nLab inconsistency

Idea

A system in formal logic is called inconsistent if it admits a proof of a contradiction (that is, usually, a proof of false, or an inhabitant of the empty type).

Accordingly an axiom is called inconsistent or to lead to an inconsistency if adding it to an (implicitly understood) ambient logical system makes that system inconsistent.

In most usual logical systems, it follows that an inconsistent system admits a proof of every proposition, by the rule ex falso quodlibet (which is just the elimination rule for the empty type). For this reason, sometimes (especially in type theory), the adjective “inconsistent” is used to mean a system with this property instead. If we want to distinguish, then a system which admits a proof of every proposition may be called trivial.

As a further complication, a paraconsistent logic is often described as ‘inconsistent but not trivial’. However, many paraconsistent logics (such as dual-intuitionistic logic?) admit ex falso quadlibet and fail to prove $\bot$; their ‘inconsistency’ is a proof of $p \wedge \neg{p}$ (or two proofs, one of $p$ and one of $\neg{p}$), which then fails to entail $\bot$. We thus get three distinct levels of inconsistency:

• triviality (proving everything),
• proving a statement identified as $\bot$,
• proving $p$ and $\neg{p}$ for some statement $p$ and using an operator identified as negation.

A logic that is trivial in this technical sense can still yield rich metatheory if one considers not only the existence of a proof but also the structure of the proof space. An example of such a logic is differential linear logic DiLL where every judgement — including the empty sequent — can be derived using the zero proof.

Examples

• Russell's paradox

• Burali-Forti's paradox

Related concepts

• consistency

• paraconsistent logic