Logic Exercises in Gensler`s Textbook

Answer Key to Logic Exercises in Gensler’s Textbook
Exercise 2.1a
1. no e is f
This is not a wff, since both letters after “no” have to be capitals.
=> no E is F
2. g is H
This is a wff.
3. J is K
This is not a wff, since wffs beginning with a letter have to begin with a small letter.
=> j is K
4. all M is not Q
This is not a wff, since the form in 4 is not among the eight forms allowed in our language.
=> all M is Q
5. some L is m
This is not a wff, since both letters after “some” have to be capitals.
=> some L is M
6. p is not Q
This is a wff.
7. R is not S
This is not a wff, since wffs beginning with a letter have to begin with a small letter.
=> r is not S
8. not all T is U
This is not a wff, since the form in 8 is not among the eight forms allowed in our language.
=> all T is U, some T is not U
9. some X is not Y
This is a wff.
Exercise 2.1b
1. This is a sentence. これは文だ。
t is S
2. This isn’t the first sentence. これは最初の文ではない。
t is not f
3. No logical positivist believes in God. どんな論理実証主義者も神の存在を信じない。
(No logical positivist is a believer in God.)
no L is B
4. The book on your desk is green. あなたの机の上のその本は緑色だ。
b is G
5. All dogs hate cats. すべての犬は猫を嫌う。
(All dogs are cat haters.)
all D is H
6. Kant is the greatest philosopher. カントは最も偉大な哲学者だ。
k is g
7. Ralph was born in Detroit. ラルフはデトロイトで生まれた。
(Ralph is a person born in Detroit.)
r is D
8. Detroit is the birthplace of Ralph. デトロイトはラルフの生誕地だ。
d is b
9. Alaska is a state. アラスカは州だ。
a is S
10. Alaska is the biggest state. アラスカは最大の州だ。
a is s
11. Carol is my only sister. キャロルは私の唯一の姉妹だ。
c is s
12. Carol lives in Big Pine Key. キャロルはビッグパインキーに住んでいる。
(Carol is a person living in Big Pine Key.)
c is L
13. The idea of goodness is itself good. 善という概念はそれ自体良いものだ。
g is G
14. All Michigan players are intelligent. すべてのミシガンの選手は頭がいい。
all M is I
1
15. Michigan’s team is awesome. ミシガンのチームはすごい。
m is A
16. Donna is Ralph’s wife. ドンナはラルフの妻だ。
d is w
p.12
Exercise 2.2a
1. all C is D
∴ some C is not E
This is not a syllogism. Each formula is a wff, but the letter D and the letter E do not occur twice.
2. g is not l
∴ l is not g
This is a syllogism. Each formula is a wff, each letter occurs twice, and the letters form a chain.
3. no Y is E
all G is Y
∴ no Y is E
This is not a wff. Each formula is a wff, but the letter Y occurs three times and the letter G occur only once.
4. ∴ all S is S
This is a syllogism. Each formula is a wff, each letter occurs twice, and each letter forms a chain.
5. k is not L
all M is L
some N is M
Z is N
∴ k is not Z
This is not a syllogism. The formula “Z is N” is not a wff, since if there is a letter in the sentence initial position, it has to be a lower
case letter.
Exercise 2.2b
1. w is not s
2. some C is B
3. no R is S
4. a is C
5.
6.
7.
8.
all P is B
r is not D
s is w
some C is not P
Exercise 2.2c
1. no P* is B*
some C is not B*
∴ some C* is P*
Invalid: B is starred twice and there are three stars on the right-hand side.
2. x is W
x is not Y*
∴ some W* is not Y Valid: W and Y are starred exactly once and there is one star on the right-hand side.
The captain of the ship is a woman.
The captain of the ship does not sleep well.
∴ Some women do not sleep well.
3. no H* is B*
no H* is D*
∴ some B* is not D Invalid: B and H are starred twice and there are two stars on the right-hand side.
