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Cengage Advantage Books: Understanding Arguments: An Introduction to Informal Logic

Walter Sinnott-Armstrong, Robert J. Fogelin

Chapter 6

Propositional Logic - all with Video Answers

Educators

Chapter Questions

01:55

Problem 1

The proposition "The night is young, and you're so beautiful" is a substitution instance of which of the following propositional forms?
1. $p$
5. $p \& q \& r$
2. $q$
6. $p \& p$
3. $p \& q$
7. $p$ or $q$
4. $p \& r$

Adriano Chikande
Adriano Chikande
Numerade Educator
01:55

Problem 2

Which of the following propositions is a substitution instance of " $p \& q \& q$ "?
1. The night is young, and you're so beautiful, and my flight leaves in thirty minutes.
2. The night is young, and you're so beautiful, and my flight leaves in thirty minutes, and my flight leaves in thirty minutes.
3. You're so beautiful, and you're so beautiful, and you're so beautiful.

Abhijith V
Abhijith V
Numerade Educator
01:00

Problem 3

For each of the following propositions, give three different propositional forms of which that proposition is a substitution instance.
1. The night is young, and you're so beautiful, and my flight leaves in thirty minutes.
2. The night is young, and you're so beautiful, and you're so beautiful.

JH
J Hardin
Numerade Educator

Problem 4

Indicate whether each of the following sentences expresses a propositional conjunction or a nonpropositional conjunction-that is, whether or not it expresses a conjunction of two propositions. If the sentence could be either, then specify a context in which it would naturally be used to express a propositional conjunction and a different context in which it would naturally be used to express a nonpropositional conjunction.
1. A Catholic priest married John and Mary.
2. Fred had pie and ice cream for dessert.
3. The winning presidential candidate rarely loses both New York and California.
4. Susan got married and had a child.
5. Jane speaks both French and English.
6. Someone who speaks both French and English is bilingual.
7. Ken and Naomi are two of my best friends.
8. Miranda and Nick cooked dinner.
9. I doubt that John is poor and happy.

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23:52

Problem 5

Are the following arguments valid by virtue of their propositional form? Why or why not?
1. Donald owns a tower in New York and a palace in Atlantic City. Therefore, Donald owns a palace in Atlantic City.
2. Tom owns a house. Therefore, Tom owns a house and a piece of land.
3. Ilsa is tall. Therefore, Ilsa is tall, and Ilsa is tall.
4. Bernie has a son and a daughter. Bernie has a father and a mother. Therefore, Bernie has a son and a mother.
5. Mary got married and had a child. Therefore, Mary had a child and got married.
6. Bess and Katie tied for MVP. Therefore, Bess tied for MVP.

Bernabe Montoya
Bernabe Montoya
Numerade Educator
00:36

Problem 6

For each of the following claims, determine whether it is true or false. Defend your answers.
1. An argument that is a substitution instance of a valid argument form is always valid.
2. An argument that is a substitution instance of an invalid argument form is always invalid.
3. An invalid argument is always a substitution instance of an invalid argument form.

Ian Shi
Ian Shi
Numerade Educator
01:14

Problem 7

Explain the differences in meaning among "Not everyone loves running," "Everyone does not love running," "Everyone loves not running," and "Everyone loves running-not!" For each, is it a negation of "Everyone loves running"? Why or why not?

Dale Sanford
Dale Sanford
Numerade Educator

Problem 8

Negative terms or prefixes can often be interpreted in more than one way. Explain two ways to interpret each of the following sentences. Describe a context in which it would be natural to interpret it in each way.
1. You may not go to the meeting.
2. I cannot recommend him too highly.
3. He never thought he'd go to the Himalayas.
4. Have you not done all of your homework?
5. All of his friends are not students.
6. I will not go to some football games next season.
7. No smoking section available.
8. The lock on his locker was unlockable.

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Problem 9

Put each of the following sentences in symbolic form. Be sure to specify exactly which sentence is represented by each capital letter, and pay special attention to the placement of the negation. If the sentence could be interpreted in more than one way, symbolize each interpretation and describe a context in which it would be natural to interpret it in each way.
1. It won't rain tomorrow.
2. It might not rain tomorrow.
3. There is no chance that it will rain tomorrow.
4. I believe that it won't rain tomorrow.
5. Joe is not too smart or else he's very clever.
6. Kristin is not smart or rich.
7. Sometimes you feel like a nut; sometimes you don't. (from an advertisement for Mounds and Almond Joy candies, which are made by the same company and are exactly alike except that only one of them has a nut)

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Problem 10

Explain why argument forms 1-2 are valid. Use common language that would be understandable to someone who has not read this chapter.

