Why is the contrapostive of an implication equivalent to its normal truth table? i.e. why is this the case:
$ \begin{array}{c|l|c} p & q & \text~p \implies \text~q \\ \hline 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \\ 0 & 0 & 1 \\ \end{array} $
Given that the nomal implication table is:
$ \begin{array}{c|l|c} B & A & B \implies A \\ \hline 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \\ 0 & 0 & 1 \\ \end{array} $
Specifically, in the first table and first row: p = 1, thus ~p = 0; q = 0, thus ~q = 1. Given these, if one enters these values ( B=0 and A=1) into the second, basic implication table, then the statement is true.
An example would help. I cannot grasp the meaning so I can't really think of any good examples.