2.3 THEOREMS OF THE PROPOSITIONAL CALCULUS 89(conjunction distributes over
disjunction)
(biconditional law; strategy in an iff
proof )
(equivalence of contrapositive)(strategy for deriving conclusion
9vr)
(strategy for using hypothesis
P v 9)
(strategy for deriving conclusion
9Ar)
(indirect approach to using
hypothesis p A 9)
(strategy for using hypothesis
P A 9)IMPORTANT IMPLICATIONS
If p and q are statement forms such that the conditional p -* q is a tau-
tology, we say that this conditional statement is an implication and that
p logically implies q. Because this situation means that q is true under all
truth conditions for which p is true (i.e., the truth of p "forces" q to be true),
we say that p is a stronger statement form than q, or that q is weaker than
p in this case.
Note the relationship between implication and equivalence. Part (m) of
Theorem 1 indicates that if p and q are equivalent, then each implies the
other. On the other hand, if it is known only that p implies q, the possibility
that p and q are equivalent is left open. In fact, the latter is true if and
only if q implies p. The relationship between implication and equivalence
is described in another way in (n) of Theorem 2; equivalence is stronger than
(mere) implication.
EXAMPLE (^4) Show that p A q is a stronger statement form than p, which is,
in turn, stronger than p v q.
Solution We need only show that both the conditionals (p A q) -+ p and
p -, (p v q) are tautologies. This conclusion should have been reached
in Exercises 2(a, b), Article 2.2. El
EXAMPLE 5 Show that p A (p -+ q) is a stronger statement form than q.
Solution This means simply that the conditional [p ~(p -+ q)] -, q is a
tautology, as may be verified easily by constructing a truth table.