Section 1.3 Quantifiers, Predicates, and Validity
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PRACTICE 19 Using the predicate symbols F(x) for “x is a fruit,” V(x) for “x is a vegetable,” and S(x,y) for
“x is sweeter than y,” write wffs that express the following statements. (The domain is the
whole world.)
a. Some vegetable is sweeter than all fruits.
b. Every fruit is sweeter than all vegetables.
c. Every fruit is sweeter than some vegetable.
d. Only fruits are sweeter than vegetables.
Negating statements with quantifiers, as in negating compound statements,
requires care. The negation of the statement, “Everything is beautiful,” is, “It is
false that everything is beautiful,” or, “Something is nonbeautiful.” Symbolically,
[(4x)A(x)]′ is equivalent to (E x)[A(x)]′
Note that, “Everything is nonbeautiful,” or (4x)[A(x)]′, says something stronger
than the negation of the original statement.
The negation of, “Something is beautiful,” is, “Nothing is beautiful,” or,
“ Everything fails to be beautiful.” Symbolically,
[(E x)A(x)]′ is equivalent to (4x)[A(x)]′
In English, the statement, “Everything is not beautiful,” would often be misinterpreted as, “Not everything is beautiful,” or, “There is something nonbeautiful.”
However, this misinterpretation, symbolized by (E x)[A(x)]′, is not as strong as the
negation of the original statement.
PRACTICE 20 Which of the following statements expresses the negation of, “Everybody loves somebody
sometime”?
a. Everybody hates somebody sometime.
b. Somebody loves everybody all the time.
c. Everybody hates everybody all the time.
d. Somebody hates everybody all the time.
PRACTICE 18 Using the predicate symbols S(x) for “x is a student,” I(x) for “x is intelligent,” and M(x)
for “x likes music,” write wffs that express the following statements. (The domain is the
collection of all people.)
a. All students are intelligent.
b. Some intelligent students like music.
c. Everyone who likes music is a stupid student.
d. Only intelligent students like music.
47
PRACTICE 19 Using the predicate symbols F(x) for “x is a fruit,” V(x) for “x is a vegetable,” and S(x,y) for
“x is sweeter than y,” write wffs that express the following statements. (The domain is the
whole world.)
a. Some vegetable is sweeter than all fruits.
b. Every fruit is sweeter than all vegetables.
c. Every fruit is sweeter than some vegetable.
d. Only fruits are sweeter than vegetables.
Negating statements with quantifiers, as in negating compound statements,
requires care. The negation of the statement, “Everything is beautiful,” is, “It is
false that everything is beautiful,” or, “Something is nonbeautiful.” Symbolically,
[(4x)A(x)]′ is equivalent to (E x)[A(x)]′
Note that, “Everything is nonbeautiful,” or (4x)[A(x)]′, says something stronger
than the negation of the original statement.
The negation of, “Something is beautiful,” is, “Nothing is beautiful,” or,
“ Everything fails to be beautiful.” Symbolically,
[(E x)A(x)]′ is equivalent to (4x)[A(x)]′
In English, the statement, “Everything is not beautiful,” would often be misinterpreted as, “Not everything is beautiful,” or, “There is something nonbeautiful.”
However, this misinterpretation, symbolized by (E x)[A(x)]′, is not as strong as the
negation of the original statement.
PRACTICE 20 Which of the following statements expresses the negation of, “Everybody loves somebody
sometime”?
a. Everybody hates somebody sometime.
b. Somebody loves everybody all the time.
c. Everybody hates everybody all the time.
d. Somebody hates everybody all the time.
PRACTICE 18 Using the predicate symbols S(x) for “x is a student,” I(x) for “x is intelligent,” and M(x)
for “x likes music,” write wffs that express the following statements. (The domain is the
collection of all people.)
a. All students are intelligent.
b. Some intelligent students like music.
c. Everyone who likes music is a stupid student.
d. Only intelligent students like music.