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Name: _____________________________________
PHIL 2303 – Introduction to Formal Logic
Professor Joshua Ellery, J.D., M.A.
Homework #1
Directions: Answer all questions to the best of your ability. Please write ALL of your answers on
this homework sheet. If you need additional space, please attach your extra page(s) to this
homework sheet. Please verify that your FULL name (first name and last name) is written on the
front page of this homework sheet and, if applicable, any extra pages you give to me.
Part I: True/False Questions – Indicate whether the statement is true or false.
______
1. According to the lecture, a premise is a statement from which an inference can be made.
______
2. All sound arguments are valid.
______
3. All valid arguments are sound.
______
______
______
4. If an argument has one false premise but all of the other premises are true, it is still a sound
argument because the majority of the premises are true.
5. An inductive argument is an argument whose conclusion is guaranteed to follow from the
premises.
6. Evaluative (or normative) language is language that passes a judgment against someone or
some position/claim.
______
7. All of these words are considered to be premise indicators: since, because, therefore.
______
8. All of these words are considered to be conclusion indicators: hence, thus, therefore.
Part II: Multiple Choice Questions – Choose the best answer.
______
9. Which of the following sentences are statements (meaning they are NOT requests,
commands, questions or inquiries)? (Chapter 1.1)
1.
2.
3.
4.
5.
Elephants are larger than human beings.
Would you like some more casserole or more dessert?
Some desserts are creamy, while others are not creamy.
Oh no! An elephant ate my dessert!
The President of the United States is not a Democrat.
A. All of the above sentences are statements.
B. Sentences #1, #2, and #3
C. Sentences #1, #3, and #5
D. Sentences #1, #3, #4, and #5
E. None of the above sentences are statements.
*** Page 1 of 3 ***
______
10. Consider the following argument: “Over the past twenty years, every crow that I have ever
observed has been black. Therefore, all crows are black.” This argument is an example of
a(n) ___________. (Chapter 1.8)
A. Deductive argument
B. Inductive argument
C. Socratic argument
D. Charitable argument
______
11. Consider the following argument: “Professor Ellery is either dashing or debonair. If he is
dashing, then he is suave. But if he is debonair, then he is also suave. Therefore, Professor
Ellery is suave.” This argument is an example of a(n) ___________. (Chapter 1.8)
A. Deductive argument
B. Inductive argument
C. Socratic argument
D. Charitable argument
______
12. Consider the following passage: “I believe that Professor Ellery is most likely the funniest
professor that I’ve ever had.”
Which rhetorical technique from Chapter 1.10 is being used here?
A. Assuring
B. Guarding
C. Discounting
D. None of the above.
______
13. Consider the following passage: “Four out of five law professors agree that taking a formal
logic class will help you perform better in law school.”
Which rhetorical technique from Chapter 1.10 is being used here?
A. Assuring
B. Guarding
C. Discounting
D. None of the above.
______
14. Consider the following passage: “Even though the Mayor is stealing money from the city,
he’s done a fantastic job revitalizing downtown and he built us a new football stadium.”
Which rhetorical technique from Chapter 1.10 is being used here?
A. Assuring
B. Guarding
C. Discounting
D. None of the above.
*** Page 2 of 3 ***
______
15. Consider the following passage: “Requiring homework assignments in my Logic class has
been met with wide-spread skepticism from my students. Nevertheless, my students
perform better on the exams because of the extra practice.”
Which rhetorical technique from Chapter 1.10 is being used here?
A. Assuring
B. Guarding
C. Discounting
D. None of the above
Part III: Additional Questions – Please show your answers/work in the space provided.
Using the informal test of validity, indicate whether or not the following arguments are valid. If the
argument is invalid, provide a counterexample. (Chapters 1.6 and 1.7)
16. Betty is a pastry chef. Therefore, Betty knows how to grill steaks.
17. All skunks are cute. All cute things are cuddly. Therefore, skunks are cuddly.
18. Shaquille O’Neal is larger than Kobe Bryant. Kobe Bryant is larger than Tony Parker.
Therefore, Shaquille O’Neal is larger than Tony Parker.
19. If I was born in Houston, then I hate Dallas and all of its citizens. I was born in Houston.
Therefore, I hate Dallas and all of its citizens.
20. Donald Trump is President of the United States. Therefore, Donald Trump was elected
President of the United States.
*** Page 3 of 3 ***
Name: _____________________________________
PHIL 2303 – Introduction to Formal Logic
Professor Joshua Ellery, J.D., M.A.
Homework #2
Directions: Answer all questions to the best of your ability. Please write ALL of your answers on
this homework sheet. If you need additional space, please attach your extra page(s) to this
homework sheet. Please verify that your FULL name (first name and last name) is written on the
front page of this homework sheet and, if applicable, any extra pages you give to me.
Part I: True/False Questions – Indicate whether the statement is true or false.
______
______
______
______
______
______
1. The statement “Professor Ellery is both FUNNY and HANDSOME.” is an example of an
atomic proposition.
2. The statement “Professor Ellery is both FUNNY and HANDSOME.” is an example of a
complex proposition.
3. A well-formed formula is a sentence in our symbolic language that has exactly one (1)
interpretation or meaning.
4. Consider this statement: “If I pass logic, then I will believe in the power of prayer.” The “If
I pass logic…” portion of the statement is known as the sufficient condition.
5. Consider this statement: “If I pass logic, then I will believe in the power of prayer.” The “If
I pass logic…” portion of the statement is known as the necessary condition.
6. The main connector in the following symbolized statement is the “tribar” (≡):
~[(~P v ~Q) ≡ (P ⊃ Q)]
Part II: Symbolization – Symbolize the following statements, using the abbreviations that have
been provided. Write your answer in the space provided.
7. I enjoy both MATH and LOGIC, but I do not enjoy SCIENCE.
(Note: M = I enjoy math; L = I enjoy logic; S = I enjoy science)
8. If I pass SCIENCE and LOGIC, then I will be HAPPY; however, the process will cost me
both MONEY and TIME.
9. School is FUN and THRILLING unless we start too EARLY, the class is BORING, and the
professor SUCKS.
*** Page 1 of 5 ***
10. I will not eat both CRAPPY food and BLAND food; however, I do enjoy SCOTCH or
VODKA.
11. Neither WIND nor RAIN will keep me from attending provided that SUZY does not attend.
(Note: W = It is windy; R = It is raining; S = Suzy does attend)
12. Exceeding the SPEED limit is a sufficient condition to get a TICKET, but if you HIT the
officer in the mouth, you’ll definitely go to JAIL.
13. Josh will take the TRIP only if he SAVES enough money and he catches neither a COLD
nor the FLU.
14. I will ACE logic if and only if I STUDY hard, complete the HOMEWORK, and pass the
EXAMS.
*** Page 2 of 5 ***
Part III: Truth Tables – Use the truth table test for validity to demonstrate whether the following
arguments are valid or invalid. Show your truth table and write your answer in the space provided.
Remember: If the argument is invalid, you must indicate which row(s) of the truth table make it so.
15. ~C • (A v B)
~A
∴ B • ~C
Valid or Invalid? _____________
16. (~C • B) ≡ (A v B)
(~A ≡ B) v (C ≡ B)
~(B v C)
∴ ~C ⊃ A
Valid or Invalid? _____________
*** Page 3 of 5 ***
Part IV: More Truth Tables – For each statement below, use a truth table to demonstrate whether
the statement is a tautology, contradiction, or contingent statement. Show your truth table and write
your answer in the space provided.
17. ~(~X ⊃ ~Y) ⊃ ~(X•Y)
_____________
18. (A ⊃ B) ≡ (A v B)
_____________
*** Page 4 of 5 ***
Part V: Even More Truth Tables – For each statement below, use a truth table to demonstrate
whether the two statements are materially equivalent or not. Show your truth table and write your
answer in the space provided.
19. (P ≡ Q) and (P ⊃ Q) • (Q ⊃ P)
_____________
20. (~P v ~Q) and ~(~P • ~Q)
_____________
*** Page 5 of 5 ***
Name: _____________________________________
PHIL 2303 – Introduction to Formal Logic
Professor Joshua Ellery, J.D., M.A.
Homework #3
Directions: Answer all questions to the best of your ability. Please write ALL of your answers on
this homework sheet. If you need additional space, please attach your extra page(s) to this
homework sheet. Please verify that your FULL name (first name and last name) is written on the
front page of this homework sheet and, if applicable, any extra pages you give to me.
Part I: True/False Questions – Indicate whether the statement is true or false.
______
______
______
______
1. The Simplification rule can be used on either conjunctive statements or disjunctive
statements.
2. The Modus Ponens rule allows you to derive the necessary condition of any conditional
statement provided that the sufficient condition is also present.
3. The Modus Tollens rule allows you to derive the necessary condition of any conditional
statement provided that the sufficient condition is also present.
4. The Hypothetical Syllogism rule and Disjunctive Syllogism rule are both rules that allow
you to manipulate or derive conditional statements.
______
5. The following is an example of the Association rule: (~P v ~Q) ∴ (~Q v ~P)
______
6. The following is an example of the Commutation rule: (~P v ~Q) ∴ (~Q v ~P)
______
7. The following is an example of the Contraposition rule: (~P v ~Q) ∴ ~(P • Q)
______
8. The following is an example of the DeMorgan’s rule: (~P v ~Q) ∴ ~(P • Q)
Part II: Multiple Choice Questions – Choose the best answer.
______
9. Consider the following argument:
(D • E) v (F ⊃ G)
~(F ⊃ G)
∴D•E
This is an example of what rule in action?
A. Hypothetical Syllogism
B. Disjunctive Syllogism
C. Modus Ponens
D. Modus Tollens
*** Page 1 of 5 ***
______
10. Consider the following argument:
(A v C) ⊃ (X • Y)
~(X • Y)
∴ ~(A v C)
This is an example of what rule in action?
A. Hypothetical Syllogism
B. Disjunctive Syllogism
C. Modus Ponens
D. Modus Tollens
______
11. Consider the following argument:
(A ≡ B) • (P ≡ Q)
∴A≡B
This is an example of what rule in action?
A. Biconditional Exchange
B. Conjunction
C. Constructive Dilemma
D. Simplification
______
12. Consider the following argument:
~B ⊃ D
∴ ~D ⊃ ~ ~B
This is an example of what rule in action?
