Triviality

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Triviality is a property of formal systems in logic. A system is trivial if its set of axioms Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): X entails every well-formed formula according to the inference rules of the system, i.e.,

Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \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.