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Discussion

"The present list of 19 rules of inference constitutes a COMPLETE system of truth-functional logic, in the sense that it permits the construction of a formal proof of validity for ANY valid truth-functional argument." (FN1) 

The first nine rules of the list are rules of inference that "correspond to elementary argument forms whose validity is easily established by truth tables." (Id, page 351). The remaining ten rules are the Rules of Replacement, "which permits us to infer from any statement the result of replacing any component of that statement by any other statement logically equivalent to the component replaced." (Id, page 359). 

Here are the 19 Rules of Inference: 
  

1. Modus Ponens (M.P.)	p  q   p      q
2.Modus Tollens (M.T.)	p  q       ~q   ~p
3.Hypothetical Syllogism (H.S.)	p  q   q  r   p  r
4.Disjunctive Syllogism (D.S.)	p v q   ~ p      q
5. Constructive Dilemma (C.D.)	(p  q) . (r  s)   p v r   q v s
6. Absorption (Abs.)	p  q   p  (p. q)
7. Simplification (Simp.)	p . q   p
8. Conjunction (Conj.)	p   q   p . q
9. Addition (Add.)	p   p v q

10.De Morgan's Theorem (De M.)	~(p . q)  (~p v ~q)   ~(p v q)  (~p . ~q)
11. Commutation (Com.)	(p v q)  (q v p)    (p . q)  (q . p)
12. Association (Assoc.)	[p v (q v r)]  [(p v q) v r]     [p . (q . r)]  [(p . q) . r]
13.Distribution (Dist)	[p . (q v r)]  [(p . q) v (p . r)]   [p v (q . r)]  [(p v q) . (p v r)]
14.Double Negation (D.N.)	p  ~ ~p
15. Transposition (Trans.)	(p  q)  (~q  ~p)
16. Material Implication (M. Imp.)	(p  q)  (~p v q)
17. Material Equivalence (M. Equiv.)	(p  q)  [(p  q) . (q  p)]   (p  q)  [(p . q) v (~p . ~q)]
18. Exportation (Exp.)	[(p . q)  r]  [p  (q  r)]
19. Tautology (Taut.)	p  (p v p)   p  (p . p)