(a) Given primitive statements p, q, r, show that the implication [(p q) (p ...

Question:

(a) Given primitive statements p, q, r, show that the implication
[(p ˆ¨ q) ˆ§ (¬p ˆ¨ r)] †’ (q ˆ¨ r)
is a tautology.
(b) The tautology in part (a) provides the rule of inference known as resolution, where the conclusion (q ˆ¨ r) is called the resolvent. This rule was proposed in 1965 by J. A. Robinson and is the basis of many computer programs designed to automate a reasoning system.
In applying resolution each premise (in the hypothesis) and the conclusion are written as clauses. A clause is a primitive statement or its negation, or it is the disjunction of terms each of which is a primitive statement or the negation of such a statement. Hence the given rule has the clauses (p ˆ¨ q) and (¬p ˆ¨ r) as premises and the clause (q ˆ¨ r) as its conclusion (or, resolvent). Should we have the premise ¬(p ˆ§ q), we replace this by the logically equivalent clause ¬p ˆ¨ ¬q, by the first of DeMorgan's Laws. The premise ¬(p ˆ¨ q) can be replaced by the two clauses ¬p, ¬ q. This is due to the second DeMorgan Law and the Rule of Conjunctive Simplification. For the premise p ˆ¨ (q ˆ§ r), we apply the Distributive Law of ˆ¨ over ˆ§ and the Rule of Conjunctive Simplification to arrive at either of the two clauses p ˆ¨ q, p ˆ¨ r. Finally, the premise p †’ q becomes the clause ¬ p ˆ¨ q.
Establish the validity of the following arguments, using resolution (along with the rules of inference and the laws of logic).
(i)

(ii)

(iii)

(a) Given primitive statements p, q, r, show that the

(iv)

(v)

(vi)

(c) Write the following argument in symbolic form, then use resolution (along with the rules of inference and the laws of logic) to establish its validity.
Jonathan does not have his driver's license or his new car is out of gas. Jonathan has his driver's license or he does not like to drive his new car. Jonathan's new car is not out of gas or he does not like to drive his new car. Therefore, Jonathan does not like to drive his new car.