How is the professor using logical argumentation
in coming up with mathematical proofs?
How can learning about deductive logic,
such as arguments based on mathematics,
help us make better-informed decisions?
WHAT’S TO COME
239 | What Is a Deductive Argument?
241 | Types of Deductive Arguments
247 | Hypothetical Syllogisms
252 | Categorical Syllogisms
257 | Translating Ordinary Arguments
into Standard Form
261 | Critical Thinking Issue: Perspectives
on the Death Penalty
n Sir Arthur Conan Doyle’s mystery
story “Silver Blaze,” detective Sherlock
Holmes uses his extraordinary powers of deductive logic to solve the
mystery of the disappearance of racehorse
Silver Blaze and the murder of the horse’s
trainer, John Straker. His head shattered by
a savage blow, Straker’s body was found a
quarter mile from the King’s Pyland stables
where Silver Blaze was kept. A search is
carried out of the surrounding moors and
of the neighboring Mapleton stables for the
After interviewing everyone who might
have been involved and collecting all the
facts, Holmes concludes that Silver Blaze is
still alive and hidden in the Mapleton stables,
■ What is a deductive argument?
■ What are some of the types of deductive
■ What is a syllogism, and how do we know if it is
even though the earlier search of the stables had failed to turn up the missing
“It’s this way, Watson,” [says Holmes]. “Now, supposing that
[Silver Blaze] broke away during or after the tragedy, where could
he have gone to? The horse is a very gregarious creature. If left to
himself his instincts would have been either to return to King’s Pyland
or go over to Mapleton. Why would he run wild upon the moor? He
surely should have been seen by now . . . He must have gone to
King’s Pyland or to Mapleton. He is not at King’s Pyland. Therefore,
he is at Mapleton.”1
As it turns out, Holmes’s deduction is right. The missing racehorse is at
Mapleton, the silver blaze on its nose covered over to disguise its appearance.
Sherlock Holmes also solves the “murder” of the horse’s trainer through
deductive logic. He learns from the stable hand that the guard dog did not bark
when Silver Blaze was “stolen” from the stables. Therefore, Holmes concludes,
the person who took Silver Blaze must have been familiar to the dog. This eliminated suspects who were strangers. Holmes then eliminates, one by one, the other
suspects, leaving only the horse. As Holmes stated in another story: “When you have eliminated the impossible, whatever remains, however improbable, must be the truth.”2 He concludes that
the horse must have accidentally killed its trainer when Straker, who was something of a scoundrel,
used a surgical knife found in his possession to nick the tendons of Silver Blaze’s ham so the horse
would develop a slight limp and lose the upcoming race. Holmes explains, “Once in the hollow,
[Straker] had got behind the horse and had struck a light; but the creature, frightened at the sudden
glare, and with the strange instinct of animals feeling that some mischief was intended, had lashed
out, and the steel shoe had struck Straker full on the forehead.”3
To generations of mystery readers, Sherlock Holmes has epitomized the skilled reasoner. In
this chapter we’ll learn how to evaluate deductive arguments and practice some of the strategies
used by Holmes and others who are skilled in deductive argumentation. In Chapter 8 we will:
• Identify the essential attributes of a deductive argument
• Distinguish between validity, invalidity, and soundness in a deductive argument
• Learn how to recognize and evaluate arguments by elimination, mathematical arguments, and argument from definition
Some deductive arguments are more involved and may
have several dependent premises and subconclusions.
• Study the different types of hypothetical syllogisms, including modus ponens, modus tollens, and chain arguments
Valid and Invalid Arguments
• Learn how to recognize standard-form categorical syllogisms
• Reevaluate categorical syllogisms using Venn diagrams
• Practice putting arguments that are in ordinary language
into standard form
Finally, we will analyze different arguments regarding the justification of the death penalty (capital punishment).
