 By (As), we have that \(\{A,\neg A\}\vdash A\) and \(\{\)A,\(\neg\)A\(\}\vdash \neg A\). So by \(\neg\)I we have that \(\{A\}\vdash \neg \neg A\). However, we do not have the converse yet. Intuitively, \(\neg \neg \theta\) corresponds to “it is not the case that it is not the case that” . One might think that this last is equivalent to \(\theta\), and we have a rule to that effect: 