54 LOGIC, PART I: THE PROPOSITIONAL CALCULUS Chapter 2
The designation T(true) or F(false), one and only one of which is assignable
to any given statement, is called the truth value of that statement.
EXAMPLE 1 The following are statements:
(a) The moon is made of green cheese.
(b) (e")2 = e2".
(c)^6 is a prime number.
(d) February 5,^1992 falls on a Wednesday.
(e) The millionth digit in the decimal expansion of a is 6.Discussion Statements (a) and (c) are clearly false (i.e., have truth value F),
whereas (b) is true. The truth values of statements (d) and (e) are not
so evident, but are determinable; item (d), for instance, is true. In con-
nection with (e), it is important to understand that we need not know
specifically whether or not a statement is true in order to label it a
statement. It is only in recent years, with the advent of high-speed com-
puters, that it has become practical to find the answer to such questions.
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EXAMPLE 2 The following are not statements:
(a) Is equal to e2"?
(b) If only every day could be like this one!
(c) Every goople is an aardling.
(d)^2 + 3i is less than^5 + 3i.
(e) x>5.
(f) This proposition is false.Discussion Items (a) and (b) fail to be statements, because they are inter-
rogative and exclamatory sentences, respectively, rather than declarative.
Item (c) fails, sinpe some of its words are not really words, but rather,
nonsense collections of letters. Item (d) fails for the same reason as (c),
but in a more subtle and purely mathematical way. We will see in
Chapter 9 that there is no notion of "less than" or "greater than" between
pairs of complex numbers, so a statement that one is less than another
is meaningless.
Item (e) is an important example. It is not a statement, but rather,
it is an open sentence or predicate, the topic of the next chapter. It is
neither true nor false since it contains a variable, essentially an "empty
place" in the sentence. A predicate becomes either true or false, and thus
a statement, when we either quantify it or substitute a specific object
for its variable. A less clearcut example related to item (e) is a sentence
such as x(x + 4) = x2 + 4x. Strictly speaking, this sentence is a pre-
dicate. Yet, because it is true for any possible substitution of a real
number for x, it is common practice to say "x(x + 4) = x2 + 4x" when