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CS344 : Introduction to Artificial
Intelligence
Pushpak Bhattacharyya
CSE Dept.,
IIT Bombay
Lecture 5- Deduction Theorem
Formalization of propositional logic (review)
Axioms :
A1
( A  ( B  A))
(( A  ( B  C ))  (( A  B)  ( A  C )))
A2
((( A  F )  F )  A)
A3
Inference rule:
Given ( A  B) 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
Example: To prove ( P  P)
i) P  ( P  P)
A1 : P for A and B
ii) P  (( P  P)  P)
A1: P for A and ( P  P) for B
iii) [( P  (( P  P)  P))  (( P  ( P  P))  ( P  P))]
A2: with P for A,
( P for
P ) B and P for C
iv)( P  ( P  P)  ( P  P))
MP, (ii), (iii)
v) ( P  P )
MP, (i), (iv)
Shorthand
1. ¬ P
2.
is written as
(( P  F )  Q)
PF
and called 'NOT P'
is written as ( P  Q) and called
'P OR Q’
3.
(( P  (Q  F ))  F )
is written as
'P AND Q'
Exercise: (Challenge)
- Prove that
A  (( A))
( P  Q)
and called
A very useful theorem (Actually a meta
theorem, called deduction theorem)
Statement
If
A1, A2, A3 ............. An ├ B
then
A1, A2, A3, ...............An-1├
An  B
├ is read as 'derives'
Given
A1
A2
A3
.
.
.
.
An
B
A1
A2
A3
.
.
.
.
An-1
Picture 1
An  B
Picture 2
Use of Deduction Theorem
Prove
A  (( A))
i.e.,
A  (( A  F )  F )
A, A ├FF
A├ ( A  F )  F
├
A  (( A  F )  F )
(M.P)
(D.T)
(D.T)
Very difficult to prove from first principles, i.e., using axioms and
inference rules only
Prove P  ( P  Q )
i.e. P  (( P  F )  Q)
P, P  F , Q  F ├ F
P, P ├F
├Q
P├
├
(Q  F )(D.T)
F
(M.P with A3)
(P  F )  Q
P  (( P  F )  Q)
More proofs
1. ( P  Q)  ( P  Q)
2. ( P  Q)  (Q  P)
3. ( P  Q)  ((Q  P)  Q)
Proof Sketch of the Deduction
Theorem
To show that
If
A1, A2, A3,… An |- B
Then
A1, A2, A3,… An-1 |- An B
Case-1: B is an axiom
One is allowed to write
A1, A2, A3,… An-1 |- B
|- B(AnB)
|- (AnB); mp-rule
Case-2: B is An
AnAn is a theorem (already proved)
One is allowed to write
A1, A2, A3,… An-1 |- (AnAn)
i.e. |- (AnB)
Case-3: B is Ai
where (i <>n)
Since Ai is one of the hypotheses
One is allowed to write
A1, A2, A3,… An-1 |- B
|- B(AnB)
|- (AnB); mp-rule
Case-4: B is result of MP
Suppose
B comes from applying MP on
Ei and Ej
Where, Ei and Ej come before B in
A1, A2, A3,… An |- B
B is result of MP (contd)
If it can be shown that
A1, A2, A3,… An-1 |- An Ei
and
A1, A2, A3,… An-1 |- (An (EiB))
Then by applying MP twice
A1, A2, A3,… An-1 |- An B
B is result of MP (contd)
This involves showing that
If
A1, A2, A3,… An |- Ei
Then
A1, A2, A3,… An-1 |- An Ei
(similarly for AnEj)
B is result of MP (contd)
Adopting a case by case analysis as
before,
We come to shorter and shorter length
proof segments eating into the body of
A1, A2, A3,… An |- B
Which is finite. This process has to
terminate. QED
Important to note





Deduction Theorem is a meta-theorem
(statement about the system)
PP is a theorem (statement
belonging to the system)
The distinction is crucial in AI
Self reference, diagonalization
Foundation of Halting Theorem, Godel
Theorem etc.
Example of ‘of-about’
confusion

“This statement is false”

Truth of falsity cannot be decided