Basic Concepts in Deductive Reasoning
As noted in Chapter 2, at the broadest level there are two types
of arguments: deductive and
inductive. The difference between these types is largely a
matter of the strength of the con-
nection between premises and conclusion. Inductive arguments
are defined and discussed in
Chapter 5; this chapter focuses on deductive arguments. In this
section we will learn about
three central concepts: validity, soundness, and deduction.
Validity
Deductive arguments aim to achieve validity, which is an
extremely strong connection
between the premises and the conclusion. In logic, the word
valid is only applied to argu-
ments; therefore, when the concept of validity is discussed in
this text, it is solely in reference
to arguments, and not to claims, points, or positions. Those
expressions may have other uses
in other fields, but in logic, validity is a strict notion that has to
do with the strength of the

connection between an argument’s premises and conclusion.
To reiterate, an argument is a collection of sentences, one of
which (the conclusion) is sup-
posed to follow from the others (the premises). A valid
argument is one in which the truth of
the premises absolutely guarantees the truth of the conclusion;
in other words, it is an argu-
ment in which it is impossible for the premises to be true while
the conclusion is false. Notice
that the definition of valid does not say anything about whether
the premises are actually
true, just whether the conclusion could be false if the premises
were true. As an example, here
is a silly but valid argument:
Everything made of cheese is tasty.
The moon is made of cheese.
Therefore, the moon is tasty.
No one, we hope, actually thinks that the moon is made of
cheese. You may or may not agree
that everything made of cheese is tasty. But you can see that if
everything made of cheese
were tasty, and if the moon were made of cheese, then the moon
would have to be tasty. The
truth of that conclusion simply logically follows from the truth
of the premises.
har85668_03_c03_059-118.indd 60 4/22/15 2:04 PM

Here is another way to better understand the strictness of the
concept of validity: You have
probably seen some far-fetched movies or read some bizarre
books at some point. Books and
movies have magic, weird science fiction, hallucinations, and
dream sequences—almost any-
thing can happen. Imagine that you were writing a weird,
bizarre novel, a novel as far removed
from reality as possible. You certainly could write a novel in
which the moon was made of
cheese. You could write a novel in which everything made of
cheese was tasty. But you could
not write a novel in which both of these premises were true, but
in which the moon turned
out not to be tasty. If the moon were made of cheese but was not
tasty, then there would be
at least one thing that was made of cheese and was not tasty,
making the first premise false.
Therefore, if we assume, even hypothetically, that the premises
are true (even in strange
hypothetical scenarios), it logically follows that the conclusion
must be as well. Therefore, the
argument is valid. So when thinking about whether an argument
is valid, think about whether
it would be possible to have a movie in which all the premises
were true but the conclusion
was false. If it is not possible, then the argument is valid.
Here is another, more realistic, example:
All whales are mammals.
All mammals breathe air.
Therefore, all whales breathe air.

Is it possible for the premises to be true and the conclusion
false? Well, imagine that the conclu-
sion is false. In that case there must be at least one whale that
does not breathe air. Let us call
that whale Fred. Is Fred a mammal? If he is, then there is at
least one mammal that does not
breathe air, so the second premise would be false. If he isn’t,
then there is at least one whale that
is not a mammal, so the first premise would be false. Again, we
see that it is impossible for the
conclusion to be false and still have all the premises be true.
Therefore, the argument is valid.
Here is an example of an invalid argument:
No whales live on land.
Therefore, no mammals live on land.
In this case we can tell that the truth of the conclusion is not
guaranteed by the premises
because the premises are actually true and the conclusion is
actually false. Because a valid
argument means that it is impossible for the premises to be true
and the conclusion false, we
can be sure that an argument in which the premises are actually
true and the conclusion is
actually false must be invalid. Here is a trickier example of the
same principle:
Some mammals live in the water.
Therefore, some whales live in the water.
har85668_03_c03_059-118.indd 61 4/22/15 2:04 PM

This one is trickier because both prem-
ises are true, and the conclusion is
true as well, so many people may be
tempted to call it valid. However, what
is important is not whether the prem-
ises and conclusion are actually true
but whether the premises guarantee
that the conclusion is true. Think about
making a movie: Could you make a
movie that made this argument’s prem-
ises true and the conclusion false?
Suppose you make a movie that is set
in a future in which whales move back
onto land. It would be weird, but not
any weirder than other ideas movies
have presented. If seals still lived in the
water in this movie, then both prem-
ises would be true, but the conclusion
would be false, because all the whales
would live on land.
Because we can create a scenario in which the premises are true
and the conclusion is false,
it follows that the argument is invalid. So even though the
conclusion isn’t actually false, it’s
enough that it is possible for it to be false in some situation that
would make the premises

true. This mere possibility means the argument is invalid.
Soundness
Once you understand what valid means in logic, it is very easy
to understand the concept of
soundness. A sound argument is just a valid argument in which
all the premises are true. In
defining validity, we saw two examples of valid arguments; one
of them was sound and the
other was not. Since both examples were valid, the one with
true premises was the one that
was sound.
We also saw two examples of invalid arguments. Both of those
are unsound simply because
they are invalid. Sound arguments have to be valid and have all
true premises. Notice that
since only arguments can be valid, only arguments can be
sound. In logic, the concept of
soundness is not applied to principles, observations, or anything
else. The word sound in logic
is only applied to arguments.
Here is an example of a sound argument, similar to one you may
recall seeing in Chapter 2:
All men are mortal.
Bill Gates is a man.
Therefore, Bill Gates is mortal.
Plusphoto/iStock/Thinkstock
Consider the following argument: “If it is raining,
then the streets are wet. The streets are wet.
Therefore, it is raining.” Is this a valid argument?
Could there be another reason why the road is wet?

har85668_03_c03_059-118.indd 62 4/22/15 2:04 PM
There is no question about the argument’s validity. Therefore,
as long as these premises are
true, it follows that the conclusion must be true as well. Since
the premises are, in fact, true,
we can reason the conclusion is too.
It is important to note that having a true conclusion is not part
of the definition of soundness.
If we were required to know that the conclusion was true before
deciding whether the argu-
ment is sound, then we could never use a sound argument to
discover the truth of the conclu-
sion; we would already have to know that the conclusion was
true before we could judge it
to be sound. The magic of how deductive reasoning works is
that we can judge whether the
reasoning is valid independent of whether we know that the
premises or conclusion are actu-
ally true. If we also notice that the premises are all true, then
we may infer, by the power of
pure reasoning, the truth of the conclusion.
Therefore, knowledge of the truth of the premises and the
ability to reason validly enable us
to arrive at some new information: that the conclusion is true as
well. This is the main way

that logic can add to our bank of knowledge.
Although soundness is central in considering whether to accept
an argument’s conclusion, we
will not spend much time worrying about it in this book. This is
because logic really deals with
the connections between sentences rather than the truth of the
sentences themselves. If some-
one presents you with an argument about biology, a logician can
help you see whether the argu-
ment is valid—but you will need a biologist to tell you whether
the premises are true. The truth
of the premises themselves, therefore, is not usually a matter of
logic. Because the premises can
come from any field, there would be no way for logic alone to
determine whether such premises
are true or false. The role of logic—specifically, deductive
reasoning—is to determine whether
the reasoning used is valid.
Deduction
You have likely heard the term deduction used in other
contexts: As Chapter 2
noted, the detective Sherlock Holmes (and others) uses
deduction to refer to any process by
which we infer a conclusion from pieces of evidence. In rhetoric
classes and other places, you
may hear deduction used to refer to the process of reasoning
from general principles to a specific
conclusion. These are all acceptable uses of the term in their
respective contexts, but they do not
reflect how the concept is defined in logic.
In logic, deduction is a technical term. Whatever other
meanings the word may have in other
contexts, in logic, it has only one meaning: A deductive

argument is one that is presented as
being valid. In other words, a deductive argument is one that is
trying to be valid. If an argu-
ment is presented as though it is supposed to be valid, then we
may infer it is deductive. If an
argument is deductive, then the argument can be evaluated in
part on whether it is, in fact,
valid. A deductive argument that is not found to be valid has
failed in its purpose of demon-
strating its conclusion to be true.
In Chapters 5 and 6, we will look at arguments that are not
trying to be valid. Those are induc-
tive arguments. As noted in Chapter 2, inductive arguments
simply attempt to establish their
conclusion as probable—not as absolutely guaranteed. Thus, it
is not important to assess
whether inductive arguments are valid, since validity is not the
goal. However, if a deductive
Plusphoto/iStock/Thinkstock
Consider the following argument: “If it is raining,
then the streets are wet. The streets are wet.
Therefore, it is raining.” Is this a valid argument?
Could there be another reason why the road is wet?
har85668_03_c03_059-118.indd 63 4/22/15 2:04 PM

argument is not valid, then it has failed in its goal; therefore,
for deductive reasoning, validity
is a primary concern.
Consider someone arguing as follows:
All donuts have added sugar.
All donuts are bad for you.
Therefore, everything with added sugar is bad for you.
Even though the argument is invalid—
exactly why this is so will be clearer
in the next section—it seems clear
that the person thinks it is valid. She
is not merely suggesting that maybe
things with added sugar might be bad
for you. Rather, she is presenting the
reasoning as though the premises
guarantee the truth of the conclusion.
Therefore, it appears to be an attempt
at deductive reasoning, even though
this one happens to be invalid.
Because our definition of validity
depends on understanding the author’s
intention, this means that deciding
whether something is a deductive argu-
ment requires a bit of interpretation—
we have to figure out what the person
giving the argument is trying to do. As
noted briefly in Chapter 2, we ought to seek to provide the most
favorable possible interpreta-
tion of the author’s intended reasoning. Once we know that an
argument is deductive, the next
question in evaluating it is whether it is valid. If it is deductive

but not valid, we really do not
need to consider anything further; the argument fails to
demonstrate the truth of its conclusion
in the intended sense.
BananaStock/Thinkstock
Interpreting the intention of the person making an
argument is a key step in determining whether the
argument is deductive.
Examine the following arguments. Then determine whether they
are deductive argu-
ments or not.
1. Charles is hard to work with, since he always interrupts
others. Therefore, I do not
want to work with Charles in the development committee.
2. No physical object can travel faster than light. An electron is
a physical object. So an
electron cannot travel faster than light.
3. The study of philosophy makes your soul more slender,
healthy, and beautiful. You
should study philosophy.
(continued)
har85668_03_c03_059-118.indd 64 4/22/15 2:04 PM

4. We should go to the beach today. It’s sunny. The dolphins are
out, and I have a bottle
of fine wine.
5. Triangle A is congruent to triangle B. Triangle A is an
equilateral triangle. Therefore,
triangle B is an equilateral triangle.
6. The farmers in Poland have produced more than 500 bushels
of wheat a year on
average for the past 10 years. This year they will produce more
than 500 bushels of
wheat.
7. No dogs are fish. Some guppies are fish. Therefore, some
guppies are not dogs.
8. Paying people to mow your lawn is not a good policy. When
people mow their own
lawns, they create self-discipline. In addition, they are able to
save a lot of money
over time.
9. If Mount Roosevelt was completed in 1940, then it’s only 73
years old. Mount Roos-
evelt is not 73 years old. Therefore, Mount Roosevelt was not
completed in 1940.
10. You’re either with me, or you’re against me. You’re not
with me. Therefore, you’re
against me.

11. The worldwide use of oil is projected to increase by 33%
over the next 5 years. How-
ever, reserves of oil are dwindling at a rapid rate. That means
that the price of oil will
drastically increase over the next 5 years.
12. A nation is only as great as its people. The people are
reliant on their leaders. Leaders
create the laws in which all people can flourish. If those laws
are not created well, the
people will suffer. This is why the people of the United States
are currently suffering.
13. If we save up money for a house, then we will have a place
to stay with our children.
However, we haven’t saved up any money for a house.
Therefore, we won’t have a
place to stay with our children.
14. We have to focus all of our efforts on marketing because
right now; we don’t have
any idea of who our customers are.
15. Walking is great exercise. When people exercise they are
happier and they feel better
about themselves. I’m going to start walking 4 miles every day.
16. Because all libertarians believe in more individual freedom,
all people who believe
in individual freedom are libertarians.
17. Our dogs are extremely sick. I have to work every day this
week, and our house is a
mess. There’s no way I’m having my family over for Festivus.
18. Pigs are smarter than dogs. Animals that are easier to train

are smarter than other
animals. Pigs are easier to train than dogs.
19. Seventy percent of the students at this university come from
upper class families.
The school budget has taken a hit since the economic downturn.
We need funding
for the three new buildings on campus. I think it’s time for us to
start a phone cam-
paign to raise funds so that we don’t plunge into bankruptcy.
20. If she wanted me to buy her a drink, she would’ve looked
over at me. But she never
looked over at me. So that means that she doesn’t want me to
buy her a drink.
har85668_03_c03_059-118.indd 65 4/22/15 2:04 PM
Section 3.2 Evaluating Deductive Arguments
3.2 Evaluating Deductive Arguments
If validity is so critical in evaluating deductive argu-
ments, how do we go about determining whether an
argument is valid or invalid? In deductive reason-
ing, the key is to look at the pattern of an argument ,
which is called its logical form. As an example, see if
you can tell whether the following argument is valid:
All quidnuncs are shunpikers.

All shunpikers are flibbertigibbets.
Therefore, all quidnuncs are flibbertigibbets.
You could likely tell that the argument is valid even
though you do not know the meanings of the words.
This is an important point. We can often tell whether
an argument is valid even if we are not in a posi-
tion to know whether any of its propositions are
true or false. This is because deductive validity typi-
cally depends on certain patterns of argument. In
fact, even nonsense arguments can be valid. Lewis
Carroll (a pen name for C. L. Dodgson) was not only
the author of Alice’s Adventures in Wonderland, but
also a clever logician famous for both his use of non-
sense words and his tricky logic puzzles.
We will look at some of Carroll’s puzzles in this
chapter’s sections on categorical logic, but for now,
let us look at an argument using nonsense words from his poem
“Jabberwocky.” See if you can
tell whether the following argument is valid:
All bandersnatches are slithy toves.
All slithy toves are uffish.
Therefore, all bandersnatches are uffish.
If you could tell the argument about quidnuncs was valid, you
were probably able to tell that
this argument is valid as well. Both arguments have the same
pattern, or logical form.
Representing Logical Form
Logical form is generally represented by using variables or
other symbols to highlight the pat-
tern. In this case the logical form can be represented by
substituting capital letters for certain

parts of the propositions. Our argument then has the form:
All S are M.
All M are P.
Therefore, all S are P.
Pantheon/SuperStock
In addition to his well-known literary
works, Lewis Carroll wrote several
mathematical works, including three
books on logic: Symbolic Logic Parts 1
and 2, and The Game of Logic, which was
intended to introduce logic to children.
har85668_03_c03_059-118.indd 66 4/22/15 2:05 PM
Any argument that follows this pattern, or form, is valid. Try it
for yourself. Think of any three
plural nouns; they do not have to be related to each other. For
example, you could use sub-
marines, candy bars, and mountains. When you have thought of
three, substitute them for the
letters in the pattern given. You can put them in any order you
like, but the same word has to
replace the same letter. So you will put one noun in for S in the
first and third lines, one noun
for both instances of M, and your last noun for both cases of P.
If we use the suggested nouns,

we would get:
All submarines are candy bars.
All candy bars are mountains.
Therefore, all submarines are mountains.
This argument may be close to nonsense, but it is logically
valid. It would not be possible to
make up a story in which the premises were true but the
conclusion was false. For example,
if one wizard turns all submarines into candy bars, and then a
second wizard turns all candy
bars into mountains, the story would not make any sense (nor
would it be logical) if, in the
end, all submarines were not mountains. Any story that makes
the premises true would have
to also make the conclusion true, so that the argument is valid.
As mentioned, the form of an argument is what you get when
you remove the specific mean-
ing of each of the nonlogical words in the argument and talk
about them in terms of variables.
Sometimes, however, one has to change the wording of a claim
to make it fit the required
form. For example, consider the premise “All men like dogs.” In
this case the first category
would be “men,” but the second category is not represented by a
plural noun but by a predi-
cate phrase, “like dogs.” In such cases we turn the expression
“like dogs” into the noun phrase
“people who like dogs.” In that case the form of the sentence is
still “All A are B,” in which B is
“people who like dogs.” As another example, the argument:
Some mammals live in the water.

Therefore, at least some whales live in the water.
can be rewritten with plural nouns as:
Some mammals are things that live in the water.
Therefore, at least some whales are things that live in the water.
and has the form:
All A are B.
Some B are C.
Therefore, at least some A are C.
The variables can represent anything (anything that fits
grammatically, that is). When we
substitute specific expressions (of the appropriate grammatical
category) for each of the vari-
ables, we get an instance of that form. So another instance of
this form could be made by
replacing A with Apples, B with Bananas, and C with
Cantaloupes. This would give us
har85668_03_c03_059-118.indd 67 4/22/15 2:05 PM
All Apples are Bananas.
Some Bananas are Cantaloupes.
Therefore, at least some Apples are Cantaloupes.

It does not matter at this stage whether the sentences are true or
false or whether the reason-
ing is valid or invalid. All we are concerned with is the form or
pattern of the argument.
We will see many different patterns as we study deductive
logic. Different kinds of deductive
arguments require different kinds of forms. The form we just
used is based on categories; the
letters represented groups of things, like dogs, whales,
mammals, submarines, or candy bars.
That is why in these cases we use plural nouns. Other patterns
will require substituting entire
sentences for letters. We will study forms of this type in
Chapter 4. The patterns you need to
know will be introduced as we study each kind of argument, so
keep your eyes open for them.
Using the Counterexample Method
By definition, an argument form is valid if and only if all of its
instances are valid. Therefore,
if we can show that a logical form has even one invalid
instance, then we may infer that
the argument form is invalid. Such an instance is called a
counterexample to the argument
form’s validity; thus, the counterexample method for showing
that an argument form is
invalid involves creating an argument with the exact same form
but in which the premises
are true and the conclusion is false. (We will examine other
methods in this chapter and in
later chapters.) In other words, finding a counterexample
demonstrates the invalidity of the
argument’s form.
Consider the invalid argument example from the prior section:

All donuts have added sugar.
All donuts are bad for you.
Therefore, everything with added sugar is bad for you.
By replacing predicate phrases with noun phrases, this argument
has the form:
All A are B.
All A are C.
Therefore, all B are C.
This is the same form as that of the following, clearly invalid
argument:
All birds are animals.
All birds have feathers.
Therefore, all animals have feathers.
Because we can see that the premises of this argument are true
and the conclusion is false, we
know that the argument is invalid. Since we have identified an
invalid instance of the form, we
know that the form is invalid. The invalid instance is a
counterexample to the form. Because
we have a counterexample, we have good reason to think that
the argument about donuts is
not valid.
har85668_03_c03_059-118.indd 68 4/22/15 2:05 PM

One of our recent examples has the form:
All A are B.
Some B are C.
Therefore, at least some A are C.
Here is a counterexample that challenges this argu-
ment form’s validity:
All dogs are mammals.
Some mammals are cats.
Therefore, at least some dogs are
cats.
By substituting dogs for A, mammals for B, and cats
for C, we have found an example of the argument’s
form that is clearly invalid because it moves from
true premises to a false conclusion. Therefore, the
argument form is invalid.
Here is another example of an argument:
All monkeys are primates.
No monkeys are reptiles.
Therefore, no primates are reptiles.
The conclusion is true in this example, so many may mistakenly
think that the reasoning is
valid. However, to better investigate the validity of the
reasoning, it is best to focus on its form.
The form of this argument is:
All A are B.
No A are C.

