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PHI101: Art of Reasoning
Review for Test 4
Section 01: May 3, Thursday, 10:30am
Section 02: May 1, Tuesday, 8:00am
Section 03: May 2, Wednesday, 3:30pm
Zhaolu Lu, PhD
Professor of Philosophy
General Features of the test
• Coverage: Chapters 5 and 6
• Number of Questions: 40
• Question Type: multiple choice
Chapter 5 Test Topics
1. The pure forms of compound propositions
2. The standard English expressions of compound
propositions
3. Various English expressions of compound
propositions.
4. The logical nature of compound propositions
5. Negations of compound propositions
6. Inclusive and exclusive disjunction
1.The pure forms of compound propositions
Sample Question:
Which of the following is the pure form of
conditional propositions?
a. P & Q
b. P Q
c. P Q
d. P Q
Answer: c
2. The standard English expressions of
compound propositions
Sample Question: Which of the following is a
disjunctive proposition?
a. Jack is both wise and crafty.
b. Jack is wise if and only if he is crafty.
c. If Jack is wise, then he is crafty.
d. Jack is either wise or crafty
Answer: d
3. Various English expressions of
compound propositions
Sample Question: Which of the following is a
conjunctive proposition?
a. The victim died before the cassette was rewound.
b. The cassette was not rewound.
c. The victim died or the cassette was rewound.
d. The cassette was rewound only if the victim died.
Answer: a
4. The logical nature of compound propositions
Sample Question 1: If “John is a knave” is true, which
of the following propositions must also be true?
a. John is a knave just in case he is a fool.
b. John is both a knave and a fool.
c. If John is a knave then he is a fool.
d. John is either a knave or a fool.
Answer: d
4. The logical nature of compound propositions
continued
Sample Question 2: If “John is a knave” is true and
“John is a fool” is false, which of the following
propositions must be false?
a. John is either a knave or a fool.
b. John is a knave just in case he is not a fool.
c. If John is a knave then he is a fool.
d. John is a knave but he is not a fool.
Answer: c
5. Negations of compound propositions
Sample Question: Which of the following propositions
is the correct negation of the proposition “John is either
a knave or a fool”?
a. John is both a knave and a fool.
b. John is a knave but not a fool.
c. John is not a knave but he is a fool.
d. John is neither a knave nor a fool.
Answer: d
6. Inclusive and exclusive disjunction
Sample Question: Which of the following is an
inclusive disjunction?
a.
b.
c.
d.
Either the American team or the Russian team will win
the gold medal.
The enemy troop will launch attack either at 7pm or at
8pm.
Britney’s husband is either at the snack bar or on the
tennis court.
Tom is either a singer or an actor.
Answer: d
Chapter 6 Test Topics
1. Recognize the pure forms of deduction
reasoning with compound proposition
2. Recognize the types of deduction
reasoning with
compound proposition in natural English.
3. Evaluate the validity of reasoning with
compound proposition
1. Recognize the pure forms of deduction
reasoning with compound proposition
Sample Question:
Which of the following is Modus Ponens?
a
a.
Answer: d
b.
c.
d.
1. Recognize the pure forms of deduction
reasoning with compound proposition continued
Sample Question: Which of the following is the form of
deductive reasoning to a disjunctive proposition?
a
a.
Answer: b
b.
c.
d.
2. Recognize the types of deduction reasoning
with compound proposition in natural English.
Sample Question: Which of the following is an example of
deductive reasoning from a disjunctive proposition?
a. If you want to be respected, you must respect others. You
want to be respected. Therefore, you must respect others.
b. Tanya will attend the party. Therefore, Tanya will either
attend the part or stay home .
c. Either Jen or Sara will win the gold medal. Jen will not win
the gold medal. Therefore, Sara will win the gold medal.
d. We will see the movie if we get tickets. We get tickets.
Therefore, we will see the movie.
Answer: c
3. Evaluate the validity of reasoning with
compound proposition
Sample Question: Which of the following is deductively valid?
a. If you want to be respected, you must respect others. you
must respect others. Therefore, you want to be respected.
b. Tanya will either attend the part or stay home. Therefore,
Tanya will attend the party.
c. Either Jen or Sara will win a gold medal. Jen will win a
gold medal. Therefore, Sara will not win a gold medal.
d. We will see the movie if we get tickets. We get tickets.
Therefore, we will see the movie.
Answer: d
3. Evaluate the validity of reasoning with
compound proposition continued
Sample Question: Which of the following is deductively invalid?
a. If Bob joins in the team, then Tim stays. If Tim stays, then the
coach is happy. So, if Bob joins in the team, the coach is happy.
b. If Bob does not join in the team, then Tim will not stay. Tim
will stay. So, Bob joins in the team.
c. If Bob join in the team, then Tim will stay. Tim will not stay.