4. some J is not P*
all J* is F
∴ some F* is not P Valid:
Some journalists do not like politics.
All journalists feel tired.
∴ Some people who feel tired do not like politics.
5. g is not s*
∴ s* is not g
Valid:
The man in green is not the smallest person in the room.
∴ The smallest person in the room is not the man in green.
6. all L* is M
g is L
∴ g* is M*
Valid:
2
All leftists are mean.
George is a leftist.
∴ George is mean.
7. all L* is M
g is not L*
∴ g* is not M
Invalid: L is starred twice.
Every living creature shows metabolism.
This toy gorilla is not a living creature.
∴ This toy gorilla does not show metabolism.
8. some N is T
some C is not T*
∴ some N* is not C
Invalid: C is not starred.
Some novelists like tea.
Some cartoonists does not like tea.
∴ Some novelists are not cartoonists.
9. all C* is K
s is K
∴ s* is C*
Invalid: C is starred twice.
All Chinese know about Confucius.
His younger son knows about Confucius.
∴ His younger son is Chinese.
10. all D* is A
∴ all A is D*
Invalid: D is starred twice, and A is not starred.
All dragons fly in the air.
∴ All creatures that fly in the air are dragons.
11. s is C
s is H
∴ some C* is H*
Valid
Superman criticizes the President of the U.S.
Superman is a hero.
∴ Some people who criticize the President of the U.S. are heroes.
12. some C is H
∴ some C* is not H Invalid: There is no star on the right-hand side.
Some children are happy.
∴some children are not happy.
13. a is b
b is c
c is d
∴ a* is d*
Valid
Adam is Brian.
Brian is Chris.
Chris is David.
∴Adam is David.
14. no A* is B*
some B is C
some D is not C*
all D* is E
∴ some E* is A*
Invalid
No Americans believe in God.
Some believers in God are cruel.
Some Dutch people are not cruel.
All Dutch people eat cheese.
∴Some people who eat cheese are Americans.
Exercise 2.3a
1. All segregation laws degrade human personality.
All laws that degrade human personality are unjust.
∴ All segregation laws are unjust. [From Dr Martin Luther King.
すべての人種隔離法は人格を貶めるものである。
人格を貶めるすべての法律は不当なものである。
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∴すべての人種隔離法は不当なものである。
all D* is U
∴all S is U*
all S* is D
2. All Communists favor the poor.
All Democrats favor the poor.
∴All Democrats are Communists.
[This reasoning could persuage if expressed emotionally in a political speech. It’s less apt to persuade if put into a clear
premise-conclusion form.]
すべての共産主義者は貧乏人の味方をする。
All C* is F
すべての民主党員は貧乏人の味方をする。
All D* is F
∴All D is C*
Invalid
∴すべての民主党員は共産主義者である。
3. All too-much-time penalties are called before play starts.
No penalty called before play starts can be refused.
∴No too-much-time penalty can be refused.
すべての「時間超過」の罰則は試合が始まる前に宣言される。
all T* is C
試合が始まる前に宣言されたいかなる罰則も拒否できない。
no C* is R*
∴no T is R Valid
∴いかなる「時間超過」の罰則も拒否できない。
4. No one under 18 is permitted to vote.
No faculty member is under 18.
The philosophy chairperson is a faculty member.
∴The philosophy chairperson is permitted to vote.
[Applying a law (like ones about voting) requires logical reasoning. Lawyers and judges need to be logical.]
いかなる１８才未満も投票することが許されていない。
no U* is P*
いかなる大学教員も１８才未満ではない。
no F* is U*
哲学部の学部長は大学教員である。
c is F
∴c* is P* Invalid
∴哲学部の学部長は投票することが許されている。
5. All acts that maximize good consequences are right.
Some punishing of the innocent maximizes good consequences.
∴Some punishing of the innocent is right.
