Clearly, this is a highly desirable property for those who are interested in argueing logically. The reason is that in standard logic any inconsistent set of statements deductively implies any statement, and its contradiction.

It is an interesting fact that Gödel proved a theorem to the effect that set theory, and many systems like it, that are strong enough to imply most of classical mathematics, cannot be proved to be consistent within the same system - that is, the statement "this theory is consistent" must be unprovable in the system if the system is consistent.

This means that the only means to establish that such a system is consistent is to represent it somehow in another system. So far, set theory is not known to be consistent, though it is widely believed to be, and should be if it is to be the basis of mathematics.