A deductive argument
is valid if the form of the
argument is such that the
conclusion must be true if
the premises are true. The
form of an argument is
determined by its layout or
pattern of reasoning. In the
above case, the form is:
syllogism A deductive argument
presented in the form of two
supporting premises and a conclusion.
valid A deductive argument where
the form is such that the conclusion
must be true if the premises are
assumed to be true.
form The pattern of reasoning in a
All X (men) are Y
WHAT IS A DEDUCTIVE
Unlike inductive arguments, in which the premises offer
only support rather than proof for the conclusion, in a
valid deductive argument the conclusion necessarily follows from the premises. Deductive arguments sometimes
contain words or phrases such as certainly, definitely,
absolutely, conclusively, must be, and it necessarily
follows that. For example:
Marilyn is definitely not a member of the swim team,
since no freshmen are members of the swim team and
Marilyn is a freshman.
Deductive Reasoning and Syllogisms
Deductive arguments are sometimes presented in the
form of syllogisms, with two supporting premises and a
conclusion. For the purpose of analysis, in this chapter
the premises and conclusion of a syllogism will usually
be presented on separate lines, with the conclusion last.
Premise: All men are mortal.
Premise: All fathers are men.
Conclusion: Therefore, all fathers are mortal.
Deductive arguments may also be diagrammed using
the guidelines we learned on pages 112–114. In the case of
a syllogism, the two premises are always dependent:
2 (Dependent premises)
All Z (fathers) are X (men).
Therefore, all Z (fathers) are Y (mortal).
This argument is a valid form no matter what terms we
use for X, Y, and Z. Because the form is valid, if we substitute different terms for men, mortal, and fathers, and the
premises are still true, then the conclusion must be true, as
in the following example.
All cats (X) are mammals (Y).
All tigers (Z) are cats (X).
Therefore, all tigers (Z) are mammals (Y).
A false conclusion does not necessarily mean that
a deductive argument is invalid. In the two arguments
we’ve examined so far, the conclusions were both true
because the premises were true and the
form was valid. The conclusion of a
valid argument can be false only if
one of the premises is false. In the
following example, which uses the
same form as our initial argument,
we end up with a false conclusion:
Hot or Not?
All men are tall people.
Tom Cruise is a man.
Therefore, Tom Cruise is a tall
The conclusion in the above
argument is false only because
there is a false premise, not
because the form of the argument
is invalid. The first premise, “All
men are tall people,” is false.
If both premises are true and the
conclusion is false, then the argument, by definition, is invalid. For
Chapter 8 | Deductive Arguments • 239
All dogs are mammals.
Some mammals are not poodles.
Therefore, some poodles are not dogs.
It is also possible to have an invalid argument in which
the premises are true and the conclusion just happens to be
true. Consider this:
No seniors are freshman.
All freshmen are college students.
Therefore, some college students are seniors.
In this argument, the premises and conclusion are
true. However, the premises do not logically support the
conclusion. The invalidity of a form can be demonstrated
by substituting different terms for senior, freshman, and
college students, and then seeing whether we can come
up with an argument using this form in which the premises are true but the conclusion false, as in the following
No fish are dogs.
All dogs are mammals.
Therefore, some mammals are fish.
Sound and Unsound Arguments
An argument is sound if (1) it is valid and (2) the premises
are true. The argument on page 239 about fathers being
mortal is a sound argument because it is valid and the
premises are true. On the other hand, although the argument about Tom Cruise on page 239 uses a valid form, it
is not a sound argument because the first premise is false.
Invalid arguments, because they do not meet the first criterion, are always unsound.
Logic is primarily concerned with the validity of arguments. As critical thinkers, we are also interested in the
soundness of our arguments and in having our premises
supported by credible evidence and good reasoning. We
have already discussed in previous chapters guidelines for
ensuring that our premises
are accurate and credible.
sound A deductive argument that is
valid and that has true premises.
In this chapter we’ll learn
how to identify the different types of deductive arguments and how to use Venn diagrams to evaluate these arguments for validity.
Sound Unsound Unsound
Valid argument: The form or layout of the argument
is such that if the premises are true, then the
conclusion must necessarily be true.
Sound argument: The form of the argument is valid
and the premises are true.
➤APPLICATION: Identify in the text an example of
an argument that is (a) valid and sound, (b) valid and
unsound, and (c) invalid.
STOP AND ASSESS YOURSELF
E XE RC I SE 8 - 1
“Some mammals are fish” is an example of a false conclusion.
