Section 1.4 Predicate Logic
67
Verbal arguments
To prove the validity of a verbal argument, we proceed much as before. We cast
the argument in symbolic form and show that the conclusion can be deduced from
the hypotheses. If the argument involves predicate wffs, then the derivation rules
of predicate logic are available.
It is useful, however, to see where a potential proof sequence goes wrong. We
begin with the two hypotheses and then remove one of the existential quantifiers.
1. (E x)P(x) hyp
2. (E x)Q(x) hyp
3. P(a)
1, ei
Now here’s the problem. The next step would be to remove the existential quantifier from the wff at step 2, but, according to the rules for ei, we have to name the
object that has property Q by some different name, not a. So we could eventually
get to a wff in the proof sequence that looks like
P(a) ` Q(b)
but this does us no good. Existential generalization could not be used to replace
both constant symbols with a single variable. At best, we could arrive at
(E y)(E x)[P(x) ` Q( y)]
which is not what we want.
eXAMPLe 35
Show that the following argument is valid: “Every laptop has an internal disk
drive. Some laptops have a DVD drive. Therefore some laptops have both an internal disk drive and a DVD drive.” Using
L(x) is “x is a laptop.”
I(x) is “x has an internal disk drive.”
D(x) is “x has a DVD drive.”
the argument is
(4x)[L(x) S I(x)] ` (E x)[L(x) ` D(x)] S (E x)[L(x) ` I(x) ` D(x)]
pRaCtiCe 26 Is the wff a valid argument? Prove or disprove.
(E x)R(x) ` [(E x)[R(x) ` S(x)] ]′ S (E x)[S(x)]′