A. Hypothetical Syllogism
B. Contraposition
C. Constructive Dilemma
D. Double Negation
______
13. Consider the following argument:
DvE
D⊃B
E⊃C
∴BvC
This is an example of what rule in action?
A. Conjunction
B. Contraposition
C. Constructive Dilemma
D. Simplification
______
14. Consider the following argument:
~(D v E)
∴ ~D • ~E
This is an example of what rule in action?
A. Double Negation
B. DeMorgan’s Rule
C. Disjunctive Syllogism
D. Addition
*** Page 2 of 5 ***
______
15. Consider the following argument:
~(D ⊃ A) v ~(E ⊃ B)
∴ ~[(D ⊃ A) • (E ⊃ B)]
This is an example of what rule in action?
A. Double Negation
B. DeMorgan’s Rule
C. Disjunctive Syllogism
D. Addition
Part III: Proofs – Construct proofs for the following valid arguments using the rules we learned in
class. Remember: For each rule you use in your proof, be sure to provide the correct name of the
rule AND line number(s) used.
#16
1.
2.
3.
4.
AvB
C⊃D
A⊃C
~D
/∴ B
1. (P v Q) ⊃ A
2. ~A
/∴ ~Q
#17
*** Page 3 of 5 ***
#18
1.
2.
3.
4.
~Y ⊃ ~(R v X)
(P v Q) ⊃ (R v S)
T • ~S
T⊃P
/∴ Y
#19
1.
2.
3.
4.
AvB
B ⊃ (C • D)
~C
B≡E
/∴ A • ~E
*** Page 4 of 5 ***
#20
1.
2.
3.
4.
(P v S) ⊃ (T v W)
~T ≡ ~(M • O)
~[W v (S v M)]
~A ⊃ P
/∴ A
*** Page 5 of 5 ***
Name: _____________________________________
PHIL 2303 – Introduction to Formal Logic
Professor Joshua Ellery, J.D., M.A.
Homework #4
Directions: Answer all questions to the best of your ability. Please write ALL of your answers on
this homework sheet. If you need additional space, please attach your extra page(s) to this
homework sheet. Please verify that your FULL name (first name and last name) is written on the
front page of this homework sheet and, if applicable, any extra pages you give to me.
Part I: True/False Questions – Indicate whether the statement is true or false.
______
______
______
______
______
1. Analogical arguments are a type of deductive argument whereby observed similarities
between two things are used to support the conclusion that some further similarity exists
between the two things.
2. A good (strong) statistical generalization is one that is taken from an unbiased group of
respondents that is both large and representative.
3. According to the class lecture, amateur gamblers and alcoholic gamblers are the two main
groups of people that fall victim to committing the “Gambler’s Fallacy.”
4. By becoming more educated or wealthy one is able to effectively insulate themselves from
committing any confirmation bias.
5. You can avoid committing the slippery slope fallacy as long as each and every step in your
chain of reasoning has a 95% or higher probability to occur.
Part II: Short Answer Questions – Read each question below and provide your answer in the
space provided.
6. Consider the following story: Jack is a die-hard Republican and wants to determine his
party’s chances in the upcoming election. Jack conducts a combination paper and
online/electronic survey in his affluent neighborhood asking all of his male neighbors
whether they plan on voting for Republicans or those “Dirty Democrats.” After collecting
32 responses, Jack concludes from his data that the Republicans will win in a landslide.
Using the standards of evaluation we discussed in class, briefly discuss the strengths and
weaknesses of Jack’s sampling methods.
*** Page 1 of 5 ***
7. Consider the following argument: “The first car I owned had four wheels, 2 seats, a
steering wheel and was painted black. It was a Subaru. It got really good gas mileage!
My new car also has 4 wheels, 2 seats, a steering wheel, and is painted green. It’s a
Chrysler. Since my new car is very similar to my old car, my new car will get really good
gas mileage too!”
Using the standards of evaluation we discussed in class, briefly discuss the weaknesses of
this analogical argument.
8. You draw three (3) cards in succession from a standard deck of fifty-two playing cards and
lay them on the table in front of you (NOT returning any of the drawn cards back into the
deck). Using our rules of calculating probabilities, calculate the odds that you first draw a
Queen of any suit, and then draw a black King, and then draw the Ace of spades. (Please
show your work.)
9. You draw three (3) cards in succession from a standard deck of fifty-two playing cards and
lay them on the table in front of you (NOT returning any of the drawn cards back into the
deck). Using our rules of calculating probabilities, calculate the odds that you draw three (3)
Queens in a row or that you draw the Ace, King, and Queen of spades. (Please show your
work.)
*** Page 2 of 5 ***
Part III: Multiple Choice Questions – Choose the best answer.
______
10. Consider the following exchange:
Teaching Assistant: “The homework assignment we gave the students was much harder
than we intended, so I think we should give out a few extra points to students who
completed it.”
Professor: “That’s a terrible idea. If we give every student a perfect score on the
homework for no reason, then students won’t bother to work hard in the future!”
Which fallacy was committed by the Professor?
A. Slippery slope
B. Strawman fallacy
C. Genetic fallacy
D. Post hoc, ergo propter hoc
______
11. Consider the following argument: “If I am in Calgary, then I am in Alberta. I am in
Alberta, therefore I am in Calgary.”
Which fallacy (if any) was committed above?
A. Affirming the consequent
B. Affirming the disjunct
C. Denying the antecedent
D. No logical fallacy was committed. This is a valid argument.
______
12. Consider the following argument: “If I am in Calgary, then I am in Alberta. I am in
Calgary, therefore I am in Alberta.”
Which fallacy (if any) was committed above?
A. Affirming the consequent
B. Affirming the disjunct
C. Denying the antecedent
D. No logical fallacy was committed. This is a valid argument.
______
13. Consider the following argument: “If I am in Calgary, then I am in Alberta. I am not in
Calgary, therefore I am not in Alberta.”
Which fallacy (if any) was committed above?
A. Affirming the consequent
B. Affirming the disjunct
C. Denying the antecedent
D. No logical fallacy was committed. This is a valid argument.
*** Page 3 of 5 ***
______
14. Consider the following argument: “If the company continues to cut job training
opportunities, it’ll significantly weaken our preparedness to compete in the global
workforce. And if we’re not prepared to take our rightful place in the world economy, all
of those jobs will eventually go overseas to people who have been properly trained. And
once that happens, the U.S. economy will stall and we’ll all be living on the streets!”
Which fallacy was committed above?
A. Slippery slope
B. Strawman fallacy
C. Genetic fallacy
D. Post hoc, ergo propter hoc
______
15. Consider the following exchange:
Professor: “So, I understand that you’re writing your term paper on global warming.
Tell me about your research thus far. Are you confident that you’re finding good
sources for your paper?”
Student: “Yes, I’m optimistic that I’m on the right path. I found a website called
www.global-warming-is-a-hoax.com that has TONS of great stuff that support my
paper’s central thesis.”
Which fallacy was committed by the Student?
A. Survivorship bias
B. Romantic bias
C. Optimism bias
D. Confirmation bias
______
16. Consider the following exchange:
Professor: “So, I understand that you’re taking my Hyper-Advanced Microbiology
class next semester. That class has a 90% failure rate. In ten years of offering the class,
I’ve only awarded one “A.” Are you sure you still want to enroll?”
Student: “Thanks for your concern, but I’ll survive. I’m a diligent student and always
get “A’s” in all my classes, so I’ll be just fine.”
Which fallacy was committed by the Student?
A. Survivorship bias
B. Romantic bias
C. Optimism bias
D. Confirmation bias
*** Page 4 of 5 ***
______
17. Consider the following argument: “Vegetarianism is an injurious and unhealthy practice.
Why? Because if everyone were vegetarians, then farmers and the U.S. economy at-large
would be seriously affected, and many people would be thrown out of work.”
Which fallacy was committed above?
A. Appeal to authority
B. Appeal to consequences
C. Ad hominem
D. Tu quoque
______
18. Consider the following argument: “My logic professor has been teaching for 15 years and
is super-smart! He says that the country’s existing tax policies are ill-advised because
they’ll undercut the labor force and eventually crash the economy. That’s why I’m voting
to change our existing tax policies.”
Which fallacy was committed above?
A. Appeal to authority
B. Appeal to consequences
C. Ad hominem
D. Tu quoque
______
19. Consider the following exchange:
Waiter: “The restaurant owner wants us to sell more of the daily specials. But, the
customers are complaining that the meal is made up of 10% fatty acids!”
Chef: “Okay, no problem. Make a sign offering a daily special that’s “90% fat-free”
and just watch those specials sell like crazy!”
Which fallacy was committed above by the Chef?
A. Survivorship bias
B. Optimism bias
C. Confirmation bias
D. Framing bias
______
20. Consider the following exchange:
Chef: “How’s the spaghetti special selling today? When I made the sauce, I used the
more expensive campari tomatoes instead of the normal roma tomatoes.”
Waiter: “It’s been selling great! We’ve already sold out!”
Chef: “Using those campari tomatoes must’ve caused the spaghetti special to sell so
well.”
Which fallacy (if any) was committed above by the Chef?
A. Genetic fallacy
B. Post hoc, ergo propter hoc
C. Conjunction fallacy
D. No logical fallacy was committed. This is a valid argument.
*** Page 5 of 5 ***
Introduction to Logic and
Critical Thinking
Version 1.4
Matthew J. Van Cleave
Lansing Community College
Introduction to Logic and Critical Thinking by Matthew J. Van Cleave is licensed under a
Creative Commons Attribution 4.0 International License. To view a copy of this license, visit
http://creativecommons.org/licenses/by/4.0/.
