3.2 QUANTIFICATION 91need a substitution for n that makes r(n) true (n is a multiple of 4), where-
as p(n) is false (n is not even). Does any such integer exist? Common
sense tells us that there is none; that is, r(n) -, p(n) is true in all cases
that can actually occur. For this reason, (Vn)(r(n) -, p(n)) is true, in cor-
respondence with the intuitively evident truth of "every multiple of 4 is
even." For another approach to the truth of (Vn)(r(n) -, p(n)), calculate
the truth set R' u P of the predicate r(n) -, p(n). ClThe most important conclusion to be drawn from Example 2 [speci-
fically, part (c)] is that a statement such as "all men are mortal" is sym-
bolized logically by the universal quantifier with an implication connective.
Letting p(x) represent "x is a man" and q(x) stand for "x is mortal," the
expression (Vx)(p(x) + q(x)) corresponds to "every man is mortal." This
fact is important because many theorems in mathematics have this form,
for example, "every differentiable function is continuous," "every cyclic
group is abelian," (from group theory, a branch of abstract algebra) and
"every function continuous on a closed and bounded interval attains a
maximum on that interval."
A second, and almost equally important, conclusion from Example 2 is
that a statement such as "some men are mortal" is symbolized by the exis-
tential quantifier and conjunction, specifically by (3x)(p(x)~ q(x)). Note
that "some men are not mortal" is represented by (3x)(p(x) A -q(x)). Can
you spot any relationship between the statements "all men are mortal" and
"some men are not mortal"? We will discuss one in the next article (see
Example 3, Article 3.3).
Another important message of Example 2 [recall part (b)] is that expres-
sions such as (Vx)(h(x)v k(x)) and (Vx)(h(x))v(Vx)(k(x)) represent dif-
ferent statements. First, their English translations are different. Second,
the role of the connective v in each is different. Namely, v is a con-
nective between propositions in (Vx)(h(x)) v (Vx)(k(x)). The truth of this
compound statement depends on the truth of the individual statements
(Vx)(h(x)) and (Vx)(k(x)) in accordance with the truth tabular definition
of v in Article 2.1. In the statement (Vx)(h(x) v k(x)), v is a connective
'\ between propositional functions, as defined in Definition 2(b), Article 3.1.
We must look at the truth set H u K of h(x) v k(x) (and ask whether it
equals U) to determine whether this quantified compound predicate is
true. A third, and most crucial, difference between (Vx)(h(x) v k(x)) and
(Vx)(h(x))v (Vx)(k(x)) is that, in some cases, it is possible for a statement
of one form to be true, whereas the corresponding statement of the other
form is false. Can you find predicates h(x) and k(x) such that one of
(Vx)(h(x) v k(x)) and (Vx)(h(x)) v (Vx)(k(x)) is true, whereas the other is
false? [see Exercise 3(e) and (f)]. Does it seem to you that the truth of one
of these two statements forces the other to be true? Your answer to both