Editing Propositional calculus (section)

==== CF grammar in BNF ====
An alternative to the syntax definitions given above is to write a [[Context-free grammar|context-free (CF) grammar]] for the language <math>\mathcal{L}</math> in [[Backus–Naur form|Backus-Naur form]] (BNF).<ref name=":41"/><ref name=":42"/> This is more common in [[computer science]] than in [[philosophy]].<ref name=":42" /> It can be done in many ways,<ref name=":41" /> of which a particularly brief one, for the common set of five connectives, is this single clause:<ref name=":42" /><ref name=":02"/>
:<math>\phi ::= a_1, a_2, \ldots  ~ | ~  \neg\phi  ~ | ~  \phi ~ \& ~ \psi  ~ | ~  \phi \vee \psi  ~ | ~  \phi \rightarrow \psi  ~ | ~  \phi \leftrightarrow \psi</math>
This clause, due to its [[Self-reference|self-referential]] nature (since <math>\phi</math> is in some branches of the definition of <math>\phi</math>), also acts as a [[recursive definition]], and therefore specifies the entire language. To expand it to add [[modal operator]]s, one need only add … <math>| ~ \Box\phi ~ | ~ \Diamond\phi</math> to the end of the clause.<ref name=":42" />