Therefore, no B are C.
To demonstrate that this form is invalid, it will suffice to
demonstrate that there is an argu-
ment of this exact form that has all true premises and a false
conclusion. Here is such a
counterexample:
All men are human.
No men are women.
Therefore, no humans are women.
Clearly, there is something wrong with this argument. Though
this is a different argument, the
fact that it is clearly invalid, even though it has the exact same
form as our original argument,
means that the original argument’s form is also invalid.
S. Harris/Cartoonstock
Can you think of a counterexample
that can prove this dog’s argument is
invalid?
har85668_03_c03_059-118.indd 69 4/22/15 2:05 PM
Section 3.3 Types of Deductive Arguments
3.3 Types of Deductive Arguments
Once you learn to look for arguments, you will see them
everywhere. Deductive arguments

play very important roles in daily reasoning. This section will
discuss some of the most impor-
tant types of deductive arguments.
Mathematical Arguments
Arguments about or involving mathematics generally use
deductive reasoning. In fact, one
way to think about deductive reasoning is that it is reasoning
that tries to establish its conclu-
sion with mathematical certainty. Let us consider some
examples.
Suppose you are splitting the check for
lunch with a friend. In calculating your
portion, you reason as follows:
I had the chicken sandwich
plate for $8.49.
I had a root beer for $1.29.
I had nothing else.
$8.49 + $1.29 = $9.78.
Therefore, my portion of the
bill, excluding tip and tax, is
$9.78.
Notice that if the premises are all true,
then the conclusion must be true also.
Of course, you might be mistaken about
the prices, or you might have forgotten
that you had a piece of pie for dessert.
You might even have made a mistake
in how you added up the prices. But
these are all premises. So long as your
premises are correct and the argument
is valid, then the conclusion is certain
to be true.

But wait, you might say—aren’t we
often mistaken about things like this?
After all, it is common for people to
make mistakes when figuring out a bill. Your friend might even
disagree with one of your
premises: For example, he might think the chicken sandwich
plate was really $8.99. How can
we say that the conclusion is established with mathematical
certainty if we are willing to
admit that we might be mistaken?
These are excellent questions, but they pertain to our certainty
of the truth of the premises.
The important feature of valid arguments is that the reasoning is
so strong that the conclu-
sion is just as certain to be true as the premises. It would be a
very strange friend indeed who
Angelinast/iStock/Thinkstock
A mathematical proof is a valid deductive
argument that attempts to prove the conclusion.
Because mathematical proofs are deductively valid,
mathematicians establish mathematical truth with
complete certainty (as long as they agree on the
premises).
har85668_03_c03_059-118.indd 70 4/22/15 2:05 PM

agreed with all of your premises and yet insisted that your
portion of the bill was something
other than $9.78. Still, no matter how good our reasoning, there
is almost always some pos-
sibility that we are mistaken about our premises.
Arguments From Definitions
Another common type of deductive argument is argument from
definition. This type of
argument typically has two premises. One premise gives the
definition of a word; the second
premise says that something meets the definition. Here is an
example:
Bachelor means “unmarried male.”
John is an unmarried male.
Therefore, John is a bachelor.
Notice that as with arguments involving math, we may disagree
with the premises, but it is very
hard to agree with the premises and disagree with the
conclusion. When the argument is set out
in standard form, it is typically relatively easy to see that the
argument is valid.
On the other hand, it can be a little tricky to tell whether the
argument is sound. Have we
really gotten the definition right? We have to be very careful, as
definitions often sound right
even though they are a little bit off. For example, the stated
definition of bachelor is not quite
right. At the very least, the definition should apply only to
human males, and probably only
adult ones. We do not normally call children or animals
“bachelors.”

Chris Madden/Cartoonstock
When crafting or evaluating a deductive
argument via definition, special attention
should be paid to the clarity of the
definition.
An interesting feature of definitions is that
they can be understood as going both ways.
In other words, if bachelor means “unmarried
male,” then we can reason either from the man
being an unmarried male to his being a bach-
elor, as in the previous example, or from the
man being a bachelor to his being an unmar-
ried male, as in the following example.
Bachelor means “unmarried
male.”
John is a bachelor.
Therefore, John is an unmar-
ried male.
Arguments from definition can be very power-
ful, but they can also be misused. This typically
happens when a word has two meanings or
when the definition is not fully accurate. We will
learn more about this when we study fallacies
in Chapter 7, but here is an example to consider:
Murder is the taking of an inno-
cent life.
Abortion takes an innocent life.
Therefore, abortion is murder.
har85668_03_c03_059-118.indd 71 4/22/15 2:05 PM

This is an argument from definition, and it is valid—the
premises guarantee the truth of the
conclusion. However, are the premises true? Both premises
could be disputed, but the first
premise is probably not right as a definition. If the word murder
really just meant “taking an
innocent life,” then it would be impossible to commit murder by
killing someone who was not
innocent. Furthermore, there is nothing in this definition about
the victim being a human or
the act being intentional. It is very tricky to get definitions
right, and we should be very care-
ful about reaching conclusions based on oversimplified
definitions. We will come back to this
example from a different angle in the next section when we
study syllogisms.
Categorical Arguments
Historically, some of the first arguments to receive a detailed
treatment were categorical
arguments, having been thoroughly explained by Aristotle
himself (Smith, 2014). Categorical
arguments are arguments whose premises and conclusions are
statements about categories
of things. Let us revisit an example from earlier in this chapter:
All mammals breathe air.

Therefore, all whales breathe air.
In each of the statements of this argument, the membership of
two categories is compared.
The categories here are whales, mammals, and air breathers. As
discussed in the previous
section on evaluating deductive arguments, the validity of these
arguments depends on the
repetition of the category terms in certain patterns; it has
nothing to do with the specific cat-
egories being compared. You can test this by changing the
category terms whales, mammals,
and air breathers with any other category terms you like.
Because this argument’s form is
valid, any other argument with the same form will be valid. The
branch of deductive reason-
ing that deals with categorical arguments is known as
categorical logic. We will discuss it in
the next two sections.
Propositional Arguments
Propositional arguments are a type of reasoning that relates
sentences to each other rather
than relating categories to each other. Consider this example:
Either Jill is in her room, or she’s gone out to eat.
Jill is not in her room.
Therefore, she’s gone out to eat.
Notice that in this example the pattern is made by the sentences
“Jill is in her room” and
“she’s gone out to eat.” As with categorical arguments, the
validity of propositional arguments
can be determined by examining the form, independent of the
specific sentences used. The
branch of deductive reasoning that deals with propositional

arguments is known as proposi-
tional logic, which we will discuss in Chapter 4.
har85668_03_c03_059-118.indd 72 4/22/15 2:05 PM
Section 3.4 Categorical Logic: Introducing Categorical
Statements
3.4 Categorical Logic: Introducing Categorical Statements
The field of deductive logic is a rich and productive one; one
could spend an entire lifetime
studying it. (See A Closer Look: More Complicated Types of
Deductive Reasoning.) Because the
focus of this book is critical thinking and informal logic (rather
than formal logic), we will only
look closely at categorical and propositional logic, which focus
on the basics of argument. If
you enjoy this introductory exposure, you might consider
looking for more books and courses
in logic.
Categorical arguments have been studied extensively for more
than 2,000 years, going back
to Aristotle. Categorical logic is the logic of argument made up
of categorical statements. It
is a logic that is concerned with reasoning about certain
relationships between categories of
things. To learn more about how categorical logic works, it will
be useful to begin by analyz-
ing the nature of categorical statements, which make up the
premises and conclusions of

categorical arguments. A categorical statement talks about two
categories or groups. Just to
keep things simple, let us start by talking about dogs, cats, and
animals.
A Closer Look: More Complicated Types of Deductive
Reasoning
As noted, deductive logic deals with a precise kind of reasoning
in which logical validity is
based on logical form. Within logical forms, we can use letters
as variables to replace English
words. Logicians also frequently replace other words that occur
within arguments—such as
all, some, or, and not—to create a kind of symbolic language.
Formal logic represented in this
type of symbolic language is called symbolic logic.
Because of this use of symbols, courses in symbolic logic end
up looking like math classes. An
introductory course in symbolic logic will typically begin with
propositional logic and then
move to something called predicate logic. Predicate logic
combines everything from categori-
cal and propositional logic but allows much more flexibility in
the use of some and all. This
flexibility allows it to represent much more complex and
powerful statements.
Predicate logic forms the basis for even more advanced types of
logic. Modal logic, for example,
can be used to represent many deductive arguments about
possibility and necessity that can-
not be symbolized using predicate logic alone. Predicate logic
can even help provide a foun-
dation for mathematics. In particular, when predicate logic is
combined with a mathematical

field called set theory, it is possible to prove the fundamental
truths of arithmetic. From there
it is possible to demonstrate truths from many important fields
of mathematics, including cal-
culus, without which we could not do physics, engineering, or
many other fascinating and use-
ful fields. Even the computers that now form such an essential
part of our lives are founded,
ultimately, on deductive logic.
har85668_03_c03_059-118.indd 73 4/22/15 2:05 PM
Statements
One thing we can say about these groups is that all dogs are
animals. Of course, all cats are
animals, too. So we have the following true categorical
statements:
All dogs are animals.
All cats are animals.
In categorical statements, the first group name is called the
subject term; it is what the
sentence is about. The second group name is called the
predicate term. In the categorical
sentences just mentioned, dogs and cats are both in the subject
position, and animals is in
the predicate position. Group terms can go in either position,

but of course, the sentence
might be false. For example, in the sentence “All animals are
dogs” the term dogs is in the
predicate position.
You may recall that we can represent the logical form of these
types of sentences by replacing
the category terms with single letters. Using this method, we
can represent the form of these
categorical statements in the following way:
All D are A.
All C are A.
Another true statement we can make about these groups is “No
dogs are cats.” Which term is
in subject position, and which is in predicate position? If you
said that dogs is the subject and
cats is the predicate, you’re right! The logical form of “No dogs
are cats” can be given as “No
D are C.”
We now have two sentences in which the category dogs is the
subject: “All dogs are animals”
and “No dogs are cats.” Both of these statements tell us
something about every dog. The first,
which starts with all, tells us that each dog is an animal. The
second, which begins with no,
tells us that each dog is not a cat. We say that both of these
types of sentences are universal
because they tell us something about every member of the
subject class.
Not all categorical statements are universal. Here are two
statements about dogs that are not

universal:
Some dogs are brown.
Some dogs are not tall.
Statements that talk about some of the things in a category are
called particular statements.
The distinction between a statement being universal or
particular is a distinction of quantity.
Another distinction is that we can say that the things mentioned
are in or not in the predi-
cate category. If we say the things are in that category, our
statement is affirmative. If we
say the things are not in that category, our statement is
negative. The distinction between
har85668_03_c03_059-118.indd 74 4/22/15 2:05 PM
Statements
a statement being affirmative or negative is a distinction of
quality. For example, when we
say “Some dogs are brown,” the thing mentioned (dogs) is in the
predicate category (brown
things), making this an affirmative statement. When we say
“Some dogs are not tall,” the
thing mentioned (dogs) is not in the predicate category (tall
things), and so this is a nega-

tive statement.
Taking both of these distinctions into account, there are four
types of categorical statements:
universal affirmative, universal negative, particular affirmative,
and particular negative. Table 3.1
shows the form of each statement along with its quantity and
quality.
Table 3.1: Types of categorical statements
Quantity Quality
All S is P Universal Affirmative
No S is P Universal Negative
Some S is P Particular Affirmative
Some S is not P Particular Negative
To abbreviate these categories of statement even further,
logicians over the millennia have
used letters to represent each type of statement. The
abbreviations are as follows:
A: Universal affirmative (All S is P)
E: Universal negative (No S is P)
I: Particular positive (Some S is P)
O: Particular negative (Some S is not P)
Accordingly, the statements are known as A propositions, E
propositions, I propositions, and

O propositions. Remember that the single capital letters in the
statements themselves are
just placeholders for category terms; we can fill them in with
any category terms we like.
Figure 3.1 shows a traditional way to arrange the four types of
statements by quantity and
quality.
har85668_03_c03_059-118.indd 75 4/22/15 2:05 PM
A
All S is P No S is P
E
I
Some S is P
Co
nt
ra
di
ct
or
ie
s Contradictories
Some S is not P
O

Statements
Now we need to get just a bit clearer on what the four
statements mean. Granted, the meaning
of categorical statements seems clear: To say, for example, that
“no dogs are reptiles” simply
means that there are no things that are both dogs and reptiles.
However, there are certain
cases in which the way that logicians understand categorical
statements may differ some-
what from how they are commonly understood in everyday
language. In particular, there are
two specific issues that can cause confusion.
Clarifying Particular Statements
The first issue is with particular statements (I and O
propositions). When we use the word
some in everyday life, we typically mean more than one. For
example, if someone says that
she has some apples, we generally think that this means that she
has more than one. How-
ever, in logic, we take the word some simply to mean at least
one. Therefore, when we say
that some S is P, we mean only that at least one S is P. For
example, we can say “Some dogs
live in the White House” even if only one does.
Clarifying Universal Statements
The second issue involves universal statements (A and E
propositions). It is often called the
“issue of existential presupposition”—the issue concerns
whether a universal statement
Figure 3.1: The square of opposition

The square of opposition serves as a quick reference point when
evaluating categorical statements.
Note that A statements and O statements always contradict one
another; when one is true, the other
is false. The same is true of E statements and I statements.
A
All S is P No S is P
E
I
Some S is P
Co
nt
ra
di
ct
or
ie
s Contradictories
Some S is not P
O
har85668_03_c03_059-118.indd 76 4/22/15 2:05 PM

Statements
implies a particular statement. For example, does the fact that
all dogs are animals imply that
some dogs are animals? The question really becomes an issue
only when we talk about things
that do not really exist. For example, consider the claim that all
the survivors of the Civil War
live in New York. Given that there are no survivors of the Civil
War anymore, is the statement
true or not?
The Greek philosopher Aristotle, the inventor of categorical
logic, would have said the state-
ment is false. He thought that “All S is P” could only be true if
there was at least one S (Parsons,
2014). Modern logicians, however, hold that that “All S is P” is
true even when no S exists. The
reasons for the modern view are somewhat beyond the scope of
this text—see A Closer Look:
Existential Import for a bit more of an explanation—but an
example will help support the
claim that universal statements are true when no member of the
subject class exists.
Suppose we are driving somewhere and stop for snacks. We
decide to split a bag of M&M’s.
For some reason, one person in our group really wants the
brown M&M’s, so you promise that
he can have all of them. However, when we open the bag, it
turns out that there are no brown
candies in it. Since this friend did not get any brown M&M’s,
did you break your promise? It
seems clear that you did not. He did get all of the brown
M&M’s that were in the bag; there just

weren’t any. In order for you to have broken your promise,
there would have to be a brown
M&M that you did not let your friend have. Therefore, it is true
that your friend got all the
brown M&M’s, even though he did not get any.
This is the way that modern logicians think about universal
propositions when there are no
members of the subject class. Any universal statement with an
empty subject class is true,
regardless of whether the statement is positive or negative. It is
true that all the brown M&M’s
were given to your friend and also true that no brown M&M’s
were given to your friend.
A Closer Look: Existential Import
It is important to remember that particular statements in logic (I
and O propositions) refer
to things that actually exist. The statement “Some dogs are
mammals” is essentially saying,
“There is at least one dog that exists in the universe, and that
dog is a mammal.” The way that
logicians refer to this attribute of I and O statements is that they
have “existential import.”
This means that for them to be true, there must be something
that actually exists that has the
property mentioned in the statement.
The 19th-century mathematician George Boole, however,
presented a problem. Boole agreed
with Aristotle that the existential statements I and O had to
refer to existing things to be true.
Also, for Aristotle, all A statements that are true necessarily
imply the truth of their corre-
sponding I statements. The same goes with E and O statements.

Boole pointed out that some true A and E statements refer to
things that do not actually exist.
Consider the statement “All vampires are creatures that drink
blood.” This is a true statement.
That means that the corresponding I statement, “Some vampires
are creatures that drink
blood,” would also be true, according to Aristotle. However,
Boole noted that there are no exist-
ing things that are vampires. If vampires do not exist, then the I
statement, “Some vampires
are creatures that drink blood,” is not true: The truth of this
statement rests on the idea that
there is an actually existing thing called a vampire, which, at
this point, there is no evidence of.
(continued)
har85668_03_c03_059-118.indd 77 4/22/15 2:05 PM
Statements
Boole reasoned that Aristotle’s ideas did not work in cases
where A and E statements refer to nonexisting classes of
objects. For example, the E statement “No vampires are time
machines” is a true statement. However, both classes in this
statement refer to things that do not actually exist. Therefore,
the statement “Some vampires are not time machines” is not
true, because this statement could only be true if vampires and
time machines actually existed.

Boole reasoned that Aristotle’s claim that true A and E state-
ments led necessarily to true I and O statements was not uni-
versally true. Hence, Boole claimed that there needed to be a
revision of the forms of categorical syllogisms that are consid-
ered valid. Because one cannot generally claim that an exis-
tential statement (I or O) is true based on the truth of the cor-
responding universal (A or E), there were some valid forms of
syllogisms that had to be excluded under the Boolean (mod-
ern) perspective. These syllogisms were precisely those that
reasoned from universal premises to a particular conclusion.
Of course, we all recognize that in everyday life we can logi-
cally infer that if all dogs are mammals, then it must be true
that some dogs are mammals. That is, we know that there is
at least one existing dog that is a mammal. However, because
our logical rules of evaluation need to apply to all instances
of syllogisms, and because there are other instances where
universals do not lead of necessity to the truth of particulars,
the rules of evaluation had to
be reformed after Boole presented his analysis. It is important
to avoid committing the exis-
tential fallacy, or assuming that a class has members and then
drawing an inference about an
actually existing member of the class.
Science and Society/SuperStock
George Boole, for whom Boolean
logic is named, challenged
Aristotle’s assertion that the
truth of A statements implies
the truth of corresponding
I statements. Boole suggested
that some valid forms of
syllogisms had to be excluded.

