 for example, have very short and obvious proofs using \(\rightarrow\)-I and the variant quantifier rules.) On the other hand, they cannot be used without change in systems for intuitionistic or modal logics, as we explain in  §6:  the classical schemes mentioned above are not intuitionistically valid, but their simple proofs go through in a system that adds the quantifier rules of Quine’s textbook to a formalization of intuitionistic propositional logic, and natural attempts to use them in formulating systems of quantified modal logic lead to simple derivations of the Barcan principle and its converse. Lack of caution in using these (and related) rules in the context of non-classical logics has led to errors in published philosophical articles.