Theorem (logic) in the context of "Dialetheism"

Dialetheism (/daɪəˈlɛθiɪzəm/; from Greek δι- di- 'twice' and ἀλήθεια alḗtheia 'truth') is the view that there are statements that are both true and false. More precisely, it is the belief that there can be a true statement whose negation is also true. Such statements are called "true contradictions", dialetheia, or nondualisms.

Dialetheism is not a system of formal logic; instead, it is a thesis about truth that influences the construction of a formal logic, often based on pre-existing systems. Introducing dialetheism has various consequences, depending on the theory into which it is introduced. A common mistake resulting from this is to reject dialetheism on the basis that, in traditional systems of logic (e.g., classical logic and intuitionistic logic), every statement becomes a theorem if a contradiction is true, trivialising such systems when dialetheism is included as an axiom. Other logical systems, however, do not explode in this manner when contradictions are introduced; such contradiction-tolerant systems are known as paraconsistent logics. Dialetheists who do not want to allow that every statement is true are free to favour these over traditional, explosive logics.