A Closer Look: Existential Import (continued)
Accounting for Conversational Implication
These technical issues likely sound odd: We usually assume that
some implies that there is
more than one and that all implies that something exists. This is
known as conversational
implication (as opposed to logical implication). It is quite
common in everyday life to make a
conversational implication and take a statement to suggest that
another statement is true as
well, even though it does not logically imply that the other must
be true. In logic, we focus on
the literal meaning.
One of the common reasons that a statement is taken to
conversationally imply another is
that we are generally expected to make the most fully
informative statement that we can in
response to a question. For example, if someone asks what time
it is and you say, “Sometime
after 3,” your statement seems to imply that you do not know
the exact time. If you knew
it was 3:15 exactly, then you probably should have given this
more specific information in
response to the question.
har85668_03_c03_059-118.indd 78 4/22/15 2:05 PM
Statements

For example, we all know that all dogs are animals. Suppose,
however, someone says, “Some
dogs are animals.” That is an odd thing to say: We generally
would not say that some dogs are
animals unless we thought that some of them are not animals.
However, that would be mak-
ing a conversational implication, and we want to make logical
implications. For the purposes
of logic, we want to know whether the statement “some dogs are
animals” is true or false. If
we say it is false, then we seem to have stated it is not true that
some dogs are animals; this,
however, would seem to mean that there are no dogs that are
animals. That cannot be right.
Therefore, logicians take the statement “Some dogs are
animals” simply to mean that there is
at least one dog that is an animal, which is true. The statement
“Some dogs are not animals” is
not part of the meaning of the statement “Some dogs are
animals.” In the language of logic, the
statement that some S are not P is not part of the meaning of the
statement that some S are P.
Of course, it would be odd to make the less informative
statement that some dogs are animals,
since we know that all dogs are animals. Because we tend to
assume someone is making the
most informative statement possible, the statement “Some dogs
are animals” may conversa-
tionally imply that they are not all animals, even though that is
not part of the literal meaning
of the statement.
In short, a particular statement is true when there is at least one
thing that makes it true, even

if the universal statement would also be true. In fact, sometimes
we emphasize that we are
not talking about the whole category by using the words at
least, as in, “At least some planets
orbit stars.” Therefore, it appears to be nothing more than
conversational implication, not lit-
eral meaning, that leads our statement “Some dogs are animals”
to suggest that some also are
not. When looking at categorical statements, be sure that you
are thinking about the actual
meaning of the sentence rather than what might be
conversationally implied.
Complete the following problems.
1. “All dinosaurs are things that are extinct.” Which of the
following is the subject term
in this statement?
a. dinosaurs
b. things that are extinct
2. “No Honda Civics are Lamborghinis.” Which of the
following is the predicate term in
this statement?
a. Lamborghinis
b. Honda Civics
3. “Some authors are people who write horror.” Which of the
following is the predicate
term in this statement?
a. authors
b. people who write horror
4. “Some politicians are not people who can be trusted.” Which

of the following is the
subject term in this statement?
a. politicians
b. people who can be trusted
(continued)
har85668_03_c03_059-118.indd 79 4/22/15 2:05 PM
Section 3.5Categorical Logic: Venn Diagrams as Pictures of
Meaning
5. “All mammals are pieces of cheese.” Which of the following
is the predicate term in
this statement?
a. pieces of cheese
b. mammals
6. What is the quantity of the following statement? “All
dinosaurs are things that are
extinct.”
a. universal
b. particular
c. affirmative
d. negative
7. What is the quality of the following statement? “No Honda
Civics are Lamborghinis.”
a. universal
b. particular
c. affirmative

d. negative
8. What is the quality of the following statement? “Some
authors are people who write
horror.”
a. universal
b. particular
c. affirmative
d. negative
9. What is the quantity of the following statement? “Some
politicians are not people
who can be trusted.”
a. universal
b. particular
c. affirmative
d. negative
10. What is the quality of the following statement? “All
mammals are pieces of cheese.”
a. universal
b. particular
c. affirmative
d. negative
3.5 Categorical Logic: Venn Diagrams as Pictures
of Meaning
Given that it is sometimes tricky to parse out the meaning and
implications of categorical
statements, a logician named John Venn devised a method that
uses diagrams to clarify the
literal meanings and logical implications of categorical claims.
These diagrams are appro-

priately called Venn diagrams (Stapel, n.d.). Venn diagrams not
only give a visual picture of
the meanings of categorical statements, they also provide a
method by which we can test the
validity of many categorical arguments.
har85668_03_c03_059-118.indd 80 4/22/15 2:05 PM
E
H
E
Meaning
Drawing Venn Diagrams
Here is how the diagramming works: Imagine we get a bunch of
people together and all go to
a big field. We mark out a big circle with rope on the field and
ask everyone with brown eyes
to stand in the circle. Would you stand inside the circle or
outside it? Where would you stand
if we made another circle and asked everyone with brown hair
to stand inside? If your eyes
or hair are sort of brownish, just pick whether you think you
should be inside or outside the
circles. No standing on the rope allowed! Remember your
answers to those two questions.

Here is an image of the brown-eye circle, labeled “E” for
“eyes”; touch inside or outside the
circle indicating where you would stand.
Here is a picture of the brown-hair circle, labeled “H” for
“hair”; touch inside or outside the
circle indicating where you would stand.
Notice that each circle divides the people into two groups:
Those inside the circle have the
feature we are interested in, and those outside the circle do not.
Where would you stand if we put both circles on the ground at
the same time?
E
H
har85668_03_c03_059-118.indd 81 4/22/15 2:05 PM
E H
E H
Meaning
As long as you do not have both brown eyes and brown hair,
you should be able to figure out
where to stand. But where would you stand if you have brown

eyes and brown hair? There is
not any spot that is in both circles, so you would have to
choose. In order to give brown-eyed,
brown-haired people a place to stand, we have to overlap the
circles.
Now there is a spot where people who have both brown hair and
brown eyes can stand: where
the two circles overlap. We noted earlier that each circle divides
our bunch of people into two
groups, those inside and those outside. With two circles, we
now have four groups. Figure 3.2
shows what each of those groups are and where people from
each group would stand.
E H
E H
har85668_03_c03_059-118.indd 82 4/22/15 2:05 PM
All S is P
S P
No S is P
S P
Some S is P

S P
Some S is not P
S P
Neither brown eyes nor brown hair
Brown eyes,
not
brown hair
Brown hair,
not
brown eyes
Brown eyes
and
brown hair
Meaning
With this background, we can now draw a picture for each
categorical statement. When we
know a region is empty, we will darken it to show there is
nobody there. If we know for sure that
someone is in a region, we will put an x in it to represent a
person standing there. Figure 3.3
shows the pictures for each of the four kinds of statements.
Figure 3.3: Venn diagrams of categorical statements

Each of the four categorical statements can be represented
visually with a Venn diagram.
All S is P
S P
No S is P
S P
Some S is P
S P
Some S is not P
S P
Figure 3.2: Sample Venn diagram
Neither brown eyes nor brown hair
Brown eyes,
not
brown hair
Brown hair,
not
brown eyes
Brown eyes
and

brown hair
har85668_03_c03_059-118.indd 83 4/22/15 2:05 PM
Meaning
In drawing these pictures, we adopt the convention that the
subject term is on the left and
the predicate term is on the right. There is nothing special about
this way of doing it, but
diagrams are easier to understand if we draw them the same way
as much as possible. The
important thing to remember is that a Venn diagram is just a
picture of the meaning of a state-
ment. We will use this fact in our discussion of inferences and
arguments.
Drawing Immediate Inferences
As mentioned, Venn diagrams help us determine what
inferences are valid. The most basic of
such inferences, and a good place to begin, is something called
immediate inference. Immedi-
ate inferences are arguments from one categorical statement as
premise to another as con-
clusion. In other words, we immediately infer one statement
from another. Despite the fact
that these inferences have only one premise, many of them are
logically valid. This section will
use Venn diagrams to help discern which immediate inferences
are valid.

The basic method is to draw a diagram of the premises of the
argument and determine if the
diagram thereby shows the conclusion is true. If it does, then
the argument is valid. In other
words, if drawing a diagram of just the premises automatically
creates a diagram of the con-
clusion, then the argument is valid. The diagram shows that any
way of making the premises
true would also make the conclusion true; it is impossible for
the conclusion to be false when
the premises are true. We will see how to use this method with
each of the immediate infer-
ences and later extend the method to more complicated
arguments.
Conversion
Conversion is just a matter of switching the positions of the
subject and predicate terms.
The resulting statement is called the converse of the original
statement. Table 3.2 shows the
converse of each type of statement.
Table 3.2: Conversion
Statement Converse
All S is P. All P is S.
No S is P. No P is S.
Some S is P. Some P is S.
Some S is not P. Some P is not S.
Forming the converse of a statement is easy; just switch the

subject and predicate terms with
each other. The question now is whether the immediate
inference from a categorical state-
ment to its converse is valid or not. It turns out that the
argument from a statement to its
converse is valid for some statement types, but not for others.
In order to see which, we have
to check that the converse is true whenever the original
statement is true.
An easy way to do this is to draw a picture of the two
statements and compare them. Let us
start by looking at the universal negative statement, or E
proposition, and its converse. If we
form an argument from this statement to its converse, we get the
following:
har85668_03_c03_059-118.indd 84 4/22/15 2:05 PM
No S is P
S P
No P is S
S P
Some S is P
S P

Some P is S
S P
Meaning
No S is P.
Therefore, no P is S.
Figure 3.4 shows the Venn diagrams for these statements.
As you can see, the same region is shaded in both pictures—the
region that is inside both
circles. It does not matter which order the circles are in, the
picture is the same. This means
that the two statements have the same meaning; we call such
statements equivalent.
The Venn diagrams for these statements demonstrate that all of
the information in the con-
clusion is present in the premise. We can therefore infer that the
inference is valid. A shorter
way to say it is that conversion is valid for universal negatives.
We see the same thing when we look at the particular
affirmative statement, or I proposition.
In the case of particular affirmatives as well, we can see that all
of the information in the
conclusion is contained within the premises. Therefore, the
immediate inference is valid. In
fact, because the diagram for “Some S is P” is the same as the
diagram for its converse,
“Some P is S” (see Figure 3.5), it follows that these two
statements are equivalent as well.

Figure 3.4: Universal negative statement and its converse
In this representation of “No S is P. Therefore, no P is S,” the
areas shaded are the same, meaning the
statements are equivalent.
No S is P
S P
No P is S
S P
Figure 3.5: Particular affirmative statement and its converse
As with the E proposition, all of the information contained in
the conclusion of the I proposition is
also contained within the premises, making the inference valid.
Some S is P
S P
Some P is S
S P
har85668_03_c03_059-118.indd 85 4/22/15 2:05 PM

Some S is P
S P
Some P is S
S P
Some S is not P
S P
Some P is not S
S P
Meaning
However, there will be a big difference when we draw pictures
of the universal affirmative
(A proposition), the particular negative (O proposition), and
their converses (see Figure 3.6
and Figure 3.7).
In these two cases we get different pictures, so the statements
do not mean the same thing. In
the original statements, the marked region is inside the S circle
but not in the P circle. In the
converse statements, the marked region is inside the P circle but
not in the S circle. Because
there is information in the conclusions of these arguments that
is not present in the premises,
we may infer that conversion is invalid in these two cases.
Figure 3.6: Universal affirmative statement and its converse

Unlike Figures 3.4 and 3.5 where the diagrams were identical,
we get two different diagrams for
A propositions. This tells us that there is information contained
in the conclusion that was not
included in the premises, making the inference invalid.
Some S is P
S P
Some P is S
S P
Figure 3.7: Particular negative statement and its converse
As with A propositions, O propositions present information in
the conclusion that was not present in
the premises, rendering the inference invalid.
Some S is not P
S P
Some P is not S
S P
har85668_03_c03_059-118.indd 86 4/22/15 2:05 PM

Non-brown
eyes
Non-brown
hair
Non-brown
eyes and
non-brown
hair
Brown eyes and brown hair
Non-brown
eyes
Non-brown
hair
Non-brown
eyes and
non-brown
hair
Meaning
Let us consider another type of immediate inference.
Contraposition
Before we can address contraposition, it is necessary to
introduce the idea of a complement

class. Remember that for any category, we can divide things
into those that are in the category
and those that are out of the category. When we imagined rope
circles on a field, we asked all
the brown-haired people to step inside one of the circles. That
gave us two groups: the brown-
haired people inside the circle, and the non-brown-haired people
outside the circle. These
two groups are complements of each other. The complement of a
group is everything that is
not in the group. When we have a term that gives us a category,
we can just add non- before
the term to get a term for the complement group. The
complement of term S is non-S, the
complement of term animal is nonanimal, and so on. Let us see
what complementing a term
does to our Venn diagrams.
Recall the diagram for brown-eyed people. You were inside the
circle if you have brown eyes,
and outside the circle if you do not. (Remember, we did not let
people stand on the rope; you
had to be either in or out.) So now consider the diagram for
non-brown-eyed people.
If you were inside the brown-eyed circle, you would be outside
the non-brown-eyed circle.
Similarly, if you were outside the brown-eyed circle, you would
be inside the non-brown-eyed
circle. The same would be true for complementing the brown-
haired circle. Complementing
just switches the inside and outside of the circle.
Do you remember the four regions from Figure 3.2? See if you
can find the regions that would
have the same people in the complemented picture. Where

would someone with blue eyes
and brown hair stand in each picture? Where would someone
stand if he had red hair and
green eyes? How about someone with brown hair and brown
eyes?
In Figure 3.8, the regions are colored to indicate which ones
would have the same people in
them. Use the diagram to help check your answers from the
previous paragraph. Notice that
the regions in both circles and outside both circles trade places
and that the region in the left
circle only trades places with the region in the right circle.
Non-brown
eyes
Non-brown
hair
Non-brown
eyes and
non-brown
hair
har85668_03_c03_059-118.indd 87 4/22/15 2:05 PM
S P Non-S Non-P

Meaning
Now that we know what a complement is, we are ready to look
at the immediate infer-
ence of contraposition. Contraposition combines conversion and
complementing; to get
the contrapositive of a statement, we first get the converse and
then find the complement
of both terms.
Let us start by considering the universal affirmative statement,
“All S is P.” First we form its
converse, “All P is S,” and then we complement both class
terms to get the contrapositive, “All
non-P is non-S.” That may sound like a mouthful, but you
should see that there is a simple,
straightforward process for getting the contrapositive of any
statement. Table 3.3 shows the
process for each of the four types of categorical statements.
Table 3.3: Contraposition
Original Converse Contrapositive
All S is P. All P is S. All non-P is non-S.
No S is P. No P is S. No non-P is non-S.
Some S is P. Some P is S. Some non-P is non-S.
Some S is not P. Some P is not S. Some non-P is not non-S.
Figure 3.9 shows the diagrams for the four statement types and
their contrapositives, colored

so that you can see which regions represent the same groups.
Figure 3.8: Complement class
S P Non-S Non-P
har85668_03_c03_059-118.indd 88 4/22/15 2:05 PM
All S is P
S P
All non-P is non-S
S P
No S is P
S P
No non-P is non-S
S P
Some S is P
S P
Some non-P is non-S
S P

Some S is not P
S P
Some non-P is not non-S
S P
Meaning
Figure 3.9: Contrapositive Venn diagrams
Using the converse and contrapositive diagrams, you can infer
the original statement.
All S is P
S P
All non-P is non-S
S P
No S is P
S P
No non-P is non-S
S P
Some S is P
S P

Some non-P is non-S
S P
Some S is not P
S P
Some non-P is not non-S
S P
har85668_03_c03_059-118.indd 89 4/22/15 2:05 PM
Meaning
As you can see, contraposition preserves meaning in universal
affirmative and particular nega-
tive statements. So from either of these types of statements, we
can immediately infer their
contrapositive, and from the contrapositive, we can infer the
original statement. In other words,
these statements are equivalent; therefore, in those two cases,
the contrapositive is valid.
In the other cases, particular affirmative and universal negative,
we can see that there is infor-
mation in the conclusion that is not present in diagram of the
premise; these immediate infer-

ences are invalid.
There are more immediate inferences that can be made, but our
main focus in this chapter is
on arguments with multiple premises, which tend to be more
interesting, so we are going to
move on to syllogisms.
Answer the following questions about conversion and
contraposition.
1. What is the converse of the statement “No humperdinks are
picklebacks”?
a. No humperdinks are picklebacks.
b. All picklebacks are humperdinks.
c. Some humperdinks are picklebacks.
d. No picklebacks are humperdinks.
2. What is the converse of the statement “Some mammals are
not dolphins”?
a. Some dolphins are mammals.
b. Some dolphins are not mammals.
c. All dolphins are mammals.
d. No dolphins are mammals.
3. What is the contrapositive of the statement “All couches are
pieces of furniture”?
a. All non-couches are non-pieces of furniture.
b. All pieces of furniture are non-couches.
c. All non-pieces of furniture are couches.
d. All non-pieces of furniture are non-couches.
4. What is the contrapositive of the statement “Some apples are
not vegetables”?

a. Some non-apples are not non-vegetables.
b. Some non-vegetables are not non-apples.
c. Some non-vegetables are non-apples.
d. Some non-vegetables are apples.
5. What is the converse of the statement “Some men are
bachelors”?
a. Some bachelors are men.
b. Some bachelors are non-men.
c. All bachelors are men.
d. No women are bachelors.
har85668_03_c03_059-118.indd 90 4/22/15 2:05 PM
Section 3.6 Categorical Logic: Categorical Syllogisms
3.6 Categorical Logic: Categorical Syllogisms
Whereas contraposition and conversion can be seen as
arguments with only one premise, a syl-
logism is a deductive argument with two premises. The
categorical syllogism, in which a conclu-
sion is derived from two categorical premises, is perhaps the
most famous—and certainly one of
the oldest—forms of deductive argument. The categorical
syllogism—which we will refer to here
as just “syllogism”—presented by Aristotle in his Prior
Analytics (350 BCE/1994), is a very spe-
cific kind of deductive argument and was subsequently studied
and developed extensively by
logicians, mathematicians, and philosophers.

Ron Morgan/Cartoonstock
Aristotle’s categorical syllogism uses
two categorical premises to form a
deductive argument.
Terms
We will first discuss the syllogism’s basic outline,
following Aristotle’s insistence that syllogisms are
arguments that have two premises and a conclu-
sion. Let us look again at our standard example:
All S are M.
All M are P.
There are three total terms here: S, M, and P. The
term that occurs in the predicate position in the
conclusion (in this case, P) is the major term. The
term that occurs in the subject position in the con-
clusion (in this case, S) is the minor term. The other
term, the one that occurs in both premises but not
the conclusion, is the middle term (in this case, M).
The premise that includes the major term is called
the major premise. In this case it is the first premise.
The premise that includes the minor term, the second
one here, is called the minor premise. The conclusion
will present the relationship between the predicate
term of the major premise (P) and the subject term
of the minor premise (S) (Smith, 2014).
There are 256 possible different forms of syllogisms, but only a
small fraction of those are
valid, which can be shown by testing syllogisms through the
traditional rules of the syllogism

or by using Venn diagrams, both of which we will look at later
in this section.
Distribution
As Aristotle understood logical propositions, they referred to
classes, or groups: sets of things.
So a universal affirmative (type A) proposition that states “All
Clydesdales are horses” refers
to the group of Clydesdales and says something about the
relationship between all of the
members of that group and the members of the group “horses.”
However, nothing at all is said
har85668_03_c03_059-118.indd 91 4/22/15 2:05 PM
about those horses that might not be Clydesdales, so not all
members of the group of horses
are referred to. The idea of referring to members of such groups
is the basic idea behind dis-
tribution: If all of the members of a group are referred to, the
term that refers to that group
is said to be distributed.
Using our example, then, we can see that the proposition “All
Clydesdales are horses” refers
to all the members of that group, so the term Clydesdales is said
to be distributed. Universal
affirmatives like this one distribute the term that is in the first,
or subject, position.

