Editing Natural deduction (section)

=== Common definition styles ===
In [[classical logic|classical]] [[propositional calculus]] the [[formal language]] <math>\mathcal{L}</math> is usually defined (here: by [[recursive definition|recursion]]) as follows:{{sfn|Allen|Hand|2022|p=12}}

# Each [[propositional variable]] is a [[Well-formed formula|formula]].
# "<math>\bot</math>" is a formula.
# If <math>\varphi</math> and <math>\psi</math> are formulae, so are <math>(\varphi \land \psi)</math>, <math>(\varphi \lor \psi)</math>, <math>(\varphi \to \psi)</math>, <math>(\varphi \leftrightarrow \psi)</math>.
# Nothing else is a formula.

[[Negation]] (<math>\neg</math>) is defined as implication to [[False (logic)#False, negation and contradiction|falsity]]
:<math>\neg \phi \; \overset{\text{def}}{=} \; \phi \to \bot</math>,
where <math>\bot</math> (falsum) represents a contradiction or absolute falsehood.{{sfn|Kleene|2002}}{{sfn|Prawitz|1965}}{{sfn|von Plato|2013|p=18}}{{sfn|Van Dalen|2013}}{{sfn|Hansson|Hendricks|2018|p=179}}

Older publications, and publications that do not focus on logical systems like [[minimal logic|minimal]], [[intuitionistic logic|intuitionistic]] or [[Hilbert system]]s, take negation as a primitive [[logical connective]], meaning it is assumed as a basic operation and not defined in terms of other connectives.{{sfn|Ayala-Rincón|de Moura|2017|pages=2,20}}{{sfn|Hansson|Hendricks|2018|p=38}} Some authors, such as [[David Bostock (philosopher)|Bostock]], use <math>\bot</math> and <math>\top</math>, and also define <math>\neg</math> as primitives.{{sfn|Bostock|1997|p=21}}<ref>This is required in [[paraconsistent logic]]s that do not treat <math>\neg</math> and <math>(\phi \to \bot)</math> as equivalents.</ref>