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What happens in a logical system if we do not state ex falso quodlibet?

$\bot\rightarrow P$

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See Paraconsistent Logic for a family of logics that reject Ex Falso (aka: Principle of Explosion).

See also: Walter Carnielli & Marcelo Esteban Coniglio, Paraconsistent Logic: Consistency, Contradiction and Negation (Springer, 2016),

as well as: Holger Andreas & Peter Verdée (editors), Logical Studies of Paraconsistent Reasoning in Science and Mathematics (2016, Springer).

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