Table of contents
Preface
Chapter 1: Reconstructing and analyzing arguments
1.1 What is an argument?
1.2 Identifying arguments
1.3 Arguments vs. explanations
1.4 More complex argument structures
1.5 Using your own paraphrases of premises and conclusions to
reconstruct arguments in standard form
1.6 Validity
1.7 Soundness
1.8 Deductive vs. inductive arguments
1.9 Arguments with missing premises
1.10 Assuring, guarding, and discounting
1.11 Evaluative language
1.12 Evaluating a real-life argument
Chapter 2: Formal methods of evaluating arguments
2.1 What is a formal method of evaluation and why do we need them?
2.2 Propositional logic and the four basic truth functional connectives
2.3 Negation and disjunction
2.4 Using parentheses to translate complex sentences
2.5 “Not both” and “neither nor”
2.6 The truth table test of validity
2.7 Conditionals
2.8 “Unless”
2.9 Material equivalence
2.10 Tautologies, contradictions, and contingent statements
2.11 Proofs and the 8 valid forms of inference
2.12 How to construct proofs
2.13 Short review of propositional logic
2.14 Categorical logic
2.15 The Venn test of validity for immediate categorical inferences
2.16 Universal statements and existential commitment
2.17 Venn validity for categorical syllogisms
Chapter 3: Evaluating inductive arguments and probabilistic and statistical
fallacies
3.1 Inductive arguments and statistical generalizations
3.2 Inference to the best explanation and the seven explanatory virtues
3.3 Analogical arguments
3.4 Causal arguments
3.5 Probability
3.6 The conjunction fallacy
3.7 The base rate fallacy
3.8 The small numbers fallacy
3.9 Regression to the mean fallacy
3.10 Gambler’s fallacy
Chapter 4: Informal fallacies
4.1 Formal vs. informal fallacies
4.1.1 Composition fallacy
4.1.2 Division fallacy
4.1.3 Begging the question fallacy
4.1.4 False dichotomy
4.1.5 Equivocation
4.2 Slippery slope fallacies
4.2.1 Conceptual slippery slope
4.2.2 Causal slippery slope
4.3 Fallacies of relevance
4.3.1 Ad hominem
4.3.2 Straw man
4.3.3 Tu quoque
4.3.4 Genetic
4.3.5 Appeal to consequences
4.3.6 Appeal to authority
Answers to exercises
Glossary/Index
Preface
Preface
This is an introductory textbook in logic and critical thinking. The goal of the
textbook is to provide the reader with a set of tools and skills that will enable
them to identify and evaluate arguments. The book is intended for an
introductory course that covers both formal and informal logic. As such, it is not
a formal logic textbook, but is closer to what one would find marketed as a
“critical thinking textbook.” The formal logic in chapter 2 is intended to give an
elementary introduction to formal logic. Specifically, chapter 2 introduces
several different formal methods for determining whether an argument is valid
or invalid (truth tables, proofs, Venn diagrams). I contrast these formal methods
with the informal method of determining validity introduced in chapter 1. What
I take to be the central theoretical lesson with respect to the formal logic is
simply that of understanding the difference between formal and informal
methods of evaluating an argument’s validity. I believe there are also practical
benefits of learning the formal logic. First and foremost, once one has
internalized some of the valid forms of argument, it is easy to impose these
structures on arguments one encounters. The ability to do this can be of use in
evaluating an argumentative passage, especially when the argument concerns a
topic with which one is not very familiar (such as on the GRE or LSAT).
However, what I take to be of far more practical importance is the skill of being
able to reconstruct and evaluate arguments. This skill is addressed in chapter 1,
where the central ideas are that of using the principle of charity to put
arguments into standard form and of using the informal test of validity to
evaluate those arguments. Since the ability to reconstruct and evaluate
arguments is a skill, one must practice in order to acquire it. The exercises in
each section are intended to give students some practice, but in order to really
master the skill, one must practice much, much more than simply completing the
exercises in the text. It makes about as much sense to say that one could
become a critical thinker by reading a critical thinking textbook as that one
could become fluent in French by reading a French textbook. Logic and critical
thinking, like learning a foreign language, takes practice because it is a skill.
While chapters 1 and 2 mainly concern deductive arguments, chapter 3
addresses inductive arguments, including probabilistic and statistical fallacies. In
a world in which information is commonly couched within probabilistic and
statistical frameworks, understanding these basic concepts, as well as some of
the common mistakes is essential for understanding our world. I have tried to
i
Preface
write chapter 3 with an eye towards this understanding. As with all the chapters,
I have tried to walk the fine line between being succinct without sacrificing
depth.
Chapter 4 picks out what I take to be some of the most common fallacies, both
formal and informal. In my experience, many critical thinking textbooks end up
making the fallacies sound obvious; one is often left wondering how anyone
could commit such a fallacy. In my discussion of the fallacies I have tried to
correct this not only in the particular examples I use in the text and exercises,
but also by discussing what makes a particular fallacy seductive.
I have used numerous different textbooks over the years that I have been
teaching logic and critical thinking courses. Some of them were very good;
others were not. Although this textbook is my attempt to improve on what I’ve
encountered, I am indebted to a number of textbooks that have shaped how I
teach logic and critical thinking. In particular, Sinnott-Armstrong and Fogelin’s
Understanding Arguments: An Introduction to Informal Logic and Copi and
Cohen’s Introduction to Logic have influenced how I present the material here
(although this may not be obvious). My interest in better motivating the
seductiveness of the fallacies is influenced by Daniel Kahneman’s work in
psychology (for which he won the Nobel Prize in economics in 2002).
This textbook is an “open textbook” that is licensed under the Creative
Commons Attribution 4.0 license (CC BY 4.0). Anyone can take this work and
alter it for their own purposes as long as they give appropriate credit to me,
noting whether or not you have altered the text. (If you would like to alter the
text but have come across this textbook in PDF format, please do not hesitate to
email me at vancleam@lcc.edu and I will send you the text in a file format that is
easier to manipulate.)
At Lansing Community College, my place of
employment, we have undertaken an initiative to reduce the cost of textbooks. I
see this as an issue of access to education and even an issue of justice. Some
studies have shown that without access to the textbook, a student’s
performance in the class will suffer. Many students lack access to a textbook
simply because they do not buy it in the first place since there are more pressing
things to pay for (rent, food, child care, etc.) and because the cost of some
textbooks is prohibitive. Moreover, both professors and students are beholden
to publishers who profit from selling textbooks (professors, because the content
of the course is set by the author of the textbook, and perhaps market forces,
rather than by the professor herself; students, because they must buy the newest
ii
Preface
editions of increasingly expensive textbooks). If education is necessary for
securing certain basic human rights (as philosophers like Martha Nussbaum have
argued), then lack of access to education is itself an issue of justice. Providing
high quality, low-cost textbooks is one, small part of making higher education
more affordable and thus more equitable and just. This open textbook is a
contribution towards that end.
Matthew J. Van Cleave
January 4, 2016
iii
Chapter 1: Reconstructing and analyzing arguments
1.1 What is an argument?
This is an introductory textbook in logic and critical thinking. Both logic and
critical thinking centrally involve the analysis and assessment of arguments.
“Argument” is a word that has multiple distinct meanings, so it is important to
be clear from the start about the sense of the word that is relevant to the study
of logic. In one sense of the word, an argument is a heated exchange of
differing views as in the following:
Sally: Abortion is morally wrong and those who think otherwise are
seeking to justify murder!
Bob: Abortion is not morally wrong and those who think so are right-wing
bigots who are seeking to impose their narrow-minded views on all the
rest of us!
Sally and Bob are having an argument in this exchange. That is, they are each
expressing conflicting views in a heated manner. However, that is not the sense
of “argument” with which logic is concerned. Logic concerns a different sense
of the word “argument.” An argument, in this sense, is a reason for thinking
that a statement, claim or idea is true. For example:
Sally: Abortion is morally wrong because it is wrong to take the life of an
innocent human being, and a fetus is an innocent human being.
In this example Sally has given an argument against the moral permissibility of
abortion. That is, she has given us a reason for thinking that abortion is morally
wrong. The conclusion of the argument is the first four words, “abortion is
morally wrong.” But whereas in the first example Sally was simply asserting that
abortion is wrong (and then trying to put down those who support it), in this
example she is offering a reason for why abortion is wrong.
We can (and should) be more precise about our definition of an argument. But
before we can do that, we need to introduce some further terminology that we
will use in our definition. As I’ve already noted, the conclusion of Sally’s
argument is that abortion is morally wrong. But the reason for thinking the
conclusion is true is what we call the premise. So we have two parts of an
argument: the premise and the conclusion. Typically, a conclusion will be
supported by two or more premises. Both premises and conclusions are
statements. A statement is a type of sentence that can be true or false and
1
Chapter 1: Reconstructing and analyzing arguments
corresponds to the grammatical category of a “declarative sentence.”
example, the sentence,
For
The Nile is a river in northeastern Africa
is a statement. Why? Because it makes sense to inquire whether it is true or
false. (In this case, it happens to be true.) But a sentence is still a statement
even if it is false. For example, the sentence,
The Yangtze is a river in Japan
is still a statement; it is just a false statement (the Yangtze River is in China). In
contrast, none of the following sentences are statements:
Please help yourself to more casserole
Don’t tell your mother about the surprise
Do you like Vietnamese pho?
The reason that none of these sentences are statements is that it doesn’t make
sense to ask whether those sentences are true or false (rather, they are requests
or commands, and questions, respectively).
So, to reiterate: all arguments are composed of premises and conclusions, which
are both types of statements. The premises of the argument provide a reason
for thinking that the conclusion is true. And arguments typically involve more
than one premise. A standard way of capturing the structure of an argument is
by numbering the premises and conclusion. For example, recall Sally’s
argument against abortion:
Abortion is morally wrong because it is wrong to take the life of an
innocent human being, and a fetus is an innocent human being.
We could capture the structure of that argument like this:
1. It is morally wrong to take the life of an innocent human being
2. A fetus is an innocent human being
3. Therefore, abortion is morally wrong
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Chapter 1: Reconstructing and analyzing arguments
By convention, the last numbered statement (also denoted by the “therefore”) is
the conclusion and the earlier numbered statements are the premises. This is
what we call putting an argument into standard argument form. We can now
give a more precise definition of an argument. An argument is a set of
statements, some of which (the premises) attempt to provide a reason for
thinking that some other statement (the conclusion) is true. Although arguments
are typically given in order to convince or persuade someone of the conclusion,
the argument itself is independent of one’s attempt to use it to convince or
persuade. For example, I have just given you this argument not in an attempt to
convince you that abortion is morally wrong, but as an illustration of what an
argument is. Later on in this chapter and in this book we will learn some
techniques of evaluating arguments, but for now the goal is to learn to identify
an argument, including its premises and conclusion(s). It is important to be able
to identify arguments and understand their structure, whether or not you agree
with conclusion of the argument. In the next section I will provide some
techniques for being able to identify arguments.
