PROPOSITIONAL LOGIC
Proposition
A proposition is a statement that is either true or false,
but not both.
 Atlanta was the site of the 1996 Summer Olympic
games.
 1+1 = 2
 3+1 = 5
 What will my CS1050 grade be?
Definition 1. Negation of p
Let p be a proposition.
The statement “It is not
the case that p” is also a
proposition, called the
“negation of p” or ¬p
(read “not p”)
p = The sky is blue.
p = It is not the case that
the sky is blue.
p = The sky is not blue.
Table 1.
The Truth Table for the
Negation of a Proposition
p
¬p
T
F
F
T
Definition 2. Conjunction of p and q
Let p and q be
propositions. The
proposition “p and q,”
denoted by pq is true
when both p and q are
true and is false
otherwise. This is called
the conjunction of p and
q.
Table 2. The Truth Table for
the Conjunction of two
propositions
p
q
pq
T
T
F
F
T
F
T
F
T
F
F
F
Definition 3. Disjunction of p and q
Table 3. The Truth Table for
the Disjunction of two
propositions
p
q
pq
T
T
F
F
T
F
T
F
T
T
T
F
Let p and q be propositions.
The proposition “p or q,”
denoted by pq, is the
proposition that is false
when p and q are both
false and true otherwise.
Definition 4. Exclusive or of p and q
Table 4. The Truth Table for
the Exclusive OR of two
propositions
p
q
pq
T
T
F
F
T
F
T
F
F
T
T
F
Let p and q be
propositions. The exclusive
or of p and q, denoted by
pq, is the proposition
that is true when exactly
one of p and q is true and
is false otherwise.
Definition 5. Implication pq
Let p and q be propositions. The
implication pq is the
proposition that is false when p
is true and q is false, and true
otherwise. In this implication p is
called the hypothesis (or
antecedent or premise) and q is
called the conclusion (or
consequence).
Table 5. The Truth Table for
the Implication of pq.
p
q
pq
T
T
F
F
T
F
T
F
T
F
T
T
Implications








If p, then q
p implies q
if p,q
p only if q
p is sufficient for q
q if p
q whenever p
q is necessary for p

Not the same as the ifthen construct used in
programming
languages such as If p
then S
Implications
How can both p and q be false, and pq be true?
•Think of p as a “contract” and q as its “obligation” that is
only carried out if the contract is valid.
•Example: “If you make more than $25,000, then you must
file a tax return.” This says nothing about someone who
makes less than $25,000. So the implication is true no
matter what someone making less than $25,000 does.
•Another example:
p: Bill Gates is poor.
q: Pigs can fly.
pq is always true because Bill Gates is not poor. Another
way of saying the implication is
“Pigs can fly whenever Bill Gates is poor” which is true
since neither p nor q is true.
Related Implications
Converse of p  q
is
qp
Contrapositive
of
pq
is the proposition
q  p
Inverse
of p  q
Is the proposition
p  q
Example


implication: “If it rains today, I will go to college
tomorrow”
Converse: I will go to college tomorrow only if it
rains today


Contrapositive : If I do not go to college tomorrow,
then it will not have rained today
Inverse : If it does not rain today, then I will not go
to college tomorrow
Definition 6. Biconditional
Table 6. The Truth Table for
the biconditional pq.
p
q
pq
T
T
F
F
T
F
T
F
T
F
F
T
Let p and q be propositions.
The biconditional pq is
the proposition that is true
when p and q have the
same truth values and is
false otherwise. “p if and
only if q, p is necessary and
sufficient for q”
Practice



