Triviality

Triviality is a property of formal systems in logic. A system is trivial if its set of axioms $$X$$ entails every well-formed formula according to the inference rules of the system, i.e.,

$$\forall \phi.\left[X\vdash \phi\right]$$

An example of triviality is a classical propositional system containing contradictory sentences for axioms; in such a system the principle of explosion can be used to infer any possible proposition or sentence.

On the other hand, a system of paraconsistent logic can be non-trivial even if it contains contradictory axioms, since its inference rules do not include the principle of explosion.