However, what if the proposition were a universal negative
(type E) proposition, such as “No
koala bears are carnivores”? Here all the members of the group
“koala bears” (the subject term)
are referred to, but all the members of the group “carnivores”
(the predicate term) are also
referred to. When we say that no koala bears are carnivores, we
have said something about all
koala bears (that they are not carnivores) and also something
about all carnivores (that they
are not koala bears). So in this universal negative proposition,
both of its terms are distributed.
To sum up distribution for the universal propositions, then:
Universal affirmative (A) proposi-
tions distribute only the first (subject) term, and universal
negative (E) propositions distrib-
ute both the first (subject) term and the second (predicate) term.
The distribution pattern follows the same basic idea for
particular propositions. A particular
affirmative (type I) proposition, such as “Some students are
football players,” refers only to
at least one member of the subject class (“students”) and only to
at least one member of the
predicate class (“football players”). Thus, remembering that
some is interpreted as meaning
“at least one,” the particular affirmative proposition distributes
neither term, for this proposi-
tion does not refer to all the members of either group.
Finally, a particular negative (type O) proposition, such as
“Some Floridians are not surfers,”
only refers to at least one Floridian—but says that at least one
Floridian does not belong to the

entire class of surfers or is excluded from the entire class of
surfers. In this way, the particular
negative proposition distributes only the term that refers to
surfers, or the predicate term.
To sum up distribution for the particular propositions, then:
particular affirmative (I) propo-
sitions distribute neither the first (subject) nor the second
(predicate) term, and particular
negative (O) propositions distribute only the second (predicate)
term. This is a lot of detail, to
be sure, but it is summarized in Table 3.4.
Table 3.4: Distribution
Proposition Subject Predicate
A Distributed Not
E Distributed Distributed
I Not Not
O Not Distributed
har85668_03_c03_059-118.indd 92 4/22/15 2:05 PM
Once you understand how distribution works, the rules for
determining the validity of syl-

logisms are fairly straightforward. You just need to see that in
any given syllogism, there are
three terms: a subject term, a predicate term, and a middle term.
But there are only two posi-
tions, or “slots,” a term can appear in, and distribution relates to
those positions.
Rules for Validity
Once we know how to determine whether a term is distributed,
it is relatively easy to learn
the rules for determining whether a categorical syllogism is
valid. The traditional rules of the
syllogism are given in various ways, but here is one standard
way:
Rule 1: The middle term must be distributed at least once.
Rule 2: Any term distributed in the conclusion must be
distributed in its corresponding
premise.
Rule 3: If the syllogism has a negative premise, it must have a
negative conclusion, and if
the syllogism has a negative conclusion, it must have a negative
premise.
Rule 4: The syllogism cannot have two negative premises.
Rule 5: If the syllogism has a particular premise, it must have a
particular conclusion, and
if the syllogism has a particular conclusion, it must have a
particular premise.
A syllogism that satisfies all five of these rules will be valid; a
syllogism that does not will be
invalid. Perhaps the easiest way of seeing how the rules work is

to go through a few examples.
We can start with our standard syllogism with all universal
affirmatives:
All M are P.
All S are M.
Rule 1 is satisfied: The middle term is distributed by the first
premise; a universal affirmative
(A) proposition distributes the term in the first (subject)
position, which here is M. Rule 2 is
satisfied because the subject term that is distributed by the
conclusion is also distributed by
the second premise. In both the conclusion and the second
premise, the universal affirmative
proposition distributes the term in the first position. Rule 3 is
also satisfied because there is
not a negative premise without a negative conclusion, or a
negative conclusion without a neg-
ative premise (all the propositions in this syllogism are
affirmative). Rule 4 is passed because
both premises are affirmative. Finally, Rule 5 is passed as well
because there is a universal
conclusion. Since this syllogism passes all five rules, it is valid.
These get easier with practice, so we can try another example:
Some M are not P.
All M are S.
Therefore, some S are not P.
Rule 1 is passed because the second premise distributes the
middle term, M, since it is the sub-
ject in the universal affirmative (A) proposition. Rule 2 is
passed because the major term, P, that

har85668_03_c03_059-118.indd 93 4/22/15 2:05 PM
is distributed in the O conclusion is also distributed in the
corresponding O premise (the first
premise) that includes that term. Rule 3 is passed because there
is a negative conclusion to go
with the negative premise. Rule 4 is passed because there is
only one negative premise. Rule 5
is passed because the first premise is a particular premise (O).
Since this syllogism passes all
five rules, it is valid; there is no way that all of its premises
could be true and its conclusion false.
Both of these have been valid; however, out of the 256 possible
syllogisms, most are invalid.
Let us take a look at one that violates one or more of the rules:
No P are M.
Some S are not M.
Rule 1 is passed. The middle term is distributed in the first
(major) premise. However, Rule 2 is
violated. The subject term is distributed in the conclusion, but
not in the corresponding second
(minor) premise. It is not necessary to check the other rules;
once we know that one of the rules
is violated, we know that the argument is invalid. (However, for

the curious, Rule 3 is violated as
well, but Rules 4 and 5 are passed).
Venn Diagram Tests for Validity
Another value of Venn diagrams is that they provide a nice
method for evaluating the validity
of a syllogism. Because every valid syllogism has three
categorical terms, the diagrams we use
must have three circles:
The idea in diagramming a syllogism is that we diagram each
premise and then check to see if
the conclusion has been automatically diagrammed. In other
words, we determine whether the
conclusion must be true, according to the diagram of the
premises.
har85668_03_c03_059-118.indd 94 4/22/15 2:05 PM
M
S P
It is important to remember that we never draw a diagram of the
conclusion. If the argu-
ment is valid, diagramming the premises will automatically
provide a diagram of the conclu-
sion. If the argument is invalid, diagramming the premises will
not provide a diagram of the

conclusions.
Diagramming Syllogisms With Universal Statements
Particular statements are slightly more difficult in these
diagrams, so we will start by looking
at a syllogism with only universal statements. Consider the
following syllogism:
All S is M.
No M is P.
Therefore, no S is P.
Remember, we are only going to diagram the two premises; we
will not diagram the conclusion.
The easiest way to diagram each premise is to temporarily
ignore the circle that is not relevant to
the premise. Looking just at the S and M circles, we diagram
the first premise like this:
M
S P
har85668_03_c03_059-118.indd 95 4/22/15 2:05 PM
M
M
S P

Here is what the diagram for the second premise looks like:
Now we can take those two diagrams and superimpose them, so
that we have one diagram of
both premises:
M
M
S P
har85668_03_c03_059-118.indd 96 4/22/15 2:05 PM
M
S P
Now we can check whether the argument is valid. To do this, we
see if the conclusion is true
according to our diagram. In this case our conclusion states that
no S is P; is this statement
true, according to our diagram? Look at just the S and P circles;
you can see that the area
between the S and P circles (outlined) is fully shaded. So we
have a diagram of the conclu-
sion. It does not matter if the S and P circles have some extra

shading in them, so long as the
diagram has all the shading needed for the truth of the
conclusion.
Let us look at an invalid argument next.
All S is M.
All P is M.
Therefore, all S is P.
Again, we diagram each premise and look to see if we have a
diagram of the conclusion. Here
is what the diagram of the premises looks like:
M
S P
har85668_03_c03_059-118.indd 97 4/22/15 2:05 PM
M
S P
Now we check to see whether the conclusion must be true,
according to the diagram. Our
conclusion states that all S is P, meaning that no unshaded part
of the S circle can be outside
of the P circle. In this case you can see that we do not have a

diagram of the conclusion. Since
we have an unshaded part of S outside of P (outlined), the
Let us do one more example with all universals.
All M are P.
No M is S.
Therefore, no S is P.
Here is how to diagram the premises:
M
S P
har85668_03_c03_059-118.indd 98 4/22/15 2:05 PM
M
S P
Is the conclusion true in this diagram? In order to know that the
conclusion is true, we would
need to know that there are no S that are P. However, we see in
this diagram that there is room
for some S to be P. Therefore, these premises do not guarantee
the truth of this conclusion, so
the argument is invalid.

Diagramming Syllogisms With Particular Statements
Particular statements (I and O) are a bit trickier, but only a bit.
The problem is that when you
diagram a particular statement, you put an x in a region. If that
region is further divided by
a third circle, then the single x will end up in one of those
subregions even though we do not
know which one it should go in. As a result, we have to adopt a
convention to indicate that the
x may be in either of them. To do this, we will draw an x in
each subregion and connect them
with a line to show that we mean the individual might be in
either subregion. To see how this
works, let us consider the following syllogism.
M
S P
har85668_03_c03_059-118.indd 99 4/22/15 2:05 PM
M
S P
Some S is not M.
All P are M.
Therefore, some S is not P.

We start by diagramming the first premise:
Then we add the diagram for the second premise:
M
S P
har85668_03_c03_059-118.indd 100 4/22/15 2:05 PM
M
S P
Notice that in diagramming the second premise, we shaded over
one of the linked x’s. This
leaves us with just one x. When we look at just the S and P
circles, we can see that the remain-
ing is inside the S circle but outside the P circle.
To see if the argument is valid, we have to determine whether
the conclusion must be true
according to this diagram. The truth of our conclusion depends
on there being at least one
S that is not P. Here we have just such an entity: The remaining
x is in the S circle but not in the
P circle, so the conclusion must be true. This shows that the
conclusion validly follows from

the premises.
Here is an example of an invalid syllogism.
Some S is M.
Some M is P.
Therefore, some S is P.
M
S P
har85668_03_c03_059-118.indd 101 4/22/15 2:05 PM
M
S P
Here is the diagram with both premises represented:
Now it seems we have x’s all over the place. Remember, our job
now is just to see if the conclu-
sion is already diagrammed when we diagram the premises. The
diagram of the conclusion
would have to have an x that was in the region between where
the S and P circles overlap. We
can see that there are two in that region, each linked to an x
outside the region. The fact that
they are linked to other x’s means that neither x has to be in the

middle region; they might
both be at the other end of the link. We can show this by
carefully erasing one of each pair of
linked x’s. In fact, we will erase one x from each linked pair,
trying to do so in a way that makes
the conclusion false. First we erase the right-hand x from the
pair in the S circle. Here is what
the diagram looks like now:
M
S P
M
S P
har85668_03_c03_059-118.indd 102 4/22/15 2:05 PM
M
S P
Here is the diagram with both premises represented:
Now it seems we have x’s all over the place. Remember, our job
now is just to see if the conclu-
sion is already diagrammed when we diagram the premises. The
diagram of the conclusion

would have to have an x that was in the region between where
the S and P circles overlap. We
can see that there are two in that region, each linked to an x
outside the region. The fact that
they are linked to other x’s means that neither x has to be in the
middle region; they might
both be at the other end of the link. We can show this by
carefully erasing one of each pair of
linked x’s. In fact, we will erase one x from each linked pair,
trying to do so in a way that makes
the conclusion false. First we erase the right-hand x from the
pair in the S circle. Here is what
the diagram looks like now:
M
S P
M
S P
Now we erase the left-hand x from the remaining pair. Here is
the final diagram:
har85668_03_c03_059-118.indd 103 4/22/15 2:05 PM
M
S P

Notice that there are no x’s remaining in the overlapped region
of S and P. This modification
of the diagram still makes both premises true, but it also makes
the conclusion false. Because
this combination is possible, that means that the argument must
be invalid.
Here is a more common example of an invalid categorical
syllogism:
All S are M.
Some M are P.
Therefore, some S are P.
This argument form looks valid, but it is not. One way to see
that is to notice that Rule 1 is vio-
lated: The middle term does not distribute in either premise.
That is why this argument form
represents an example of the common deductive error in
reasoning known as the “undistrib-
uted middle.”
M
S P
har85668_03_c03_059-118.indd 104 4/22/15 2:05 PM
M

S P
A perhaps more intuitive way to see why it is invalid is to look
at its Venn diagram. Here is how
we diagram the premises:
The two x’s represent the fact that our particular premise states
that some M are P and does
not state whether or not they are in the S circle, so we represent
both possibilities here. Now
we simply need to check if the conclusion is necessarily true.
We can see that it is not, because although one x is in the right
place, it is linked with another x
in the wrong place. In other words, we do not know whether the
x in “some M are P” is inside
or outside the S boundary. Our conclusion requires that the x be
inside the S boundary, but
we do not know that for certain whether it is. Therefore, the
argument is invalid. We could,
for example, erase the linked x that is inside of the S circle, and
we would have a diagram that
makes both premises be true and the conclusion false.
M
S P
har85668_03_c03_059-118.indd 105 4/22/15 2:05 PM

M
S P
Because this diagram shows that it is possible to make the
premises true and the conclusion
false, it follows that the argument is invalid.
A final way to understand why this form is invalid is to use the
counterexample method and
consider that it has the same form as the following argument:
Therefore, some dogs are cats.
This argument has the same form and has all true premises and a
false conclusion. This coun-
terexample just verifies that our Venn diagram test got the right
answer. If applied correctly,
the Venn diagram test works every time. With this example, all
three methods agree that our
Moral of the Story: The Venn Diagram Test for Validity
Here, in summary, are the steps for doing the Venn diagram test
for validity:
1. Draw the three circles, all overlapping.
2. Diagram the premises.
a. Shade in areas where nothing exists.

b. Put an x for areas where something exists.
c. If you are not sure what side of a line the x should be in, then
put two linked x’s,
one on each side.
3. Check to see if the conclusion is (must be) true in this
diagram.
a. If there are two linked x’s, and one of them makes the
conclusion true and the
other does not, then the argument is invalid because the
premises do not guar-
antee the truth of the conclusion.
b. If the conclusion must be true in the diagram, then the
argument is valid; other-
wise it is not.
M
S P
har85668_03_c03_059-118.indd 106 4/22/15 2:05 PM
Because this diagram shows that it is possible to make the
premises true and the conclusion
false, it follows that the argument is invalid.

A final way to understand why this form is invalid is to use the
counterexample method and
consider that it has the same form as the following argument:
Therefore, some dogs are cats.
This argument has the same form and has all true premises and a
false conclusion. This coun-
terexample just verifies that our Venn diagram test got the right
answer. If applied correctly,
the Venn diagram test works every time. With this example, all
three methods agree that our
Moral of the Story: The Venn Diagram Test for Validity
Here, in summary, are the steps for doing the Venn diagram test
for validity:
1. Draw the three circles, all overlapping.
2. Diagram the premises.
a. Shade in areas where nothing exists.
b. Put an x for areas where something exists.
c. If you are not sure what side of a line the x should be in, then
put two linked x’s,
one on each side.
3. Check to see if the conclusion is (must be) true in this
diagram.
a. If there are two linked x’s, and one of them makes the
conclusion true and the
other does not, then the argument is invalid because the

premises do not guar-
antee the truth of the conclusion.
b. If the conclusion must be true in the diagram, then the
argument is valid; other-
wise it is not.
Answer the following questions. Note that some questions may
have more than one
answer.
1. Which rules does the following syllogism pass?
All M are P.
Some M are S.
Therefore, some S are P.
a. Rule 1: The middle term must be distributed at least once.
b. Rule 2: Any term distributed in the conclusion must be
distributed in its corre-
sponding premise.
c. Rule 3: If the syllogism has a negative premise, it must have
a negative conclusion,
and if the syllogism has a negative conclusion, it must have a
negative premise.
d. Rule 4: The syllogism cannot have two negative premises.
e. Rule 5: If the syllogism has a particular premise, it must have
a particular con-
clusion, and if the syllogism has a particular conclusion, it must
have a particular
premise.

f. All the rules
2. Which rules does the following syllogism fail?
No P are M.
All S are M.
sponding premise.
negative premise.
(continued)
har85668_03_c03_059-118.indd 107 4/22/15 2:05 PM
a particular con-
have a particular

premise.
f. All the rules
Some M are P.
Some S are not M.
sponding premise.
a negative conclu-
sion, and if the syllogism has a negative conclusion, it must
have a negative
premise.
a particular con-
have a particular
premise.
f. All the rules
No P are M.
No M are S.

Therefore, no S are P.
sponding premise.
a negative conclu-
have a negative
premise.
a particular con-
have a particular
premise.
f. All the rules
All M are P.
Some M are not S.
sponding premise.

a negative conclu-
have a negative
premise.
(continued)
har85668_03_c03_059-118.indd 108 4/22/15 2:05 PM
a particular con-
have a particular
premise.
f. All the rules
All humans are dogs.
Some dogs are mammals.
Therefore, no humans are mammals.

sponding premise.
a negative conclu-
have a negative
premise.
a particular con-
have a particular
premise.
f. None of the rules
Some books are hardbacks.
All hardbacks are published materials.
Therefore, some books are published materials.
sponding premise.
a negative conclu-

have a negative
premise.
a particular con-
have a particular
premise.
No politicians are liars.
Some politicians are men.
Therefore, some men are not liars.
sponding premise.
negative premise.
(continued)
har85668_03_c03_059-118.indd 109 4/22/15 2:05 PM

a particular con-
have a particular
premise.
Some Macs are computers.
No PCs are Macs.
Therefore, all PCs are computers.
sponding premise.
a negative conclu-
have a negative
premise.

a particular con-
have a particular
premise.
All media personalities are people who manipulate the masses.
No professors are media personalities.
Therefore, no professors are people who manipulate the masses.
sponding premise.
a negative conclu-
have a negative
premise.
a particular con-
have a particular
premise.

11. Examine the following syllogisms. In the first pair, the
terms that are distributed are
marked in bold. Can you explain why? The second pair is left
for you to determine
which terms, if any, are distributed.
Some P are M.
Some M are not S.
No P are M.
All M are S.
All M are P.
All M are S.
Some P are not M.
No S are M.
har85668_03_c03_059-118.indd 110 4/22/15 2:05 PM
Section 3.7 Categorical Logic: Types of Categorical Arguments
3.7 Categorical Logic: Types of Categorical Arguments
Many examples of deductively valid arguments that we have
considered can seem quite sim-

ple, even if the theory and rules behind them can be a bit
daunting. You might even wonder
how important it is to study deduction if even silly arguments
about the moon being tasty are
considered valid. Remember that this is just a brief introduction
to deductive logic. Deductive
arguments can get quite complex and difficult, even though they
are built from smaller pieces
such as those we have covered in this chapter. In the same way,
a brick is a very simple thing,
interesting in its form, but not much use all by itself. Yet
someone who knows how to work
with bricks can make a very complex and sturdy building from
them.
Thus, it will be valuable to consider some of the more complex
types of categorical argu-
ments, sorites and enthymemes. Both of these types of
arguments are often encountered in
everyday life.
Sorites
A sorites is a specific kind of argument that strings together
several subarguments. The word
sorites comes from the Greek word meaning a “pile” or a
“heap”; thus, a sorites-style argu-
ment is a collection of arguments piled together. More
specifically, a sorites is any categorical
argument with more than two premises; the argument can then
be turned into a string of
categorical syllogisms. Here is one example, taken from Lewis
Carroll’s book Symbolic Logic
(1897/2009):
The only animals in this house are cats;
Every animal is suitable for a pet, that loves to gaze at the

moon;
When I detest an animal, I avoid it;
No animals are carnivorous, unless they prowl at night;
No cat fails to kill mice;
No animals ever take to me, except what are in this house;
Kangaroos are not suitable for pets;
None but carnivora kill mice;
I detest animals that do not take to me;
Animals, that prowl at night, always love to gaze at the moon.
Therefore, I always avoid kangaroos. (p. 124)
Figuring out the logic in such complex sorites can be
challenging and fun. However, it is easy
to get lost in sorites arguments. It can be difficult to keep all
the premises straight and to make
sure the appropriate relationships are established between each
premise in such a way that,
ultimately, the conclusion follows.
Carroll’s sorites sounds ridiculous, but as discussed earlier in
the chapter, many of us develop
complex arguments in daily life that use the conclusion of an
earlier argument as the premise
of the next argument. Here is an example of a relatively short
one:
har85668_03_c03_059-118.indd 111 4/22/15 2:05 PM
All of my friends are going to the party.