Exercise 1: Which of the following sentences are statements and which are
not?
1. No one understands me but you.
2. Alligators are on average larger than crocodiles.
3. Is an alligator a reptile or a mammal?
4. An alligator is either a reptile or a mammal.
5. Don’t let any reptiles into the house.
6. You may kill any reptile you see in the house.
7. East Africans are not the best distance runners.
8. Obama is not a Democrat.
9. Some humans have wings.
10. Some things with wings cannot fly.
11. Was Obama born in Kenya or Hawaii?
12. Oh no! A grizzly bear!
13. Meet me in St. Louis.
14. We met in St. Louis yesterday.
15. I do not want to meet a grizzly bear in the wild.
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Chapter 1: Reconstructing and analyzing arguments
1.2 Identifying arguments
The best way to identify whether an argument is present is to ask whether there
is a statement that someone is trying to establish as true by basing it on some
other statement. If so, then there is an argument present. If not, then there
isn’t. Another thing that can help in identifying arguments is knowing certain
key words or phrases that are premise indicators or conclusion indicators. For
example, recall Sally’s abortion argument:
Abortion is morally wrong because it is wrong to take the life of an
innocent human being, and a fetus is an innocent human being.
The word “because” here is a premise indicator. That is, “because” indicates
that what follows is a reason for thinking that abortion is morally wrong. Here is
another example:
I know that the student plagiarized since I found the exact same
sentences on a website and the website was published more than a year
before the student wrote the paper.
In this example, the word “since” is a premise indicator because what follows it
is a statement that is clearly intended to be a reason for thinking that the
student plagiarized (i.e., a premise). Notice that in these two cases, the premise
indicators “because” and “since” are interchangeable: I could have used
“because” in place of “since” or “since” in the place of “because” and the
meaning of the sentences would have been the same. In addition to premise
indicators, there are also conclusion indicators. Conclusion indicators mark that
what follows is the conclusion of an argument. For example,
Bob-the-arsonist has been dead for a year, so Bob-the-arsonist didn’t set
the fire at the East Lansing Starbucks last week.
In this example, the word “so” is a conclusion indicator because what follows it
is a statement that someone is trying to establish as true (i.e., a conclusion).
Here is another example of a conclusion indicator:
A poll administered by Gallup (a respected polling company) showed
candidate x to be substantially behind candidate y with only a week left
before the vote, therefore candidate y will probably not win the election.
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Chapter 1: Reconstructing and analyzing arguments
In this example, the word “therefore” is a conclusion indicator because what
follows it is a statement that someone is trying to establish as true (i.e., a
conclusion). As before, in both of these cases the conclusion indicators “so”
and “therefore” are interchangeable: I could have used “so” in place of
“therefore” or “therefore” in the place of “so” and the meaning of the
sentences would have been the same.
Table 1 contains a list of some common premise and conclusion indicators:
Premise indicators
since
because
for
as
given that
seeing that
for the reason that
is shown by the fact that
Conclusion indicators
therefore
so
hence
thus
implies that
consequently
it follows that
we may conclude that
Although these words and phrases can be used to identify the premises and
conclusions of arguments, they are not failsafe methods of doing so. Just
because a sentence contains them does not mean that you are dealing with an
argument. This can easily be shown by examples like these:
I have been running competitively since 1999.
I am so happy to have finally finished that class.
Although “since” can function as a premise indicator and although “so” can
function as a conclusion indicator, neither one is doing so here. This shows that
you can’t simply mindlessly use occurrences of these words in sentences to show
that there is an argument being made. Rather, we have to rely on our
understanding of the English sentence in order to determine whether an
argument is being made or not. Thus, the best way to determine whether an
argument is present is by asking the question: Is there a statement that
someone is trying to establish as true or explain why it is true by basing it on
some other statement? If so, then there is an argument present. If not, then
there isn’t. Notice that if we apply this method to the above examples, we will
5
Chapter 1: Reconstructing and analyzing arguments
see that there is no argument present because there is no statement that
someone is trying to establish as true by basing it on some other statement. For
example, the sentence “I have been running competitively since 1999” just
contains one statement, not two. But arguments always require at least two
separate statements—one premise and one conclusion, so it cannot possibly be
an argument.
Another way of explaining why these occurrences of “so” and “since” do not
indicate that an argument is present is by noting that both premise indicators
and conclusion indicators are, grammatically, conjunctions. A grammatical
conjunction is a word that connects two separate statements. So, if a word or
term is truly being used as a premise or conclusion indicator, it must connect
two separate statements. Thus, if “since” were really functioning as a premise
indicator in the above example then what followed it would be a statement. But
“1999” is not a statement at all. Likewise, in the second example “so” is not
being used as a conclusion indicator because it is not conjoining two separate
statements. Rather, it is being used to modify the extent of “happy.” In
contrast, if I were to say “Tom was sleeping, so he couldn’t have answered the
phone,” then “so” is being used as a conclusion indicator. In this case, there
are clearly two separate statements (“Tom was sleeping” and “Tom couldn’t
have answered the phone”) and one is being used as the basis for thinking that
the other is true.
If there is any doubt about whether a word is truly a premise/conclusion
indicator or not, you can use the substitution test. Simply substitute another
word or phrase from the list of premise indicators or conclusion indicators and
see if the resulting sentence still makes sense. If it does, then you are probably
dealing with an argument. If it doesn’t, then you probably aren’t. For example,
we can substitute “it follows that” for “so” in the Bob-the-arsonist example:
Bob-the-arsonist has been dead for a year, it follows that Bob-the-arsonist
didn’t set the fire at the East Lansing Starbucks last week.
However, we cannot substitute “because” for “so” in the so-happy-I-finishedthat-class example:
I am because happy to have finally finished that class.
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Chapter 1: Reconstructing and analyzing arguments
Obviously, in the latter case the substitution of one conclusion indicator for
another makes the sentence meaningless, which means that the “so” that
occurred originally wasn’t functioning as a conclusion indicator.
Exercise 2: Which of the following are arguments?
identify the conclusion of the argument.
If it is an argument,
1. The woman in the hat is not a witch since witches have long noses and
she doesn’t have a long nose.
2. I have been wrangling cattle since before you were old enough to tie
your own shoes.
3. Albert is angry with me so he probably won’t be willing to help me wash
the dishes.
4. First I washed the dishes and then I dried them.
5. If the road wasn’t icy, the car wouldn’t have slid off the turn.
6. Albert isn’t a fireman and he isn’t a fisherman either.
7. Are you seeing that rhinoceros over there? It is huge!
8. The fact that obesity has become a problem in the U.S. is shown by the
fact that obesity rates have risen significantly over the past four decades.
9. Bob showed me a graph with the rising obesity rates and I was very
surprised to see how much they’ve risen.
10. Albert isn’t a fireman because Albert is a Greyhound, which is a kind of
dog, and dogs can’t be firemen.
11. Charlie and Violet are dogs and since dogs don’t sweat, it is obvious that
Charlie and Violet don’t sweat.
12. The reason I forgot to lock the door is that I was distracted by the clown
riding a unicycle down our street while singing Lynyrd Skynyrd’s “Simple
Man.”
13. What Bob told you is not the real reason that he missed his plane to
Denver.
14. Samsung stole some of Apple’s patents for their smartphones, so Apple
stole some of Samsung’s patents back in retaliation.
15. No one who has ever gotten frostbite while climbing K2 has survived to
tell about it, therefore no one ever will.
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Chapter 1: Reconstructing and analyzing arguments
1.3 Arguments vs. explanations
So far I have defined arguments in terms of premises and conclusions, where the
premises are supposed to provide a reason (support, evidence) for accepting
the conclusion. Many times the goal of giving an argument is simply to establish
that the conclusion is true. For example, when I am trying to convince someone
that obesity rates are rising in the U.S. I may cite evidence such as studies from
the Center for Disease Control (CDC) and the National Institute of Health (NIH).
The studies I cite would function as premises for the conclusion that obesity
rates are rising. For example:
We know that obesity is on the rise in the U.S. because multiple studies
carried out by the CDC and NIH have consistently shown a rise in obesity
over the last four decades.
We could put this simple argument into standard form like this:
1. Multiple studies by the CDC and NIH have consistently shown a rise in
obesity over the last four decades.
2. Therefore, obesity is on the rise in the U.S.
The standard form argument clearly distinguishes the premise from the
conclusion and shows how the conclusion is supposed to be supported by the
evidence offered in the premise. Again, the goal of this simple argument would
be to convince someone that the conclusion is true. However, sometimes we
already know that a statement or claim is true and we are trying to establish why
it is true rather than that it is true. An argument that attempts to show why its
conclusion is true is an explanation. Contrast the previous example with the
following:
The reason that the rate of obesity is on the rise in the U.S. is that the
foods we most often consume over the past four decades have
increasingly contained high levels of sugar and low levels of dietary fiber.
Since eating foods high in sugar and low in fiber triggers the insulin
system to start storing those calories as fat, it follows that people who
consume foods high in sugar and low in fiber will tend to store more of
the calories consumed as fat.
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Chapter 1: Reconstructing and analyzing arguments
This passage gives an explanation for why obesity is on the rise in the U.S.
Unlike the earlier example, here it is taken for granted that obesity is on the rise
in the U.S. That is the claim whose truth we are trying to explain. We can put
the obesity explanation into standard form just like any other argument. In
order to do this, I will make some paraphrases of the premises and conclusion of
the argument (for more on how to do this, see section 1.5 below).
1. Over the past four decades, Americans have increasingly consumed
foods high in sugar and low in fiber.
2. Consuming foods high in sugar and low in fat triggers the insulin
system to start storing those calories as fat.
3. When people store more calories as fat, they tend to become obese.
4. Therefore, the rate of obesity is on the rise in the U.S.
Notice that in this explanation the premises (1-3) attempt to give a reason for
why the conclusion is true, rather than a reason for thinking that the conclusion is
true. That is, in an explanation we assume that what we are trying to explain
(i.e., the conclusion) is true. In this case, the premises are supposed to show
why we should expect or predict that the conclusion is true. Explanations often
give us an understanding of why the conclusion is true. We can think of
explanations as a type of argument, we just have to distinguish two different
types of argument: those that attempt to establish that their conclusion is true
(arguments), and those that attempt to establish why their conclusion is true
(explanations).
