# Rules of implication

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"Rules of implication"

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Rules of implication

1. Modus ponens (MP)                        5. Constructive dilemma (CD)

pq                                            (p  q)  (r  s)
p                                              pr
q                                              qs
2. Modus tollens (MT)
6. Simplification (Simp)
pq
pq
~q
p
~p
3. Hypothetical syllogism (HS)                      7. Conjunction (Conj)
pq                                            p
qr                                            q
pr                                            pq
4. Disjunctive syllogism (DS)
pq
p
~p
pq
q

Rules of Replacement

9. DeMorgan’s rule (DM)                        14. Transposition (Trans)
~(p  q) :: (~p  ~q)                           (p  q) :: (~q  ~p)
~(p  q) :: (~p  ~q)
10. Commutativity (Com)
15. Material implication (Impl)
(p  q) :: (q  p)
(p  q) :: (~p  q)
(p  q) :: (q  p)
11. Associativity (Assoc)                   16. Material equivalence (Equiv)

[p  (q  r)] :: [(p  q)  r]                  (p  q) :: [(p  q)  (q  p)]
[p  (q  r)] :: [(p  q)  r]                (p  q) :: [(p  q)  (~p  ~q)]
12. Distribution (Dist)
17. Exportation (Exp)
[p  (q  r)] :: [(p  q)  (p  r)]
[(p  q)  r] :: [p  (q  r)]
[p  (q  r)] :: [(p  q)  (p  r)]

18. Tautology (Taut)
13. Double negation (DN)
p :: (p  p)
p :: ~~p
p :: (p  p)

```
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