LOGIC, PART 11: THE PREDICATE CALCULUS
CHAPTER 3
There are many kinds of statements that we wish to make in mathematics
(and in everyday life) that cannot be symbolized and logically analyzed
solely in terms of the propositional calculus. In addition to the external
complexity introduced by the need to link statements by using connectives,
there is an internal complexity in statements containing words such as "all,"
"every," and "some," which requires logical analysis beyond that afforded
by the propositional calculus. Such an analysis is the subject of the predicate
calculus, the topic of this chapter.
The following example demonstrates the difficulties that can arise if only
the propositional calculus is available to analyze statements.
EXAMPLE 1 Let P and Q be sets, let p represent the statement "x is an
element of P" and q the statement "x is an element of Q." Analyze, in
terms of the propositional calculus, the statement (p -, q) v (q -, p).
iscussion Perhaps the best place to begin the analysis is with the truth
table for the statement form (p -, q) v (q -+ p). Before reading further,
you should construct that table. The results may be somewhat surprising,
for this statement form is a tautology (all possible tautologies were not
exhausted in Chapter 2!). Hence, in particular, if we make the indicated
substitutions for p and q, the statement form p --+ q becomes "x E P
implies x E Q," while q -, p is "x E Q implies x E P." The disjunction
then says "(x E P implies x E Q) or (x E Q implies x E P)" and is true, since
any statement of the form (p -, q) v (q -, p) is true under all possible
truth conditions. But "x E P implies x E Q x E Q implies x E P" seems