2.1 BASIC CONCEPTS OF THE PROPOSITIONAL CALCULUS 55we mean "for every x, x(x + 4) = x2 + 4x," and treat the former as a
(true) statement. You should adopt the strict point of view whenever
this issue arises in the exercises for this article.
Item (f) may remind you of Exercise 10, Article 1.1, since it involves
a paradox. At first glance, it may appear to be a statement. But if it's
true, then it must be false (Why?), and if it's false, it must be true. In
other words, if it's either true or false, then it must be both true and
false; but this violates the definition of "statement." 0Another situation worth mentioning in connection with Example 2 is
sentences whose truth or falsehood depends on the time at which they are
uttered. Tn the strictest sense, a sentence like "today is Monday" might
be regarded not to be a statement, because "today" is a variable (like x in
the inequality x > 7). The same can be said for "it is raining," and "the
current Speaker of the U.S. House of Representatives is a Democrat." But
often in practice, when such sentences are used in everyday discourse, there
is a great deal of unspoken, but solidly understood, background material
(involving implicitly either substitution or quantification) that renders the
sentence clearly true or false, and thus "a statement when used in context."
Therefore when you say "it is sunny outside," you are usually saying
something that, in that specific time and place, can be confirmed or refuted.
It is interesting to note that this problem doesn't arise in sentences with
purely mathematical content; perhaps this can be regarded as a manifesta-
tion of the timelessness of mathematics.
COMPOUND STATEMENTS AND LOGICAL CONNECTIVES
All the statements in Example 1 were simple statements not composed in
any way of other statements. The propositional calculus is about compound
statements consisting of two or more component statements, joined by one
or more logical connectives.
The propositional calculus is to statements what ordinary algebra is to
numbers. In algebra we use variables x, y, z, etc., to represent numbers;
in the propositional calculus we use letters in lower case, such as p, q, and
r to represent statements. In algebra we have operations such as "plus" and
"times" that allow us to combine numbers to get a new number; in the
propositional calculus we have logical connectives, represented by symbols
such as v, A, and +, by which we can combine statements to get a new
statement. Thus if p and q are statements, then p v q and q -, p, for instance,
will also be statements, compound statements in fact. It is important to
realize that the truth value of a compound statement will depend on the
truth values of its component statements only (in a manner prescribed by
the connective involved) and not on the compound statements themselves.
Thus to know whether a statement of the form p A q is true, we need only