Suppose that Carr is out. We will show that this assumption produces a contradiction. If Carr is out, then we know this: ""If Allen is out, then Brown is in"", because there has to be someone in "to mind the shop". But, we also know that whenever Allen goes out he takes Brown with him, so as a general rule, ""If Allen is out, then Brown is out"". The two statements we have arrived at are incompatible, because if Allen is out then Brown cannot be both In (according to one) and Out (according to the other). There is a contradiction. So we must abandon our hypothesis that Carr is Out, and conclude that Carr must be In.