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Problem 11

Give other instances of argument forms 3-4 that are not valid. Explain why these instances are invalid and why they show that the general argument form is invalid.

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Problem 12

Give other instances of argument forms 3-4 that are not valid. Explain why these instances are invalid and why they show that the general argument form is invalid.
1. $\sim X \vee Y$
2. $\sim(X \vee Y)$
3. $\sim(Z \vee Z)$
4. $\sim(Z \vee \sim Z)$
5. $\sim \sim(A \vee B)$
6. $(A \vee Z) \& B$
7. $(A \vee X) \&(B \vee Z)$
8. $(A \& Z) \vee(B \& Z)$
9. $\sim(A \vee(Z \vee X))$
10. $\sim(A \vee \sim(Z \vee X))$
11. $\sim A \vee \sim(Z \vee X)$
12. $\sim Z \vee(Z \& A)$
13. $\sim(Z \vee(Z \& A))$
14. $\sim((Z \vee Z) \& A)$
15. $A \vee((\sim B \& C) \vee \sim(\sim B \vee \sim(Z \vee B)))$
16. $A \&((\sim B \& C) \mathrm{n} \sim(\sim B \vee \sim(Z \vee B)))$

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Problem 13

Using the truth table technique outlined above, show that argument forms 1-2 in the above section on process of elimination are valid and that argument forms $3-4$ in the same section are invalid.

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02:33

Problem 14

Is the following argument valid in our technical sense? Explain why or why not. Could it be sound? Explain why or why not.
(1) Frogs are green.
(2) Frogs are not green.
$\therefore$ (3) I am president. (from $1-2$ )

Adithya Ramanujam
Adithya Ramanujam
Numerade Educator

Problem 15

Using the truth table technique outlined above, test the following argument forms for validity:
$$
\begin{aligned}
& \text { 1. } \sim p \vee q \\
& \frac{p}{\therefore \sim q}
\end{aligned}
$$
$$
\begin{aligned}
& \therefore \sim(p \vee q) \\
& \therefore \sim q
\end{aligned}
$$
$\begin{aligned} & \text { 3. } \sim(p \vee q) \\ & \therefore q \\ & \text { 4. } \sim(p \vee q) \\ & \text { p } \\ & \therefore r \\ & \text { 5. } \sim(p \& q) \\ & \quad \therefore \\ & \therefore \sim p \\ & \text { 6. } \sim(p \& q) \\ & \sim q \\ & \therefore p\end{aligned}$
$$
\begin{aligned}
& 7 .(p \& q) \vee(p \& r) \\
& \therefore \overline{p \&(q \vee r)} \\
& 8 .(p \vee q) \&(p \vee r) \\
& \therefore p \&(q \vee r) \\
& \text { 9. } p \& q \\
& \therefore(p \vee r) \&(q \vee r) \\
& \text { 10. } p \vee q \\
& \therefore(p \& r) \vee(q \& r)
\end{aligned}
$$

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00:23

Problem 16

Construct a truth table analysis of the expression on the right side of the preceding definition, and compare it with the truth table definition of exclusive disjunction.

Amy Jiang
Amy Jiang
Numerade Educator

Problem 17

Use truth tables to test the following argument forms for validity:
1. $p$
$$
\therefore p \underline{p}
$$
$$
\begin{aligned}
& \text { 2. } p \vee q \\
& \qquad p \\
& \therefore \sim q \\
& \text { 3. } p \& q \\
& \therefore \sim(p \subseteq q)
\end{aligned}
$$
4. $\sim(p \& q)$
$$
\therefore p \vee q
$$
5. $p \vee q$
$$
\therefore p \vee q
$$
6. $p \vee q$

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03:12

Problem 18

Use truth tables to test which of the following propositional forms are truthfunctionally equivalent to each other:
1. $\sim(p \vee q)$
2. $\sim(\sim p \vee \sim q)$
3. $\sim p \& \sim q$
4. $p \& q$

