Section 1.3 Quantifiers, Predicates, and Validity
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7. For each wff, find an interpretation in which it is true and one in which it is false.
a. (4x)([A(x) ~ B(x)] ` [A(x) ` B(x)]′)
b. (4x)(4y)[P(x, y) S P( y, x)]
c. (4x)[P(x) S (E y)Q(x, y)]
8. For each wff, find an interpretation in which it is true and one in which it is false.
a. (E x)[A(x) ` (4y)B(x, y)]
b. [(4x)A(x) S (4x)B(x)] S (4x)[A(x) S B(x)]
c. (E x)[P(x) ~ Q(x)] ` (4x)[P(x) S Q(x)]
9. Identify the scope of each of the quantifiers in the following wffs and indicate any free variables.
a. (4x)[P(x) S Q( y)]
c. (E x)[(4y)P(x, y) ` Q(x, y)]
b. (E x)[A(x) ` (4y)B( y)]
d. (E x)(E y)[A(x, y) ` B( y, z) S A(a, z)]
10. Explain why each of the following expressions is written incorrectly.
a. (E )(Q(x) ` P(x)
b. (4y)(Q( y) P( y))
c. (4x)(4y)Q(x) S P( y)
11. Which of the following sentences are equivalent to the statement
All circles are round.
a. If it’s round, it’s a circle.
b. Roundness is a necessary property of circles.
c. Something that isn’t round can’t be a circle.
d. Some round things are circles.
12. Which of the following sentences are equivalent to the statement
Cats are smarter than dogs.
a. Some cats are smarter than some dogs.
b. There is a cat that is smarter than all dogs.
c. All cats are smarter than all dogs.
d. Only cats are smarter than dogs.
e. All cats are smarter than any dog.
13. Using the predicate symbols shown and appropriate quantifiers, write each English language statement as
a predicate wff. (The domain is the whole world.)
D(x): x is a day
S(x): x is sunny
R(x): x is rainy
M: Monday
T: Tuesday
a. All days are sunny.
b. Some days are not rainy.
c. Every day that is sunny is not rainy.
d. Some days are sunny and rainy.