Use propositional logic to prove the validity of the arguments in Exercises 25β33. These will become additional derivation rules for propositional logic, summarized in Table 1.14.
25. (P β¨ Q) β§ P' β Q
26. (P β Q) β (Q' β P')
27. (Q' β P') β (P β Q)
28. P β P β§ P
29. P β¨ P β P (Hint: Instead of assuming the hypothesis, begin with a version of Exercise 28; also make use of Exercise 27.)
30. [(P β§ Q) β R] β [P β (Q β R)]
31. P β§ P' β Q
32. P β§ (Q β¨ R) β (P β§ Q) β¨ (P β§ R) (Hint: First rewrite the conclusion.)
33. P β¨ (Q β§ R) β (P β¨ Q) β§ (P β¨ R) (Hint: Prove both P β¨ (Q β§ R) β (P β¨ Q) and P β¨ (Q β§ R) β (P β¨ R); for each proof, first rewrite the conclusion.)
TABLE 1.14
More Inference Rules
From
Can Derive
Name/Abbreviation for Rule
P β Q, Q β R
P β R [Example 16]
Hypothetical syllogism β hs
P β¨ Q, P'
Q [Exercise 25]
Disjunctive syllogism β ds
P β Q
Q' β P' [Exercise 26]
Contraposition β cont
Q' β P'
P β Q [Exercise 27]
Contraposition β cont
P
P β§ P [Exercise 28]
Self-reference β self
P β¨ P
P [Exercise 29]
Self-reference β self
(P β§ Q) β R
P β (Q β R) [Exercise 30]
Exportation β exp
P, P'
Q [Exercise 31]
Inconsistency β inc
P β§ (Q β¨ R)
(P β§ Q) β¨ (P β§ R) [Exercise 32]
Distributive β dist
P β¨ (Q β§ R)
(P β¨ Q) β§ (P β¨ R) [Exercise 33]
Distributive β dist