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
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p→q
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p ˄ (p→q)
|
(p ˄ (p→q)) →q)
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T
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T
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T
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T
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T
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T
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F
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F
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F
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T
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F
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T
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T
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F
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T
|
|
F
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F
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T
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F
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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
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q→r
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p→r
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(p→q) ˄ (q→r)
|
((p→q) ˄ (q→r))
→(p→r))
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F
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F
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F
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T
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T
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T
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T
|
T
|
|
F
|
F
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T
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T
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T
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T
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T
|
T
|
|
F
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T
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F
|
T
|
F
|
T
|
F
|
T
|
|
F
|
T
|
T
|
T
|
T
|
T
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T
|
T
|
|
T
|
F
|
F
|
F
|
T
|
F
|
F
|
T
|
|
T
|
F
|
T
|
F
|
T
|
T
|
F
|
T
|
|
T
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T
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F
|
T
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F
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F
|
F
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T
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|
T
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T
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T
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T
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T
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T
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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
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F
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T
|
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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
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F
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T
|
|
F
|
T
|
F
|
T
|
|
T
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F
|
F
|
T
|
|
T
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T
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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
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T
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F
|
F
|
T
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F
|
T
|
|
T
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F
|
F
|
T
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F
|
F
|
T
|
|
F
|
T
|
T
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F
|
T
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F
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T
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|
F
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F
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T
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T
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T
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T
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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
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F
|
T
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F
|
T
|
|
T
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F
|
F
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F
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F
|
T
|
|
F
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T
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T
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F
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F
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T
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|
F
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F
|
T
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F
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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)]
|
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T
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T
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T
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F
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T
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T
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T
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T
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T
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T
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T
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F
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F
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T
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F
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T
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F
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T
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T
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F
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T
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F
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T
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T
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T
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T
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T
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T
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F
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F
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F
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T
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F
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F
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F
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T
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F
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T
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T
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T
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T
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T
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T
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T
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T
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F
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T
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F
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T
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T
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F
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T
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F
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T
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F
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F
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T
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T
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F
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T
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T
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F
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T
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F
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F
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F
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T
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F
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F
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F
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F
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T
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