From Godel’s first incompleteness theorem, any formal axiomatic system which is consistent must necessarily be incomplete. It follows that any complete system must be inconsistent, and by the principle of explosion (i.e. “from contradiction comes anything”), any sentence – true or false – under these axioms can be derived. In other words, the system becomes trivial under conditions that allow for inconsistency.

Paraconsistent logic attempts to override formal inconsistency by introducing a method to discriminate between contradictions, in order to tolerate systems of logic that are inconsistent. Importantly, any paraconsistent system is propositionally weaker than consistent systems; this allows paraconsistent systems to be more expressive than consistent systems.

In order to override the principle of explosion, either 1) disjunction introduction (\(A \vdash A \vee B\)) and/or 2) disjunctive syllogism (\(A \vee B, \neg A \vdash B\)) must be rejected. Typically, disjunctive syllogism is easier to reject, particularly from a dialetheist perspective, as arguments can be made that both \(A\) and \(\neg A\) can be valid under certain circumstances. Under this assumption, proof by contradiction becomes inconsistency non-robust, as contradictory statements are able to coexist.

A valid criticism of paraconsistent logic is that any circumstance in which \(\vdash A\) and \(\vdash \neg A\) merely represents subcontrary-forming operators. One potential way of refraining from rejecting the principle of explosion is to adopt multi-valued logic to allow for multiple truth values to coexist.