50
Formal Logic
S e c t I o n 1 . 3 Review
tecHnIQueS
• Determine the truth value of a predicate wff in a
given interpretation.
• Translate English language statements into predicate wffs, and vice versa.
• Recognize a valid wff and explain why it is valid.
• Recognize a nonvalid wff and construct an interpretation in which it is false or has no truth value.
MAIn IdeAS
• The truth value of predicate wffs depends on the
interpretation considered.
• Valid predicate wffs are “intrinsically true”—true
in all interpretations.
W
W
W
eXeRcISeS 1.3
1. What is the truth value of each of the following wffs in the interpretation where the domain consists of the
integers, O(x) is “x is odd,” L(x) is “x < 10,” and G(x) is “x > 9”?
a. (E x)O(x)
c. (E x)[L(x) ` G(x)]
b. (4x)[L(x) S O(x)]
d. (4x)[L(x) ~ G(x)]
2. What is the truth value of each of the following wffs in the interpretation where the domain consists of the
integers, A(x) is “x < 5” and B(x) is “x < 7”?
a. (E x)A(x)
c. (4x)[A(x) S B(x)]
b. (E x)[A(x) ` B(x)]
d. (4x)[B(x) S A(x)]
3. What is the truth value of each of the following wffs in the interpretation where the domain consists of the
integers?
a. (4x)(E y)(x + y = x)
e. (4x)(4y)(x < y ~ y < x)
b. (E y)(4x)(x + y = x)
f. (4x)[x < 0 S (E y)( y > 0 ` x + y = 0)]
c. (4x)(E y)(x + y = 0)
g. (E x)(E y)(x
2
= y)
d. (E y)(4x)(x + y = 0)
h. (4x)(x
2
> 0)
4. What is the truth value of each of the following wffs in the interpretation where the domain consists of the
real numbers?
a. (4x)(E y)(x = y
2
)
c. (E x)(4y)(x = y
2
)
b. (4x)(4y)(x = y
2
)
d. (E x)(E y)(x = y
2
)
5. Give the truth value of each of the following wffs in the interpretation where the domain consists of
the states of the United States, Q(x, y) is “x is north of y,” P(x) is “x starts with the letter M,” and a is
“Massachusetts.”
a. (4x)P(x)
b. (4x)(4y)(4z)[Q(x, y) ` Q( y, z) S Q(x, z)]
c. (E y)(E x)Q( y, x)
d. (4x)(E y)[P( y) ` Q(x, y)]
e. (E y)Q(a, y)
f. (E x)[P(x) ` Q(x, a)]
6. Give the truth value of each of the following wffs in the interpretation where the domain consists of
people, M(x, y) is “x is the mother of y”, F(x) is “x is female”, M(x) is “x is male.”
a. (4x)(E y)(M( y, x))
d. (E x)(E y)(M(x, y) ` M( y))
b. (E x)(4y)(M(x, y))
e. (E x)(4y)(M(x, y) S F( y))
c. (4x)(4y)(M(x, y) S M( y))
Formal Logic
S e c t I o n 1 . 3 Review
tecHnIQueS
• Determine the truth value of a predicate wff in a
given interpretation.
• Translate English language statements into predicate wffs, and vice versa.
• Recognize a valid wff and explain why it is valid.
• Recognize a nonvalid wff and construct an interpretation in which it is false or has no truth value.
MAIn IdeAS
• The truth value of predicate wffs depends on the
interpretation considered.
• Valid predicate wffs are “intrinsically true”—true
in all interpretations.
W
W
W
eXeRcISeS 1.3
1. What is the truth value of each of the following wffs in the interpretation where the domain consists of the
integers, O(x) is “x is odd,” L(x) is “x < 10,” and G(x) is “x > 9”?
a. (E x)O(x)
c. (E x)[L(x) ` G(x)]
b. (4x)[L(x) S O(x)]
d. (4x)[L(x) ~ G(x)]
2. What is the truth value of each of the following wffs in the interpretation where the domain consists of the
integers, A(x) is “x < 5” and B(x) is “x < 7”?
a. (E x)A(x)
c. (4x)[A(x) S B(x)]
b. (E x)[A(x) ` B(x)]
d. (4x)[B(x) S A(x)]
3. What is the truth value of each of the following wffs in the interpretation where the domain consists of the
integers?
a. (4x)(E y)(x + y = x)
e. (4x)(4y)(x < y ~ y < x)
b. (E y)(4x)(x + y = x)
f. (4x)[x < 0 S (E y)( y > 0 ` x + y = 0)]
c. (4x)(E y)(x + y = 0)
g. (E x)(E y)(x
2
= y)
d. (E y)(4x)(x + y = 0)
h. (4x)(x
2
> 0)
4. What is the truth value of each of the following wffs in the interpretation where the domain consists of the
real numbers?
a. (4x)(E y)(x = y
2
)
c. (E x)(4y)(x = y
2
)
b. (4x)(4y)(x = y
2
)
d. (E x)(E y)(x = y
2
)
5. Give the truth value of each of the following wffs in the interpretation where the domain consists of
the states of the United States, Q(x, y) is “x is north of y,” P(x) is “x starts with the letter M,” and a is
“Massachusetts.”
a. (4x)P(x)
b. (4x)(4y)(4z)[Q(x, y) ` Q( y, z) S Q(x, z)]
c. (E y)(E x)Q( y, x)
d. (4x)(E y)[P( y) ` Q(x, y)]
e. (E y)Q(a, y)
f. (E x)[P(x) ` Q(x, a)]
6. Give the truth value of each of the following wffs in the interpretation where the domain consists of
people, M(x, y) is “x is the mother of y”, F(x) is “x is female”, M(x) is “x is male.”
a. (4x)(E y)(M( y, x))
d. (E x)(E y)(M(x, y) ` M( y))
b. (E x)(4y)(M(x, y))
e. (E x)(4y)(M(x, y) S F( y))
c. (4x)(4y)(M(x, y) S M( y))