Editing Propositional calculus (section)

==== Schematic form of P<sub>2</sub> ====
One may avoid using the rule of substitution by giving the axioms in schematic form, using them to generate an infinite set of axioms. Hence, using Greek letters to represent schemata (metalogical variables that may stand for any [[well-formed formula]]s), the axioms are given as:<ref name="BostockIntermediate" /><ref name=":48" />

* <math>\varphi \to (\psi \to \varphi)</math>
* <math>(\varphi \to (\psi \to \chi)) \to ((\varphi \to \psi) \to (\varphi \to \chi))</math>
* <math>(\neg \varphi \to \neg \psi) \to (\psi \to \varphi)</math>

The schematic version of P<sub>2</sub> is attributed to [[John von Neumann]],<ref name=":44" /> and is used in the [[Metamath]] "set.mm" formal proof database.<ref name=":48" /> It has also been attributed to [[David Hilbert|Hilbert]],<ref name=":49"/> and named <math>\mathcal{H}</math> in this context.<ref name=":49" />