Section 1.3 Quantifiers, Predicates, and Validity
45
In addition, it is legitimate to “slide” a quantifier over a predicate that does not
involve the variable of that quantifier. Because D(x) does not involve y, we could
slide the universal quantifier in wff (2) to the front (but not past the existential
quantifier), giving the equivalent wff
(E x)(4y)[D(x) ` (R( y) S C(x, y))]
(a)
In wff (a), we still have grouping symbols around R( y) S C(x,y). Without the
grouping symbols, this wff becomes
(E x)(4y)[D(x) ` R( y) S C(x, y)]
(b)
which, according to the order of precedence of connectives, is equivalent to
(E x)(4y)[(D(x) ` R( y)) S C(x, y)]
A quick truth table exercise shows that D ` (R S C ) is not equivalent to
(D ` R) S C, so wff (b) is not equivalent to wff (a) and thus does not represent
statement (2) in the table.
Translation from an English statement into a predicate wff is a little harder
than translation into a propositional wff, partly because of the added expressiveness of the verbal form and partly because there can be multiple correct predicate
wffs. Here is a summary of translation tips:
• Look for the key words that signify the type of quantifier:
for all, for every, for any, for each: use a universal quantifier
for some, there exists: use an existential quantifier.
• English sometimes uses “understood” universal quantifiers. For example,
“Dogs chase rabbits,” is understood to mean, “All dogs chase all rabbits.”
• If you use a universal quantifier, the connective that goes with it is almost
always implication.
• If you use an existential quantifier, the connective that goes with it is
almost always conjunction.
• Whatever comes after the word “only” is the conclusion of an implication;
that is, it comes after “then” in an “if–then” statement.
• You are most apt to arrive at a correct translation if you follow the order
of the English words.
eXAMPLe 23
Let’s do a couple of examples in great detail. The first is
All giraffes are tall.
The property of being a giraffe and the property of being tall are unary predicates.
We’ll use G(x) for “x is a giraffe” and T(x) for “x is tall”. Following the sentence
structure, we first see “All,” which tells us that there’s a universal quantifier, so the
wff begins with
(4x)( … )
45
In addition, it is legitimate to “slide” a quantifier over a predicate that does not
involve the variable of that quantifier. Because D(x) does not involve y, we could
slide the universal quantifier in wff (2) to the front (but not past the existential
quantifier), giving the equivalent wff
(E x)(4y)[D(x) ` (R( y) S C(x, y))]
(a)
In wff (a), we still have grouping symbols around R( y) S C(x,y). Without the
grouping symbols, this wff becomes
(E x)(4y)[D(x) ` R( y) S C(x, y)]
(b)
which, according to the order of precedence of connectives, is equivalent to
(E x)(4y)[(D(x) ` R( y)) S C(x, y)]
A quick truth table exercise shows that D ` (R S C ) is not equivalent to
(D ` R) S C, so wff (b) is not equivalent to wff (a) and thus does not represent
statement (2) in the table.
Translation from an English statement into a predicate wff is a little harder
than translation into a propositional wff, partly because of the added expressiveness of the verbal form and partly because there can be multiple correct predicate
wffs. Here is a summary of translation tips:
• Look for the key words that signify the type of quantifier:
for all, for every, for any, for each: use a universal quantifier
for some, there exists: use an existential quantifier.
• English sometimes uses “understood” universal quantifiers. For example,
“Dogs chase rabbits,” is understood to mean, “All dogs chase all rabbits.”
• If you use a universal quantifier, the connective that goes with it is almost
always implication.
• If you use an existential quantifier, the connective that goes with it is
almost always conjunction.
• Whatever comes after the word “only” is the conclusion of an implication;
that is, it comes after “then” in an “if–then” statement.
• You are most apt to arrive at a correct translation if you follow the order
of the English words.
eXAMPLe 23
Let’s do a couple of examples in great detail. The first is
All giraffes are tall.
The property of being a giraffe and the property of being tall are unary predicates.
We’ll use G(x) for “x is a giraffe” and T(x) for “x is tall”. Following the sentence
structure, we first see “All,” which tells us that there’s a universal quantifier, so the
wff begins with
(4x)( … )