Propositional logic

Rushdi Shams, Dept of CSE, KUET, Bangladesh 1
Knowledge Representation
Propositional Logic
Artificial Intelligence
Version 2.0
There are 10 types of people in this world- who understand binary
and who do not understand binary

Propositional Logic

Introduction
 Need formal notation to represent knowledge,
allowing automated inference and problem solving.
 One popular choice is use of logic.
 Propositional logic is the simplest.
 Symbols represent facts: P, Q, etc..
 These are joined by logical connectives (and, or,
implication) e.g., P Λ Q; Q R
 Given some statements in the logic we can deduce new
facts (e.g., from above deduce R)

Syntactic Properties of
Propositional Logic
 If S is a sentence, S is a sentence (negation)
 If S1 and S2 are sentences, S1 S2 is a sentence
(conjunction)
(disjunction)
(implication)
(bi-conditional)

Semantic Properties of
Propositional Logic
S is true iff S is false
S1 S2 is true iff S1 is true and S2 is true
S1 S2 is true iff S1is true or S2 is true
S1 S2 is true iff S1 is false or S2 is true
i.e., is false iff S1 is true and S2 is false
S1 S2 is true iff S1 S2 is true and
S2 S1 is true

Truth Table for Connectives

Model of a Formula
 If the value of the formula X holds 1 for the
assignment A, then the assignment A is called model
for formula X.
 That means, all assignments for which the formula X
is true are models of it.

Model of a Formula

Model of a Formula:
Can you do it?

Satisfiable Formulas
 If there exist at least one model of a formula then the
formula is called satisfiable.
 The value of the formula is true for at least one
assignment. It plays no rule how many models the
formula has.

Satisfiable Formulas

Valid Formulas
 A formula is called valid (or tautology) if all
assignments are models of this formula.
 The value of the formula is true for all assignments. If
a tautology is part of a more complex formula then
you could replace it by the value 1.

Valid Formulas

Unsatisfiable Formulas
 A formula is unsatisfiable if none of its
assignment is true in no models

Logical equivalence
 Two sentences are logically equivalent iff true in same models: α ≡ ß
iff α╞ β and β╞ α

Deduction: Rule of Inference
1. Either cat fur was found at the scene of the crime, or dog fur was
found at the scene of the crime. (Premise)
 C v D

2. If dog fur was found at the scene of the crime, then officer
Thompson had an allergy attack. (Premise)
 D → A

3. If cat fur was found at the scene of the crime, then Macavity is
responsible for the crime. (Premise)
 C → M

4. Officer Thompson did not have an allergy attack. (Premise)
 ¬ A

5. Dog fur was not found at the scene of the crime. (Follows from 2
D → A and 4. ¬ A). When is ¬ A true? When A is false- right?
Now, take a look at the implication truth table. Find what is the
value of D when A is false and D → A is true
 ¬ D

Rules for Inference:
Modus Tollens
 If given α → β
and we know ¬β
Then ¬α

6. Cat fur was found at the scene of the crime. (Follows from 1
C v D and 5 ¬ D). When is ¬ D true? When D is false- right?
Now, take a look at the OR truth table. Find what is the value of
C when D is false and C V D is true
 C

Disjunctive Syllogism
 If given α v β
and we know ¬α
then β
 If given α v β
and we know ¬β
then α

7. Macavity is responsible for the crime. (Conclusion. Follows from
3 C → M and 6 C). When is C → M true given that C is true?
Take a look at the Implication truth table.
 M

Modus Ponens
 If given α → β
and we know α
Then β

References
 Artificial Intelligence: A Modern Approach (2nd
Edition)
by Russell and Norvig
Chapter 7
 http://www.iep.utm.edu/p/prop-log.htm#H5

Propositional logic

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Propositional logic