Concept
Dialetheism (from Greek δι- 'twice' and ἀλήθεια '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. Graham Priest defines dialetheism as the view that there are true contradictions. Jc Beall is another advocate; his position differs from Priest's in advocating constructive (methodological) deflationism regarding the truth predicate. The liar's paradox and Russell's paradox deal with self-contradictory statements in classical logic and naïve set theory, respectively. Contradictions are problematic in these theories because they cause the theories to explode—if a contradiction is true, then every proposition is true. The classical way to solve this problem is to ban contradictory statements, to revise the axioms of the logic so that self-contradictory statements do not appear. Dialetheists, on the other hand, respond to this problem by accepting the contradictions as true. Dialetheism allows for the unrestricted axiom of comprehension in set theory, claiming that any resulting contradiction is a theorem.