Prove that implication is transitive in the propositional calculus, that is, that ((P → Q) ∧ (Q → R)) → (P → R). a. Prove that modus ponens is sound for propositional calculus. Hint: use truth tables...

Prove that implication is transitive in the propositional calculus, that is, that ((P → Q) ∧ (Q → R)) → (P → R).

a. Prove that modus ponens is sound for propositional calculus. Hint: use truth tables to enumerate all possible interpretations.

b. Abduction is an inference rule that infers P from P → Q and Q. Show that abduction is not sound (see Chapter 7).

c. Show modus tollens ((P → Q) ∧ ¬ Q) → ¬ P is sound.