Some of the rules of inference can be proven using the other rules of inference and the laws of propositional logic.

(a)

One of the rules of inference is Modus tollens:

p q q p

Prove that Modus tollens is valid using the laws of propositional logic and any of the other rules of inference besides Modus tollens. (Hint: you will need one of the conditional identities from the laws of propositional logic).

(b)

One of the rules of inference is Modus ponens:

p q p q

Prove that Modus ponens is valid using the laws of propositional logic and any of the other rules of inference besides Modus ponens. (Hint: you will need one of the conditional identities from the laws of propositional logic).

(c)

One of the rules of inference is Disjunctive syllogism :

p q p q

Prove that Disjunctive syllogism is valid using the laws of propositional logic and any of the other rules of inference besides Disjunctive syllogism. (Hint: you will need one of the conditional identities from the laws of propositional logic).

(d)

One of the rules of inference is Resolution:

p q p r q r

Prove that Resolution is valid using the laws of propositional logic and any of the other rules of inference besides Resolution. (Hint: you will need one of the conditional identities from the laws of propositional logic).