Exercise 3: Which of the following is an explanation and which is an
argument? Identify the main conclusion of each argument or explanation.
(Remember if the premise(s) seems to be establishing that the conclusion
is true, it is an argument, but if the premise(s) seems to be establishing
why the conclusion is true, it is an explanation.)
1. Wanda rode the bus today because her car was in the shop.
2. Since Wanda doesn’t have enough money in her bank account, she
has not yet picked up her car from the shop.
3. Either Bob or Henry rode the bus to work today. But it wasn’t Henry
because I saw him riding his bike to work. Therefore, it was Bob.
4. It can’t be snowing right now since it only snows when it is 32 degrees
or below and right now it is 40 degrees.
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Chapter 1: Reconstructing and analyzing arguments
5. The reason some people with schizophrenia hear voices in their head
is that the cognitive mechanism that monitors their own self-talk is
malfunctioning and they attribute their own self-talk to some external
source.
6. Fracking should be allowed because, although it does involve some
environmental risk, it reduces our dependence on foreign oil and
there is much greater harm to the environment due to foreign oil
drilling than there is due to fracking.
7. Wanda could not have ridden the bus today because today is a citywide holiday and the bus service is not operating.
8. The Tigers lost their star pitcher due to injury over the weekend,
therefore the Tigers will not win their game against the Pirates.
9. No one living in Pompeii could have escaped before the lava from Mt.
Vesuvius hit. The reason is simple: the lava was flowing too fast and
there was nowhere to go to escape it in time.
10. The reason people’s allergies worsen when they move to Cincinnati is
that the pollen count in Cincinnati is higher than almost anywhere else
in the surrounding area.
1.4 More complex argument structures
So far we have seen that an argument consists of a premise (typically more than
one) and a conclusion. However, very often arguments and explanations have a
more complex structure than just a few premises that directly support the
conclusion. For example, consider the following argument:
No one living in Pompeii could have survived the eruption of Mt.
Vesuvius. The reason is simple: the lava was flowing too fast and there
was nowhere to go to escape it in time. Therefore, this account of the
eruption, which claims to have been written by an eyewitness living in
Pompeii, was not actually written by an eyewitness.
The main conclusion of this argument—the statement that depends on other
statements as evidence but doesn’t itself provide any evidence for any other
statement—is:
A. This account of the eruption of Mt. Vesuvius was not actually written
by an eyewitness.
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Chapter 1: Reconstructing and analyzing arguments
However, the argument’s structure is more complex than simply having a couple
of premises that provide evidence directly for the conclusion. Rather, some
statement provides evidence directly for the main conclusion, but that statement
itself is supported by another statement. To determine the structure of an
argument, we must determine which statements support which. We can use our
premise and conclusion indicators to help with this. For example, the passage
contains the phrase, “the reason is…” which is a premise indicator, and it also
contains the conclusion indicator, “therefore.” That conclusion indicator helps
us to identify the main conclusion, but the more important thing to see is that
statement A does not itself provide evidence or support for any of the other
statements in the argument, which is the clearest reason why statement A is the
main conclusion of the argument. The next question we must answer is: which
statement most directly supports A? What most directly supports A is:
B. No one living in Pompeii could have survived the eruption of Mt.
Vesuvius.
However, there is also a reason offered in support of B. That reason is that:
C. The lava from Mt. Vesuvius was flowing too fast and there was
nowhere for someone living in Pompeii to go in order to escape it in
time.
So the main conclusion (A) is directly supported by B, and B is supported by C.
Since B acts as a premise for the main conclusion but is also itself the conclusion
of further premises, we refer to B as an intermediate conclusion. The important
thing to recognize here is that one and the same statement can act as both a
premise and a conclusion. Statement B is a premise that supports the main
conclusion (A), but it is also itself a conclusion that follows from C. Here is how
we would put this complex argument into standard form (using numbers this
time, as we always do when putting an argument into standard form):
1. The lava from Mt. Vesuvius was flowing too fast and there was
nowhere for someone living in Pompeii to go in order to escape it in
time.
2. Therefore, no one living in Pompeii could have survived the eruption
of Mt. Vesuvius. (from 1)
11
Chapter 1: Reconstructing and analyzing arguments
3. Therefore, this account of the eruption of Mt. Vesuvius was not
actually written by an eyewitness. (from 2)
Notice that at the end of statement 2 I have written in parentheses “from 1”
(and likewise at the end of statement 3 I have written “from 2”). This is a
shorthand way of saying: “this statement follows from statement 1.” We will use
this convention as a way of keeping track of the structure of the argument. It
may also help to think about the structure of an argument spatially, as figure 1
shows:
The main argument here (from 2 to 3) contains a subargument, in this case the
argument from 1 to 2. A subargument, as the term suggests, is a part of an
argument that provides indirect support for the main argument. The main
argument is simply the argument whose conclusion is the main conclusion.
Another type of structure that arguments can have is when two or more
premises provide direct but independent support for the conclusion. Here is an
example of an argument with that structure:
I know that Wanda rode her bike to work today because when she arrived
at work she had her right pant leg rolled up (which cyclists do in order to
keep their pants legs from getting caught in the chain). Moreover, our
coworker, Bob, who works in accounting, saw her riding towards work at
7:45 am.
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Chapter 1: Reconstructing and analyzing arguments
The conclusion of this argument is “Wanda rode her bike to work today” and
there are two premises that provide independent support for it: the fact that
Wanda had her pant leg cuffed and the fact that Bob saw her riding her bike.
Here is the argument in standard form:
1. Wanda arrived at work with her right pant leg rolled up.
2. Cyclists often roll up their right pant leg.
3. Bob saw Wanda riding her bike towards work at 7:45.
4. Therefore, Wanda rode her bike to work today. (from 1-2, 3
independently)
Again, notice that next to statement 4 of the argument I have written the
premises from which that conclusion follows. In this case, in order to avoid any
ambiguity, I have noted that the support for the conclusion comes
independently from statements 1 and 2, on the one hand, and from statement 3,
on the other hand. It is important to point out that an argument or subargument
can be supported by one or more premises. We see this in the present
argument since the conclusion (4) is supported jointly by 1 and 2, and singly by
3. As before, we can represent the structure of this argument spatially, as figure
2 shows:
There are endless different argument structures that can be generated from
these few simple patterns. At this point, it is important to understand that
arguments can have these different structures and that some arguments will be
longer and more complex than others. Determining the structure of very
complex arguments is a skill that takes some time to master. Even so, it may
help to remember that any argument structure ultimately traces back to some
combination of these.
13
Chapter 1: Reconstructing and analyzing arguments
Exercise 4: Write the following arguments in standard form and show how
the argument is structured using a diagram like the ones I have used in
this section.
1. There is nothing wrong with prostitution because there is nothing
wrong with consensual sexual and economic interactions between
adults. Moreover, since there’s no difference between a man who
goes on a blind date with a woman, buys her dinner and then has sex
with her and a man who simply pays a woman for sex, that is another
reason for why there is nothing wrong with prostitution.
2. Prostitution is wrong because it involves women who have typically
been sexually abused as children. We know that most of these
women have been abused from multiple surveys done with women
who have worked in prostitution and that show a high percentage of
self-reported sexual abuse as children.
3. There was someone in this cabin recently because there was warm
water in the tea kettle and because there was wood still smoldering in
the fireplace. But the person couldn’t have been Tim because Tim has
been with me the whole time. Therefore, there must be someone else
in these woods.
4. It is possible to be blind and yet run in the Olympic Games since
Marla Runyan did it at the 2000 Sydney Olympics.
5. The train was late because it had to take a longer, alternate route
since the bridge was out.
6. Israel is not safe if Iran gets nuclear missiles since Iran has threatened
multiple times to destroy Israel and if Iran had nuclear missiles it would
be able to carry out this threat. Moreover, since Iran has been
developing enriched uranium, they have the key component needed
for nuclear weapons—every other part of the process of building a
nuclear weapon is simple compared to that. Therefore, Israel is not
safe.
7. Since all professional hockey players are missing front teeth and
Martin is a professional hockey player, it follows that Martin is missing
front teeth. And since almost all professional athletes who are missing
their front teeth have false teeth, it follows that Martin probably has
false teeth.
8. Anyone who eats the crab rangoon at China Food restaurant will
probably have stomach troubles afterward. It has happened to me
every time, which is why it will probably happen to other people as
14
Chapter 1: Reconstructing and analyzing arguments
well. Since Bob ate the crab rangoon at China Food restaurant, he will
probably have stomach troubles afterward.
9. Albert and Caroline like to go for runs in the afternoon in Hyde Park.
Since Albert never runs alone, we know that any time Albert is
running, Caroline is running too. But since Albert looks like he has just
run (since he is panting hard), it follows that Caroline must have ran
too.
10. Just because Jeremy’s prints were on the gun that killed Tim and the
gun was registered to Jeremy, it doesn’t follow that Jeremy killed Tim
since Jeremy’s prints would certainly be on his own gun and someone
else could have stolen Jeremy’s gun and used it to kill Tim.
1.5
Using your own paraphrases of premises and conclusions to
reconstruct arguments in standard form
Although sometimes we can just lift the premises and conclusion verbatim from
the argument, we cannot always do this. Paraphrases of premises or conclusions
are sometimes needed in order to make the standard form argument as clear as
possible. A paraphrase is the use of different words to capture the same idea in
a clearer way. There will always be multiple ways of paraphrasing premises and
conclusions and this means that there will never be just one way of putting an
argument into standard form. In order to paraphrase well, you will have to rely
on your understanding of English to come up with what you think is the best way
of capturing the essence of the argument. Again, typically there is no single
right way to do this, although there are certainly better and worse ways of doing
it. For example, consider the following argument:
Just because Jeremy’s prints were on the gun that killed Tim and the gun
was registered to Jeremy, it doesn’t follow that Jeremy killed Tim since
Jeremy’s prints would certainly be on his own gun and someone else
could have stolen Jeremy’s gun and used it to kill Tim.
What is the conclusion of this argument? (Think about it before reading on.)
Here is one way of paraphrasing the conclusion:
The fact that Jeremy’s prints were on the gun that killed Tim and the gun
was registered to Jeremy doesn’t mean that Jeremy killed Tim.
