The law of non-contradiction

https://en.wikipedia.org/wiki/Law_of_noncontradiction

The law of non-contradiction, ¬(P ∧ ¬P)

In logic, the law of non-contradiction (LNC) (also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction)

states that contradictory propositions cannot both be true in the same sense at the same time.

e.g. the two propositions "A is B" and "A is not B" are mutually exclusive.

Formally this is expressed as the tautology ¬(p ∧ ¬p).

One reason to have this law is the principle of explosion, which states that anything follows from a contradiction. The law is employed in a reductio ad absurdum proof.

To express the fact that the law is tenseless and to avoid equivocation, sometimes the law is amended to state that the contradictory propositions cannot both be true at the same time and in the same sense.

It is one of the so called three laws of thought, along with its complement, the law of excluded middle and the law of identity.

The law of noncontradiction is logically equivalent to the law of excluded middle by De Morgan's laws.

However, no system of logic is built on just these laws, and none of these laws provide inference rules, such as modus ponens or De Morgan's laws.

The law of non-contradiction and the law of excluded middle create a dichotomy in "logical space", wherein the two parts are "mutually exclusive" and "jointly exhaustive".

The law of non-contradiction is merely an expression of the mutually exclusive aspect of that dichotomy, and the law of excluded middle, an expression of its jointly exhaustive aspect.