Bad mistakes and technical mistakes

Some types of mistake are bad mistakes, in that they are liable to facilitate the "proving" of things that are incorrect. Failure to observe restrictions on the designators in the application of the "xj and $xj rules constitutes a bad mistake. The application of the "xj rule to formulae which do not begin """, and the application of the $xj rule to formulae which do not begin "$" are also bad mistakes. On the other hand some mistakes are just technical mistakes, in that they do not facilitate incorrect proofs. The application of the ¬"xj rule to formulae which do not begin "¬"" and the application of the ¬$xj rule to formulae which do not begin "¬$" are technical mistakes. This is because "¬"xj", for example, is equivalent to "$x¬j"; so, replacing the former by the latter wherever it occurs in a formula results in a formula which is equivalent to the formula one started with. However, technical mistakes are still mistakes!

Note the following:

Reference back to the interpretation

As with the propositional tableaux, so with predicate tableaux, it is always possible that reference back to one's interpretation may reveal inconsistency in a branch even when the branch has not closed.