Akash M
Akash M
Numerade Educator
06:11

Problem 19

Use truth tables to determine whether the expressions in each of the following pairs are truth-functionally equivalent:
1. "p" and "p \& p"
2. " $p$ " and " $p \vee p$ "
3. " $p \vee \sim p "$ " and " $\sim(p \& \sim p) "$
4. " $p$ " and " $p \&(q \vee \sim q)$ "
5. "p" and "p \& $(q \& \sim q)$ "
6. " $p$ " and " $p \vee(q \& \sim q)$ "
7. "p\& $(q \vee r)$ " and " $p \vee(q \& r)$ "
8. "p \& $(q \& r)$ " and " $(p \& q) \& r$ "
9. " $\sim(p \vee q)$ " and " $\sim p \vee q$ "
10. " $\sim(p \vee q)$ " and " $\sim p \& \sim q$ "
11. " $\sim \sim(p \vee q)$ " and " $\sim p \& \sim \sim q$ "
12. " $\sim(p \& q)$ " and " $\sim p \vee q$ "
13. " $\sim \sim(p \& q)$ " and " $\sim p \vee \sim \sim q$ "
14. " $\sim \sim p \sim \sim q$ " and " $\sim(\sim p \& \sim q)$ "
15. " $\sim \sim p \& \sim \sim q$ " and " $\sim(\sim p \vee \sim q)$ "
16. " $p \& \sim \sim q$ " and " $\sim p \& q^{\prime \prime}$

RO
Reynald Oliveria
Numerade Educator

Problem 20

The argument form called modus tollens looks like this:
$$
\begin{aligned}
& p \supset q \\
& \therefore \sim q
\end{aligned}
$$

Use truth tables to show that this argument form is valid.

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Problem 21

A second standard fallacy is called affirming the consequent. It looks like this:
$$
\begin{aligned}
& p \supset q \\
& q p
\end{aligned}
$$

Use truth tables to show that this argument form is invalid.

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02:41

Problem 22

In his radio address to the nation on April 17, 1982, President Ronald Reagan argued that the United States should not accept a treaty with the Soviet Union that would mutually freeze nuclear weapons at current levels, because he believed that the United States had fallen behind. Here is a central part of his argument:
It would be wonderful if we could restore the balance of power with the Soviet Union without increasing our military power. And, ideally, it would be a long step towards assuring peace if we could have significant and verifiable reductions of arms on both sides. But let's not fool ourselves. The Soviet Union will not come to any conference table bearing gifts. Soviet negotiators will not make unilateral concessions. To achieve parity, we must make it plain that we have the will to achieve parity by our own effort.

Put Reagan's central argument into standard form. Then symbolize it and its form. Does his argument commit any fallacy? If so, identify it.

Alejandro Ruiz
Alejandro Ruiz
Numerade Educator

Problem 23

Two more classic, common, and useful argument forms combine conditionals with disjunction. Using truth tables, test them for validity.
$$
\begin{array}{ll}
\text { Constructive Dilemma } & \text { Destructive Dilemma } \\
p \vee q & \sim p \vee \sim q \\
p \supset r & r \supset p \\
q \supset r & r \supset q \\
\hline \therefore r & \therefore \sim r
\end{array}
$$

