LOGIC, PART I: THE PROPOSITIONAL CALCULUS
CHAPTER 2
Consider the sentence "Mathematics is a complicated subject, and in order
to study mathematics, we must accept the fact that many mathematical
concepts cannot be formulated in a simple manner." In addition to ex-
pressing ideas that are true (a reason students of junior-senior level mathe-
matics must be well grounded in logic), this sentence provides an example
of the object that is central to the study of logic, the statement, and, in
particular, the compound statement.
Statements, or declarative sentences, are basic to all human communica-
tion, and are not specific to mathematics. But there are few areas of en-
deavor in which precise command over the structure of statements is as
critical as it is in mathematics (the law might be one example). Mathematics
is full of subtleties, of fine distinctions and complicated formulations. Take,
for example, the epsilon-delta definition of the limit of a function: Iff is
defined on an open interval containing a real number a, and if L is a real
number, then we say
L = lim f(x) if and only if, for every E > 0, there exists 6 > 0
x-'a
such that whenever 0 < Ix - a1 < 6, then (f(x) - LI < EThis definition is of crucial importance in mathematical analysis. But even
though most students are exposed to it early in their first calculus course,
few come to appreciate its meaning until much later, and many never do.
Why is this definition perceived as formidable? Why, for instance, would
it be beyond most calculus students to describe, in terms of epsilons and
deltas, what it means for L not to equal lim f(x) as x tends to a? [Can you,