1. What do you mean when you say that you can prove something with certainty?
Give a specific example of a proof from your everyday experience (keep it as brief
as possible). What type of logic does the proof use—inductive or deductive?
2. In the story “Silver Blaze,” Sherlock Holmes tells Watson that when it comes
to the art of reasoning, many people rely on opinion and unsupported assumptions.
E X E RCI SE 8- 1 c on t .
The difficulty, he maintains, is to detach the framework of undeniable fact from the embellishments
of hearsay and reporters. What do you think he meant by this? Explain using examples from your
3. Using substitution, show that the form of each of the following deductive arguments is invalid. Remember: To establish invalidity, your premises must be true when you are substituting new terms for the
ones in the original argument.
*a. All fraternity members are men.
No women are fraternity members.
Therefore, no women are men.
b. If it is raining, then it is cloudy.
It is cloudy.
Therefore, it is raining.
c. No mice are humans.
Some mice are rodents.
Therefore, no humans are rodents.
*d. Some married people are college students.
All wives are married people.
Therefore, some wives are college students.
e. All flowers are plants.
All orchids are plants.
Therefore, all orchids are flowers.
f. If my baby sister is a college student, then she is a high school graduate.
My baby sister is not a college student.
Therefore, my baby sister is not a high school graduate.
4. The following arguments are all valid arguments. Determine whether each argument is sound or
*a. No mammals are birds. Some penguins are mammals. Therefore, some penguins are not birds.
b. Some twins are sisters. All twins are siblings. Therefore, some siblings are sisters.
c. All students are dormitory residents. No dormitory residents are birds. Therefore, no birds are
*d. If Mexico is in South America, then Mexico is not a country bordering the United States. Mexico
is in South America. Therefore, Mexico is a not country bordering the United States.
e. All people living in England are citizens of the European Union.
All members of the British royal family are people living in England.
Therefore, all members of the British royal family are citizens of the European Union.
f. All millionaires are rich people. Some Americans are not rich people. Therefore, some
Americans are not millionaires.
TYPES OF DEDUCTIVE
There are several types of deductive arguments. In this
section, we’ll be looking at three types of deductive arguments used in everyday reasoning:
Arguments by elimination
Arguments based on mathematics
Arguments from definition
Arguments by Elimination
An argument by elimination rules out different possibilities
until only one possibility remains. In the introduction to
this chapter, we saw Sherlock Holmes using an arguargument by elimination A
ment by elimination. He
deductive argument that rules out
different possibilities until only one
reasoned that Silver Blaze
had to be at one of the two
stables. Since it wasn’t at
King’s Pyland, it must be at Mapleton. In “Thinking Outside the Box: Bo Dietl, Top Cop” on page 242, we profile
Chapter 8 | Deductive Arguments • 241
a New York City detective who is skilled in this type of
Like detectives, physicians are trained in this type of
deductive logic. In diagnosing an illness, a physician starts
by doing a physical examination and, often, by ordering
tests. If the examination and test results eliminate the most
common explanations of the symptoms, then the physician moves on to check out less obvious possibilities
until the mystery is solved. Indeed, Dr. Joseph Bell, one
of Sir Arthur Conan Doyle’s professors at the University
of Edinburgh Medical School, was the inspiration for the
character Sherlock Holmes.
Arguments by elimination are frequently used in everyday life. For instance, suppose it is the first day of the semester and you arrive on campus with 10 minutes to spare. You
check your schedule and see that your first class, Introduction to Psychology, is in Winthrop Hall. However, on your
schedule the room number is smudged and you can’t read
Outside the Box
B O DI E T L , Top Cop
DI SC U SSI O N Q UE STI ONS
Bo Dietl is a modern Sherlock Holmes. Born in Queens, New York, in 1950, Dietl wanted a job
where he could make a real difference in people’s lives. When he learned about the test to get into
the police academy, he decided to give it a try.
