Inconsistency

The property of a formal system opposite to that of consistency. A formal system is called inconsistent if it fails to be consistent. If a certain class of formulas in the language of the given formal system fails to be consistent, then one says that the class is inconsistent with the given formal system. In particular, if a class consisting of a single formula is inconsistent with the formal system, then the formula is said to be inconsistent with the given system. Inconsistency of a formula means that if it is included among the axioms, an inconsistent formal system results.

Inconsistent formal systems have no meaningful interpretation.

The discovery that the negation of a certain formula is inconsistent with a given formal system constitutes the method of so-called proof by reductio ad absurdum: For the usual formal systems it follows from the inconsistency of a formula that its negation is deducible.

References

[a1]	Yu.I. Manin, "A course in mathematical logic" , Springer (1977) (Translated from Russian)