Tuesday, May 28, 2013

What are the rules of inference?


To reach to the conclusion from given hypothesis certain valid steps are applied which is called rules of inference.
1.   Modus ponens: Whenever p and p→q are both true then we confirm q is also true. This is valid because (p ˄ (p→q)) →q) is a tautology.
p
q
p→q
p ˄ (p→q)
(p ˄ (p→q)) →q)
T
T
T
T
T
T
F
F
F
T
F
T
T
F
T
F
F
T
F
T

2.   Transitive rule (Hypothetical syllogism): Whenever the two proposition p→q and q→r both are true then we can confirm the implication p→r is true. This is valid because (((p→q) ˄ (q→r)) →(p→r)) is a tautology.
p
q
r
p→q
q→r
p→r
(p→q) ˄ (q→r)
((p→q) ˄ (q→r)) →(p→r))
F
F
F
T
T
T
T
T
F
F
T
T
T
T
T
T
F
T
F
T
F
T
F
T
F
T
T
T
T
T
T
T
T
F
F
F
T
F
F
T
T
F
T
F
T
T
F
T
T
T
F
T
F
F
F
T
T
T
T
T
T
T
T
T

3.   Addition rule: We can conclude the disjunction p ∨ q is true whenever p is true. This is valid because p→(p ∨ q) is a tautology.
p
q
p ∨ q
p→(p ∨ q)
F
F
F
T
F
T
T
T
T
F
T
T
T
T
T
T

4.   Simplification rule: Whenever p ˄ q is true we can conclude p is true. This is valid because (p ˄ q) →p is a tautology.
p
q
p ˄ q
(p ˄ q) →p
F
F
F
T
F
T
F
T
T
F
F
T
T
T
T
T

5.   Conjunction rule: Whenever p and q are true we can conclude p ˄ q is true. This is valid because [((p) ˄ (q))→(p ˄ q)] is a tautology.

6.   Modus Tollens: Whenever ¬q and p→q are true then we can conclude ¬p is true. This is valid because (¬q  ˄ (p→q)) → ¬p is a tautology.
p
q
¬p
¬q
p→q
¬q  ˄ (p→q)
¬q  ˄ (p→q)) → ¬p
T
T
F
F
T
F
T
T
F
F
T
F
F
T
F
T
T
F
T
F
T
F
F
T
T
T
T
T

7.   Disjunction Syllogism: Whenever (p ∨ q) and ¬p are true we can conclude q is true. This is because [(p ∨ q) ˄ ¬p]→q is a tautology.
p
q
¬p
p ∨ q
(p ∨ q) ˄ ¬p
[(p ∨ q) ˄ ¬p]→q
T
T
F
T
F
T
T
F
F
F
F
T
F
T
T
F
F
T
F
F
T
F
F
T

8.   Resolution rule: Whenever (p ∨ q) and (¬p ∨ r) are both true then we can conclude (q ∨ r). This is valid because [((p ∨ q) ˄ (¬p ∨ r)) →(q ∨ r)] is a tautology.
p
q
r
¬p
p ∨ q
¬p ∨ r
q ∨ r
(p ∨ q) ˄ (¬p ∨ r)
[((p ∨ q) ˄ (¬p ∨ r)) →(q ∨ r)]
T
T
T
F
T
T
T
T
T
T
T
F
F
T
F
T
F
T
T
F
T
F
T
T
T
T
T
T
F
F
F
T
F
F
F
T
F
T
T
T
T
T
T
T
T
F
T
F
T
T
F
T
F
T
F
F
T
T
F
T
T
F
T
F
F
F
T
F
F
F
F
T