Question

How do you create a truth table to prove that for any statement, p,~(~p) equals p?

Answer 1

If $p$ is true, then $\sim p$ is false, which in turn means $\sim(\sim p)$ is true.

If $p$ is false, then $\sim p$ is true, and so $\sim(\sim p)$ is false.

So, because $p\equiv\sim(\sim p)$ in both cases, the statement is a tautology (always true).

If you were to put this in a table, you would have one column each for $p,\sim p,\sim(\sim p)$. In the first column ($p$) you can think of $p$ as an independent variable that can only take two values, true and false. In the next column ($\sim p$), you would negate the value in the previous column. And so on.

It should roughly look like this:

p ... ~p ... ~(~p)
T ...  F  ...    T
F ...  T  ...    F