Logic 2
The conditional and biconditional
The Conditional
• Consider two statements, p and q.
• If p being true always leads to q being true, we have a
conditional relationship
“if p is true then q is true”, “if p then q”, “p q”. See
Waner and Conenoble for other equivalent phrases.
These all say the same thing, and have the following
truth table:
• P is the antecedent of hypothesis, q is the consequent
or conclusion.
Rules of the conditional
• The only way p q is false is if p is true and q is
false. Called a “broken promise.”
• If p and q are both false, p q is true.
• p q is true when p is false, no matter what the
truth value of q.
• True always implies true. If p and q are both true,
then pq is true.
• True can’t imply false. If p is true and q is false
then pq is false.
• False implies anything. If p is false, then pq is
true no matter if q is true or not.
The “switcheroo” law
Show that p q  (p)q
This is often important for proofs.
This implies that p q is true if either p is false or if q is
true. Using DeMorgan’s Law, this also means that p q
 (p(q))
Important equivalencies of p q
if p then q
q follows from p
q if p
whenever p, q
p is sufficient for q
is the same as
is the same as
is the same as
is the same as
is the same as
p implies q
not p unless q
p only if q
q whenever p
q is necessary for p
Notice how p being a sufficient condition for q makes q a
necessary condition for p.
Exercise:
Show that the commutative law does not hold for the
conditional, that is that p q is not equivalent to q p
using a truth table.
The statement p q is called the converse of q p, and
q p is the converse of p q. They are not equivalent.
Exercises
Give an example where the converse of a conditional is
not true. (Think advertising).
Give an example when the converse of a conditional is
true.
Contrapositive
The contrapositive of p q is (q) (p)
Exercise 1: Use truth tables to show a statement and its
contrapositive are equivalent.
Exercise 2: Use switcheroo and other properties to show a
statement and its contrapositive are equivalent.
p q  (p)q  q(p)  (q)(p)  (q) (p)
switcheroo; commutative; double neg; switcheroo
Exercise 3:
Give the converse and contrapositive of the statement
If you study hard you will get a good grade. Explain
how the contrapositive can be interpreted as true, but
not necessarily the converse.
Converse: If you get a good grade you studied hard.
Contrapositive: If you don’t get a good grade, you did
not study hard.
Biconditional
The biconditional, written pq, is defined to be the
statement (p q)(q p). With the true table
Exercise: Use truth tables to show that
pq (p q)(q p).
Common phrasings for the biconditional
• p if and only if q
• p is necessary and equivalent for q
• p is equivalent to q