No one who goes to the party is boring.
People that are not boring interest me.
Therefore, all of my friends interest me.
Here is another example that we might reason through when
thinking about biology:
All lizards are reptiles.
No reptiles are mammals.
Only mammals nurse their young.
Therefore, no lizards nurse their young.
There are many examples like these. It is possible to break them
into smaller syllogistic sub-
arguments as follows:
All lizards are reptiles.
No reptiles are mammals.
Therefore, no lizards are mammals.
No lizards are mammals.
Only mammals nurse their young.
Therefore, no lizards nurse their young.
Breaking arguments into components like this can help improve
the clarity of the overall
reasoning. If a sorites gets too long, we tend to lose track of
what is going on. This is part
of what can make some arguments hard to understand. When
constructing your own argu-
ments, therefore, you should beware of bunching premises
together unnecessarily. Try to
break a long argument into a series of smaller arguments
instead, including subarguments,
to improve clarity.
Enthymemes

While sorites are sets of arguments strung together into one
larger argument, a related argu-
ment form is known as an enthymeme, a syllogistic argument
that omits either a premise or
a conclusion. There are also many nonsyllogistic arguments that
leave out premises or con-
clusions; these are sometimes also called enthymemes as well,
but here we will only consider
enthymemes based on syllogisms.
A good question is why the arguments are missing premises.
One reason that people may
leave a premise out is that it is considered to be too obvious to
mention. Here is an example:
All dolphins are mammals.
Therefore, all dolphins are animals.
Here the suppressed premise is “All mammals are animals.”
Such a statement probably does
not need to be stated because it is common knowledge, and the
reader knows how to fill it in
to get to the conclusion. Technically speaking, we are said to
“suppress” the premise that does
not need to be stated.
har85668_03_c03_059-118.indd 112 4/22/15 2:05 PM
Sometimes people even leave out conclusions if they think that

the inference involved is so
clear that no one needs the conclusion stated explicitly.
Arguments with unstated conclusions
are considered enthymematic as well. Let us suppose a baseball
fan complains, “You have to
be rich to get tickets to game 7, and none of my friends is rich.”
What is the implied conclu-
sion? Here is the argument in standard form:
Everyone who can get tickets to game 7 is rich.
None of my friends is rich.
Therefore, ???
In this case we may validly infer that none of the fan’s friends
can get tickets to game 7.
To be sure, you cannot always assume your audience has the
required background knowl-
edge, and you must attempt to evaluate whether a premise or
conclusion does need to be
stated explicitly. Thus, if you are talking about math to
professional physicists, you do not
need to spell out precisely what the hypotenuse of an angle is.
However, if you are talking to
third graders, that is certainly not a safe assumption.
Determining the background knowledge
of those with whom one is talking—and arguing—is more of an
art than a science.
Validity in Complex Arguments
Recall that a valid argument is one whose premises guarantee
the truth of the conclusion.
Sorites are illustrations of how we can “stack” smaller valid
arguments together to make
larger valid arguments. Doing so can be as complicated as
building a cathedral from bricks,

but so long as each piece is valid, the structure as a whole will
be valid.
How do we begin to examine a complex argument’s validity?
Let us start by looking at another
example of sorites from Lewis Carroll’s book Symbolic Logic
(1897/2009):
Babies are illogical.
Nobody is despised who can manage a crocodile.
Illogical persons are despised.
Therefore, no babies can manage a crocodile. (p. 112)
Is this argument valid? We can see that it is by breaking it into
a pair of syllogisms. Start by
considering the first and third premises. We will rewrite them
slightly to show the All that
Carroll has assumed. With those two premises, we can build the
following valid syllogism:
All babies are illogical.
All illogical persons are despised.
Therefore, all babies are despised.
har85668_03_c03_059-118.indd 113 4/22/15 2:05 PM
Using the tools from this chapter (the rules, Venn diagrams, or
just by thinking it through
carefully), we can check that the syllogism is valid. Now we can

use the conclusion of our
syllogism along with the remaining premise and conclusion
from the original argument to
construct another syllogism.
All babies are despised.
No despised persons can manage a crocodile.
Therefore, no babies can manage a crocodile.
Again, we can check that this syllogism is valid using the tools
from this chapter. Since both
of these arguments are valid, the string that combines them is
valid as well. Therefore, the
original argument (the one with three premises) is valid.
This process is somewhat like how we might approach adding a
very long list of numbers. If
you need to add a list of 100 numbers (suppose you are
checking a grocery bill), you can do
it by adding them together in groups of 10, and then adding the
subtotals together. As long
as you have done the addition correctly at each stage, your final
answer will be the correct
total. This is one reason validity is important. It allows us to
have confidence in complex
arguments by examining the smaller arguments from which they
are, or can be, built. If one
of the smaller arguments was not valid, then we could not have
complete confidence in the
larger argument.
But what about soundness? What use is the argument about
babies managing crocodiles
when we know that babies are not generally despised? Again,
let us make a comparison to
adding up your grocery bill. Arithmetic can tell you if your bill

is added correctly, but it can-
not tell you if the prices are correct or if the groceries are really
worth the advertised price.
Similarly, logic can tell you whether a conclusion validly
follows from a set of premises, but it
cannot generally tell you whether the premises are true, false, or
even interesting. By them-
selves, random deductive arguments are as useful as sums of
random numbers. They may be
good practice for learning a skill, but they do not tell us much
about the world unless we can
somehow verify that their premises are, in fact, true. To learn
about the world, we need to
apply our reasoning skills to accurate facts (usually outside of
arithmetic and logic) known to
be true about the world.
This is why logicians are not as concerned with soundness as
they are with validity, and why
a mathematician is only concerned with whether you added
correctly, and not with whether
the prices were correctly recorded. Logic and mathematics give
us skills to apply valid reason-
ing to the information around us. It is up to us, and to other
fields, to make sure the informa-
tion that we use in the premises is correct.
har85668_03_c03_059-118.indd 114 4/22/15 2:05 PM

Answer the following questions.
1. This is the name that is given to an argument that has two
premises and one conclusion.
a. syllogism
b. creative syllogism
c. enthymeme
d. sorites
e. none of the above
2. The discovery of categorical logic is often attributed to this
philosopher.
a. Plato
b. Boole
c. Aristotle
d. Kant
e. Hume
3. Which of the following is a type of deductive argument?
a. generalization
b. categorical syllogism
c. argument by analogy
d. modus spartans
4. All categorical statements have which of the following?
a. mood and placement
b. figure and form
c. number and validity
d. quantity and quality
e. all of the above
5. The premise that contains the predicate term of the
conclusion in a categorical syl-

logism is __________.
a. the minor premise
b. the major premise
c. the necessary premise
d. the conclusion
Chapter Summary
Validity is the central concept of deductive reasoning. An
argument is valid when the truth
of the premises absolutely guarantees the truth of the
conclusion. For valid arguments, if
the premises are true, then the conclusion must be true also.
Valid arguments need not have
true premises, but if they do, then they are sound arguments.
Because they use valid reason-
ing and have true premises, sound arguments are guaranteed to
have true conclusions.
har85668_03_c03_059-118.indd 115 4/22/15 2:05 PM
Deductive arguments can include mathematical arguments,
arguments from definitions, cat-
egorical arguments, and propositional arguments. Categorical
arguments allow us to reason
about things based on their properties. Categorical arguments
with two premises are called

syllogisms. The validity of syllogisms can be evaluated either
with a system of rules or by
using Venn diagrams.
Syllogisms often leave one premise or the conclusion unstated.
These are called
enthymemes. Sometimes strings of syllogisms are combined into
a larger argument called a
sorites. If we have a string of valid arguments that are combined
to make a larger argument,
then we may infer that the long argument composed of these
parts is valid as well.
The process of using subarguments to create longer ones allows
us to make rather complex
valid arguments out of simple parts. This is an important
motivation for studying deductive
logic. As with arithmetic, computer programming, and structural
engineering, combining
smaller steps in a careful way allows us to create complex
structures that are fully reliable
because they are built out of reliable parts.
1. How does the logical definition of validity differ from the
way that the term valid is
used in everyday speech? How do you plan on differentiating
the two as you con-
tinue studying logic?
2. In the chapter, you read a section about the importance of
having evidence that sup-
ports your arguments. Is it important to claim to believe things
only when one has
evidence, or are there some things that people can justifiably

believe without evi-
dence? Why?
3. How would you describe what a deductive argument is to
someone who does not
know the technical terms that apply to arguments? What
examples would you use to
demonstrate deduction?
4. What is the point of being able to understand if a deductive
argument is valid or
sound? Why is it important to be able to determine these things?
If you do not think
it is important, how would you justify your claims that it is not
important to be able
to determine validity?
5. Has there ever been a time that you presented an argument in
which you had little
or no evidence to support your claims? What types of claims did
you use in the place
of premises? What types of techniques did you use to try to
present an argument
with no information to back up your conclusion(s)? What is a
better method to use
in the future?
Web Resources
http://www.philosophyexperiments.com/validorinvalid/Default.a
spx
This game at the Philosophy Experiments website tests your
ability to determine whether
an argument is valid.
http://www.thefirstscience.org/syllogistic-machine
This professor’s blog includes an online syllogism solver that

allows you to explore fallacies,
figures, terms, and modes of syllogisms. Click on “Notes on
Syllogistic Logic” for more cover-
age of topics discussed in this chapter.
har85668_03_c03_059-118.indd 116 4/22/15 2:05 PM
http://www.philosophyexperiments.com/validorinvalid/Default.a
spx
http://www.thefirstscience.org/syllogistic-machine
Key Terms
argument from definition An argument in
which one premise is a definition.
categorical argument An argument
entirely composed of categorical statements.
categorical logic The branch of deduc-
tive logic that is concerned with categorical
arguments.
categorical statement A statement that
relates one category or class to another. Spe-
cifically, if S and P are categories, the cate-
gorical statements relating them are: All S is
P, No S is P, Some S is P, and Some S is not P.
complement class For a given class, the

complement class consists of all things that
are not in the given class. For example, if S is
a class, its complement class is non-S.
contraposition The immediate inference
obtained by switching the subject and predi-
cate terms with each other and complement-
ing them both.
conversion The immediate inference
obtained by switching the subject and predi-
cate terms with each other.
counterexample method The method of
proving an argument form to be not valid
by constructing an instance of it with true
premises and a false conclusion.
deductive argument An argument that is
presented as being valid—if the primary
evaluative question about the argument is
whether it is valid.
distribution Referring to members of
groups. If all the members of a group are
referred to, the term that refers to that
group is said to be distributed.
enthymeme An argument in which one or
more claims are left unstated.
immediate inferences Arguments from one
categorical statement as premise to another
as conclusion. In other words, we immedi-
ately infer one statement from another.

instance A term in logic that describes
the sentence that results from replacing
each variable within the form with specific
sentences.
logical form The pattern of an argument or
claim.
predicate term The second term in a cat-
egorical proposition.
quality In logic, the distinction between a
statement being affirmative or negative.
quantity In logic, the distinction between a
statement being universal or particular.
sorites A categorical argument with more
than two premises.
sound Describes an argument that is valid
and in which all of the premises are true.
subject term The first term in a categorical
proposition.
syllogism A deductive argument with
exactly two premises.
valid An argument in which the premises
absolutely guarantee the conclusion, such
that is impossible for the premises to be true
while the conclusion is false.
Venn diagram A diagram constructed of
overlapping circles, with shaded areas or x’s,

which shows the relationships between the
represented groups.
har85668_03_c03_059-118.indd 117 4/22/15 2:05 PM
1. not deductive
2. deductive
3. not deductive
4. not deductive
5. deductive
6. not deductive
7. deductive
8. not deductive
9. deductive
10. deductive
11. not deductive
12. not deductive
13. deductive
14. not deductive
15. not deductive
16. deductive
17. not deductive
18. deductive
19. not deductive

20. deductive
1. a
2. a
3. b
4. a
5. a
6. a
7. d
8. c
9. b
10. c
1. d
2. b
3. d
4. b
5. a
1. e
2. c
3. b
4. d
5. b
6. a, b, and c
7. e
8. e

9. c
10. b
11. All M are P.
All M are S.
Therefore, all S
are P.
Some P are not M.
No S are M.
Therefore, no S
are P.
1. a
2. c
3. b
4. d
5. a
har85668_03_c03_059-118.indd 118 4/22/15 2:05 PM
25
2The Argument

Rolphot/iStock/Thinkstock
Learning Objectives
1. Articulate a clear definition of logical argument.
2. Name premise and conclusion indicators.
3. Extract an argument in the standard form from a speech or
essay with the aid of paraphrasing.
4. Diagram an argument.
5. Identify two kinds of arguments—deductive and inductive.
6. Distinguish an argument from an explanation.
har85668_02_c02_025-058.indd 25 4/9/15 1:30 PM
Section 2.1 Arguments in Logic
Chapter 1 defined logic as the study of arguments that provides
us with the tools for arriving
at warranted judgments. The concept of argument is indeed
central to this definition. In this
chapter, then, our focus shall be entirely on defining
arguments—what they are, how their
component parts function, and how learning about arguments
helps us lead better lives. Most
especially, in this chapter we will introduce the standard

argument form, which is the struc-
ture that helps us identify arguments and distinguish good ones
from bad ones.
2.1 Arguments in Logic
Chapter 1 provisionally defined argument as a methodical
defense of a position. We referred
to this as the commonsense understanding of the way the word
argument is employed in
logic. The commonsense definition is very useful in helping us
recognize a unique form of
expression in ordinary human communication. It is part of the
human condition to differ in
opinion with another person and, in response, to attempt to
change that person’s opinion.
We may attempt, for example, to provide good reasons for
seeing a particular movie or to
show that our preferred kind of music is the best. Or we may try
to show others that smoking
or heavy drinking is harmful. As you will see, these are all
arguments in the commonsense
understanding of the term.
In Chapter 1 we also distinguished the commonsense
understanding of argument from
the meaning of argument in ordinary use. Arguments in ordinary
use require an exchange
between at least two people. As clarified in Chapter 1,
commonsense arguments do not neces-
sarily involve a dialogue and therefore do not involve an
exchange. In fact, one could develop
a methodical defense of a position—that is, a commonsense
argument—in solitude, simply to
examine what it would require to advocate for a particular
position. In contrast, arguments,
as understood in ordinary use, are characterized by verbal

disputes between two or more
people and often contain emotional outbursts. Commonsense
arguments are not character-
ized by emotional outbursts, since unbridled emotions present
an enormous handicap for the
development of a methodical defense of a position.
In logic an argument is a set of claims in which some, called the
premises, serve as support for
another claim, called the conclusion. The conclusion is the
argument’s main claim. For the most
part, this technical definition of argument is what we shall
employ in the remainder of this book,
though we may use the commonsense definition when talking
about less technical examples.
Table 2.1 should help clarify which meanings are acceptable
within logic. Take a moment to
review the table and fix these definitions in your mind.
Table 2.1: Comparing meanings for the term argument
Meaning in ordinary use Commonsense meaning Technical
meaning in logic
A verbal quarrel or disagree-
ment, often characterized
by raised voices and flaring
emotions.
The methodical and well-
researched defense of a position
or point of view advanced in
relation to a disputed issue.
A set of claims in which some,
called premises, serve as support

for another claim, called the
conclusion.
har85668_02_c02_025-058.indd 26 4/9/15 1:30 PM
Arguments in the technical sense are a primary way in which we
can defend a position.
Accordingly, we can find the structure of logical arguments in
commonsense arguments all
around us: in letters to the editor, social media, speeches,
advertisements, sales pitches, pro-
posals submitted for grant funds or bank loans, job applications,
requests for a raise, commu-
nications of values to children, marriage proposals, and so on.
Arguments often provide the
basis on which most of our decisions are made. We read or hear
an argument, and if we are
convinced by it, then we accept its conclusion. For example,
consider the following argument:
“I’m just not a math person.” We hear this all the time from
anyone who found
high school math challenging. . . . In high school math at least,
inborn talent
is less important than hard work, preparation, and self-
confidence. This is
what high school math teachers, college professors, and private
tutors have
observed as the pattern of those who become good in high

school math. They
point out that in any given class, students fall in a wide range of
levels of math
preparation. This is not due to genetic predisposition. What is
rarely observed
is that some children come from households in which parents
introduce them
to math early on and encourage them to practice it. These
students will imme-
diately obtain perfect scores while the rest do not. As a result,
the students
without previous preparation in math immediately assume that
those with
perfect scores have a natural math talent, without knowing
about the prepa-
ration that these students had in their homes. In turn, the
students who obtain
perfect scores assume that they have a natural math talent given
their scores
relative to the rest of the class, so they are motivated to
continue honing their
math skills and, by doing this, they cement their top of the class
standing. Thus,
the belief that math ability cannot change becomes a self-
fulfilling prophecy.
(Kimball & Smith, 2013)
In this argument, the position defended by the authors is that
the belief that math ability can-
not change becomes a self-fulfilling prophecy. The authors
support this claim with reasons
that show good performance in math is not typically the result
of a natural ability but of hav-
ing a family support system for learning, a prior preparation in
math from home, and continu-
ous practice. It makes the case that it is hard work and

preparation that lead to a person’s
proficiency in math and other subjects, not genetic
predisposition. This argument helps us
recognize that we frequently accept oft-repeated information as
fact without even question-
ing the basis. As you can see, an argument such as this can
provide a solid basis for our every-
day decisions, such as encouraging our children to work hard
and practice in the subjects they
find most difficult or deciding to obtain a university degree
with confidence later in life.
To understand the more technical definition of an argument as a
set of premises that support
a conclusion, consider the following presentation of the
reasoning from the commonsense
argument we have just examined.
Good performance in math is not due to genetics.
Good performance in math only requires preparation and
continuous practice.
Students who do well initially assume they have natural talent
and practice
more.
har85668_02_c02_025-058.indd 27 4/9/15 1:30 PM

Students who do less well initially assume they do not have
natural talent and
practice less.
Therefore, believing that one’s math ability cannot change
becomes a self-
fulfilling prophecy.
Presenting the reasoning this way can do a great deal to clarify
the argument and allow us to
examine its central claims and reasoning. This is why the field
of logic adopts the more techni-
cal definition of argument for much of its work.
Regardless of what we think about math, an important
contribution of this argument is that it
makes the case that it is hard work and preparation that lead to
our proficiency in math, and
not the factor of genetic predisposition. Logic is much the same
way. If you find some concepts
difficult, don’t assume that you just lack talent and that you
aren’t a “logic person.” With prac-
tice and persistence, anyone can be a logic person.
On your way to becoming a logic person, it is important to
remember that not everything that
presents a point of view is an argument (see Table 2.2 for
examples of arguments and nonar-
guments). Consider that when one expresses a complaint,
command, or explanation, one is
indeed expressing a point of view. However, none of these
amount to an argument.
Table 2.2: Is it an argument?
Argument Not an argument

