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PHI 109 Homework Solutions, Chapter 4 ~ É º · Ú
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Exercise 1:
b) Humans are descended either from small primates or great apes.
P = Humans are descended from small primates. A = Humans are descended from great apes
Either P or A P Ú A
d) People like to drive automobiles because it gives them a sense of freedom.
Not a truth functional compound. Must be symbolized as a simple sentence.
f) If high-speed trains are developed, then automobile use will decrease.
H = High-speed trains are developed. A = Automobile use will decrease.
If H, then A. H É A
Exercise 3:
b) Mary will make supper if John is working late.
M if L. L É M
d) John works late if and only if Mary does not.
L if and only if not W. L iff not W. L º not W L º ~W
f) John will not work late unless his boss requires it.
Not L unless B. Not L if not B. ~B É ~L
h) John or Mary will work late, but not both.
L or W, but not both L and W. (L or W) · (not both L and W). (L Ú W) · ~(L and W). (L Ú W) · ~(L · W)
j) Only if John works late and it is not a holiday will Mary make supper.
Only if L and not O will M. M É (L and not O). M É (L · not O). M É (L · ~O)
Exercise 4:
b) I will diet only if I am motivated, and I will exercise only if I don’ get too tired.
D only if M, and E only if not T. (D only if M) · (E only if not T) (D É M) · (E É not T) (D É M) · (E É ~T)
d) I will exercise if I am motivated, provided I am not either lazy or depressed.
E if M, provided not either L or P. (Not either L or P) É (E if M) ~(Either L or P) É (M É E) ~(L Ú P) É (M É E)
f) I will be depressed if and only if I am lazy and neither diet nor exercise.
P if and only if L and neither D nor E. P º (L and Neither D nor E) P º (L · (Neither D nor E)) P º (L · ~ (Either D or E)) P º (L · ~ (D Ú E))
h) I gain weight if and only if I don’t diet, provided I am not motivated and neither swim nor run.
G if and only if not D, provided not M and neither S nor R. (not M and neither S nor R) É (G if and only if not D) (not M · neither S nor R) É (Gº not D) (~M · (neither S nor R)) É (Gº ~D) (~M · ~ (either S or R)) É (Gº ~D) (~M · ~ (S Ú R)) É (Gº ~D)
Exercise 5:
b) If I don’t either buy a new car or keep my gas guzzler, I’ll get either a used car or a bicycle.
If not either K or N, either U or B. (not either K or N) É (either U or B) ~ (either K or N) É (U Ú B) ~ (K Ú N) É (U Ú B)
d) The city will build a bike path only if tax revenues do not fall and student enrollment does not drop, but student enrollment will drop.
P only if not T and not E, but E. (P only if not T and not E) · E ((P É (not T and not E)) · E ((P É (not T· not E)) · E ((P É (~T· ~E)) · E
f) I’ll buy a bike and get fit provided the city builds a bike path and I don’t keep my gas guzzler or buy a new or used car.
B and F provided P and not K or N or U. P and not (K or N or U) É B and F (P ·not (K or N or U)) É (B · F) (P · ~ (K or N or U)) É (B · F) (P · ~ (K Ú (N or U))) É (B · F) (P · ~ (K Ú (N Ú U))) É (B · F)
Exercise 6:
b) The economy will not improve and interest rates will not rise if either consumer spending falls or unemployment rises.
Not E and not R if either S or U. (Either S or U) É (Not E and Not R). (S Ú U) É (Not E · Not R). (S Ú U) É (~E · ~R)
d) Interest rates will not rise if the economy improves, provided consumers do not increase borrowing.
Not R if E, provided not C. Not C É (Not R if E). ~C É (E É Not R) ~C É (E É ~R)
h) Stock prices will fall and the economy will fail to improve if interest rates rise and the deficit is not reduced, unless either more jobs are created or there is a boom in housing.
F and not E if R and D, unless either J or H. F and not E if R and D, if not either J or H. (Not either J or H) É (F and not E if R and D) ~(either J or H) É ((R and D) É (F and not E)) ~(J Ú H) É ((R · D) É (F · not E)) ~(J Ú H) É ((R · D) É (F · ~E))
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