This document discusses logic and inferencing in artificial intelligence. It covers: - Deduction, induction, and abduction as forms of reasoning. Deduction reasons from general to specific, induction from specific to general, and abduction concludes a hypothesis from an observation. - Propositional and predicate logic. Propositional logic uses propositions and operators like AND and OR. Predicate logic adds predicates and quantifiers. - Methods for proving tautologies like semantic trees and Hilbert's formalization of propositional logic using axioms and the modus ponens inference rule.

1. CS344 : Introduction to Artificial
Intelligence
Pushpak Bhattacharyya
CSE Dept.,
IIT Bombay
Lecture 4- Logic

2. Logic and inferencing
Vision NLP
Expert
Systems
Planning
Robotics
 Search
 Reasoning
 Learning
 Knowledge
Obtaining implication of given facts and rules -- Hallmark of
intelligence

3. Inferencing through
− Deduction (General to specific)
− Induction (Specific to General)
− Abduction (Conclusion to hypothesis in absence of any other evidence
to contrary)
Deduction
Given: All men are mortal (rule)
Shakespeare is a man (fact)
To prove: Shakespeare is mortal (inference)
Induction
Given: Shakespeare is mortal
Newton is mortal (Observation)
Dijkstra is mortal
To prove: All men are mortal (Generalization)

4. If there is rain, then there will be no picnic
Fact1: There was rain
Conclude: There was no picnic
Deduction
Fact2: There was no picnic
Conclude: There was no rain (?)
Induction and abduction are fallible forms of reasoning. Their conclusions are
susceptible to retraction
Two systems of logic
1) Propositional calculus
2) Predicate calculus

5. Propositions
− Stand for facts/assertions
− Declarative statements
− As opposed to interrogative statements (questions) or imperative
statements (request, order)
Operators
=> and ¬ form a minimal set (can express other operations)
- Prove it.
Tautologies are formulae whose truth value is always T, whatever the
assignment is
)
(
(~),
),
(
),
( 

 N
IMPLIC
NOT
OR
AND

6. Model
In propositional calculus any formula with n propositions has 2n models
(assignments)
- Tautologies evaluate to T in all models.
Examples:
1)
2)
-e Morgan with AND
P
P 

)
(
)
( Q
P
Q
P 






7. Semantic Tree/Tableau method of proving tautology
Start with the negation of the formula
α-formula
β-formula
α-formula
p
q
¬q
¬ p
- α - formula
- β - formula
)]
(
)
(
[ Q
P
Q
P 






)
( Q
P 

)
( Q
P 



- α - formula

8. Example 2:
B C B C
Contradictions in all paths
X
α-formula
¬ A ¬C
¬ A
¬B ¬ A ¬B
A
B∨ C
A
B∨ C A
B∨ C
A
B∨ C
(α - formulae)
(β - formulae)
(α - formula)
)]
(
)
(
)
(
[ C
A
B
A
C
B
A 






)
( C
B
A 

))
(
)
(( C
A
B
A 



)
( B
A 

))
( C
A 


9. Exercise:
Prove the backward implication in the previous
example

10. Formal Systems
 Rule governed
 Strict description of structure and rule application
 Constituents
 Symbols
 Well formed formulae
 Inference rules
 Assignment of semantics
 Notion of proof
 Notion of soundness, completeness, consistency,
decidability etc.

11. Hilbert's formalization of propositional calculus
1. Elements are propositions : Capital letters
2. Operator is only one :  (called implies)
3. Special symbol F (called 'false')
4. Two other symbols : '(' and ')'
5. Well formed formula is constructed according to the grammar
WFF P|F|WFFWFF
6. Inference rule : only one
Given AB and
A
write B
known as MODUS PONENS

12. 7. Axioms : Starting structures
A1:
A2:
A3
This formal system defines the propositional calculus
))
(
( A
B
A 

)))
(
)
((
))
(
(( C
A
B
A
C
B
A 





)
)
)
((( A
F
F
A 



13. Notion of proof
1. Sequence of well formed formulae
2. Start with a set of hypotheses
3. The expression to be proved should be the last line in the
sequence
4. Each intermediate expression is either one of the hypotheses
or one of the axioms or the result of modus ponens
5. An expression which is proved only from the axioms and
inference rules is called a THEOREM within the system

14. Example of proof
From P and and prove R
H1: P
H2:
H3:
i) P H1
ii) H2
iii) Q MP, (i), (ii)
iv) H3
v) R MP, (iii), (iv)
Q
P 
Q
P 
Q
P 
R
Q 
R
Q 
R
Q 

15. Prove that is a THEOREM
i) A1 : P for A and B
ii) A1: P for A and for B
iii)
A2: with P for A, for B and P for C
iv) MP, (ii), (iii)
v) MP, (i), (iv)
)
( P
P 
)
)
(( P
P
P
P 


)
( P
P
P 

))]
(
))
(
((
))
)
((
[( P
P
P
P
P
P
P
P
P 







)
( P
P 
))
(
)
(
( P
P
P
P
P 



)
( P
P 
)
( P
P 

16. Formalization of propositional logic (review)
Axioms :
A1
A2
A3
Inference rule:
Given and A, write B
A Proof is:
A sequence of
i) Hypotheses
ii) Axioms
iii) Results of MP
A Theorem is an
Expression proved from axioms and inference rules
))
(
( A
B
A 

)))
(
)
((
))
(
(( C
A
B
A
C
B
A 





)
)
)
((( A
F
F
A 


)
( B
A 

17. Example: To prove
i) A1 : P for A and B
ii) A1: P for A and for B
iii)
A2: with P for A, for B and P for C
iv) MP, (ii), (iii)
v) MP, (i), (iv)
)
( P
P 
)
)
(( P
P
P
P 


)
( P
P
P 

))]
(
))
(
((
))
)
((
[( P
P
P
P
P
P
P
P
P 







)
( P
P 
))
(
)
(
( P
P
P
P
P 



)
( P
P 
)
( P
P 

18. Shorthand
1. is written as and called 'NOT P'
2. is written as and called
'P OR Q’
3. is written as and called
'P AND Q'
Exercise: (Challenge)
- Prove that
¬P F
P 
)
)
(( Q
F
P 
 )
( Q
P 
)
))
(
(( F
F
Q
P 

 )
( Q
P 
))
(
( A
A 



19. A very useful theorem (Actually a meta
theorem, called deduction theorem)
Statement
If
A1, A2, A3 ............. An ├ B
then
A1, A2, A3, ...............An-1├
├ is read as 'derives'
Given
A1
A2
A3
.
.
.
.
An
B
Picture 1
A1
A2
A3
.
.
.
.
An-1
Picture 2
B
An 
B
An 

20. Use of Deduction Theorem
Prove
i.e.,
├ F (M.P)
A├ (D.T)
├ (D.T)
Very difficult to prove from first principles, i.e., using axioms and
inference rules only
))
(
( A
A 


)
)
(( F
F
A
A 


F
A
A 
,
F
F
A 
 )
(
)
)
(( F
F
A
A 



21. Prove
i.e.
├ F
├ (D.T)
├ Q (M.P with A3)
P├
├
)
( Q
P
P 

)
)
(( Q
F
P
P 


F
Q
F
P
P 
 ,
,
F
P
P 
, F
F
Q 
 )
(
Q
F
P 
 )
(
)
)
(( Q
F
P
P 

