"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

q 2.Modus Tollens (M.T.)

p 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

p . q 9. Addition (Add.)p

p v q

Any of the following logically equivalent expressions can replace each other wherever they occur:

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)

FN1: Introduction to Logic, Irving M. Copi and Carl Cohen, Prentice Hall, Eleventh Edition, 2001, page 361. The book contains the following footnote after this paragraph: "A method of proving this kind of completeness for a set of rules of inference can be found in I. M. Copi, Symbolic Logic, 5th Edition. (New York: Macmillian, 1979), chap 8, See also John A. Winnie, "The Completeness of Copi's System of Natural Deduction," Notre Dame Journal of Formal Logic 11 (July 1970), 379-382."

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