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Chapter 1: Reconstructing and analyzing arguments
This statement seems to capture the essence of the main conclusion in the
above argument. The premises of the argument would be:
1. Jeremy’s prints would be expected to be on a gun that was registered
to him
2. Someone could have stolen Jeremy’s gun and then used it to kill Tim
Notice that while I have paraphrased the first premise, I have left the second
premise almost exactly as it appeared in the original paragraph. As I’ve said,
paraphrases are needed in order to try to make the standard form argument as
clear as possible and this is what I’ve tried to do in capturing premise 1 as well
as the conclusion of this argument. So here is the reconstructed argument in
standard form:
1. Jeremy’s prints would be expected to be on a gun that was registered
to him
2. Someone could have stolen Jeremy’s gun and then used it to kill Tim
3. Therefore, the fact that Jeremy’s prints were on the gun that killed Tim
and the gun was registered to Jeremy doesn’t mean that Jeremy killed
Tim. (from 1-2)
However, as I have just noted, there is more than one way of paraphrasing the
premises and conclusion of the argument. To illustrate this, I will give a second
way that one could accurately capture this argument in standard form. Here is
another way of expressing the conclusion:
We do not know that Jeremy killed Tim.
That is clearly what the above argument is trying to ultimately establish and it is
a much simpler (in some ways) conclusion than my first way of paraphrasing the
conclusion. However, it also takes more liberties in interpreting the argument
than my original paraphrase. For example, in the original argument there is no
occurrence of the word “know.” That is something that I am introducing in my
own paraphrase. That is a totally legitimate thing to do, as long as introducing
new terminology helps us to clearly express the essence of the premise or
conclusion that we’re trying to paraphrase.1 Since my second paraphrase of the
How do we know that a paraphrase is accurate? Unfortunately, there is no simple way to
answer this question. The only answer is that you must rely on your mastery and understanding
of English in order to determine for yourself whether the paraphrase is a good one or not. This
1
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conclusion differs from my first paraphrase, you can expect that my premises will
differ also. So how shall I paraphrase the premises that support this conclusion?
Here is another way of paraphrasing the premises and putting the argument into
standard form:
1. Tim was killed by a gun that was registered to Jeremy and had
Jeremy’s prints on it.
2. It is possible that Jeremy’s gun was stolen from him.
3. If Jeremy’s gun was stolen from him, then Jeremy could not have
killed Tim.
4. Therefore, we do not know that Jeremy killed Tim. (from 1-3)
Notice that this standard form argument has more premises than my first
reconstruction of the standard form argument (which consisted of only three
statements). I have taken quite a few liberties in interpreting and paraphrasing
this argument, but what I have tried to do is to get down to the most essential
logic of the original argument. The paraphrases of the premises I have used are
quite different from the wording that occurs in the original paragraph. I have
introduced phrases such as “it is possible that” as well as conditional
statements (if…then statements), such as premise 3. Nonetheless, this
reconstruction seems to get at the essence of the logic of the original argument.
As long as your paraphrases help you to do that, they are good paraphrases.
Being able to reconstruct arguments like this takes many years of practice in
order to do it well, and much of the material that we will learn later in the text
will help you to better understand how to capture an argument in standard form,
but for now it is important to recognize that there is never only one way of
correctly capturing the standard form of an argument. And the reason for this is
that there are multiple, equally good, ways of paraphrasing the premises and
conclusion of an argument.
1.6. Validity
So far we have discussed what arguments are and how to determine their
structure, including how to reconstruct arguments in standard form. But we
have not yet discussed what makes an argument good or bad. The central
concept that you will learn in logic is the concept of validity. Validity relates to
is one of those kinds of skills that is difficult to teach, apart from just improving one’s mastery of
the English language.
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how well the premises support the conclusion, and it is the golden standard that
every argument should aim for. A valid argument is an argument whose
conclusion cannot possibly be false, assuming that the premises are true.
Another way of putting this is as a conditional statement: A valid argument is an
argument in which if the premises are true, the conclusion must be true. Here is
an example of a valid argument:
1. Violet is a dog
2. Therefore, Violet is a mammal (from 1)
You might wonder whether it is true that Violet is a dog (maybe she’s a lizard or
a buffalo—we have no way of knowing from the information given). But, for the
purposes of validity, it doesn’t matter whether premise 1 is actually true or false.
All that matters for validity is whether the conclusion follows from the premise.
And we can see that the conclusion, Violet is a mammal, does seem to follow
from the premise, Violet is a dog. That is, given the truth of the premise, the
conclusion has to be true. This argument is clearly valid since if we assume that
“Violet is a dog” is true, then, since all dogs are mammals, it follows that “Violet
is a mammal” must also be true. As we’ve just seen, whether or not an
argument is valid has nothing to do with whether the premises of the argument
are actually true or not. We can illustrate this with another example, where the
premises are clearly false:
1. Everyone born in France can speak French
2. Barack Obama was born in France
3. Therefore, Barak Obama can speak French (from 1-2)
This is a valid argument. Why? Because when we assume the truth of the
premises (everyone born in France can speak French, Barack Obama was born in
France) the conclusion (Barack Obama can speak French) must be true. Notice
that this is so even though none of these statements is actually true. Not
everyone born in France can speak French (think about people who were born
there but then moved somewhere else where they didn’t speak French and
never learned it) and Obama was not born in France, but it is also false that
Obama can speak French. So we have a valid argument even though neither
the premises nor the conclusion is actually true. That may sound strange, but if
you understand the concept of validity, it is not strange at all. Remember:
validity describes the relationship between the premises and conclusion, and it
means that the premises imply the conclusion, whether or not that conclusion is
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true. In order to better understand the concept of validity, let’s look at an
example of an invalid argument:
1. George was President of the United States
2. Therefore, George was elected President of the United States (from 1)
This argument is invalid because it is possible for the premise to be true and yet
the conclusion false. Here is a counterexample to the argument. Gerald Ford
was President of the United States but he was never elected president, since
Ford Replaced Richard Nixon when Nixon resigned in the wake of the
Watergate scandal.2 So it doesn’t follow that just because someone is President
of the United States that they were elected President of the United States. In
other words, it is possible for the premise of the argument to be true and yet the
conclusion false. And this means that the argument is invalid. If an argument is
invalid it will always be possible to construct a counterexample to show that it is
invalid (as I have done with the Gerald Ford scenario). A counterexample is
simply a description of a scenario in which the premises of the argument are all
true while the conclusion of the argument is false.
In order to determine whether an argument is valid or invalid we can use what
I’ll call the informal test of validity. To apply the informal test of validity ask
yourself whether you can imagine a world in which all the premises are true and
yet the conclusion is false. If you can imagine such a world, then the argument
is invalid. If you cannot imagine such a world, then the argument is valid.
Notice: it is possible to imagine a world where the premises are true even if the
premises aren’t, as a matter of actual fact, true. This is why it doesn’t matter for
validity whether the premises (or conclusion) of the argument are actually true.
It will help to better understand the concept of validity by applying the informal
test of validity to some sample arguments.
1. Joan jumped out of an airplane without a parachute
2. Therefore, Joan fell to her death (from 1)
To apply the informal test of validity we have to ask whether it is possible to
imagine a scenario in which the premise is true and yet the conclusion is false (if
so, the argument is invalid). So, can we imagine a world in which someone
As it happens, Ford wasn’t elected Vice President either since he was confirmed by the Senate,
under the twenty fifth amendment, after Spiro Agnew resigned. So Ford wasn’t ever elected by
the Electoral College—as either Vice President or President.
2
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jumped out of an airplane without a parachute and yet did not fall to her death?
(Think about it carefully before reading on.) As we will see, applying the
informal test of validity takes some creativity, but it seems clearly possible that
Joan could jump out of an airplane without a parachute and not die—she could
be perfectly fine, in fact. All we have to imagine is that the airplane was not
operating and in fact was on the ground when Joan jumped out of it. If that
were the case, it would be a) true that Joan jumped out of an airplane without a
parachute and yet b) false that Joan fell to her death. Thus, since it is possible
to imagine a scenario in which the premise is true and yet the conclusion is false,
the argument is invalid. Let’s slightly change the argument, this time making it
clear that the plane is flying:
1. Joan jumped out of an airplane travelling 300 mph at a height of
10,000 ft without a parachute
2. Joan fell to her death (from 1)
Is this argument valid? You might think so since you might think that anyone
who did such a thing would surely die. But is it possible to not die in the
scenario described by the premise? If you think about it, you’ll realize that there
are lots of ways someone could survive. For example, maybe someone else who
was wearing a parachute jumped out of the plane after them, caught them and
attached the parachute-less person to them, and then pulled the ripcord and
they both landed on the ground safe and sound. Or maybe Joan was
performing a stunt and landed in a giant net that had been set up for that
purpose. Or maybe she was just one of those people who, although they did
fall to the ground, happened to survive (it has happened before). All of these
scenarios are consistent with the information in the first premise being true and
also consistent with the conclusion being false. Thus, again, any of these
counterexamples show that this argument is invalid. Notice that it is also
possible that the scenario described in the premises ends with Joan falling to
her death. But that doesn’t matter because all we want to know is whether it is
possible that she doesn’t. And if it is possible, what we have shown is that the
conclusion does not logically follow from the premise alone. That is, the
conclusion doesn’t have to be true, even if we grant that the premise is. And
that means that the argument is not valid (i.e., it is invalid).
Let’s switch examples and consider a different argument.
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Chapter 1: Reconstructing and analyzing arguments
1. A person can be President of the United States only if they were born
in the United States.
2. Obama is President of the United States.
3. Kenya is not in the United States.
4. Therefore, Obama was not born in Kenya (from 1-3)
In order to apply the informal test of validity, we have to ask whether we can
imagine a scenario in which the premises are both true and yet the conclusion is
false. So, we have to imagine a scenario in which premises 1, 2, and 3 are true
and yet the conclusion (“Obama was not born in Kenya”) is false. Can you
imagine such a scenario? You cannot. The reason is that if you are imagining
that it is a) true that a person can be President of the United States only if they
were born in the United States, b) true that Obama is president and c) true that
Kenya is not in the U.S., then it must be true that Obama was not born in Kenya.
