Bridge to Abstract Mathematics: Mathematical Proof and Structures

92 LOGIC, PART II: THE PREDICATE CALCULUS Chapter 3

questions should be "yes." Before going on to the next article, try to decide

which of the two preceding statement forms is "stronger" than the other.

EXAMPLE 3 Examples involving quantifiers can shed further light on the

definition of the conditional connective +. Consider the statement

"every square is a rectangle," where U = the set of all quadrilaterals in

a given plane. This statement is evidently true, and may be symbolized

(Vx)(S(x) -P r(x)), where s(x) represents "x is a square" and r(x)

stands for "x is a rectangle." If (Vx)(s(x) + r(x)) is true, as our in-

tuition dictates, the conditional s(q) + r(q) must be true for every sub-

stitution of a specific quadrilateral q for the variable x. Note, however,

that q,, q2, and 9, are possible substitutions, where:

Name Physical model Description

q, is a square and T

a rectangle

q2 is a rectangle but F

not a square

q3 is neither a

rectangle nor a

Hence if (Vx)(s(x) + r(x)) is a true statement, the truth tabular defini-

tion of s(x) + r(x) must be such that s(q) -P r(q) is true in all preced-

ing cases (which it is!). The fourth case, where s(q) is T and r(q) is F,

cannot occur, since a physical model would have to be a square which

is not a rectangle. The fact that s(q) + r(q) would be false in such a

situation (by the truth tabular definition of +) has no bearing on the

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    • -... truth of (Vx)(s(x) + r(x)) precisely because this situation can never occur.
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Exercises

1. Let U be the set of all problems on a comprehensive list of problems in science.
   Define four predicates over U by:
   p(x): x is a mathematics problem.
   q(x): x is difficult. (according to some well-defined criterion)
   r(x): x is easy. (according to some well-defined criterion)
   s(x): x is unsolvable.