  We would have to say when an introduction or elimination rule counts as “purely inferential,” to exclude rules like these:  \begin{equation*} \frac{a~\text{is red}}{Ra} \quad \frac{A, B, \text{water is}~H_{2}O}{A * B} \end{equation*}  The strictest criterion would allow only rules in which every sign, besides a single instance of the constant being characterized, is either structural (like the comma) or schematic (like \(\dq{A}\)). But although this condition is met by the standard rules for conjunction, it is not met by the natural deduction introduction rule for negation, which must employ either another logical constant (\(\dq{\bot}\)) or another instance of the negation sign than the one being introduced. Thus one must either relax the condition for being “purely inferential” or add more structure (see especially Belnap 1982).