The principle of explosion states that a single contradictory statement among a logical set is enough to prove all propositions wrong. So, for a set to be considered valid, it must be consistent*.
What is the name of the rule, axiom or principle that demands such consistency for a logical set of propositions to be valid (and so avoid falling into the previous)? Aristotle's law of non contradiction solves the issue for a single statement, but not for a set.
I know I'm possibly looking for glasses I'm already wearing, but I just can't find it... Thanks in advance.
* UPDATED (thanks, @MauroAllegranza): (was)"a set must be logically consistent": I didn't mean that all sets MUST be consistent. I seek for the axiom/rule/condition that determines the consistency of a set.
A set of sentences is called inconsistent if one can derive a contradiction from it and consistent otherwise. There's no law requiring every set to be consistent.