Reprinted with permission from The Hill Times.
Why? This presents a defense of a position. But not
all letters to the editor contain arguments.
©Bettmann/Corbis
Why not? This only reports a news story. It informs
us of the role of the university but does not offer a
defense of a position.
(continued)
har85668_02_c02_025-058.indd 28 4/9/15 1:30 PM
Argument Not an argument
Greg Gibson/Associated Press
Why? This is a photo of former president Bill Clin-
ton making a speech, in which he defends his posi-
tion that the facts are different than those reported
by the media. Not all speeches contain arguments,
only those that defend a position.
©MIKE SEGAR/Reuters/Corbis
Why not? This is a debate between two presidential

candidates. Although each candidate may present
various arguments, the debate as a whole is not an
argument. It is not a defense of a position; it is an
exchange between two people on various subjects.
Emmanuel Dunand/AFP/Getty Images
Why? This ad makes a claim and offers a reason for
why viewers should take notice.
©James Lawrence/Transtock/Corbis
Why not? This ad has no words, so it makes no
specific claim. Even if we try to interpret it to make
a claim, no defense is offered.
To help us properly identify logical arguments, we need clear
criteria for what a logical argu-
ment is. Let us start unpacking what is involved in arguments
by addressing their smallest
element: the claim.
Claims
A claim is an assertion that something is or is not the case.
Claims take the form of declara-
tive sentences. It is important to note that each premise or
conclusion consists of one single
claim. In other words, each premise or conclusion consists of
one single declarative sentence.
Claims can be either true or false. This means that if what is
asserted is actually the case, then
the claim is true. If the claim does not correspond to what is
actually the case, then the claim
is false. For example, the claim “milk is in the refrigerator”
predicates that the subject of the

Table 2.2: Is it an argument? (continued)
har85668_02_c02_025-058.indd 29 4/9/15 1:31 PM
claim, milk, is in the refrigerator. If this claim corresponds to
the facts (if the refrigerator con-
tains milk), then this claim is true. If it does not correspond to
the facts (if the refrigerator
does not contain milk), then the claim is false.
Not all claims, however, can be easily
checked for truth or falsity. For exam-
ple, the truth of the claim “Jacob has
the best wife in the world” cannot be
settled easily, even if Jacob is the one
asserting this claim (“I have the best
wife in the world”). In order to under-
stand what he could possibly mean by
“best wife in the world,” we would have
to propose the criteria for what makes
a good wife in the first place, and as if
this were not challenging enough, we
would then have to establish a method
or procedure to make comparisons
among good wives. Of course, Jacob
could merely mean “I like being mar-
ried to my wife,” in which case he is not
stating a claim about his wife being the

best in the world but merely stating a feeling. It is not
uncommon to hear people state things
that sound like claims but are actually just expressions of
preference or affection, and distin-
guishing between these is often challenging because we are not
always clear in the way we
employ language. Nonetheless, it is important to note that we
often make claims from a par-
ticular point of view, and these claims are different from factual
claims. Claims that advance a
point of view, such as the example of Jacob’s wife—and
especially claims about morality and
ethicality—are indeed more challenging to settle as true or false
than factual claims, such as
“The speed limit here is 55.”
The important point is that both kinds of claims—the factual
claim and the point-of-view
claim—assert that something is or is not the case, affirm or
deny a particular predicate of a
subject, and can be either true or false. The following sentences
are examples of claims that
meet these criteria.
• There is a full moon tonight.
• Pecans are better than peanuts.
• All flights to Paris are full.
• BMWs are expensive to maintain.
• Lola is my sister.
The following are not claims:
• Is it raining? Why? Because questions are not, and cannot be,
assertions that some-
thing is the case.

• Oh, to be in Paris in the springtime! Why? Because this
expresses a sentiment but
does not state that anything might be true or false.
• Buy a BMW! Why? Because a command is not an assertion
that something is the case.
Image Source Pink/Image Source/Thinkstock
What factual claims can you make about this image?
har85668_02_c02_025-058.indd 30 4/9/15 1:31 PM
We often intend to advance claims in ways that do not present
our claims clearly and properly—
for example, by means of rhetorical questions, vague
expressions of affection, and commands
or metaphors that demand interpretation. But it is important to
recognize that intention is not
sufficient when communicating with others. In order for our
intended claims to be identified as
claims, they should meet the three criteria previously
mentioned.
Claims are sometimes called propositions. We will use the terms
claims and propositions inter-
changeably in this book. In this chapter we will stick to the
word claim, but in subsequent

chapters, we will move to the more formal terminology of
propositions.
The Standard Argument Form
In informal logic the main method for identifying, constructing,
or examining arguments is
to extract what we hear or read as arguments and put this in
what is known as the standard
argument form. It consists of claims, some of which are called
premises and one of which
is called the conclusion. In the standard argument form,
premises are listed first, each on a
separate line, with the conclusion on the line after the last
premise. There are various meth-
ods for displaying standard form. Some methods number the
premises; others separate the
conclusion with a line. We will generally use the following
method, prefacing the conclusion
with the word therefore:
Premise
Premise
Therefore, Conclusion
The number of premises can be as few as one and as many as
needed. We must approach
either extreme with caution given that, on the one hand, a single
premise can offer only very
limited support for the conclusion, and on the other hand, many
premises risk error or confu-
sion. However, there are certain kinds of arguments that,
because of their formal structure,
may contain only a limited number of premises.
In the standard argument form, each premise or conclusion
should be only one sentence long,

and premises and conclusions should be stated as clearly and
briefly as possible. Accordingly,
we must avoid premises or conclusions that have multiple
sentences or single sentences with
multiple claims. The following example shows what not to do:
I live in Boston, and I like clam chowder.
My family also lives in Boston. They also like clam chowder.
My friends live in Boston. They all like clam chowder, too.
Therefore, everyone I know in Boston likes clam chowder.
If you want to make more than one claim about the same
subject, then you can break your
declarative sentences into several sentences that each contain
only one claim. The clam chow-
der argument can then be rewritten as follows:
I live in Boston.
I like clam chowder.
My family lives in Boston.
My family likes clam chowder.
har85668_02_c02_025-058.indd 31 4/9/15 1:31 PM
My friends live in Boston.
My friends like clam chowder.
Therefore, everyone I know in Boston likes clam chowder.
The relationship between premises and the conclusion is that of

inference—the process of
drawing a claim (the conclusion) from the reasons offered in the
premises. The act of reason-
ing from the premises serves as the glue connecting the
premises with the conclusion.
Determine whether the following sentences are claims
(propositions) or nonclaims
(nonpropositions).
1. Moby Dick is a great novel.
2. Computers have made our lives easier.
3. If we go to the movies, we will need to drive the minivan.
4. Do you want to drive the minivan to the movies?
5. Drive the minivan.
6. Either I am a human or I am a dog.
7. Michael Jordan was a great football player.
8. Was it time for you to leave?
9. Private property is a right of every American.
10. Universalized health care is communism.
11. Don’t you dare vote for universalized health care.
12. Nietzsche collapsed in a square upon seeing a man beat a
horse.

13. Hooray!
14. Those who reject equality seek tyranny.
15. How many feet are in a mile?
16. If you cannot understand the truth value of a claim, then it
is not a claim.
17. Something is a claim if and only if it has a truth value.
18. Treat your boss with respect.
19. Men are much less likely to have osteoporosis than women
are.
20. Why are women less likely to have heart attacks?
21. Do as we say.
22. I believe that you should do as your parents say.
23. Socrates is mortal.
(continued)
har85668_02_c02_025-058.indd 32 4/9/15 1:31 PM
Section 2.2 Putting Arguments in the Standard Form
24. Why did Freud hold such strange beliefs about parent–child

relationships?
25. A democracy exists if and only if its citizens participate in
autonomous elections.
26. Do your best.
27. The unexamined life is not worth living.
28. Ayn Rand believed that selfishness was a virtue.
29. Is selfishness a virtue?
30. What people love is not the object of desire, but desire
itself.
31. Hey!
32. Those who cannot support themselves should not be
supported by taxpayer dollars.
33. Particle and wave behavior are properties of light.
34. Why do we feed so many pounds of plants to animals each
year?
35. Go and give your brother a kiss.
36. Because the mind conditions reality, it is impossible to
know the thing as such.
37. The library at the local university has more than 300,000
books.
38. Does the nature of reality consist of an ultimately creative
impulse?

39. You are taking a quiz.
40. Are you taking a quiz?
2.2 Putting Arguments in the Standard Form
Presenting arguments in the standard argument form is crucial
because it provides us with a
dispassionate method that will allow us to find out whether the
argument is good, regardless
of how we feel about the subject matter. The first step is to
identify the fundamental argument
being presented.
At first it might seem a bit daunting to identify an argument,
because arguments typically do
not come neatly presented in the standard argument form.
Instead, they may come in confus-
ing and unclear language, much like this statement by Special
Prosecutor Francis Schmitz of
Wisconsin regarding Governor Scott Walker:
Governor Walker was not a target of the investigation. At no
time has he been
served with a subpoena. . . . While these documents outlined the
prosecutor’s
legal theory, they did not establish the existence of a crime;
rather, they were
arguments in support of further investigation to determine if
criminal charges
against any person or entity are warranted. (Crocker, 2014,
para. 7 & 10)
har85668_02_c02_025-058.indd 33 4/9/15 1:31 PM

This was a position presented in regard to the investigation of
an alleged illegal campaign
finance coordination during the 2011–2012 recall elections
(Stein, 2014). Does it claim a vin-
dication of Walker? Or does it suggest that there may be
sufficient evidence to make Walker a
central figure in the investigation? How would you even begin
to make heads or tails of such
a confusing argument? Do not despair. The remainder of this
section will show you exactly
what to look for in order to make sense of the most complicated
argument. With a little prac-
tice, you will be able to do this without much effort.
Find the Conclusion First
Although the conclusion is last in the
standard form, the conclusion is the
first thing to find because the conclu-
sion is the main claim in an argument.
The other claims—the premises—are
present for the sole purpose of support-
ing the conclusion. Accordingly, if you
are able to find the conclusion, then you
should be able to find the premises.
The good news is that language is not
only a means for expressing ideas; it
also offers a road map for the ideas

presented. Chapter 1 underscored the
fundamental importance of clear, pre-
cise, and correct language in logical
reasoning. When used properly, lan-
guage also offers structures and direc-
tions for communicating meaning,
thus facilitating our understanding of
what others are saying. One punctua-
tion mark—the question mark—tells
us that we are confronting a question. A different punctuation
mark—the parentheses—tells
us that we are being given relevant information but only as an
aside or afterthought to the
main point; if removed, the parenthetical information would not
alter the main point. In the
case of arguments, some words serve as signposts identifying
conclusions. Take the following
example of an argument in the standard argument form:
All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.
The word therefore indicates that the sentence is a conclusion.
In fact, the word therefore is
the standard conclusion indicator we will use when constructing
arguments in the stan-
dard argument form. However, there are other conclusion
indicators that are used in ordinary
arguments, including:
• Consequently . . .
• So . . .
• Hence . . .
• Thus . . .

Xtock Images/iStock/Thinkstock
Punctuation, parentheses, and conclusion
indicators all serve as signposts to assist us when
deconstructing an argument. They provide important
clues about where to find the conclusion as well as
supporting claims.
har85668_02_c02_025-058.indd 34 4/9/15 1:31 PM
• Wherefore . . .
• As a result . . .
• It follows that . . .
• For these reasons . . .
• We may conclude that . . .
When a conclusion indicator is present, it can help identify the
conclusion in an argument.
Unfortunately, many arguments do not come with conclusion
indicators. In such cases start
by trying to identify the main point. If you can clearly identify
a single main point, then that
is likely to be the conclusion. But sometimes you will have to
look at a passage closely to find
the conclusion. Suppose you encounter the following argument:
Don’t you know that driving without a seat belt is dangerous?
Statistics show
that you are 10 times more likely to be injured in an accident if

you are not
wearing one. Besides, in our state you can get fined $100 if you
are caught not
wearing one. You ought to wear one even if you are driving a
short distance.
Arguments are often longer and more complicated than this one,
but let us work with this
simple case before trying more complicated examples. You
know that the first thing you need
to do is to look for the conclusion. The problem is that the
author of the argument does not
use a conclusion indicator. Now what? Nothing to worry about.
Just remember that the con-
clusion is the main claim, so the thing to look for is what the
author may be trying to defend.
Although the first sentence is stated as a question—remember,
punctuation marks give us
important clues—the author seems to intend to assert that
driving without a seat belt is dan-
gerous. In fact, the second sentence offers evidence in support
of this claim. On the other
hand, the third sentence seems to be important, yet it does not
speak to driving without a seat
belt being dangerous, only expensive. In the final sentence, we
find a claim that is supported
by all the others. Because of this, the final sentence presents the
conclusion.
Now, it so happens that in this case, the conclusion is at the end
of this short argument, but
keep in mind that conclusions can be found in various places in
essays, such as the beginning
or sometimes in the middle. Now that you have identified your
first piece of the puzzle, we
have this:

Premise 1: ?
Premise 2: ?
Premise 3: ?
Therefore, you ought to wear a seat belt whenever you drive.
You might have noticed that the conclusion does not appear as
it did in the essay. The origi-
nal sentence is “You ought to wear one even if you are driving a
short distance.” Why did we
modify it? Once again, clarity is of the essence in logical
reasoning. Conclusions should make
the subject clear, so the pronoun one was replaced with the
actual subject to which the author
is referring: seat belt. In addition, the predicate “even if you are
driving a short distance” was
rewritten to reflect the more inclusive point that the author
seems to be making: that you
should wear a seat belt whenever you drive.
har85668_02_c02_025-058.indd 35 4/9/15 1:31 PM
This modification of language, known as paraphrasing, is part
of the construction of argu-
ments in the standard argument form. The act of extracting an
argument from a longer piece
to its fundamental claims in the standard argument form
necessarily involves paraphrasing
the original language to the clearest and most precise form

possible. This concept will be
addressed in greater detail later in this section.
Find the Premises Next
After identifying the conclusion, the next thing to do is look for
the reasons the author offers
in defense of his or her position. These are the premises. As
with conclusions, there are prem-
ise indicators that serve as signposts that reasons are being
offered for the main claim or
conclusion. Some examples of premise indicators are:
• Since . . .
• For . . .
• Given that . . .
• Because . . .
• As . . .
• Owing to . . .
• Seeing that . . .
• May be inferred from . . .
To practice identifying premises, let us return to our seat belt
example:
Don’t you know that driving without a seat belt is dangerous?
Statistics show
that you are 10 times more likely to be injured in an accident if
you are not
wearing one. Besides, in our state you can get fined $100 if you
are caught not
wearing one. You ought to wear one even if you are driving a
short distance.
Notice again that this argument starts
with a question: “Don’t you know that
driving without a seat belt is danger-

ous?” The author is not really asking
whether you know that driving with-
out a seat belt is dangerous. Rather,
the author seems to be asking a rhe-
torical question—a question that does
not actually demand an answer—to
assert that driving without a seat belt
is dangerous. You should avoid asking
rhetorical questions in the essays that
you write, because the outcome can be
highly uncertain. The success of a rhe-
torical question depends on the reader
or listener first understanding the hid-
den meaning behind the rhetorical
question and then correctly articulat-
ing the answer you have in mind. This
does not always work.
Hkeita/iStock/Thinkstock
Much like a map will get you from point A to point
B, putting an argument into the standard argument
form will help you navigate from the conclusion to
the premises and vice versa.
har85668_02_c02_025-058.indd 36 4/9/15 1:31 PM
For the sake of this example, however, let us do our best to try
to get at the author’s inten-

tion. We could paraphrase the first premise to the following
claim: Driving without a seat
belt is dangerous. Does this paraphrased claim serve as a
premise in support of the conclu-
sion? In order to answer this, we need to put the conclusion in
the form of a question. Again,
premises are reasons offered in support of the conclusion, so if
we have a well-constructed
argument, then the premises should answer why the conclusion
is the case. This is what we
would have:
Question: Why must you wear a seat belt whenever you drive?
Answer: Because driving without a seat belt is dangerous.
This works, so the paraphrased claim that we drew from the
author’s rhetorical question is
indeed a reason in defense of the conclusion. So now we have
one more piece of the puzzle:
Premise 1: Driving without a seat belt is dangerous.
Premise 2: ?
Premise 3: ?
Let us now move to the next sentence: “Statistics show that you
are 10 times more likely to be
injured in an accident if you are not wearing one.” Is this a
claim that can be a support for the
conclusion? In other words, if we put the conclusion in the form
of a question again as we did
before, would this sentence be an adequate reason in response?
Let us see.

Answer: Because statistics show that you are 10 times more
likely to be
injured in an accident if you are not wearing one.
The answer provides a reason in support of the conclusion, and
thus, we have another prem-
ise. Now we have most of the puzzle completed, as follows:
Premise 1: Driving without a seat belt is dangerous.
Premise 2: Statistics show that you are 10 times more likely to
be injured in an
accident if you are not wearing one.
Premise 3: ?
We have one more sentence left in the argument, which reads:
“Besides, in our state you
can get fined $100 if you are caught not wearing one.” Is this a
premise? Well, it is uncer-
tain, since the sentence is not presented in the form of a claim.
So let us paraphrase it as a
claim as follows: “Not wearing a seat belt can result in a $100
fine.” This is now a claim, and
the paraphrasing has not altered the meaning, so we can proceed
to our question: Is this a
har85668_02_c02_025-058.indd 37 4/9/15 1:31 PM

premise for the argument that we are examining? Once again,
let us put the conclusion into
a question:
Answer: Because not wearing a seat belt can result in a $100
fine.
This is a claim that can be a support for the conclusion, and
thus, we have another premise.
We can now see the argument presented more formally as
follows:
Driving without a seat belt is dangerous.
Statistics show that you are 10 times more likely to be injured
in an accident
if you are not wearing one.
Not wearing a seat belt can result in a $100 fine.
The Necessity of Paraphrasing
As we have discussed, extracting the fundamental claims from a
written or a spoken argument
often involves paraphrasing. Paraphrasing is not merely an
option but rather a necessity in order
to uncover the intended argument in the best way possible. Most
other arguments presented
to you (especially those in the media) will not consist of only
premises and the conclusion in
clearly identifiable language. Furthermore, many arguments will
be much longer and compli-
cated than the seat belt argument example. Often, arguments are
presented with many other
sentences that do not serve the purposes of an argument, such as
empty rhetorical devices,

filler sentences that aim to manipulate your emotions, and so
on. So your task in extracting an
argument from such sources is akin to that of a surgeon—
removing all those linguistic tumors
that obscure the argument in order to reveal the basic claims
presented and their supporting
evidence. In other words, you should expect to do paraphrasing
as a necessary task when you
attempt to draw an argument in the standard form from almost
any source.
It is important to recognize that not everyone who advances an
argument does so clearly or
even coherently. This is precisely why the structure of the
standard argument form is such a
powerful tool to command. It offers you the machinery to
distinguish arguments from what are
not arguments. It also helps you unearth the elements of an
argument that are buried under
complicated prose and rhetoric. And it helps you evaluate the
worthiness of the argument
presented once it has been fully clarified. You should
paraphrase all claims when presenting
them in the standard argument form, whether the claims are
implied in a long argumentative
essay or speech or in shorter arguments that may be ambiguous
or unclear. (To understand
the added benefits, see Everyday Logic: Modesty and Charity.)
har85668_02_c02_025-058.indd 38 4/9/15 1:31 PM