Bob does not join in the team.
d. If Bob join in the team, then Tim will stay. Bob does not join in
the team. Tim will not stay.
Answer: d
PHI101: Art of Reasoning
Chapter 6. Deduction with
Compound Proposition
Zhaolu Lu, PhD
Professor of Philosophy
Objectives and Tools
• Two Objectives
1) Understand reasoning processes from and to
a compound proposition.
2) Learn to evaluate reasoning processes
analytically and critically.
• Symbols: P, Q, R, ( ), [ ]
The basic notions
What is reasoning from a compound proposition?
Example
A compound proposition
Reasoning from a compound proposition is a reasoning
process that takes a compound proposition as its major
premise (often, the background assumption or theory).
What is reasoning to a compound proposition?
Example: a collective thinking process
Alan: What is a kingfisher?
Sara: If a bird is a kingfisher, then it is brightly colored.
Tony: If a bird is brightly colored, then it is a kingfisher.
Alan: So, a bird is a kingfisher if and only if it is brightly colored.
A compound proposition
Reasoning to a compound proposition is a reasoning
process that that arrives at a compound proposition as its
conclusion.
Combination of the two types
Reasoning both from and to a compound proposition
Example
Individual task
Is the following a reasoning from a compound
proposition or to a compound proposition?
We will see the movie if we
get tickets.
We get tickets.
Therefore,
we will see the movie.
It is reasoning from a compound proposition.
Individual task
Is the following a reasoning from a compound
proposition or to a compound proposition?
Tim visited Paris last year.
He visited London the year
before last year.
Hence, Tim has been to
both Paris and London.
It is reasoning to a compound proposition.
Individual task
Is the following a reasoning from a compound
proposition or to a compound proposition?
Carol will either rent the
movie or buy the book.
Carol will buy the book.
Therefore,
she will not rent the movie.
It is reasoning from a compound proposition.
Individual task
Is the following a reasoning from a compound
proposition or to a compound proposition?
Dan went swimming. The
water was freezing cold.
So, Dan went swimming
even though the water was
freezing cold.
It is reasoning to a compound proposition.
Our Focus
We focus on three types:
Reasoning from and to a conjunction
Reasoning from and to a disjunction
Reasoning from and to a conditional
1. Conjunctive Deduction
(1) Reasoning to a conjunctive proposition
This type of reasoning requires the ability to pick up
and combine crucial data among multiple ones.
Group task: Find the solution to a bank robbery case
Video (1:40): https://www.youtube.com/watch?v=Yv5CPyKrUIA
Group discussion:
How did the detective know Dave is the robber?
Answer:
Dave runs a digital photo store and he doesn’t
need a darkroom for digital photos.
The pure form of reasoning to a conjunctive proposition
The Form
Example
“and”
Jen is running.
Jen is singing.
Jen is singing while she is running.
Deductively valid form
A complex example of reasoning to a conjunctive proposition
A detective entered a crime scene where a dead body lay.
The victim was holding a gun. A tape recorder lay by his
side. The detective played the recorder and heard words “I
am tired of this life and hence I have decided to relieve
myself from the worldly pains” followed by a gunshot.
P: The victim was dead.
Q: The cassette was rewound.
P & Q: The victim died before the cassette was rewound.
and
(2) Reasoning from a conjunctive proposition
The pure form
Example 1
Example 2
Deductively
valid forms
The bowl of squid eyeball stew is hot and
delicious.
The bowl of squid eyeball stew is hot.
The bowl of squid eyeball stew is hot and
delicious.
The bowl of squid eyeball stew is delicious.
A complex example of
reasoning from a conjunctive proposition
Example 1
P & Q: Snakes, which are reptiles without
legs, kill a large number of rodents.
P: Snakes are reptiles without legs.
Example 2
P & Q: Snakes, which are reptiles without
legs, kill a large number of rodents.
Q: Snakes kill a large number of rodents.
Implicit conjunction
Application of conjunctive deduction
Reasoning from a conjunctive proposition requires the
ability to analyze a package of data.
Individual task:
A. One of following four people killed one of the others.
1. The murderer had his leg amputated last month.
2. Mike was a farmer before he moved to the city.
3. Dan grew up together with the murderer.
4. Jeff wants to install a new computer next week.
5. Ben met Dan for the first time six months ago.
6. Dan ran marathon yesterday.
Who killed whom?
Solution to the riddle: use conjunctive reasoning
A: One of following four people killed one of the others.