[This argumet and the next give a mini-debate on utilitarianism, which holds that all acts that maximize the total of good
consequences for everyone are right. Moral philosophy would try to evaluate the premises; logic just focuses on whehter the
conclusion follows.]
良い結果を最大限にもたらす行為はすべて正しい。
all M* is R
無実の人間を罰する行為のあるものは良い結果を最大限にもたらす。
some P is M
∴some P* is R*
Valid
∴無実の人間を罰する行為のあるものは正しい。
6. No punishing of the innocent is right.
Some punishing of the innocent maximizes good consequneces.
∴Some acts that maximize good consequences aren’t right.
無実の人間を罰するいかなる行為も正しくない。
no P* is R*
無実の人間を罰する行為のあるものは良い結果を最大限にもたらす。
some P is M
∴some M* is not R Valid
∴良い結果を最大限にもたらす行為のあるものは正しくない。
7. All huevos revueltos are buenos para el desayuno.
All café con leche is bueno para el desayuno.
∴All café con leche is huevos revueltos.
[To test whether this argument is valid, you don't have to understand its meaning; you only have to grasp the form. In doing
formal (deductive) logic, you don't have to know what you're talking about; you only have to know the logical form of what
you're talking about.]
all H* is B
all C* is B
∴all C is H* Invalid
This means “all scrambled eggs are good for breakfast, all coffee with milk is good for breakfast, therefore all coffee with milk is
scrambled eggs.”
8. The belief that there's a God is unnecessary to explain our experience.
All beliefs unnecessary to explain our experience ought to be rejected.
:. The belief that there's a God ought to be rejected.
[St Thomas Aquinas mentioned this argument in order to dispute the first premise.]
神が存在するという信念は我々の経験を説明するのに必要ない。
我々の経験を説明するのに必要ないすべての信念は否定されるべきだ。
∴神が存在するという信念は否定されるべきだ。
b is U
all U* is O
4
∴b* is O*
Valid
The belief in God gives practical life benefits (courage, peace, zeal, love, . . .).
All beliefs that give practical life benefits are pragmatically justifiable.
∴ The belief in God is pragmatically justifiable.
[From William James, an American pragmatist philosopher.]
神の存在を信じることは人生における役に立つ利益（勇気・平
る。
安・熱情・愛など）を与えてくれる。
b is G
人生における役に立つ利益を与えてくれるすべての信念は実
all G* is P
用主義の観点から正当化できる。
∴b* is P*
Valid
∴神の存在を信じることは実用主義の観点から正当化でき
10. All sodium salt gives a yellow flame when put into the flame of a Bunsen burner.
This material gives a yellow flame when put into the flame of a Bunsen burner.
∴ This material is sodium salt.
∴この物質はナトリウム塩である。
すべてのナトリウム塩はブンゼンバーナーの炎に入れると黄
色の炎を出す。
all S* is G
この物質はブンゼンバーナーの炎に入れると黄色の炎を出
m is G
す。
∴m* is S* Invalid
11.
All abortions kill innocent human life.
No killing of innocent human life is right.
∴ No abortions are right.
すべての妊娠中絶は罪のない人間の命を絶つ行為だ。
all A* is K
罪のない人間の命を絶つどのような行為も正しくない。
no K* is R*
∴どのような妊娠中絶も正しくない。
∴no A is R Valid
12. All acts that maximize good consequences are right.
All socially useful abortions maximize good consequences.
∴ All socially useful abortions are right.
良い結果を最大化するような行為はすべて正しい。
all M* is R
社会的に有益な中絶はすべて良い結果を最大化する。
all A* is M
∴社会的に有益な中絶はすべて正しい。
∴all A is R* Valid
13. That drink is transparent.
That drink is tasteless.
All vodka is tasteless.
∴ Some vodka is transparent.
あの飲み物は透明である。
d is P
あの飲み物は無味である。
d is L
すべてのウオッカは無味である。
all V* is L
∴some V* is P*
Invalid
∴ウオッカのいくらかは透明である。
14. Judy isn't the world's best cook.
The world's best cook lives in Detroit.