Thinking Analytically
Identifying an argument’s components as we have just done is
an example of analytical think-
ing. When we analyze something, we examine its architectural
structure—that is, the relation
of the whole to its parts—to identify its parts and to see how the
parts fit together as a whole.
Let us examine an excerpt from President Barack Obama’s
(2014) speech on Ebola as a way
of bringing the new skills from this section all together:
Everyday Logic: Modesty and Charity
The goal of paraphrasing is to find the best presentation of the
premises and conclusions
intended. By presenting the argument offered in its best
possible light, this will help you see
not only how far off the argument is from an optimal defense,
but also how good it is despite
its bad presentation. Why should you be so charitable?
First we must keep in mind that ideas are important, even if the
ideas are not ours. So we
must always give our utmost due diligence to the examination of
ideas. Sometimes even the
roughest presentation of ideas can contain the most impressive
pearls of insight. If we are not
charitable to the ideas of others, then we might miss out on
hidden wisdom.
Second, modesty is a good intellectual habit to develop. It is
very easy to fall into the trap
of thinking that our thoughts are the best ones around. This is

generally far from the truth.
The most fruitful innovations of mankind have been quite
unexpected, often as the result of
someone paying attention to others’ ideas and coming up with a
new way of putting them to
use. This applies to all sorts of things, including everything
from the ways in which cooking
methods turned into regional cuisines, to scientific discoveries,
product innovations, and the
emergence of the Internet.
That modesty has advantages is not a new idea. In the 1980s
Peter Drucker wrote the book Inno-
vation and Entrepreneurship, in which he recounts, among many
other stories, the story of how
Ray Kroc founded the burger chain McDonald’s®. As the well-
known story goes, Kroc bought
a hamburger stand from the McDonald brothers, along with their
invention of a milkshake
machine. Although Kroc never invented anything, his
entrepreneurial genius was in seeing the
potential of a hamburger, fries, and milkshake business that
catered to mothers with little chil-
dren and turning this vision into a billion-dollar standardized
operation (Drucker, 1985/2007).
Even if you dislike McDonald’s, the point is that Kroc noticed
the potential for something that
many, including the McDonald brothers themselves, had
overlooked. Gems are everywhere
in the world of ideas, but we often have to dust them off,
remove all the excess baggage, and
extract what is good in them. Intellectual modesty allows us to
do this; we don’t blind our-
selves by assuming our own ideas are best. Once we seek to
fully understand others’ ideas

and allow them to challenge our own, we can do all sorts of
good things: understand an idea
more clearly, understand someone better, and understand
ourselves (our values, what we find
important, and so on) better as well.
Given that our human social world is characterized by diversity
of ideas, modesty also marks
the path of cooperation, harmony, and respect among human
beings. This is one of the many
small ways in which the application of logical reasoning can
help us all have better lives and
better relations with other people. If we could all use logical
reasoning on a regular basis, per-
haps we would not have as many wars and atrocities as we have
today.
har85668_02_c02_025-058.indd 39 4/9/15 1:31 PM
In West Africa, Ebola is now an epidemic of the likes that we
have not seen
before. It’s spiraling out of control. It is getting worse. It’s
spreading faster and
exponentially. Today, thousands of people in West Africa are
infected. That num-
ber could rapidly grow to tens of thousands. And if the outbreak
is not stopped
now, we could be looking at hundreds of thousands of people
infected, with pro-

found political and economic and security implications for all of
us. So this is an
epidemic that is not just a threat to regional security—it’s a
potential threat to
global security if these countries break down, if their economies
break down, if
people panic. That has profound effects on all of us, even if we
are not directly
contracting the disease. (para. 8)
We have identified “The West African Ebola epidemic is a
potential threat to global security”
as the conclusion. What are the premises? Read the passage a
few times while asking yourself,
“Why should I think the epidemic is a global threat?” Obama
says that the epidemic is not
like others, that it is growing faster and exponentially. He
moves from there being thousands
of people infected, to tens of thousands, to the possibility of
hundreds of thousands. So far,
everything is about how fast the epidemic is growing.
In the middle of the seventh sentence, the president switches
from talking about the growth of
the epidemic to claiming that it has profound economic and
security implications. What is the
basis for the claim that the growth will have these effects?
Notice that it is not in the seventh
sentence, at least not explicitly. However, the last part of the
eighth sentence does address
this. In that sentence, Obama suggests three conditions that
might lead to a global security
threat: “if these countries break down, if their economies break
down, if people panic.” So the
extreme growth of the epidemic may lead to the breakdown of
economies or countries, or it

may lead to widespread panic. If any of these things happen,
there are “profound effects on
all of us.” Therefore, the epidemic is a potential threat to global
security. We can now list the
premises, and indeed the entire argument, in standard form as
follows:
The West African Ebola epidemic is growing extremely fast.
If the growth isn’t stopped, the countries may break down.
If the growth isn’t stopped, the economies may break down.
If the growth isn’t stopped, people may panic.
Any of these things would have profound effects on people
outside of the region.
Therefore, the West African Ebola epidemic is a potential threat
to global security.
Notice that putting the argument in standard form may lose
some of the fluidity of the origi-
nal, but it more than makes up for it in increased clarity.
Identify the premises and conclusions in the following
arguments.
1. Every time I turn on the radio, all I hear is vulgar language
about sex, violence, and
drugs. Whether it’s rock and roll or rap, it’s all the same. The
trend toward vulgarity
has to change. If it doesn’t, younger children will begin
speaking in these ways, and
this will spoil their innocence.
2. Letting your kids play around on the Internet all day is like
dropping them off in
downtown Chicago to spend the day by themselves. They will

find something that
gets them into trouble.
(continued)
har85668_02_c02_025-058.indd 40 4/9/15 1:31 PM
3. Too many intravenous drug users continue to risk their lives
by sharing dirty nee-
dles. This situation could be changed if we were to supply drug
addicts with a way
to get clean needles. This would lower the rate of AIDS in this
high-risk population
as well as allow for the opportunity to educate and attempt to
aid those who are
addicted to heroin and other intravenous drugs.
4. I know that Stephen has a lot of money. His parents drive a
Mercedes. His dogs wear
cashmere sweaters, and he paid cash for his Hummer.
5. Dogs are better than cats, since they always listen to what
their masters say. They
also are more fun and energetic.
6. All dogs are warm-blooded. All warm-blooded creatures are
mammals. Hence, all
dogs are mammals.

7. Chances are that I will not be able to get in to see Slipknot
since it is an over-21 show,
and Jeffrey, James, and Sloan were all carded when they tried to
get in to the club.
8. This is not the best of all possible worlds, because the best of
all possible worlds
would not contain suffering, and this world contains much
suffering.
9. Some apples are not bananas. Some bananas are things that
are yellow. Therefore,
some things that are yellow are not apples.
10. Since all philosophers are seekers of truth, it follows that no
evil human is a seeker
after truth, since no philosophers are evil humans.
11. All squares are triangles, and all triangles are rectangles. So
all squares are rectangles.
12. Deciduous trees are trees that shed their leaves. Maple trees
are deciduous trees.
Thus, maple trees will shed their leaves at some point during
the growing season.
13. Joe must make a lot of money teaching philosophy, since
most philosophy professors
are rich.
14. Since all mammals are cold-blooded, and all cold-blooded
creatures are aquatic, all
mammals must be aquatic.
15. If you drive too fast, you will get into an accident. If you
get into an accident, your

insurance premiums will increase. Therefore, if you drive too
fast, your insurance
premiums will increase.
16. The economy continues to descend into chaos. The stock
market still moves down
after it makes progress forward, and unemployment still hovers
around 10%. It is
going to be a while before things get better in the United States.
17. Football is the best sport. The athletes are amazing, and it is
extremely complex.
18. We should go to see Avatar tonight. I hear that it has
amazing special effects.
19. All doctors are people who are committed to enhancing the
health of their patients.
No people who purposely harm others can consider themselves
to be doctors. It fol-
lows that some people who harm others do not enhance the
health of their patients.
20. Guns are necessary. Guns protect people. They give people
confidence that they can
defend themselves. Guns also ensure that the government will
not be able to take
over its citizenry.
har85668_02_c02_025-058.indd 41 4/9/15 1:31 PM

There’s snow
on the ground.
It’s cold outside.
Section 2.3 Representing Arguments Graphically
2.3 Representing Arguments Graphically
In the preceding section, we discussed the component parts of
an argument and how we can
identify each when we encounter them in writing. Although the
standard argument form is
useful and will be used throughout this book, you may find it
easier to display the structure
of an argument by drawing the connections between the parts of
an argument. We will start
by learning some simple techniques for diagramming arguments.
An argument diagram (also
called an argument map) is just a drawing that shows how the
various pieces of an argument
are related to each other.
Representing Reasons That Support a Conclusion
The simplest argument consists of two claims, one of which
supports the other—which means
that one is the premise and the other is the conclusion. For
example:
There is snow on the ground, so it must be cold outside.
To represent this argument, we put each claim in a box and
draw an arrow to show which one
supports the other. We can diagram this argument in the
following way:

Notice that the claims are represented by simple, complete
sentences. The premise is at
the start of the arrow, and the conclusion is at the end. The
arrow represents the process of
inferring the conclusion from the premise. Seeing snow on the
ground is indeed a reason for
believing that it is cold.
But arguments can be more complex. First, consider that an
argument may have more than
one line of support. For example:
There’s snow
on the ground.
har85668_02_c02_025-058.indd 42 4/9/15 1:31 PM
There’s snow
on the ground.
It’s February
in Idaho.
It’s February
in Idaho.

It’s a very
cold year.
There’s snow
on the ground.
The important thing here is that the two lines of support are
independent of each other.
Knowing that it is February in Idaho is a reason for thinking
that it is cold outside, even if you
do not see snow. Similarly, seeing snow outside is a reason for
thinking it is cold regardless of
when or where you see it.
Second, it can also be the case that a single line of support
contains multiple premises that
work together. For example, although February in Idaho offers
good grounds for thinking it is
cold outside, this reason is strengthened if it also happens to be
a particularly cold year. A year
being particularly cold is not by itself much of a reason to think
it is cold outside. Even a cold
year will be warm in the summer. But a February day in a cold
year is even more likely to be
cold than a February day in a warm one. We represent this by
starting the arrow at a group of
premises (bottom):
It’s February
in Idaho.
It’s a very

cold year.
There’s snow
on the ground.
There’s snow
on the ground.
It’s February
in Idaho.
har85668_02_c02_025-058.indd 43 4/9/15 1:31 PM
It’s February
in Idaho.
It’s a very
cold year.
There’s snow
on the ground.
John came in with
snow on his boots.

Although arrows can sometimes start at a group of claims, they
always end at a single claim.
This is because every simple argument or inference has only one
conclusion, no matter how
many premises it may have.
Finally, arguments can form chains with some claims being used
as a conclusion for one infer-
ence and a premise for another. For example, if your reason for
thinking that there is snow on
the ground is that your friend John just came in with snow on
his boots, this can be indicated
in a diagram as follows:
Notice that the claim “There is snow on the ground” is a
conclusion for one inference and a
premise for another. From these basic patterns we can build
extremely complicated arguments.
It’s February
in Idaho.
It’s a very
cold year.
There’s snow
on the ground.
John came in with
snow on his boots.
har85668_02_c02_025-058.indd 44 4/9/15 1:31 PM

It’s February
in Idaho.
It’s a very
cold year.
There’s snow
on the ground.
Most people outside
aren’t wearing coats.
John came in with
snow on his boots.
Representing Counterarguments
We will discuss one more refinement, and then we will have all
of the basic tools we need for
constructing argument maps. Sometimes lines of reasoning
count against a conclusion rather
than support it. If we look out the window and notice that most
of the students outside are
not wearing coats, that might lead us to believe that it is not
very cold even though it is Febru-
ary and we see snow. We will represent this sort of contrary
argument by using a red arrow

with a slash through it:
Just as with supporting lines of reasoning, opposing lines may
have multiple premises or
chains. From the point of view of logic, these lines of opposing
reasoning are not really part
of the argument. However, such reasoning is often included
when presenting an argument, so
it is useful to have a way to represent it. This is especially true
when you are trying to under-
stand an argument in order to write an essay about it. It is good
practice to note what objec-
tions an author has already considered so that you do not just
repeat them.
It’s February
in Idaho.
It’s a very
cold year.
There’s snow
on the ground.
Most people outside
aren’t wearing coats.
John came in with
snow on his boots.
har85668_02_c02_025-058.indd 45 4/9/15 1:31 PM

2 1
4
3
With that, you have all the basic tools you need to create
argument diagrams. In principle,
arguments of any complexity can be represented with diagrams
of this sort. In practice, as
arguments get more complex, there are many interpretational
choices about how to repre-
sent them.
Diagramming Efficiently
One issue that arises when creating argument diagrams is that
including each premise and
conclusion can make diagrams large and cumbersome. A
common practice is to number each
statement in an argument and make the diagram with circled
numbers representing each
premise and conclusion. See Figure 2.1 for an illustration of the
seat belt example from the
previous section.
The seat belt example is not a complex argument, but the
diagram in Figure 2.1 is able to
show how the hidden assertion in the first question is supported
by the second statement
and how, together with the third assertion, the conclusion is
supported. Sketching diagrams

that show the relationship among the premises and their
connections to the conclusion
is very helpful in understanding complex arguments. Yet you
must keep in mind that the
diagramming is the second stage of the process, since you will
have to first identify the ele-
ments of the argument.
Figure 2.1: Diagramming the structure of an argument
This diagram shows the relationship between each of the
sentences in the seat belt example. Here are
the claims: 1. Don’t you know that driving without a seat belt is
dangerous? 2. Statistics show that you
are 10 times more likely to be injured in an accident if you are
not wearing one. 3. Besides, in our
state you can get fined $100 if you are caught not wearing one.
4. You ought to wear one even if you
are driving a short distance. Notice how numbering the
individual components of each argument and
diagramming them will help you see the relationship among the
pieces and how the pieces work
together to support the conclusion.
2 1
4
3
har85668_02_c02_025-058.indd 46 4/9/15 1:31 PM

Section 2.4 Classifying Arguments
2.4 Classifying Arguments
There are many ways of classifying arguments. In logic, the
broadest division is between
deductive and inductive arguments. Recall that Section 2.1
introduced the notion of inference,
the process of drawing a judgment from the reasons offered in
the premises. The distinction
between deductive and inductive arguments is based on the
strength of that inference. A con-
clusion can follow from the premises very tightly or very
loosely, and there is a wide range
in between. For deductive arguments, the expectation is that the
conclusion will follow from
the premises necessarily. For inductive arguments, the
expectation is that the conclusion will
follow from the premises probably but not necessarily. We shall
explore these two kinds of
arguments in greater depth in subsequent chapters. In this
section our goal is to achieve a
basic grasp of their respective definitions and understand how
the two types differ from one
another. Finally, we will improve our understanding of the
concept of an argument by com-
paring arguments to explanations, which are often mistaken for
arguments.
Draw an argument map of each of the following arguments,
using the described method
of numbering each statement and making a diagram with circled
numbers representing
each premise and conclusion.

1. (1) I know that Stephen has a lot of money. (2) His parents
drive a Mercedes. (3) His
dogs wear cashmere sweaters, and (4) he paid cash for his
Hummer.
2. (1) Guns are necessary. (2) Guns protect people, because (3)
they give people con-
fidence that they can defend themselves. (4) Guns also ensure
that the government
will not be able to take over its citizenry.
3. (1) If you drive too fast, you will get into an accident. (2) If
you get into an accident,
your insurance premiums will increase. Therefore, (3) if you
drive too fast, your
insurance premiums will increase.
4. Since (1) all philosophers are seekers of truth, it follows that
(2) no evil human is a
seeker after truth, since (3) no philosophers are evil humans.
5. (1) This cat can experience pain. So (2) it has the right to not
suffer. (3) Since we
shouldn’t cause suffering, (4) we should not harm the cat.
6. (1) If we change the construction of the conveyer belt, then
the timing of the line will
change. (2) Thus, if the timing of the line doesn’t change, then
we didn’t change the
construction of the conveyor belt. (3) In fact, the timing of the
line hasn’t changed.
(4) So that means we didn’t change the conveyer belt.
7. (1) The affordable health care act is becoming less popular.
(2) Cultural sentiment

is increasingly negative, and (3) the Senate and House are
progressively moving
toward opposition to it. (4) Just last week five Democratic
senators joined their
Republican counterparts to attempt to block certain aspects of
the act.
8. (1) Everyone should have to study logic. (2) It is becoming
more important to be able
to adapt to changes and (3) to evaluate information in today’s
workplace. (4) Logic
enhances these abilities. (5) Plus, logic helps protect us against
manipulators who try
to pawn off their fallacious arguments as truth.
har85668_02_c02_025-058.indd 47 4/9/15 1:31 PM
Deductive Arguments
In logic the terms deductive and inductive are used
in a technical sense that is somewhat different than
the way the terms may be used in other contexts.
For example, Sherlock Holmes, the protagonist in Sir
Arthur Conan Doyle’s detective novels, often referred
to his own style of reasoning as deductive. In fact, the
popularity of Sherlock Holmes introduced deductive
reasoning into ordinary speech and made it a com-
monplace term. Unfortunately, deductive reasoning
is often misunderstood, and in the case of Sherlock
Holmes, his clever style of reasoning is actually more

inductive than deductive. For example, in The Adven-
ture of the Cardboard Box, he says:
Let me run over the principal steps. We
approached the case, you remember, with
an absolutely blank mind, which is always an
advantage. We had formed no theories. We
were simply there to observe and to draw
inferences from our observations. (Doyle,
1892/2008, para. 114)
The foregoing does not describe deductive reasoning as it is
employed in logic. In fact, Sher-
lock Holmes mostly uses inductive rather than deductive
reasoning. For now, the simplest
way to present deductive arguments is to say that deductive
reasoning is the sort of reasoning
that we normally encounter in mathematical proofs. In a
mathematical proof, as long as you
do not make a mistake, you can count on the conclusion being
true. If the conclusion is not
true, you have either made an error in the proof or assumed
something that was false. The
same is true of deductive reasoning, because good deductive
arguments are characterized by
their truth-preserving nature—if the premises are true, then the
conclusion is guaranteed to
be true also. Consider the following deductive argument:
All married men are husbands.
Jacob is a married man.
Therefore, Jacob is a husband.
In this example, the conclusion necessarily follows from the
given premises. In other words, if
it is true that all married men are husbands and, moreover, that

