2.2 TAUTOLOGY, EQUIVALENCE, THE CONDITIONAL, AND BICONDITIONAL 83I would do if I finished my work" and made no commitment otherwise;
surely it cannot be labeled "false" in this case. Since it must have a truth
value, the only reasonable value is true. There are other good arguments
for the suitability of "F -, T is T." One is found in Exercise 9, Article 2.3,
another in Example 3, Article 3.2.
One final remark along this line is related to row 4 of the table defining
the biconditional. When we assert that a statement such as (5) is true, we
are in no way asserting that either of its component statements is necessarily
true, only that the entire "if and only if" statement is true. A similar remark
could be made in reference to row 4 of the table defining the conditional.
The truth of a statement such as (2) does not mean that the conclusion
("(0, 1) E (0, 1)" in this case) is necessarily true. Indeed, the conclusion is
false in this particular example!
You may have noticed earlier that none of the conditional statements
(1 through 3) seems similar to the conditionals in mathematical theorems.
What is missing in these examples? Statements such as "if n is odd, then
n + 1 is even" or "iff is differentiable at 2, then f is continuous at 2" or
"x2 - 5x + 6 = 0 if and only if x = 2 or x = 3" are what we mainly have
in mind when thinking of "if... then" theorems or "if and only if" theorems.
Note, however, that the component sentences in these examples (e.g., "n is
odd," "f is continuous at 2") are not statements, but rather, open sentences.
Also, implicit in each of these "if... then" sentences is an unwritten "for
every," that is, a quantifier. A nonmathematical example is the sentence,
"If today is Monday, then tomorrow is Tuesday." In this statement, which
happens to be true, the words "today" and "tomorrow" are both variables.
In Chapter 3, where we study open sentences and quantifiers, you will see
the "if... then" and "iff" connectives used in their most natural and mathe-
matically useful setting. For now, you should focus on mastering certain
mechanical properties of the connectives.
Exercises
- Four binary connectives A, v, +, and o have been defined, each by means of
a truth table with four rows. Show that there are exactly 16 possible definitions of
binary connectives [recall Exercise 12(a), Article 1.51. - Construct a truth table for each of the following statement forms. Label each a
tautology, contradiction, or contingency?