p: You learn the simple things well.
q: The difficult things become easy.
You do not learn the simple  The difficult things become
easy but you did not learn
things well.
p
the simple things well.
If you learn the simple
things
things well then the difficult  You learn the simple
q  p
well but the difficult things
things become easy.
did not become easy.
If you do not
learn
the
pq
simple things well, then the
difficult things will not
become easy.
p  q
p  q
Truth Table Puzzle
Steve would like to determine the relative salaries of
three coworkers using two facts (all salaries are
distinct):
 If Fred is not the highest paid of the three, then
Janice is.
 If Janice is not the lowest paid, then Maggie is
paid the most.
Who is paid the most and who is paid the least?
p : Janice is paid the most.
q: Maggie is paid the most.
r: Fred is paid the most.
s: Janice is paid the least.
p
T
F
F
F
F
q
F
T
F
T
F
r
F
F
T
F
T
s
F
T
T
F
F
Fred, Maggie, Janice
rp
T
F
T
F
T
•If Fred is not the highest paid
of the three, then Janice is.
•If Janice is not the lowest paid,
then Maggie is paid the most.
s q (rp) (sq)
F
F
T
F
T
T
T
F
F
F
p : Janice is paid the most.
q: Maggie is paid the most.
r: Fred is paid the most.
s: Janice is paid the least.
p
T
F
F
F
F
q
F
T
F
T
F
r
F
F
T
F
T
s
F
T
T
F
F
rp
T
F
T
F
T
•If Fred is not the highest paid
of the three, then Janice is.
•If Janice is the lowest paid,
then Maggie is paid the most.
s q
T
T
F
T
T
(rp) (sq)
T
F
F
F
T
Fred, Janice, Maggie or Janice, Maggie, Fred
or Janice, Fred, Maggie
Well formed Formula (WFF)





A well formed formula can be produced using following rules:
Rule 1 : A statement variable itself is a WFF
Rule 2 : If p is WFF, then p is WFF
Rule 3 : If p and q are WFF then (p q), (p  q), (p  q) and
(p  q) are also WFF
Rule 4 : A string of symbols consisting of statement variables,
connectives and parentheses is said to be WFF iff it can be
produced by applying rule 1, 2 and 3 finitely many times
Bit Operations
A computer bit has two possible values: 0 (false) and 1
(true). A variable is called a Boolean variable is its value is
either true or false.
Bit operations correspond to the logical connectives:
 OR
 AND
 XOR
Information can be represented by bit strings, which are
sequences of zeros and ones, and manipulated by
operations on the bit strings.
Truth tables for the bit operations OR,
AND, and XOR

0
0 1
0
0 1
1 1
1
1 0

0
0
1
1

0
0
0 0
1
0 1
1
1
Logical Equivalence



An important technique in proofs is to replace a
statement with another statement that is “logically
equivalent.”
Tautology: compound proposition that is always true
regardless of the truth values of the propositions in
it. Eg. p  p
Contradiction: Compound proposition that is always
false regardless of the truth values of the
propositions in it. Eg. p  p
Logically Equivalent


Compound propositions P and Q are logically
equivalent if PQ is a tautology. In other words, P
and Q have the same truth values for all
combinations of truth values of simple propositions.
This is denoted: PQ (or by P Q)

Example: DeMorgans
Prove that (pq)  (p  q)
pq
(pq) (pq) p q (p  q)

TT
TF
FT
FF
T
F
F
F
F
T
F
F
T
F
T
F
F
T
T
T
F
T
F
T
Illustration of De Morgan’s Law
(pq)
p
q
Illustration of De Morgan’s Law
p
p
Illustration of De Morgan’s Law
q
q
Illustration of De Morgan’s Law
p  q
p
q
Example: Distribution
Prove that: p  (q  r)  (p  q)  (p  r)
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r
T
F
T
F
T
F
T
F
qr p(qr) pq pr
T
T
T
T
F
T
T
T
F
T
T
T
F
T
T
T
T
T
T
T
F
F
T
F
F
F
F
T
F
F
F
F
(pq)(pr)
T
T
T
T
T
F
F
F
Prove: pq(pq)  (qp)
p
T
T
F
F
q
T
F
T
F
pq
T
F
F
T
pq qp
T
T
F
T
T
F
T
T
We call this biconditional equivalence.
(pq)(qp)
T
F
F
T
List of Logical Equivalences
pT  p;
pF  p
Identity Laws
pT  T;
pF  F
Domination Laws
pp  p;
pp  p
Idempotent Laws
(p)  p
Double Negation Law
pq  qp; pq  qp
Commutative Laws
(pq) r  p (qr); (pq)  r  p  (qr)
Associative Laws
List of Equivalences
p(qr)  (pq)(pr)
p(qr)  (pq)(pr)
Distribution Laws
(pq)(p  q)
(pq)(p  q)
De Morgan’s Laws
p  p  T
p  p  F
(pq)  (p  q)
Miscellaneous
Or Tautology
And Contradiction
Implication Equivalence
pq(pq)  (qp)
Biconditional Equivalence