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Symbolic Notation

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Rules of Replacement

The nine rules of inference on the last page cannot prove all valid arguments. There are ten additional rules of inference, called rules of replacement, because the logical expressions on each side of the equation can replace each other.

Rule 1: De Morgan's Theorems

~(p·q)(~p~q)
~(pq)(~p·~q)

Rule 2: Commutation

(pq(qp)
(p·q)(q·p)

Rule 3: Association

[p(qr)][(pq)r]
[p·(q·r)][(p·q)·r)

Rule 4: Distribution

[p·(qr)][(p·q)(p·r)]
[p(q·r)][(pq)·(pr)]

Rule 5: Double Negation

p~~p

Rule 6: Transposition

(pq)(~q~p)

Rule 7: Material Implication

(pq)(~pq)

Rule 8: Material Equivalence

(pq)[(pq)·(qp)]
(pq)[(p·q)(~p·~q)]

Rule 9: Exportation

[(p·q)r][p(qr)]

Rule 10: Tautology

p(pp)
p(p·p)

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