TRUTH TABLES
continued
Recall:
• A truth table is a listing of all possible
combinations of the individual statements
as true or false, along with the resulting
truth value of the compound statements.
• Truth tables are an aide in distinguishing
valid and invalid arguments.
Number of Rows
• If a compound statement consists of n
individual statements, each represented
by a different letter, the number of rows
required in the truth table is 2n.
Truth Table for p  q
• Recall that conditional is a
compound statement of the
form “if p then q”.
• Think of a conditional as a
promise.
• If I don’t keep my promise, in
other words q is false, then the
conditional is false if the
premise is true.
• If I keep my promise, that is q
is true, and the premise is true,
then the conditional is true.
• When the premise is false (i.e.
p is false), then there was no
promise. Hence by default the
conditional is true.
p
q
pq
T
T
F
F
T
F
T
F
T
F
T
T
Truth Table for p  q
• Another way to think
about this is with the
Law of Detchment.
• In order for the
conditional statement
to be true, a true
hypothesis must lead
to a true conclusion.
p
q
pq
T
T
F
F
T
F
T
F
T
F
T
T
Truth Table for q  p
• Here we have the
converse, or if q then
p.
• Notice that the
second and thirds
rows switch place as
we are “going
backwards.”
p
q
qp
T
T
F
F
T
F
T
F
T
T
F
T
Equivalent Expressions
• Equivalent
expressions are
symbolic expressions
that have identical
truth values for each
corresponding entry
in a truth table.
• Hence ~(~p) ≡ p.
• The symbol ≡ means
equivalent to.
p ~p ~(~p)
T F
F T
T
F
Negation of the Conditional
• Here we look
at the
negation of
the
conditional.
• Note that the
4th and 6th
columns are
identical.
• Hence p ^ ~q
is equivalent
to ~(p  q).
p
q
~q p ^ ~q p  q ~(p  q)
T
T
F
F
T
F
T
F
T
T
F
T
F
T
F
F
T
F
F
F
T
F
T
F
De Morgan’s Laws
• The negation of the conjunction p ^ q
is given by ~(p ^ q) ≡ ~p v ~q.
“Not p and q” is equivalent to “not p or not
q.”
• The negation of the disjunction p v q
is given by ~(p v q) ≡ ~p ^ ~q.
“Not p or q” is equivalent to “not p and not
q.”
Truth Table for q ↔ p
• Here we have a
biconditional, or p if
and only if q.
• Recall that for a
biconditional to be
true the original
conditional statement
and it converse must
be true, (see cases 1
& 4.)
p
q
q↔p
T
T
F
F
T
F
T
F
T
F
F
T