Jacob is a married man, then
it must be necessarily true that Jacob is a husband.
But suppose that Jacob is a 3-year-old boy, so he is not a
married man. Would the argument
still be a good deductive argument and, thereby, truth
preserving? The answer is yes, because
deductive reasoning reflects the relations between premises and
the conclusion such that if it
were to be the case that the premises were true, then it would be
impossible for the conclu-
sion to be false. If it so happens that Jacob is a 3-year-old boy,
then the second premise would
not be true, and thus, the necessity for the conclusion to be true
is broken.
Wiley Miller/Cartoonstock
har85668_02_c02_025-058.indd 48 4/9/15 1:31 PM
However, this does not mean that all we need are true premises
and a true conclusion. Good
deductive arguments are not free form; rather, they use specific
patterns that must be followed
strictly in the inferential operation. Although this might sound
rigid, the greatest advantage
of good deductive arguments is that their precise structure
guides us into grasping a truth
that we might not otherwise have recognized with the same

certainty. The use of deductive
reasoning is quite broad—in science, mathematics, and the
examination of moral problems,
to name a few examples. Subsequent chapters will demonstrate
more about the powerful
machinery of deductive arguments.
Inductive Arguments
In contrast to deductive arguments, good inductive arguments
do not need to be truth pre-
serving. Even those that have true premises do not guarantee the
truth of their conclusion. At
best, true premises in inductive arguments make the conclusion
highly probable. The prem-
ises of good inductive arguments offer good grounds for
accepting the conclusion, but they do
not guarantee its truth. Consider the following example:
The produce at my corner store is stocked by local farmers
every day.
They have a bakery, too, and they refill their shelves with fresh-
baked bread
twice a day.
I have been shopping at my corner store continuously for 5
years, and every
day is the same.
Therefore, my corner store will have fresh produce and baked
goods every
day of the week.
Let us suppose that all the premises are true. After 5 years of
going to the corner store and
getting to know its practices and the quality of its daily
offerings, the conclusion would seem
to be highly probable. But is it necessarily true? At some point
the store may change hands,

close, or experience something else that interrupts its normal
operations. Such cases show
that even though the reasoning is good, the conclusion is not
guaranteed to be true just
because the premises are true.
Another way to think of what is going on here is to address a
likely familiar fact of the human
condition: Past experience does not guarantee that the future
will be the same. Think of that
great car you loved that did not require any expensive
maintenance—and then suddenly one
day it started to break down bit by bit with age. Time changes
the performance of things.
Or think of the great quality of a clothing brand you counted on
year after year that one day
was no longer as good. Products also change with time as the
leaders of the manufacturing
company change or the standards become somewhat relaxed.
Things change. Sometimes the
changes are for the better, sometimes for the worse. But our
observation of how things are
now and have been in the past does not guarantee that things
will remain the same in the
future. Accordingly, even if the conclusion in our corner store
example seems sufficiently jus-
tified for us to venture saying that it is true, the fact is that at
some point it could change. At
best, we can say that the premises give us good grounds to
assert that it is probably true that
the store will have good produce and baked goods this coming
week.
har85668_02_c02_025-058.indd 49 4/9/15 1:31 PM

Despite having a weaker connection between premises and
conclusion, inductive arguments
are more widely used than deductive arguments. In fact, you
have likely been using inductive
reasoning your entire life without knowing it. Think about the
expectation you have that your
car, house, or other object will be in the location you last left it.
This expectation is based on
good inductive reasoning. You have good reasons for expecting
your car to be sitting in the
parking space where you left it. We can represent your
reasoning as follows:
I left my car in that spot.
I have always found my car in the same parking spot I left it in.
Therefore, my car will be in that spot when I return.
Of course, having good reason is not the same as having a
guarantee, as anyone who has expe-
rienced having their vehicle stolen can attest. This is the
difference between deductive and
inductive arguments. Because inductive arguments only
establish that their conclusions are
probable, the conclusions can turn out to be false even when the
premises are all true. The
chance may be small, but there is always a chance. By contrast,
a good deductive argument is
airtight; it is absolutely impossible for the conclusion to be
false when the premises are true.
Of course, if one of the premises is false, then neither kind of

argument can establish its con-
clusion. If you misremember which spot you parked in, then you
are not likely to find your car
immediately, even if it is right where you left it.
Arguments Versus Explanations
Mastering logical reasoning requires not only understanding
what arguments are, but also
being able to distinguish arguments from their closest
conceptual neighbors. Although it
might be clear by now why news articles, debates, and
commands are not considered argu-
ments, we should take a closer look at explanations, because
they are commonly mistaken
for arguments and present a similar framework. Arguments
provide a methodical defense of
a position, presenting evidence by means of premises in support
of a conclusion that is dis-
puted. Explanations, in contrast, tell why or how something is
the case.
Suppose that we have the following claim:
We have to travel by train instead of by plane.
If you disagree with this decision, then you might question this
claim, thus presenting a
request for evidence. Accordingly, an argument would be the
appropriate response. We could
then have the following:
The total cost for plane tickets is $2,000.
The total cost for train tickets is $1,000.
We have a budget of $1,200 for this trip.
Therefore, we have to travel by train instead of by plane.

har85668_02_c02_025-058.indd 50 4/9/15 1:31 PM
Now, suppose that you do not question the claim, but you still
want to know why we have to
travel by train. This is not a request for evidence for the
conclusion. Rather, this is a request
for the cause that leads to the conclusion. This is thus a request
for an explanation, which may
be as simple as this:
Because we do not have enough money for plane tickets.
The point of an argument is to establish its main claim as true.
The point of an explanation is to
say how or why its main claim is true. In arguments, the
premises will likely be less controver-
sial than the conclusion. It is difficult to convince someone that
your conclusion is true if they
are even less likely to agree with your premises. In
explanations, the thing being explained is
likely to be less controversial than the explanation given. There
is little reason to explain why
or how something is true if the listener does not already accept
that it is true. Unlike argu-
ments, then, explanations do not involve contested conclusions
but, instead, accepted ones.
Their point is to say why or how the primary claim is true, not
to provide reasons for believing
that it is true. This explanation might be fairly straightforward,

but distinguishing between
arguments and explanations in real life may seem a bit more
blurry.
As an example, suppose you try to start your car one morning
and it will not start. You recall
that your son drove the car last night and know that he has a bad
habit of leaving the lights on.
You see the light switch is on. You now understand why the car
will not start. In our scenario,
you found out your car would not start and then looked around
for the reason. After noticing
that the light switch was on, you came up with the following
explanation:
Your son left the lights on.
Leaving the lights on will drain the battery.
A drained battery will prevent the car from starting.
That’s why your car won’t start.
It is an explanation because you already know that your car will
not start; you just want to
know why.
On the other hand, suppose that after your son got home last
night, you noticed that he left the
lights on. Rather than turn them off or tell him to do it, you
decide to teach him a lesson by let-
ting the battery go dead. In the morning you have the following
conversation with your son:
You: I hope you don’t need to go anywhere with the car this
morning.
Son: Why?

You: You left the car’s lights on last night.
Son: So?
You: The lights will have completely drained the battery. The
car won’t start
with a dead battery, so it’s not going to start this morning.
har85668_02_c02_025-058.indd 51 4/9/15 1:31 PM
In this case the thing you are most sure of is that your son left
the lights on. You reason from
that to the conclusion that the car will not start. In this scenario,
knowing that the lights were
left on is a reason for believing that the car will not start. You
are trying to convince your son
that the car will not start, and the fact that he left the lights on
last night is the starting point
for doing so. We can show the structure of your argument as
follows:
Your son left the lights on.
Leaving the lights one will drain the battery.
A drained battery will prevent the car from starting.
Therefore, your car won’t start.
Notice that the structure of this argument is the same as the
structure of the explanation
example. The only difference is whether you are trying to show

that the car will not start or to
understand why it will not start after already realizing that it
will not. Finding the structure
will help you understand the details of the argument or
explanation, but it will not, by itself,
help you determine which one you are dealing with. For that,
you have to determine what the
author is trying to accomplish and what the author sees as
common ground with the reader.
Understanding the structure of what is said can help you become
clearer about what the
author is doing, so it is a good thing to look for, but
understanding the structure is not enough.
Determining whether a passage is an argument or an explanation
is thus often a matter of inter-
preting the intention of the speaker or writer of the claim. A
good first step is to identify the
main point or central focus of the passage. What you are
looking for is the sentence that will be
either the conclusion to the argument or the claim being
explained. If the author has not done
so, paraphrase the main claim as a single, simple sentence. Try
to avoid including words like
because or therefore in your paraphrase. Ask yourself, if this is
an argument, what is its conclu-
sion? Once you have identified the potential conclusion, try to
determine whether the author is
attempting to convince you that that sentence is true, or whether
the author assumes you agree
with the sentence and is trying to help you understand why or
how the sentence is true. If the
author is trying to convince you, then the author is advancing an
argument. If the author is try-
ing to help you get a deeper understanding, the author is
providing an explanation.

It is important to be able to tell the difference between
arguments and explanations both
when listening to others and when crafting our own arguments
and explanations. This is
because arguments and explanations are trying to accomplish
different goals; what makes an
effective argument may not make an effective explanation.
Moral of the Story: Arguments Versus Explanations
If the main claim is accepted as true from the beginning, then
the speaker or writer may be
advancing an explanation, not an argument. If the point of a
passage is to convince the reader
that the main claim is true, then it is most likely an argument.
Of course, you may question an
explanation, thus requesting an argument that the explanation is
correct.
har85668_02_c02_025-058.indd 52 4/9/15 1:31 PM
Chapter Summary
This chapter introduced the standard argument form, which is
the principal tool that we will
employ in the ensuing chapters. We examined the elements of
an argument in standard form,
starting from the fundamental notion of claim to an argument’s

proper parts—premises and
conclusion—and the relationship between these, or what we call
inference. Although the
standard argument form is simple, the relationship between
those claims we call premises and
those we call conclusions is crucial to distinguishing between
different kinds of arguments.
Diagramming these relationships is merely one way we can
analyze arguments more fully.
In this chapter we also briefly discussed two kinds of
arguments—deductive and induc-
tive. However, each one of these will be addressed individually
in subsequent chapters as we
employ them in more sophisticated applications. Additionally,
we explored how to identify
arguments in the sources we encounter, as well as how to
extract what we find and paraphrase
it so that it can be presented in the standard form. Finally, we
discussed how to distinguish
arguments from explanations and presented a simple method for
making such a distinction.
As you continue to read this book, remember that logic is not
learned by reading alone.
Learning logic demands taking notes of structures and
terminology, and it requires practice.
Accordingly, practice the exercises provided in each chapter.
Once you gain mastery of the
standard argument form, you will be able to recognize good
arguments from bad arguments,
and you will be able to present good arguments in defense of
your views. This is a powerful
skill to have, and it is now in your hands.

1. Try to find a political commercial, and outline the argument
that is presented in the
commercial. Is it easy or difficult to find premises and
conclusions in the content
of the commercial? Does the argument relate to politics or to
something outside of
politics? Are there components of the ad that you think attempt
to manipulate the
viewer? Why or why not?
2. How can you utilize what you have learned in this chapter
about arguments in your
own life? At work? At home? How does an understanding of
being able to outline
and structure arguments translate into your everyday activities?
3. Now that you understand the components of an argument,
think back to a time that
someone you know attempted to provide an argument but failed
to do so in a con-
vincing fashion. What were the mistakes that this person made
in his or her reason-
ing? What were the structural or content errors that weakened
the argument?
4. Suppose that your child refuses to go to bed. You want to
convince your child that he
or she needs to get to sleep. You feel the urge to say, “You have
to go to bed because
I said so.” However, you are now trying to use what you are
learning in this course.
What argument would you present to your child to try to
convince him or her to go
to sleep? Do you think that a strong argument would be
effective in convincing your

child? Why or why not?
5. Suppose you have a coworker who refuses to help you with a
mandatory project. You
want to convince him that he needs to help you. What premises
would you use to
support the conclusion that he ought to help you with the
project? Assuming that he
fails to find your argument convincing, what would you do
next? Why?
har85668_02_c02_025-058.indd 53 4/9/15 1:31 PM
Web Resources
http://austhink.com/critical/pages/argument_mapping.html
The group Austhink provides a number of resources on
argument mapping, including tutori-
als on how to diagram arguments.
http://www.manyworldsof logic.com/index.html
The Many Worlds of Logic website discusses many of the topics
that will be covered in this
book.
Key Terms
argument The methodical defense of a
position advanced in relation to a disputed
issue; a set of claims in which some, called

premises, serve as support for another
claim, called the conclusion.
claim A sentence that presents an assertion
that something is the case. In logic, claims
are often referred to as propositions in order
to recognize that these may be true or false.
conclusion The main claim of an argument;
the claim that is supported by the premises
but does not itself support any other claims
in the argument.
conclusion indicators The words that
signal the appearance of a conclusion in an
argument.
explanations Statements that tell why or
how something is the case. Unlike argu-
ments, explanations do not involve contested
conclusions but, instead, accepted ones.
inference The process of drawing the nec-
essary judgment or, at least, the judgment
that would follow from the reasons offered
in the premises.
premise indicators The words that signal
the appearance of a premise in an argument.
premises Claims in an argument that serve
as support for the conclusion.
standard argument form The structure of
an argument that consists of premises and
a conclusion. This structure displays each

premise of an argument on a separate line,
with the conclusion on a line following all
the premises.
1. claim
2. claim
3. claim
4. nonclaim
5. nonclaim
6. claim
7. claim
8. nonclaim
9. claim
10. claim
11. nonclaim
12. claim
13. nonclaim
14. claim
15. nonclaim
16. claim
17. claim
18. nonclaim
har85668_02_c02_025-058.indd 54 4/9/15 1:31 PM
http://austhink.com/critical/pages/argument_mapping.html
http://www.manyworldsoflogic.com/index.html

19. claim
20. nonclaim
21. nonclaim
22. claim
23. claim
24. nonclaim
25. claim
26. nonclaim
27. claim
28. claim
29. nonclaim
30. claim
31. nonclaim
32. claim
33. claim
34. nonclaim
35. nonclaim
36. claim
37. claim
38. nonclaim
39. claim
40. nonclaim
1. There are three premises: (1) “Every time I turn on the radio,
all I hear is vulgar lan-
guage about sex, violence, and drugs,” (2) “Whether it’s rock
and roll or rap, it’s all
the same,” and (3) “The trend toward vulgarity has to change.”
The conclusion is “If it
doesn’t, younger children will begin speaking in these ways,
and this will spoil their

innocence.” Notice that the final sentence is an “If . . . , then . .
.” statement. Remem-
ber that these forms of sentences are single statements. The
entire final sentence is
the conclusion of this argument.
2. The premise is “Letting your kids play around on the Internet
all day is like dropping
them off in downtown Chicago to spend the day by themselves.”
The conclusion is
“They will find something that gets them into trouble.”
3. There are three premises: (1) “Too many intravenous drug
users continue to risk
their lives by sharing dirty needles,” (2) “This would lower the
rate of AIDS in this
high-risk population,” and (3) “allow for the opportunity to
educate and attempt to
aid those who are addicted to heroin and other intravenous
drugs.” The conclusion is
“This situation could be changed if we were to supply drug
addicts with a way to get
clean needles.”
4. There are three premises: (1) “His parents drive a Mercedes,”
(2) “His dogs wear
cashmere sweaters,” and (3) “he paid cash for his Hummer.”
The conclusion is “I
know that Stephen has a lot of money.” Notice that there are
two premises in the
final sentence. Remember that words like and and as well as
usually indicate that
there are multiple statements being made in a single sentence.
5. There are two premises: (1) “they always listen to what their
masters say” and (2)

“They also are more fun and energetic.” The conclusion is
“Dogs are better than cats.”
Remember that the word since is often a premise indicator. That
means that the state-
ment that follows the word since is often a premise.
6. There are two premises: (1) “All dogs are warm-blooded” and
(2) “All warm-blooded
creatures are mammals.” The conclusion is “all dogs are
mammals.” Remember that
the word hence is a conclusion indicator. It often comes before
the conclusion of an
argument.
7. There are two premises: (1) “it is an over-21 show” and (2)
“Jeffrey, James, and Sloan
were all carded when they tried to get in to the club.” The
conclusion is “Chances
are that I will not be able to get in to see Slipknot.” Remember
that the word since is
har85668_02_c02_025-058.indd 55 4/9/15 1:31 PM
often a premise indicator. That means that the statement that
follows the word since
is often a premise.
8. There are two premises: (1) “the best of all possible worlds
would not contain suf-

fering” and (2) “this world contains much suffering.” The
conclusion is “This is not
the best of all possible worlds.” Remember that the word
because is often a premise
indicator. That means that the statement that follows the word
because is often a
premise.
9. There are two premises: (1) “Some apples are not bananas”
and (2) “Some bananas
are things that are yellow.” The conclusion is “some things that
are yellow are not
apples.” Remember that the word therefore is a conclusion
indicator. It often comes
before the conclusion of an argument.
10. There are two premises: (1) “all philosophers are seekers of
truth” and (2) “no
philosophers are evil humans.” The conclusion is “no evil
human is a seeker after
truth.” Remember that the words it follows that are conclusion
indicators. They
often come before the conclusion of an argument. Also,
remember that the word
since is often a premise indicator.
11. The premise is “All squares are triangles and all triangles
are rectangles.” The con-
clusion is “all squares are rectangles.” Remember that the word
so is a conclusion
indicator. It often comes before the conclusion of an argument.
12. There are two premises: (1) “Deciduous trees are trees that
shed their leaves” and
(2) “Maple trees are deciduous trees.” The conclusion is “maple
trees will shed their

leaves at some point during the growing season.” Remember
that the word thus is a
conclusion indicator. It often comes before the conclusion of an
argument.
13. The premise is “most philosophy professors are rich.” The
conclusion is “Joe must
make a lot of money teaching philosophy.” Remember that the
word since is often a
premise indicator. That means that the statement that follows
the word since is often
a premise.
14. There are two premises: (1) “all mammals are cold-blooded”
and (2) “all cold-blooded
creatures are aquatic.” The conclusion is “all mammals must be
aquatic.” Notice that
there are two premises in the first sentence. Remember that
words like and and as well
as usually indicate that there are multiple statements being
made in a single sentence.
15. There are two premises: (1) “If you drive too fast, you will
get into an accident” and
(2) “If you get into an accident your insurance premiums will
increase.” The conclu-
sion is “if you drive too fast, your insurance premiums will
increase.” Remember that
the word therefore is a conclusion indicator. It often comes
before the conclusion of
an argument.
16. There are three premises: (1) “The economy continues to
descend into chaos,” (2) “The
stock market still moves down after it makes progress forward,”
and (3) “unemployment

still hovers around 10%.” The conclusion is “It is going to be a
while before things get
better in the United States.” Notice that there are two premises
in the second sentence.
Remember that words like and and as well as usually indicate
that there are multiple
statements being made in a single sentence.
17. There are two premises: (1) “The athletes are amazing” and
(2) “it is extremely com-
plex.” The conclusion is “Football is the best sport.” Notice that
there are two premises
in the second sentence. Remember that words like and and as
well as usually indicate
that there are multiple statements being made in a single
sentence.
har85668_02_c02_025-058.indd 56 4/9/15 1:31 PM
2
1
43
2
1
4

3
1
3
2+
1
2
3+
18. The premise is “I hear that it has amazing special effects.”
The conclusion is “We
should go to see Avatar tonight.”
19. There are two premises: (1) “All doctors are people who are
committed to enhancing
the health of their patients” and (2) “No people who purposely
harm others can con-
sider themselves to be doctors.” The conclusion is “some people
who harm others
do not enhance the health of their patients.” Remember that the
words it follows that
are a conclusion indicator. When you see these words, think “a
conclusion is coming.”
20. There are three premises: (1) “Guns protect people,” (2)
“They give people confi-
dence that they can defend themselves,” and (3) “Guns also
ensure that the gov-
ernment will not be able to take over its citizenry.” The

conclusion is “Guns are
necessary.”
1.
2
1
43
2.
2
1
4
3
3.
1
3
2+
4.
1
2
3+

har85668_02_c02_025-058.indd 57 4/9/15 1:31 PM
2
4
3+
1
2
4
3+
1
2
1
3
4
1
2 3+
5

4+
5.
2
4
3+
1
6.
2
4
3+
1
7.
2
1
3
4
8.

1
2 3+
5
4+
har85668_02_c02_025-058.indd 58 4/9/15 1:31 PM

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