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:
Prove that the following is a valid deduction rule:
\(P \imp Q\) | |
\(Q \imp R\) | |
\(\therefore\) | \(P \imp R\) |
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.