Thus we know that on the assumption of the truth of the premises, the
conclusion must be true. And that means the argument is valid. In this
example, however, premises 1, 2, and 3 are not only assumed to be true but are
actually true. However, as we have already seen, the validity of an argument
does not depend on its premises actually being true. Here is another example
of a valid argument to illustrate that point.
1. A person can be President of the United States only if they were born
in Kenya
2. Obama is President of the United States
3. Therefore, Obama was born in Kenya (from 1-2)
Clearly, the first premise of this argument is false. But if we were to imagine a
scenario in which it is true and in which premise 2 is also true, then the
conclusion (“Obama was born in Kenya”) must be true. And this means that the
argument is valid. We cannot imagine a scenario in which the premises of the
argument are true and yet the conclusion is false. The important point to
recognize here—a point I’ve been trying to reiterate throughout this section—is
that the validity of the argument does not depend on whether or not the
premises (or conclusion) are actually true. Rather, validity depends only on the
logical relationship between the premises and the conclusion. The actual truth
of the premises is, of course, important to the quality of the argument, since if
the premises of the argument are false, then the argument doesn’t provide any
reason for accepting the conclusion. In the next section we will address this
topic.
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Exercise 5: Determine whether or not the following arguments are valid
by using the informal test of validity. If the argument is invalid, provide a
counterexample.
1. Katie is a human being.
Therefore, Katie is smarter than a
chimpanzee.
2. Bob is a fireman. Therefore, Bob has put out fires.
3. Gerald is a mathematics professor. Therefore, Gerald knows how to
teach mathematics.
4. Monica is a French teacher. Therefore, Monica knows how to teach
French.
5. Bob is taller than Susan. Susan is taller than Frankie. Therefore, Bob
is taller than Frankie.
6. Craig loves Linda. Linda loves Monique. Therefore, Craig loves
Monique.
7. Orel Hershizer is a Christian. Therefore, Orel Hershizer communicates
with God.
8. All Muslims pray to Allah. Muhammad is a Muslim. Therefore,
Muhammad prays to Allah.
9. Some protozoa are predators. No protozoa are animals. Therefore,
some predators are not animals.
10. Charlie only barks when he hears a burglar outside. Charlie is barking.
Therefore, there must be a burglar outside.
1.7 Soundness
A good argument is not only valid, but also sound. Soundness is defined in
terms of validity, so since we have already defined validity, we can now rely on it
to define soundness. A sound argument is a valid argument that has all true
premises. That means that the conclusion of a sound argument will always be
true. Why? Because if an argument is valid, the premises transmit truth to the
conclusion on the assumption of the truth of the premises. But if the premises
are actually true, as they are in a sound argument, then since all sound
arguments are valid, we know that the conclusion of a sound argument is true.
Compare the last two Obama examples from the previous section. While the
first argument was sound, the second argument was not sound, although it was
valid. The relationship between soundness and validity is easy to specify: all
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sound arguments are valid arguments, but not all valid arguments are sound
arguments.
Although soundness is what any argument should aim for, we will not be talking
much about soundness in this book. The reason for this is that the only
difference between a valid argument and a sound argument is that a sound
argument has all true premises. But how do we determine whether the premises
of an argument are actually true? Well, there are lots of ways to do that,
including using Google to look up an answer, studying the relevant subjects in
school, consulting experts on the relevant topics, and so on. But none of these
activities have anything to do with logic, per se. The relevant disciplines to
consult if you want to know whether a particular statement is true is almost never
logic! For example, logic has nothing to say regarding whether or not protozoa
are animals or whether there are predators that aren’t in the animal kingdom. In
order to learn whether those statements are true, we’d have to consult biology,
not logic. Since this is a logic textbook, however, it is best to leave the question
of what is empirically true or false to the relevant disciplines that study those
topics. And that is why the issue of soundness, while crucial for any good
argument, is outside the purview of logic.
1.8 Deductive vs. Inductive arguments
The concepts of validity and soundness that we have introduced apply only to
the class of what are called “deductive arguments”. A deductive argument is
an argument whose conclusion is supposed to follow from its premises with
absolute certainty, thus leaving no possibility that the conclusion doesn’t follow
from the premises. For a deductive argument to fail to do this is for it to fail as a
deductive argument. In contrast, an inductive argument is an argument whose
conclusion is supposed to follow from its premises with a high level of
probability, which means that although it is possible that the conclusion doesn’t
follow from its premises, it is unlikely that this is the case. Here is an example of
an inductive argument:
Tweets is a healthy, normally functioning bird and since most healthy,
normally functioning birds fly, Tweets probably flies.
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Notice that the conclusion, Tweets probably flies, contains the word “probably.”
This is a clear indicator that the argument is supposed to be inductive, not
deductive. Here is the argument in standard form:
1. Tweets is a healthy, normally functioning bird
2. Most healthy, normally functioning birds fly
3. Therefore, Tweets probably flies
Given the information provided by the premises, the conclusion does seem to
be well supported. That is, the premises do give us a strong reason for
accepting the conclusion. This is true even though we can imagine a scenario in
which the premises are true and yet the conclusion is false. For example,
suppose that we added the following premise:
Tweets is 6 ft tall and can run 30 mph.
Were we to add that premise, the conclusion would no longer be supported by
the premises, since any bird that is 6 ft tall and can run 30 mph, is not a kind of
bird that can fly. That information leads us to believe that Tweets is an ostrich or
emu, which are not kinds of birds that can fly. As this example shows, inductive
arguments are defeasible arguments since by adding further information or
premises to the argument, we can overturn (defeat) the verdict that the
conclusion is well-supported by the premises. Inductive arguments whose
premises give us a strong, even if defeasible, reason for accepting the
conclusion are called, unsurprisingly, strong inductive arguments. In contrast,
an inductive argument that does not provide a strong reason for accepting the
conclusion are called weak inductive arguments.
Whereas strong inductive arguments are defeasible, valid deductive arguments
aren’t. Suppose that instead of saying that most birds fly, premise 2 said that all
birds fly.
1. Tweets is a healthy, normally function bird.
2. All healthy, normally functioning birds can fly.
3. Therefore, Tweets can fly.
This is a valid argument and since it is a valid argument, there are no further
premises that we could add that could overturn the argument’s validity. (True,
premise 2 is false, but as we’ve seen that is irrelevant to determining whether an
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argument is valid.) Even if we were to add the premise that Tweets is 6 ft tall
and can run 30 mph, it doesn’t overturn the validity of the argument. As soon as
we use the universal generalization, “all healthy, normally function birds can
fly,” then when we assume that premise is true and add that Tweets is a healthy,
normally functioning bird, it has to follow from those premises that Tweets can
fly. This is true even if we add that Tweets is 6 ft tall because then what we have
to imagine (in applying our informal test of validity) is a world in which all birds,
including those that are 6 ft tall and can run 30 mph, can fly.
Although inductive arguments are an important class of argument that are
commonly used every day in many contexts, logic texts tend not to spend as
much time with them since we have no agreed upon standard of evaluating
them. In contrast, there is an agreed upon standard of evaluation of deductive
arguments. We have already seen what that is; it is the concept of validity. In
chapter 2 we will learn some precise, formal methods of evaluating deductive
arguments. There are no such agreed upon formal methods of evaluation for
inductive arguments. This is an area of ongoing research in philosophy. In
chapter 3 we will revisit inductive arguments and consider some ways to
evaluate inductive arguments.
1.9 Arguments with missing premises
Quite often, an argument will not explicitly state a premise that we can see is
needed in order for the argument to be valid. In such a case, we can supply the
premise(s) needed in order so make the argument valid. Making missing
premises explicit is a central part of reconstructing arguments in standard form.
We have already dealt in part with this in the section on paraphrasing, but now
that we have introduced the concept of validity, we have a useful tool for
knowing when to supply missing premises in our reconstruction of an argument.
In some cases, the missing premise will be fairly obvious, as in the following:
Gary is a convicted sex-offender, so Gary is not allowed to work with
children.
The premise and conclusion of this argument are straightforward:
1. Gary is a convicted sex-offender
2. Therefore, Gary is not allowed to work with children (from 1)
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Chapter 1: Reconstructing and analyzing arguments
However, as stated, the argument is invalid. (Before reading on, see if you can
provide a counterexample for this argument. That is, come up with an imaginary
scenario in which the premise is true and yet the conclusion is false.) Here is just
one counterexample (there could be many): Gary is a convicted sex-offender but
the country in which he lives does not restrict convicted sex-offenders from
working with children. I don’t know whether there are any such countries,
although I suspect there are (and it doesn’t matter for the purpose of validity
whether there are or aren’t). In any case, it seems clear that this argument is
relying upon a premise that isn’t explicitly stated. We can and should state that
premise explicitly in our reconstruction of the standard form argument. But
what is the argument’s missing premise? The obvious one is that no sexoffenders are allowed to work with children, but we could also use a more
carefully statement like this one:
Where Gary lives, no convicted sex-offenders are allowed to work with
children.
It should be obvious why this is a more “careful” statement. It is more careful
because it is not so universal in scope, which means that it is easier for the
statement to be made true. By relativizing the statement that sex-offenders are
not allowed to work with children to the place where Gary lives, we leave open
the possibility that other places in the world don’t have this same restriction. So
even if there are other places in the world where convicted sex-offenders are
allowed to work with children, our statements could still be true since in this
place (the place where Gary lives) they aren’t. (For more on strong and weak
statements, see section 1.10). So here is the argument in standard form:
1. Gary is a convicted sex-offender.
2. Where Gary lives, no convicted sex-offenders are allowed to work with
children.
3. Therefore, Gary is not allowed to work with children. (from 1-2)
This argument is now valid: there is no way for the conclusion to be false,
assuming the truth of the premises. This was a fairly simple example where the
missing premise needed to make the argument valid was relatively easy to see.
As we can see from this example, a missing premise is a premise that the
argument needs in order to be as strong as possible. Typically, this means
supplying the statement(s) that are needed to make the argument valid. But in
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addition to making the argument valid, we want to make the argument
plausible. This is called “the principle of charity.” The principle of charity
states that when reconstructing an argument, you should try to make that
argument (whether inductive or deductive) as strong as possible.
When it
comes to supplying missing premises, this means supplying the most plausible
premises needed in order to make the argument either valid (for deductive
arguments) or inductively strong (for inductive arguments).
