What if we don't accept ex falso quodlibet? What happens in a logical system if we do not state ex falso quodlibet?
$\bot\rightarrow P$ 

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).
