Editing False (logic) (section)

== False, negation and contradiction ==
In most logical systems, [[negation]], [[material conditional]] and false are related as:
: {{math|¬{{mvar|p}} ⇔ ({{mvar|p}} → ⊥)}}
In fact, this is the definition of negation in some systems,<ref>Dov M. Gabbay and Franz Guenthner (eds), ''Handbook of Philosophical Logic, Volume 6'', 2nd ed, Springer, 2002, {{ISBN|1-4020-0583-0}}, [https://books.google.com/books?id=JyewdfGhNAsC&pg=PA12 p. 12.]</ref> such as [[intuitionistic logic]], and can be proven in propositional calculi where negation is a fundamental connective. Because {{math|{{mvar|p}} → {{mvar|p}}}} is usually a theorem or axiom, a consequence is that the negation of false ({{math|¬ ⊥}}) is true.

A [[contradiction]] is the situation that arises when a [[statement (logic)|statement]] that is assumed to be true is shown to [[entailment|entail]] false (i.e., {{math|φ ⊢ ⊥}}). Using the equivalence above, the fact that φ is a contradiction may be derived, for example, from {{math|⊢ ¬φ}}. A statement that entails false itself is sometimes called a contradiction, and contradictions and false are sometimes not distinguished, especially due to the [[Latin]] term ''[[wikt:falsum#English|falsum]]'' being used in English to denote either, but false is one specific [[proposition]].

Logical systems may or may not contain the [[principle of explosion]] (''ex falso quodlibet'' in [[Latin]]), {{math|⊥ ⊢ φ}} for all {{math|φ}}. By that principle, contradictions and false are equivalent, since each entails the other.