Law of noncontradiction
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 "p is the case" and "p is not the case" are mutually exclusive. Formally, this is expressed as the tautology ¬(p ∧ ¬p). The law is not to be confused with the law of excluded middle which states that at least one, "p is the case" or "p is not the case", holds.
For the Fargo episode, see The Law of Non-Contradiction.
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 say "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. 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 is an expression of its jointly exhaustive aspect.
Alleged impossibility of its proof or denial[edit]
The law of non-contradiction is alleged to be neither verifiable nor falsifiable, on the ground that any proof or disproof must use the law itself prior to reaching the conclusion. In other words, in order to verify or falsify the laws of logic one must resort to logic as a weapon, an act that is argued to be self-defeating.[27] Since the early 20th century, certain logicians have proposed logics that deny the validity of the law.
Logics known as "paraconsistent" are inconsistency-tolerant logics in that there, from P together with ¬P, it does not imply that any proposition follows. Nevertheless, not all paraconsistent logics deny the law of non-contradiction and some such logics even prove it.[28][29]
Some, such as David Lewis, have objected to paraconsistent logic on the ground that it is simply impossible for a statement and its negation to be jointly true.[30] A related objection is that "negation" in paraconsistent logic is not really negation; it is merely a subcontrary-forming operator.[31][32]
In popular culture[edit]
The Fargo episode "The Law of Non-Contradiction", which takes its name from the law, was noted for its several elements relating to the law of non-contradiction, as the episode's main character faces several paradoxes. For example, she is still the acting chief of police while having been demoted from the position, and tries to investigate a man that both was and was not named Ennis Stussy, and who both was and was not her stepfather. It also features the story of a robot who, after having spent millions of years unable to help humanity, is told that he greatly helped mankind all along by observing history.[33]