Bridge to Abstract Mathematics: Mathematical Proof and Structures

2.4 ANALYSIS OF ARGUMENTS FOR LOGICAL VALIDITY, PART 1 (OPTIONAL) 77

partial premises must logically imply the conclusion. Stated differently,
the truth of the conclusion must "follow from" the assumed truth of all the
premises, in order for an argument to be valid. In particular, the conclusion
of an argument need not be true in order for an argument to be valid (see
Exercise 5). The following example illustrates a rather standard format in
which premise and conclusion of an argument are presented.

EXAMPLE 1 Test the validity of the argument

Therefore r

Solution The statement
premises, whereas the

forms above the horizontal line are the partial

one below the line is the conclusion. The issue is

whether the conditional [p A (p -, q) A (- q v r)] -+ r is a tautology. You

should verify, by using truth table, that the answer is "yes," so that the

given argument is logically valid.

In Examples 3 and 4 we discuss a method of avoiding the need to con-
struct a truth table in order to determine whether the conditional arising
from an argument is actually an implication. In the next example we deal
with an argument involving specific statements in which we must first assign
a letter to each simple statement involved and then represent symbolically
each of the partial premises and the conclusion. As a rule, the premise ends
and the conclusion begins with a word like "therefore," or "hence," or "thus."

EXAMPLE 2 Express symbolically and analyze for validity the argument "If

interest rates fall, the economy improves. If the economy improves, un-

employment drops. In order for incumbents to win reelection, it is

necessary that unemployment drop. Hence a sufficient condition for in-

cumbents to win reelection is that interest rates fall."

Solution First, we must symbolize each simple statement involved in the
argument. We do this by
p: interest rates fall
q: the economy improves
r: unemployment drops
s: incumbents are reelected
The partial premises, then, have the form p -+ q, q -+ r, and s -+ r [recall,
e.g., from Remark 1, part l(i), Article 2.3, that "r is necessary for s" trans-
lates to s -, r]. The conclusion has the form p -, s. Hence the argument,