1. The murderer had his leg amputated last month.
2. Mike was a farmer before he moved to the city.
3. Dan grew up together with the murderer.
4. Jeff wants to install a new computer next week.
5. Ben met Dan for the first time six months ago.
6. Dan ran marathon yesterday.
(1) & (6)
(3) & (5)
Not Dan & Not Ben
Mike or Jeff
(4) & (A)
Jeff killed Mike
Deductively invalid conjunctive reasoning
• Individual task: What’s wrong with the following
reasoning?
Amy likes peach ice cream. Therefore, she likes
peach ice cream and brownie sundae.
• Answer: The conclusion can be false while the
premise is true.
• The invalid form
P
P&Q
Individual Practice: Deductively valid or invalid?
(a) Spiders that are
poisonous are the black
widow and the brown
recluse. Therefore, some
spiders are poisonous.
(b) Spiders that are
poisonous are the black
widow and the brown
recluse. Therefore, all
spiders are poisonous.
Individual Practice: Deductively valid or invalid?
(c) Many schools are
professional institutions.
Therefore, many schools
that are professional
institutions do not offer
liberal arts education.
(d) Many schools are
professional institutions.
professional schools offer
liberal arts education..
Therefore, many schools
that are professional
institutions offer liberal
arts education.
2. Disjunctive Deduction
Introduction
Disjunctive deduction is also called “disjunctive
syllogism”.
A disjunctive reasoning (inference) may be a
reasoning to or from a disjunctive proposition.
Not all forms of disjunctive reasoning are
deductively valid.
(1) Reasoning to a disjunctive proposition
• The deductively valid form
If P is true, then
P Q must be true
• Here P Q can be either inclusive or
exclusive.
Examples of deductive reasoning
to a disjunctive proposition
Reason to an exclusive one
Reason to an inclusive one
Tanya will attend the party. Tanya will attend the party.
Tanya will either attend the Either Tanya or Megan will
part or stay home to
attend the party.
complete the homework.
Individual Tasks: Reason from each of
the following atomic propositions to a
disjunction in the form on the right
a.
d.
b.
c.
e.
Group Task: a riddle
Cindy, Andy, Keith and Mia were in a house when a
package was delivered. Each of them guessed what was
in the box, but only one of them was right. Who is right?
Cindy: “It’s a laptop computer.”
Andy: “I’ll bet it’s a pizza.”
Mia: “I think it’s a laptop computer or a picture.”
Keith: “It’s a picture, for sure.”
Reasoning:
If Cindy is right, then Mia is also right.
If Keith is right, then Mia is also right.
If Mia is right, then at least one of Cindy and Keith is.
(2) Reasoning from a disjunctive proposition
• The deductively valid forms
• These are deductively valid in both the case of
inclusive P Q and the case of exclusive P Q.
• Overview Video (2:42):
Examples of deductive reasoning
from a disjunctive proposition
Reason from an exclusive one
Either Jen or Sara will win
the gold medal.
Jen will not win the gold
medal.
Sara will win the gold medal.
Reason from an inclusive one
Either Jen or Sara will win a
gold medal.
Sara will not win a gold
medal.
Jen will win a gold medal.
Individual Tasks: Compose a
reasoning from each of the
following disjunctive propositions
in one of the forms on the right.
a.
c.
b.
d.
The special forms of
reasoning from a disjunctive proposition
• The forms that are valid in the case of
reasoning from exclusive disjunction but invalid
in the case of reasoning from inclusive
disjunction:
• Overview video (2:42):
Examples of using
these forms
Reason from an exclusive one
Reason from an inclusive one
Either Leslie is in the
theatre or she is at home.
Leslie is at home.
Leslie is not in the theatre.
Either Leslie is not sad or
she is a good actor.
Leslie is not sad.
Leslie is not a good actor.
Individual Task:
Is the following reasoning deductively valid?
The new budget will either create a huge deficit
or stimulate the economy.
The new budget will create a huge deficit.
The new budget will not stimulate the economy.
Individual Task:
Is the following reasoning deductively valid?
I can’t be both a spineless
extrovert and a neurotic introvert.
I am a spineless extrovert.
So, I am not a neurotic introvert.
I can be either a
spineless extrovert or
a neurotic introvert,
exclusively.
Group Task: a riddle
One of the triplets, who wear the same size shoes, left
muddy footprints all over the kitchen floor. “Who did
it?” asked their mom. “I didn’t do it,” said Annie.
“Danny did it,” said Fanny. “Fanny is lying,” said
Danny. If only one of them told the truth, who did it?
Reasoning:
Either Annie or Danny or Fanny did it. (A D) F
If D did it, then both A and F told the truth. So, ~D.