∴ Judy doesn't live in Detroit.
ジュディは世界で一番のコックではない。
j is not w*
世界で一番のコックはデトロイトに住んでいる。
w is L
∴ジュディはデトロイトに住んでいない。
∴j* is not L Invalid
15. All men are mortal.
My mother is a man.
∴ My mother is mortal.
すべての人間/男性は死を免れない。
all N* is M
私の母は男性である。
o is N
∴私の母は死を免れない。
∴o* is M*
Valid
16. All gender-neutral terms can be applied naturally to individual women.
The term "man" can't be applied naturally to individual women.
(We can't naturally say "My mother is a man" - see the previous argument.)
∴ The term "man" isn't a gender-neutral term.
[This is from the philosopher Janice Molton's discussion of sexist language.]
性別に中立的な言葉はすべて女性個人に自然に適用されう
all G* is A
る。
t is not A*
∴t* is not G
Valid
｢男性｣という言葉は女性個人に自然に適用されえない。
∴｢男性｣という言葉は性別に中立的な言葉ではない。
17. Some moral questions are controversial.
No controversial question has a correct answer.
9.
5
∴ Some moral questions don't have a correct answer.
道徳上の疑問のいくつかは意見の分かれるものがある。
some M is C
意見の分かれる疑問はどれも正しい答えを持たない。
no C* is A*
∴道徳上の疑問のいくつかは正しい答えを持たない。
∴some M* is not A
Valid
18. The idea of a perfect circle is a human concept.
The idea of a perfect circle doesn't derive from sense experience.
All ideas gained in our earthly existence derive from sense experience.
∴ Some human concepts aren't ideas gained in our earthly existence.
[This argument is from Plato. It led him to think that the soul gained ideas in a previous existence apart from the body, and so
can exist apart from matter.]
完全な円という概念は人間が持ちうる概念である。
られるものではない。
完全な円という概念は感覚を通じた経験からは出てこない。
p is H
我々のこの世の存在で得られる概念はすべて感覚を通じた経
p is not D*
験により出てくる。
All G* is D
∴人間が持ちうる概念のいくつかは我々のこの世の存在で得
∴some H* is not G
Valid
19. All beings with a right to life are capable of desiring continued existence.
All beings capable of desiring continued existence have a concept of themselves as a continuing subject of experiences.
No human fetus has a concept of itself as a continuing subject of experiences.
∴ No human fetus has a right to life. [From Michael Tooley.]
∴どの人間の胎児も生存権を持っていない。
生存権を持つ生き物はすべて生きながらえたいと希求する能
力がある。
all B* is C
生きながらえたいと希求する能力のある生き物はすべて自分
all C* is S
が経験の持続的主体であるという概念を持っている。
no H* is S*
∴no H is B Valid
どの人間の胎児も自分が経験の持続的主体であるという概念
を持っていない。
20. The bankrobber wears size-twelve hiking boots.
You wear size-twelve hiking boots.
∴ You're the bankrobber. [This is circumstantial evidence.]
その銀行強盗はサイズが12のハイキングブーツを履いている。
b is W
あなたはサイズが12のハイキングブーツを履いている。
y is W
∴あなたはその銀行強盗である。
∴y* is b* Invalid
21. All moral beliefs are products of culture.
No products of culture express objective truths.
∴ No moral beliefs express objective truths.
すべての道徳上の信念は文化の産物である。
all M* is P
いかなる文化の産物も客観的真理を表さない。
no P* is E*
∴いかなる道徳上の信念も客観的真理を表さない。
∴no M is E
Valid
22. Some books are products of culture.
Some books express objective truths.
∴ Some products of culture express objective truths.
[How could we change this argument to make it valid?