One of the most highly decorated detectives in the history of the New York Police
Department, Dietl investigated numerous high-profile murders and other felonies, obtaining
evidence through research, interviews, and other investigative techniques. He attributes much of
his success in solving more than 1,500 felonies to what he calls his “sixth sense—a nontangible
feeling good detectives use in solving cases.”*
One of the most famous crimes he solved was the 1981 rape and torture of
a Catholic nun in an East Harlem convent. Dietl concluded from the evidence
that the crime was a burglary gone wrong, rather than a sex crime, thus
narrowing his search to people with burglary records. He also knew,
from interviewing witnesses, that one of the men was probably tall
and that the other had a limp. Days later he received a tip that the
two men who committed the crime lived somewhere on 125th Street
in Harlem. However, there were hundreds of buildings and thousands
of people living on this street. He began the process of elimination by
going to the local hangouts and tenements, knocking on doors, giving
a brief description of the suspects, and asking questions. He also passed
out hundreds of business cards. His efforts paid off, and the two suspects
were apprehended and arrested. The 1998 movie One Tough Cop is based on
Dietl’s autobiography of the same name.
Discuss how Dietl’s method of solving the murder of the nun in the
East Harlem convent demonstrates deductive reasoning using an
argument by elimination.
In Chapter 2 we learned that much of reasoning is unconscious
and automatic and that scientists and mathematicians, as well
as great detectives, often resolve complex problems without
any conscious deliberation. However, to develop this ability,
they have spent years consciously resolving problems and
mentally rehearsing solutions. Think of a type of problem
in your life that you find easy to resolve with little
or no conscious deliberation. Discuss what factors,
such as your familiarity and experience with the
problem, contributed to your ease of resolution.
*Conversation with Bo Dietl on August 8, 2005.
it. What do you do? It would take too long to get a
new schedule. Instead, you head over to Winthrop
Hall and check out the building directory. It lists
twelve room numbers. Nine of them are faculty
offices, so you eliminate those nine. The remaining three are classrooms A, B, and C. You go into
classroom A and ask some students what class it
is. They tell you that it’s English Literature. You
proceed to classroom B and repeat the process;
it turns out to be a course in Business Statistics.
When you get to classroom C, you just go inside
and take a seat. How do you know this is the correct classroom? Through the use of an argument
by elimination. Assuming that your premises are
true (that your psychology course is being taught
somewhere in Winthrop Hall), the third classroom
by necessity must be your classroom.
My class is either in room A, B, or C.
My class is not in room A.
My class is not in room B.
Therefore, my class must be in room C.
In the previous example, there were three alternatives. If there are only two alternatives, the argument is referred to as a disjunctive syllogism. A disjunctive syllogism takes one of two forms:
Either A or B.
Either A or B.
A mouse locates the prize at the end of the maze through the deductive process of
In determining the whereabouts of Silver Blaze, Sherlock Holmes used a disjunctive syllogism:
Either Silver Blaze is at King’s Pyland or Silver Blaze
is at Mapleton.
Silver Blaze is not at King’s Pyland.
Therefore, Silver Blaze is at Mapleton.
Either you finished cleaning your room or you’re
staying in tonight.
You are not staying in tonight.
Therefore, you finished cleaning your room.
In a disjunctive syllogism, the two alternatives presented in the first premise—clean your room or stay in
tonight—must be the only two possibilities. If there are
other possible alternatives that have not been stated, then
the argument commits the fallacy of false dilemma. For
Either we fight the war on terrorism in Iraq, or we’ll
have to fight the terrorists here in America on our
We’re fighting the war on terrorism in Iraq.
Therefore, we won’t have to fight the terrorists here in
America on our own soil.
In this argument, the two alternatives in the first premise do not exhaust all possible alternatives. Perhaps
disjunctive syllogism A type of
deductive argument by elimination in
we could go after individwhich the premises present only two
ual terrorist cells instead of
attacking a country or even
argument based on
negotiate a truce with some
mathematics A deductive argument
of the organizations or govin which the conclusion depends on a
ernments associated with
terrorist activities. Because
the argument commits the
fallacy of false dilemma, it is not a sound argument.
Arguments Based on Mathematics
In an argument based on mathematics, the conclusion
depends on a mathematical or geometrical calculation. For
My dormitory room is rectangular in shape.
One side measures 11 feet and the side adjacent to it
measures 14 feet in length.
Therefore, my room is 154 square feet.
You can also draw conclusions about your new roommate, Chris, even before you meet, using this type of
deductive reasoning ...
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