If F did it, then both D and A told the truth. So, ~F.
If A did it, then D was the only one who told the truth.
Therefore, Annie did it.
3. Conditional Deduction
Review of conditional proposition
The form: P Q
Logical meaning: If P is true, then Q is true.
P is sufficient for Q;
Q is necessary for P.
Conditional Deduction is a set of deductively valid
forms of reasoning from or to a conditional
proposition.
(1) Modus Ponens
• Modus Ponens is a deductively valid form of
reasoning from a conditional proposition.
• An overview video (3:53):
• The logic form of Modus Ponens
P Q
P
Q
• Individual task: Compose an example of Modus
Ponens by replacing “P” and “Q” with real
propositions.
Reasoning in the form of Modus Ponens
P Q: If it is heavily snowing,
then the ground is white.
P:
It is heavily snowing.
Q:
The ground is white.
P Q: If Jack applies for a job,
then he will be hired.
P:
Jack applies for a job.
Q:
Jack will be hired.
P Q
P
Q
Individual Tasks: Compose reasoning in P Q
the form of Modus Ponens from each of P
Q
the following conditional propositions.
a.
b.
c.
(2) Modus Tollens
• Modus Tollens is a deductively valid form of
reasoning from a conditional proposition.
• An overview video (1:49):
• The logic form of Modus Ponens
P Q
~Q
~P
• Individual task: Compose an example of Modus
Tollens by replacing “P” and “Q” with real
propositions.
Reasoning in the form of Modus Tollens
P Q: If it is heavily snowing,
then the ground is white.
~Q:
The ground is not white.
~P:
It is not heavily snowing.
P Q: If Jack applies for a job,
then he will be hired.
~Q:
Jack will not be hired.
~P:
Jack does not apply for a job.
Individual Tasks: Compose reasoning in
the form of Modus Tollens from each of
the following conditional propositions.
a.
b.
c.
(3) Affirming the consequent: A deductively
invalid form of conditional reasoning
• An overview video (3:27):
• Illustration of this invalid form
P Q If Bob stays home, Tim stays home.
Tim stays home.
Q
Bob stays home.
P
Correcting an invalid reasoning
• You can correct an invalid reasoning from a
conditional proposition by converting it into either
a Modus Ponens or a Modus Tollens.
If it is heavily snowing,
then the ground is white.
The ground is white.
It is heavily snowing.
If it is heavily snowing,
then the ground is white.
The ground is not white.
It is not heavily snowing.
If it is heavily snowing,
then the ground is white.
It is heavily snowing.
The ground is white.
Individual Tasks: Convert the following
instances of invalid reasoning into valid ones
A. If you want to be respected, you must respect others.
You must respect others. So, you want to be respected.
B. If you can’t live longer, then you should live deeper.
You should live deeper. Therefore, you can’t live longer.
C. If you are too lazy to plow, then you cannot expect a
harvest. You cannot expect a harvest. Therefore, you
are too lazy to plow.
(4) Denying the antecedent: A deductively
invalid form of conditional reasoning
• An overview video (3:35):
• Illustration of this invalid form
P Q
~P
~Q
If Bob stays home, Tim stays home.
Bob does not stay home.
Tim does not stay home.
Correcting an invalid reasoning
• To correct an invalid reasoning from a conditional
proposition is to convert it into either a Modus
Ponens or a Modus Tollens.
If it is raining, then the
ground is wet.
It is not raining.
The ground is not wet.
If it is raining, then the
ground is wet.
The ground is not wet.
It is not raining.
If it is raining, then the
ground is wet.
It is raining.
The ground is wet.
Individual Tasks: Convert the following
instances of invalid reasoning into valid ones
A. If you want to walk fast, you should walk alone. You
don’t want to walk fast. So, you shouldn’t walk alone.
B. If you want to walk far, you should walk with others.
You don’t want to walk far. Therefore, you shouldn’t
walk with others.
C. If everyone is moving forward together, then success
takes care of itself. Not everyone is moving forward
together. Therefore, success does not take care of itself.
(5) Pure Hypothetical Reasoning
• Also called “hypothetical syllogism”.
• It is the basic link for a reasoning chain.
• An overview video (3:49):
• The logic form of pure hypothetical reasoning
P Q
Q R
P R
Reasoning in the form of hypothetical syllogism
P Q: If a blizzard comes, classes are cancelled.
Q R: If classes are cancelled, I stay home.
P R: If a blizzard comes, I stay home.
Individual Tasks: Filling in the blanks to make a
pure hypothetical reasoning.
A. If you walk fast, then you w …
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