2.1 BASIC CONCEPTS OF THE PROPOSITIONAL CALCULUS 53
at this point, write a positive statement corresponding to L # lim,,, f(x)?
If not, you will find a detailed consideration of this problem in Article 4.3.1
One reason is that this definition contains three instances of quantijiers,
that is, the phrases "for every" and "there exists." Another reason is that a
major part of the definition is a conditional, that is, a statement of the form
"if... then." The heavy use of quantifiers and the use of connectives (such
as "and," "or," and "if... then") are part of the normal vocabulary of
mathematicans, but these are not nearly so prevalent in everyday usage.
Facility with this mode of expression, then, doesn't come naturally; it must
be acquired.
Fortunately, this ability is more science than art. Gaining it is not
strictly a matter of experience and intuition, although both help. There
are specific rules for dealing with compound statements, which fall under
the heading of the propositional calculus and the predicate calculus. The
former is about compound statements (e.g., "either 2 # 3 or 2 + 3 = 4")
and connectives, and will be studied in this chapter. The latter studies
open sentences, or predicates (sentences containing an unknown, as in
"1x2 - 161 < 4" or "f is a continuous function") and the modification of
such expressions with various combinations of the two quantifiers described
earlier. This is the subject of Chapter 3. Many of the theorems of logic
presented in these two chapters are the basis of theorem-proving strategies
in mathematics we use throughout the remainder of the text. We will take
specific note of such strategies as they arise in these two chapters. Then,
in Chapter 4 we will focus on several immediate applications of principles
of logic, as we begin our emphasis on the writing of proofs.
Historically, the development of symbolic logic traces back primarily to
the work of the nineteenth-century British mathematician George Boole,
after whom Boolean algebra, an important branch of symbolic logic, is
named.
Basic Concepts of the Propositional Calculus
Consider the assertions "sin 2n = 1" and "the function f(x) = sin x is
periodic." From trigonometry, we know that the first is false, whereas the
second is true. But what about sentences such as "sin 2n = 1 and the sine
function is periodic" or "if sin 2n = 1, then the function f(x) = sin x is
periodic"? The truth or falsehood of these compound sentences, it turns out,
depends on the truth or falsehood of their component simple sentences and
on characteristics of the connective involved in the compound sentence. In
this article we begin to study the precise nature of this dependence.
STATEMENTS OR PROPOSITIONS
DEFINITION 1
A statement, or proposition, is a declarative sentence that is either true or
false, but is not both true and false.
