Introduction
Predicates and Quantifiers
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics  Predicates and Quantifiers
72
Conjunctive Normal Form
A
literal
is a primitive statement (propositional variable) or its
negation
p,
¬
p,
q,
¬
q
A
clause
is a disjunction of one or more literals
p
∨
q,
p
∨ ¬
q
∨
r,
¬
q,
¬
s
∨
s
∨ ¬
r
∨ ¬
q
A statement is said to be a
Conjunctive Normal Form (CNF)
if it is
a conjunction of clauses
p
∧
(p
∨ ¬
q)
∧
(
¬
r
∨ ¬
p)
p
∧
q
∧
(
¬
r
∨ ¬
p)
(
¬
r
∨
q)
∧
(p
∨ ¬
q
∨ ¬
s
∨
r)
∧
(
¬
r
∨ ¬
p)
¬
r
Discrete Mathematics  Predicates and Quantifiers
73
CNF Theorem
Theorem
Every statement is logically equivalent to a certain CNF.
Proof
(sketch)
Step 1.
Express all logic connectives in
Φ
through negation,
conjunction, and disjunction.
Let
Ψ
be the obtained statement.
Let
Φ
be a (compound) statement.
Step 2.
Using DeMorgan’s laws move all the negations in
Ψ
to
individual primitive statements. Let
Θ
denote the updated statement
Step 3.
Using distributive laws transform
Θ
into a CNF.
Discrete Mathematics  Predicates and Quantifiers
74
Example
Find a CNF logically equivalent to
(p
→
q)
→
r
Step 1.
¬
(
¬
p
∨
q)
∨
r
Step 2.
(p
∧ ¬
q)
∨
r
Step 3.
(p
∨
r)
∧
(
¬
q
∨
r)
Discrete Mathematics  Predicates and Quantifiers
75
Rule of Resolution
p
∨
q
¬
p
∨
r
∴
q
∨
r
The corresponding tautology
((p
∨
q)
∧
(
¬
p
∨
r))
→
(q
∨
r)
``Jasmine is skiing or it is not snowing.
It is snowing or Bart is playing hockey.’’
p
 `it is snowing’
q
 `Jasmine is skiing’
r
 `Bart is playing hockey’
``Therefore, Jasmine is skiing or Bart is playing hockey’’
q
∨
r is called
resolvent
Discrete Mathematics  Predicates and Quantifiers
76
Computerized Logic Inference
Convert the premises into CNF
Convert the
negation
of the conclusion into CNF
Consider the collection consisting of all the clauses that occur in
the obtained CNFs
Use the rule of resolution to obtain the
empty clause
(
∅
).
If it is
possible, then the argument is valid.
Otherwise, it is not.
Why empty clause?
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 Spring '10
 AndreiBulatov
 Logic, Quantification, Universal quantification, Existential quantification

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