Exercise 15.

Can you chain implications together? That is, if \(P \imp Q\) and \(Q \imp R\text{,}\) does that means the \(P \imp R\text{?}\) Can you chain more implications together? Let's find out:

  1. Prove that the following is a valid deduction rule:

    \(P \imp Q\)
    \(Q \imp R\)
    \(\therefore\) \(P \imp R\)

  2. Prove that the following is a valid deduction rule for any \(n \ge 2\text{:}\)

    \(P_1 \imp P_2\)
    \(P_2 \imp P_3\)
    \(\vdots\)
    \(P_{n-1} \imp P_n\)
    \(\therefore\) \(P_1 \imp P_n\text{.}\)

    I suggest you don't go through the trouble of writing out a \(2^n\) row truth table. Instead, you should use part (a) and mathematical induction.

Hint.
in-context