Although in the last example figuring out the missing premise was relatively easy
to do, it is not always so easy. Here is an argument whose missing premises are
not as easy to determine:
Since children who are raised by gay couples often have psychological
and emotional problems, the state should discourage gay couples from
raising children.
The conclusion of this argument, that the state should not allow gay marriage, is
apparently supported by a single premise, which should be recognizable from
the occurrence of the premise indicator, “since.” Thus, our initial reconstruction
of the standard form argument looks like this:
1. Children who are raised by gay couples often have psychological and
emotional problems.
2. Therefore, the state should discourage gay couples from raising
children.
However, as it stands, this argument is invalid because it depends on certain
missing premises. The conclusion of this argument is a normative statement—
a statement about whether something ought to be true, relative to some
standard of evaluation.
Normative statements can be contrasted with
descriptive statements, which are simply factual claims about what is true. For
example, “Russia does not allow gay couples to raise children” is a descriptive
statement. That is, it is simply a claim about what is in fact the case in Russia
today. In contrast, “Russia should not allow gay couples to raise children” is a
normative statement since it is not a claim about what is true, but what ought to
be true, relative to some standard of evaluation (for example, a moral or legal
standard). An important idea within philosophy, which is often traced back to
the Scottish philosopher David Hume (1711-1776), is that statements about what
ought to be the case (i.e., normative statements) can never be derived from
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statements about what is the case (i.e., descriptive statements). This is known
within philosophy as the is-ought gap. The problem with the above argument
is that it attempts to infer a normative statement from a purely descriptive
statement, violating the is-ought gap. We can see the problem by constructing
a counterexample. Suppose that in society x it is true that children raised by gay
couples have psychological problems. However, suppose that in that society
people do not accept that the state should do what it can to decrease harm to
children. In this case, the conclusion, that the state should discourage gay
couples from raising children, does not follow. Thus, we can see that the
argument depends on a missing or assumed premise that is not explicitly stated.
That missing premise must be a normative statement, in order that we can infer
the conclusion, which is also a normative statement. There is an important
general lesson here: Many times an argument with a normative conclusion will
depend on a normative premise which is not explicitly stated. The missing
normative premise of this particular argument seems to be something like this:
The state should always do what it can to decrease harm to children.
Notice that this is a normative statement, which is indicated by the use of the
word “should.” There are many other words that can be used to capture
normative statements such as: good, bad, and ought. Thus, we can reconstruct
the argument, filling in the missing normative premise like this:
1. Children who are raised by gay couples often have psychological and
emotional problems.
2. The state should always do what it can to decrease harm to children.
3. Therefore, the state should discourage gay couples from raising
children. (from 1-2)
However, although the argument is now in better shape, it is still invalid because
it is still possible for the premises to be true and yet the conclusion false. In
order to show this, we just have to imagine a scenario in which both the
premises are true and yet the conclusion is false. Here is one counterexample to
the argument (there are many). Suppose that while it is true that children of gay
couples often have psychological and emotional problems, the rate of
psychological problems in children raised by gay couples is actually lower than
in children raised by heterosexual couples. In this case, even if it were true that
the state should always do what it can to decrease harm to children, it does not
follow that the state should discourage gay couples from raising children. In
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Chapter 1: Reconstructing and analyzing arguments
fact, in the scenario I’ve described, just the opposite would seem to follow: the
state should discourage heterosexual couples from raising children.
But even if we suppose that the rate of psychological problems in children of
gay couples is higher than in children of heterosexual couples, the conclusion
still doesn’t seem to follow. For example, it could be that the reason that
children of gay couples have higher rates of psychological problems is that in a
society that is not yet accepting of gay couples, children of gay couples will face
more teasing, bullying and general lack of acceptance than children of
heterosexual couples. If this were true, then the harm to these children isn’t so
much due to the fact that their parents are gay as it is to the fact that their
community does not accept them. In that case, the state should not necessarily
discourage gay couples from raising children. Here is an analogy: At one point
in our country’s history (if not still today) it is plausible that the children of black
Americans suffered more psychologically and emotionally than the children of
white Americans. But for the government to discourage black Americans from
raising children would have been unjust, since it is likely that if there was a
higher incidence of psychological and emotional problems in black Americans,
then it was due to unjust and unequal conditions, not to the black parents, per
se. So, to return to our example, the state should only discourage gay couples
from raising children if they know that the higher incidence of psychological
problems in children of gay couples isn’t the result of any kind of injustice, but is
due to the simple fact that the parents are gay.
Thus, one way of making the argument (at least closer to) valid would be to add
the following two missing premises:
A. The rate of psychological problems in children of gay couples is
higher than in children of heterosexual couples.
B. The higher incidence of psychological problems in children of gay
couples is not due to any kind of injustice in society, but to the fact
that the parents are gay.
So the reconstructed standard form argument would look like this:
1. Children who are raised by gay couples often have psychological and
emotional problems.
2. The rate of psychological problems in children of gay couples is
higher than in children of heterosexual couples.
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Chapter 1: Reconstructing and analyzing arguments
3. The higher incidence of psychological problems in children of gay
couples is not due to any kind of injustice in society, but to the fact
that the parents are gay.
4. The state should always do what it can to decrease harm to children.
5. Therefore, the state should discourage gay couples from raising
children. (from 1-4)
In this argument, premises 2-4 are the missing or assumed premises. Their
addition makes the argument much stronger, but making them explicit enables
us to clearly see what assumptions the argument relies on in order for the
argument to be valid. This is useful since we can now clearly see which premises
of the argument we may challenge as false. Arguably, premise 4 is false, since
the state shouldn’t always do what it can to decrease harm to children. Rather,
it should only do so as long as such an action didn’t violate other rights that the
state has to protect or create larger harms elsewhere.
The important lesson from this example is that supplying the missing premises
of an argument is not always a simple matter. In the example above, I have
used the principle of charity to supply missing premises. Mastering this skill is
truly an art (rather than a science) since there is never just one correct way of
doing it (cf. section 1.5) and because it requires a lot of skilled practice.
Exercise 6: Supply the missing premise or premises needed in order to
make the following arguments valid. Try to make the premises as
plausible as possible while making the argument valid (which is to apply
the principle of charity).
1. Ed rides horses. Therefore, Ed is a cowboy.
2. Tom was driving over the speed limit. Therefore, Tom was doing
something wrong.
3. If it is raining then the ground is wet. Therefore, the ground must be
wet.
4. All elves drink Guinness, which is why Olaf drinks Guinness.
5. Mark didn’t invite me to homecoming. Instead, he invited his friend
Alexia. So he must like Alexia more than me.
6. The watch must be broken because every time I have looked at it, the
hands have been in the same place.
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Chapter 1: Reconstructing and analyzing arguments
7. Olaf drank too much Guinness and fell out of his second story
apartment window. Therefore, drinking too much Guinness caused
Olaf to injure himself.
8. Mark jumped into the air. Therefore, Mark landed back on the
ground.
9. In 2009 in the United States, the net worth of the median white
household was $113,149 a year, whereas the net worth of the median
black household was $5,677. Therefore, as of 2009, the United States
was still a racist nation.
10. The temperature of the water is 212 degrees Fahrenheit. Therefore,
the water is boiling.
11. Capital punishment sometimes takes innocent lives, such as the lives
of individuals who were later found to be not guilty. Therefore, we
should not allow capital punishment.
12. Allowing immigrants to migrate to the U.S. will take working class jobs
away from working class folks. Therefore, we should not allow
immigrants to migrate to the U.S.
13. Prostitution is a fair economic exchange between two consenting
adults. Therefore, prostitution should be allowed.
14. Colleges are more interested in making money off of their football
athletes than in educating them. Therefore, college football ought to
be banned.
15. Edward received an F in college Algebra. Therefore, Edward should
have studied more.
1.10 Assuring, guarding and discounting
As we have seen, arguments often have complex structures including
subarguments (recall that a subargument is an argument for one of the premises
of the main argument). But in practice people do not always give further
reasons or argument in support of every statement they make. Sometimes they
use certain rhetorical devices to cut the argument short, or to hint at a further
argument without actually stating it. There are three common strategies for
doing this:
Assuring: informing someone that there are further reasons although one
is not giving them now
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Chapter 1: Reconstructing and analyzing arguments
Guarding: weakening one’s claims so that it is harder to show that the
claims are false
Discounting: anticipating objections that might be raised to one’s claim
or argument as a way of dismissing those objections.3
We will discuss these in order, starting with assuring. Why would we want to
assure our audience? Presumably when we make a claim that isn’t obvious and
that the audience may not be inclined to believe. For example, if I am trying to
convince you that the United States is one of the leading producers of CO2
emissions, then I might cite certain authorities such as the Intergovernmental
Panel on Climate Change (IPCC) as saying so. This is one way of assuring our
audience: by citing authorities. There are many ways to cite authorities, some
examples of which are these:
Dentists agree that…
Recent studies have shown…
It has been established that…
Another way of assuring is to comment on the strength of one’s own
convictions. The rhetorical effect is that by commenting on how sure you are
that something is true, you imply, without saying, that there must be very strong
reasons for what you believe—assuming that the audience believes you are a
reasonable person, of course. Here are some ways of commenting on the
strength of one’s beliefs:
I’m certain that…
I’m sure that…
I can assure you that…
Over the years, I have become convinced that…
3
This characterization and discussion draws heavily on chapter 3, pp. 48-53 of SinnottArmstrong and Fogelin’s Understanding Arguments, 9th edition (Cengage Learning).
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Chapter 1: Reconstructing and analyzing arguments
I would bet a million dollars that…
Yet another way of assuring one’s audience is to make an audience member feel
that it would be stupid, odd, or strange to deny the claim one is making. One
common way to do this is by implying that every sensible person would agree
with the claim. Here are some examples:
Everyone with any sense agrees that…
Of course, no one will deny that…
There is no question that…
No one with any sense would deny that…
Another common way of doing this is by implying that no sensible person would
agree with a claim that we are trying to establish as false:
It is no longer held that…
No intelligent person would ever maintain that…
You would have to live under a rock to think that…
Assurances are not necessarily illegitimate, since the person may be right and
may in fact have good arguments to back up the claims, but the assurances are
not themselves arguments and a critical thinker will always regard them as
somewhat suspect. This is especially so when the claim …
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