Symbolic Logic

The Following slide were written using
materials from the Book:
Discrete mathematics With
Applications
Third Edition
By Susanna S. Epp
Symbolic Logic
The main purpose of logic is to build the
thinking methods.
 Provide rules,techniques, for making
decision in an argument, validating a
deduction.
 In classical logic, only phrases , assertions
with one truth value are allowed: TRUE or
FALSE, without ambiguity
 Such boolean assertions are called
propositions.

Symbolic Logic
A proposition is a statement that is either true
or false.
 The purpose of propositional logic is to
provide complex construction of rules, from
anonymous propositions called propositional
variables
 If p is a proposition ,the negation of p,
denoted by ¬p, is a proposition which means "
it is false that". Then if p is true , ¬p is false,
and if p is false , ¬p is true.

Symbolic Logic
A proposition consisting of only a single
propositional variable or single constant
(true or false) is called an atomic
proposition.
 All nonatomic propositions are called
compound propositions. All compound
propositions contain at least one logical
connective.

Symbolic Logic


If p and q are propositions, the conjunction of p
and q is the proposition " p and q” denoted by
pq.
The proposition pq is true if p and q are both
true, and false otherwise; this is describe by the
following truth table:
p
q
pq
0
0
0
0
1
0
1
0
0
1
1
1
Symbolic Logic


The disjunction of p and q is the proposition " p
or q” denoted by pq.
The proposition pq is true if at least one of the
two propositions p and p is true, and false when p
and q are both false; this is described by the
following truth table:
p
q
pq
0
0
0
0
1
1
1
0
1
1
1
1
Symbolic Logic
By combining,¬,, we can build
compound propositions and construct their
truth tables.
 Truth table for: (p q ) ¬r

(p q ) ¬r
p
1
1
1
1
0
0
0
0
q
1
1
0
0
1
1
0
0
r
1
0
1
0
1
0
1
0
pq
1
1
0
0
0
0
0
0
¬r
0
1
0
1
0
1
0
1
(p q ) ¬r
1
1
0
1
0
1
0
1
Logical equivalence

Two statements are logically equivalent if they
have equivalent truth tables.
The symbol for Logical equivalence is 

Example: p  q  q  p

1
p q
1
q p
1
1
0
0
0
0
1
0
0
0
0
0
0
P
Q
1
Double negative property
The negation of the negation of a statement
is logically equivalent to the statement
 ¬(¬p)  p

p
¬p
¬(¬p)
1
0
1
0
1
0
Showing nonequivalence

Show that the statement forms ¬(pq) and
¬p¬q are not logically equivalent.
p
q
¬p ¬q
pq ¬(pq) ¬p¬q
1
1
0
0
1
0
0
1
0
0
1
0
1
0
0
1
1
0
0
1
0
0
0
1
1
0
1
1
De Morgan’s laws
The negation of a conjunction of two
statements is logically equivalent to the
disjunction of their negations.
 ¬(pq)  ¬p  ¬q
 The negation of the disjunction of two
statements is logically equivalent to the
conjunction of their negation.
 ¬(p  q)  ¬p  ¬q

De Morgan’s laws
p
q
¬p ¬q pq ¬(pq) ¬p  ¬q
1
1
0
0
1
0
0
1
0
0
1
0
1
1
0
1
1
0
0
1
1
0
0
1
1
0
1
1
De Morgan’s laws
p
q
¬p ¬q pq ¬(pq) ¬p¬q
1
1
0
0
1
0
0
1
0
0
1
1
0
0
0
1
1
0
1
0
0
0
0
1
1
0
1
1
Tautologies and Contradiction
A tautology is a statement form that is
always true regardless of the truth values of
individual statements substituted for its
statement variables.
 A contradiction is a statement form that is
always false regardless of the truth values
of individual statements substituted for its
statement variables.

Tautologies and Contradiction
p
¬p p¬p p¬p
1
0
1
0
0
1
1
0
Tautology
Contradiction
Logical Equivalences
Given any statement variables p, q, and r, a tautology
t and a contradiction c the following logical
equivalences hold.
1.
Commutative law
pq  qp p  q  q  p
2.
Associative law
(pq)r  p (qr )
(pq)r  p(qr )
3.
Distributive law
p(qr)(pq)(pr )
p(q r)(pq) (pr )
4.
Identity
ptp
pcp

Logical Equivalences
Negation laws
p¬p t
p ¬p c
6.
Double negative law
¬ (¬ p)p
7.
Idempotent
pp p
pp p
8.
Universal bounds laws
pt t
pc c
9.
De Morgan’s laws
5.
¬(pq)  ¬p  ¬q
¬(p  q)  ¬p  ¬q
Logical Equivalences
10.
Absorption laws
p(pq)  p
p  (pq)  p
11. Negations of t and c
¬t  c
¬c  t
Simplifying statements












Verify the equivalence
¬(¬pq)(p q)  p
By De Morgan’s
(¬(¬p)¬q)(p q)
By double negative law
(p¬q)(p q)
By distributive law
 p(¬qq)
By negation law
 pc
By identity law
p
Simplifying statements
Verify the equivalence
 ¬(p¬q)(¬p¬ q)  ¬p

Simplify the following expressions. State the rule
you are using at each stage.
((pq) (pq))(pq)
 ( (pq) (pq))(pq) De Morgan’s law
 ( ()pq)( ()p()q))(pq) De
Morgan’s law
 (pq)(pq)(pq)
Double negative law

p (qq)(pq)
Distributive laws

(p t )(pq)
Negation laws

p(pq)
Universal bounds laws
 pq