For each of the following, write out what atomic proposition each constant stands for. Then translate the sentences using the constants you have defined. Finally, after you have translated the sentence, identify which truth-functional connective is the main operator of the sentence.

Question

1. Coral is not both a plant and an animal. (P, A)
2. Although protozoa and chimpanzees are both eukaryotes, they are not both animals. (There are four atomic propositions here; just use A,
B, C, and D for each different proposition.)
3. Neither chimpanzees nor protozoa are prokaryotes. (C, P)
4. China has not signed the Kyoto Protocol and neither has the United States. (C, U)
5. Either Chevrolet or McDonald’s will support the Olympic team, but they won’t both support it. (C, M)
6. Peter Jennings is either a liar or has a really bad memory. (L, M)
7. Peter Jennings is neither a liar nor has a really bad memory. (L, M)
8. Peter Jennings is both a liar and has a really bad memory. (L, M)
9. Peter Jennings is not both a liar and a person with a really bad memory. (L, M)
10.Chevrolet won’t support the Olympic team this year, and McDonald’s won’t either. (C, M)
11.Mother Theresa may be a saint. Even so, she has not been canonized yet by the Catholic Church. (S, C)
12.The best distance runner of the last two decades is either Paul Tergat or Haile Gebrselassie, but it certainly isn’t Jim Ryun. (T, G, R)
13.Jim Ryun was the best high school miler of all time, but he ran a slower time than Alan Webb. (R, W)
14. Neither Paul Tergat nor Haile Gebrselassie knows how to play hockey, but they both know how to play soccer. (A, B, C, D)
15.Ethiopians are neither good bobsledders nor tennis players, but they are excellent distance runners. (B, T, D)
16.Before Helen Keller met Annie Sullivan, she could neither speak, read, nor communicate. (S, R, C)
17.Although Helen Keller learned to communicate, she never learned to play soccer or baseball. (C, S, B)
18.Tom is allowed to play football or soccer, but not both. (F, S)
19.Tom will major in either engineering and physics, or business and sociology. (E, P, B, S)
20.Cartman is both xenophobic and racist, but he isn’t a murderer or a thief. (X, R, M, T)

Answer 1

1.
a. P = Coral is a plant; A = Coral is an animal
b. ~(P ? A)
c. Main operator is the negation
2.
a. A = Protozoa are eukaryotes; B = Chimpanzees are eukaryotes; C
= Protozoa are animals; D = Chimpanzees are animals
b. (A ? B) ? ~(C ? D)
c. Main operator is the second dot
3.
a. C = Chimpanzees are prokaryotes; P = Protozoa are prokaryotes
b. ~(C v P)
c. Main operator is the negation
4.
a. C = China has not signed the Kyoto Protocol; U = The United
States has not signed the Kyoto Protocol
b. ~(C v U)
c. Main operator is the negation
5.
a. C = Chevrolet will support the Olympic team; M = McDonald’s will
support the Olympic team
b. (C v M) ? ~(C ? M)
c. Main operator is the first dot
6.
a. L = Peter Jennings is a liar; M = Peter Jennings has a really bad
memory
b. L v M
c. Main (and only) operator is the wedge
7.
a. L = Peter Jennings is a liar; M = Peter Jennings has a really bad
memory
b. ~(L v M)
c. Main operator is the negation
8.
a. L = Peter Jennings is a liar; M = Peter Jennings has a really bad
memory
b. L ? M
c. Main (and only) operator is the dot
9.
a. L = Peter Jennings is a liar; M = Peter Jennings has a really bad
memory
b. ~(L ? M)
Answers to exercises
214
c. Main operator is the negation
10.
a. C = Chevrolet will support the Olympic team this year; M =
McDonald’s will support the Olympic team this year
b. ~(C v M)
c. Main operator is the negation
11.
a. S = Mother Theresa is a saint; C = Mother Theresa has been
canonized by the Catholic Church
b. S ? ~C
c. Main operator is the dot
12.
a. T = Paul Tergat was the best distance runner of the last two
decades; G = Haile Gebrselassie was the best distance runner of
the last two decades; R = Jim Ryun was the best distance runner of
the last two decades
b. (T v G) ? ~R
c. Main operator is the dot
13.
a. R = Jim Ryun was the best high school miler of all time; W = Jim
Ryun ran a slower time than Alan Webb
b. R ? W
c. Main (and only) operator is the dot
14.
a. A = Paul Tergat knows how to play hockey; B = Haile Gebrselassie
knows how to play hockey; C = Paul Tergat knows how to play
soccer; D = Haile Gebrselassie knows how to play soccer
b. ~(A v B) ? (C ? D)
c. Main operator is the first dot
15.
a. B = Ethiopians are good bobsledders; T = Ethiopians are good
tennis players; D = Ethiopians are good distance runners
b. ~(B v T) ? D
c. Main operator is the dot
16.
a. S = Before Helen Keller met Annie Sullivan she could speak; R =
Before Helen Keller met Annie Sullivan she could read; C = Before
Helen Keller met Annie Sullivan she could communicate
b. (~S ? ~R) ? ~C
c. Main operator is the second dot
17.
a. C = Helen Keller learned to communicate; S = Helen Keller learned
to play soccer; B = Helen Keller learned to play baseball
b. C ? (~S ? ~B)
c. Main operator is the first dot
18.
a. F = Tom is allowed to play football; S = Tom is allowed to play
soccer
b. (F v S) ? ~(F ? S)
c. Main operator is the first dot
19.
a. E = Tom will major in engineering; P = Tom will major in physics; B
= Tom will major in business; S = Tom will major in sociology
b. (E ? P) v (B ? S)
c. Main operator is the wedge
20.
a. X = Cartman is xenophobic; R = Cartman is a racist; M = Cartman
is a murderer; T = Cartman is the thief
b. (X ? R) ? (~M ? ~T)
c. Main operator is the second dot