=> 最初の言明をall books are products of culture.に変える。]
本のなかには文化の産物もある。
some B is P
本のなかには客観的真理を表すものもある。
some B is E
∴文化の産物の中には客観的真理を表すものもある。
∴some P is E Invalid
23. Dr Martin Luther King believed in objective moral truths (about the wrongness of racism).
Dr Martin Luther King disagreed with the moral beliefs of his culture.
No people who disagree with the moral beliefs of their culture are absolutizing the moral beliefs of their own culture.
∴ Some who believed in objective moral truths aren't absolutizing the moral beliefs of their own culture.
マーチン・ルーサー・キング博士は(人種差別の誤りについての)客観的な道徳的信念の正しさを信じていた。
マーチン・ルーサー・キング博士は自分の文化の道徳的信念に賛同しなかった。
自分の文化の道徳的信念に賛同しないどんな人も自分自身の文化の道徳的信念を絶対化していない。
∴客観的な道徳的信念の正しさを信じた人のなかには自分自身の文化の道徳的信念を絶対化していない人もいる。
24. All claims that would still be true even if no one believed them are objective truths.
"Racism is wrong" would still be true even if no one believed it.
"Racism is wrong" is a moral claim.
∴ Some moral claims are objective truths.
誰一人それを信じていなかったとしてもなお真であると思われる主張はすべて客観的な真実である。
｢人種差別は悪である｣は誰一人それを信じていなかったとしてもなお真であると思われる。
｢人種差別は悪である｣は道徳上の主張である。
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∴道徳上の主張の中には客観的な真実もある。
25. Some shivering people with uncovered heads have warm heads.
All shivering people with uncovered heads are losing much body heat through their heads.
All who are losing much body heat through their heads ought to put on a hat to stay warm.
∴ Some people who have warm heads ought to put on a hat to stay warm. [This was from a ski magazine.]
頭に何もかぶらずに震えている人の中には頭部が温かい人もいる。
頭に何もかぶらずに震えている人はすべて頭から多くの体温が奪われている。
頭から多くの体温を奪われている人はすべて帽子をかぶって保温すべきだ。
∴頭部が温かい人の中には帽子をかぶって保温すべき人がいる。
2.4a
1. Only free actions can justly be punished.
all J is F (All actions that are justly be punished are free.)
2. Not all actions are determined.
some A is not D (Some actions are not determined.)
3. Socially useful actions are right.
all S is R
4. None but Democrats favor the poor.
all F is D (All people who favor the poor are Democrates.)
5. At least some of the shirts are on sale.
some S is O (Some shirts are on sale.)
6. Not all of the shirts are on sale.
some S are not O (Some shirts are not on sale.)
7. No one is happy unless they are rich.
all H is R (All happy people are rich.)
8. Only rich people are happy.
all H is R
9. Every rich person is happy.
all R is H
10. Not any selfish people are happy.
no S is H
11. Whoever is happy is not selfish.
no H is S (No happy people are selfish.)
12. Altruistic people are happy.
all A is H (All altruistic people are happy.)
13. All of the shirts (individually) cost $20.
all S is C (All shirts are a thing that cost $20.)
14. All of the shirts (together) cost $20.
s is C (The set of all shirts is a thing that cost $20.)
15. Blessed are the merciful. (慈悲深き人は幸いなるかな。/祝福されているのは慈悲深い人である。)
all M is B (All merciful people are blessed.)
16. I mean whatever I say. (私の言うことはすべて本気だ。)
all S is M (All things that I say are things that I mean.)
17. I say whatever I mean. (私が本気であることはすべて言う。)
all M is S (All things that I mean are things that I say.)
18. Whoever hikes the Appalachian Trail (AT) loves nature.
all H is L (All hikers who hikes the AT are nature lovers.)
19. No person hikes the AT unless he or she likes to walk.
all H is W (All hikers who hikes the AT are people who likes to walk.)
20. Not everyone who hikes the AT is in great shape.
some H is not S (Some hikers who hikes at the AT are not in great shape.)
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