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Problem 24

Using the truth table techniques employed above, test the following argument forms for validity. (For your own entertainment, guess whether the argument form is valid or invalid before working it out.)
$$
\begin{aligned}
& \text { 1. } p \supset q \\
& \therefore q \supset p
\end{aligned}
$$
$$
\begin{aligned}
& \text { 2. } p \supset q \\
& \therefore \sim q \supset \sim p \\
& \text { 3. } \sim q \supset \sim p \\
& \therefore p \supset q
\end{aligned}
$$
$$
\begin{aligned}
& \text { 3. } \sim q \supset \sim p \\
& \therefore p \supset q
\end{aligned}
$$
4.
$$
\begin{array}{rl}
\text { 4. } p & p q \\
q & \supset r \\
\therefore & p \supset(q \& r)
\end{array}
$$
5.
$$
\text { 5. } \begin{aligned}
& p \supset q \\
& q \supset r \\
& \sim r \\
\therefore & \sim p
\end{aligned}
$$
$$
\begin{aligned}
& \text { 6. } p \supset q \\
& \quad q \supset r \\
& \therefore \sim r \supset \sim p
\end{aligned}
$$
$$
\text { 7. } \begin{aligned}
p \vee q \\
p \supset q \\
q \supset r
\end{aligned} \quad \therefore r
$$
8. $p \supset(q \vee r)$
$$
\begin{aligned}
& \sim q \\
& \sim r \\
\therefore & \sim p
\end{aligned}
$$
$$
\text { 9. }(p \vee q) \supset r
$$
10
$$
\text { 10. }(p \& q) \supset r
$$
$$
\begin{aligned}
& \text { 11. } p \supset(q \supset r) \\
& \therefore(p \& q) \supset r \\
&
\end{aligned}
$$
12
12. $(p \& q) \supset r$
$$
\therefore p \supset(q \supset r)
$$
13.
$$
\begin{aligned}
& \text { 3. } p \supset(q \supset r) \\
& q \\
& \therefore \sim r \\
&
\end{aligned}
$$
14.
$$
\begin{aligned}
& \text { 14. } p \supset(q \supset r) \\
& \therefore \supset r \\
& \therefore r
\end{aligned}
$$
$\begin{aligned} & \text { 15. }(p \vee q) \&(p \vee r) \\ & \quad \sim r \\ & \therefore \sim q \\ & \text { 16. }(p \supset q) \&(p \supset \sim r) \\ & \quad q \& r \\ & \therefore \sim p \\ & \text { 17. }(p \vee q) \supset p \\ & \therefore \sim q\end{aligned}$
18. $(p \vee q) \supset(p \& q)$
$$
\therefore(p \supset q) \&(q \supset p)
$$
19. $(p \& q) \supset(p \cap q)$
$\therefore(p \supset q) \mathrm{n}(q \supset p)$
20. $r$
$$
\therefore \overline{(p \supset q) \vee(q \supset p)}
$$

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Problem 25

Check the validity of the argument forms above using truth tables.

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Problem 26

To appreciate the complexities of the little word "only," it is useful to notice that it fits at every point in the sentence "I hit him in the eye":
Only I hit him in the eye.
I only hit him in the eye.
I hit only him in the eye.
I hit him only in the eye.
I hit him in only the eye.
I hit him in the only eye.
I hit him in the eye only.

Explain what each of these sentences means.

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Problem 27

Translate each of the following sentences into symbolic notation, using the suggested symbols as abbreviations.
1. The Reds will win only if the Dodgers collapse. $(R, D)$
2. The Steelers will win if their defense holds up. $(S, D)$
3. If it rains or snows, the game will be called off. $(R, S, O)$
4. If she came home with a trophy and a prize, she must have won the tournament. ( $T, P, W)$
5. If you order the dinner special, you get dessert and coffee. ( $S, D, C)$
6. If you order the dinner special, you get dessert; but you can have coffee whether or not you order the dinner special. ( $S, D, C)$
7. If the house comes up for sale, and if I have the money in hand, I will bid on it. $(S, M, B)$
8. If you come to dinner, I will cook you a lobster, if you want me to. (D, L, W)
9. You can be a success if only you try. $(S, T)$
10. You can be a success only if you try. $(S, T)$
11. Only if you try can you be a success. $(S, T)$
12. You can be a success if you are the only one who tries. $(S, O)$
13. Unless there is a panic, stock prices will continue to rise. $(P, R)$
14. I won't scratch your back unless you scratch mine. ( $I, Y$ )
15. You will get a good bargain provided you get there early. $(B, E)$
16. You cannot lead a happy life without friends. (Let $H=$ You can lead a happy life, and let $F=$ You have friends.)
17. The only way that horse will win the race is if every other horse drops dead. (Let $W=$ That horse will win the race, and let $D=$ Every other horse drops dead.)
18. You should take prescription drugs if, but only if, they are prescribed for you. $(T, P)$
19. The grass will die without rain. ( $D, R=$ It rains.)
20. Given rain, the grass won't die. ( $R, D=$ The grass will die.)
21. Unless it doesn't rain, the grass won't die. ( $R, D=$ The grass will die.)

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Problem 28

(a) Translate each of the following arguments into symbolic notation. Then
(b) test each argument for truth-functional validity using truth table techniques, and (c) comment on any violations of conversational rules.

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