Triviality

Cogito ergo sum Logic and rhetoric

Key articles

Logical fallacy Syllogism Argument

General logic

Tautology Mistaking the map for the territory Negative conclusion from affirmative premises Informal fallacy Shoehorning دوراهی اشتباهی

Bad logic

Quote mining Negative conclusion from affirmative premises Nirvana fallacy Informal fallacy Hyperbole Appeal to probability v - t - e

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.

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