Exercise 3: Is p -> q Equivalent to Its Converse q -> p? — Possible Solution
====================================================================

BUILDING BOTH TRUTH TABLES
------------------------------
  p | q | p -> q | q -> p
  --|---|--------|--------
  T | T |   T    |   T
  T | F |   F    |   T
  F | T |   T    |   F
  F | F |   T    |   T

Working through q -> p specifically: per this chapter's own truth
table for IMPLIES, "x -> y is false only when x is true and y is
false." For q -> p: row 2 (p=T, q=F) means q=F, p=T, so q -> p has a
false premise (q=F) and is therefore true, per the "false premise
means true" rule from this chapter. Row 3 (p=F, q=T) means q=T
(premise true) and p=F (conclusion false) - this is exactly the one
case where an implication is false, so q -> p = F here.

COMPARING THE TWO COLUMNS
------------------------------
p -> q gives: T, F, T, T
q -> p gives: T, T, F, T

These disagree in rows 2 and 3 - exactly two of the four possible
combinations. Since logical equivalence requires matching on every
row, p -> q and q -> p are NOT logically equivalent.

A CONCRETE REAL-WORLD EXAMPLE
------------------------------
"If it is raining (p), then the ground is wet (q)" is a true
statement in the ordinary sense - rain reliably wets the ground.

Its converse, "If the ground is wet (q), then it is raining (p)," is
false - the ground could be wet from a sprinkler, a spilled bucket, or
someone hosing down a driveway, none of which involve rain at all.
This is exactly row 3 from the truth table: q is true (the ground is
wet) but p is false (it isn't raining), which is precisely the one
case where an implication comes out false.

WHY THIS DISTINCTION MATTERS BEYOND THIS ONE EXAMPLE
------------------------------
Confusing a true implication with its converse is a common real
reasoning error - just because "if p then q" holds doesn't license
concluding "if q then p" also holds. The rain/wet-ground example
makes this concrete: observing a wet ground is not, by itself,
sufficient evidence that it rained.

WHY THIS WORKS AS AN ANSWER
------------------------------
It builds and explains both truth tables explicitly rather than just
stating the conclusion, identifies precisely which rows disagree,
and supplies a concrete, verifiable real-world scenario that
demonstrates why a true implication doesn't guarantee its converse is
also true.