This short article presents a more developed subject in: Mathematical Logic, Principle of Explosion and Consistency (Logic).

A trivial logic or more exactly trivial theory is a theory expressed in a logic from which any statement is demonstrable. Typically a theory of logic with the principle of explosion becomes trivial as soon as a contradiction is deduced from axioms of the theory. Triviality is a property that makes logic a bad logic, since everything and its opposite become demonstrable. I am the pope and I am not the pope, in logics that allow these statements, or The system of equation A has solutions and the system of equation A has no solutions or this bridge can support the load as well as this bridge will collapse under load. Reasoning in such a logic would therefore not be very useful for a mathematician or an engineer. Certain logics, called paracohérent, have the property of being able to reason with contradictions without becoming trivial. This is not, however, the most popular strategy among mathematicians in discovering contradictions that would make trivial theories, a well-known example is the crisis of foundations. This crisis was triggered by the discovery of paradoxes in set theory, such as Russell's paradox, which makes the theory trivial using the usual logical reasoning. The reaction of mathematicians was not necessarily to change the methods of reasoning by seeking exotic logic but rather to modify (successfully) the theory itself to remove inconsistencies. This process gave rise to consistent axiomatic theory such as ZFC set theory. Notes and edit the code

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