18 Rules of Inference

1. 18 Rules of Inference Rules of inference specify conditions under which certain inferences are truth preserving (valid) 8 rules of implication 10 rules of replacement

2. Rules of Implication These rules apply to substitution instances of argument forms The rules apply to entire lines, not parts of lines They are �one-way�: We can infer a statement (conclusion) from a set of given statements (premises), but we cannot infer the premises from the conclusion

3. For Example 1. F?(J?K) 2. J 3. K (1, 2 MP) This is an incorrect application of modus ponens, since it is applying the rule to a part of line 1 (the consequent), not the whole line. Notice that the above argument is not a substitution instance of p?q p q The antecedent in 1 is not the same statement as in line 2

4. Continued 1. p?q 2. p 3. q Though lines 1 and 2 imply line 3, line 3 does not imply lines 1 and 2. This is the sense in which rules of implication are one-way

5. Rules of Replacement These rules tell us when we can replace a statement with another statement that is logically equivalent to the original statement The rules can apply to parts of lines They are �two-way�

6. For Example Consider the replacement rule called Material implication: (p?q) :: (~p v q) (the double colon denotes logical equivalence) Consider this argument: G v (D?L) G v (~D v L) (1, Impl) We are applying Implication to a part of line 1 (the right disjunct)

7. Continued G?H ~G v H (1, Impl) 1. ~G v H 2. G?H (1, Impl) Because these statements are logically equivalent, they imply each other. It is in this sense that rules of replacement are two-way

8. More Rules of Implication So far we have looked at 5 rules of implication: modus ponens, modus tollens, disjunctive syllogism, hypothetical syllogism, and constructive dilemma There are 3 additional rules of implication, all of which are rather intuitive

9. Continued Conjunction: Simplification p p * q q p p * q Addition: p p v q

10. Be Strategic When doing derivations, first try to �find� the conclusion in the premises Set up sub-goals whose satisfaction will enable you to reach your ultimate conclusion Think in terms of �if-then:� if I can get X, then I can get Y; if I can get Y, then I can get Z, and so forth

11. Let�s Try Some Derive L from P C v L (P v G)?~C

12. See? Derive L from P C v L (P v G)?~C P v G (1, Add) ~C (3, 4 MP) L (2, 5 DS)

13. Another Derive L * G from G * M G?S S?L

14. See? Derive G * L from G * M G?S S?L G (1, Simp) S (2, 4 MP) L (3, 5 MP) G * L (4, 6 Conj)

15. One More Derive (H v L)?(M * N) from D v ((H v L)?R) (R?(M * N)) * F ~D

16. See? Derive (H v L)?(M * N) from D v ((H v L)?R) (R?(M * N)) * F ~D 4. (H v L)?R (1, 3 DS) 5. R?(M * N) (2, Simp) 6. (H v L)?(M * N) (4, 5 HS)

17. Rules of Replacement DeMorgan�s Rule: ~(p * q) :: (~p v ~q) ~(p v q) :: (~p * ~q) Commutativity: (p v q) :: (q v p) (p * q) :: (q * p) Associativity: [p v (q v r)] :: [(p v q) v r] [p * (q * r)] :: [(p * q) * r]

18. Continued Distribution: [p * (q v r)] :: [(p * q) v (p * r)] [p v (q * r)] :: [(p v q) * (p v r)] Double Negation: p :: ~~p

19. Let�s Try Some More Derive ~(G * J) from F?~((G * J) * K) F * K

20. See? Derive ~(G * J) from F?~((G * J) * K) F * K F (2, Simp) ~((G * J) * K) (1, 3 MP) ~(G * J) v ~K (4, DM) K * F (2, Com) K (6, Simp) ~K v ~(G * J) (5, Com) ~~K (7, DN) ~(G * J) (8, 9 DS)

21. Another Derive M from L * (K * J) (G v K)?M

22. See? Derive M from L * (K * J) (G v K)?M 3. (L * K) * J (1, Assoc) 4. L * K (3, Simp) 5. K * L (4, Com) 6. K (5, Simp) 7. K v G (6, Add) 8. G v K (7, Com) 9. M (2, 8 MP)

23. One More Derive S * K from (M v S) * (M v K) M?G ~G * ~F

24. See? Derive S * K from (M v S) * (M v K) M?G ~G * ~F ~G (3, Simp) ~M (2, 4 MT) M v (S * K) (1 Dist) S * K (5, 6 DS)

25. Or� Derive S * K from (M v S) * (M v K) M?G ~G * ~F ~G (3, Simp) ~M (2, 4 MT) M v S (1, Simp) S (5, 6 DS) (M v K) * (M v S) (1, Com) M v K (8, Simp) K (5, 9 DS) S * K (7, 10 Conj)