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Paraconsistent logic

Paraconsistent logics have been developed that allow for subcontrary -forming operators. Model-theoretic paraconsistent logicians often deny the assumption that there can be no model of { ϕ, ¬ ϕ } {\displaystyle \{\phi,\lnot \phi \}} and devise semantical systems in which there are such models. Alternatively, they reject the idea that propositions can be classified as true or false. Proof-theoretic paraconsistent logics usually deny the validity of one of the steps necessary for deriving an explosion, typically including disjunctive syllogism, disjunction introduction, and reductio ad absurdum.