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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 "the house is white" and "the house is not white" are mutually exclusive.
Formally the law of non-contradiction is written as ¬(P ∧ ¬P) and read as "it is not the case that a proposition is both true and false". The law of non-contradiction neither follows nor is implied by the principle of Proof by contradiction. The laws of excluded middle and non-contradiction together mean that exactly one of P and ¬P is true.
The expressions "law of non-contradiction" and "law of excluded middle" are also used for semantic principles of model theory concerning sentences and interpretations: (NC) under no interpretation is a given sentence both true and false, (EM) under any interpretation, a given sentence is either true or false.
The law of non-contradiction (alternately the 'law of contradiction' [20]): 'Nothing can both be and not be.' [19] The law of excluded middle: 'Everything must either be or not be.' [19] In accordance with the law of excluded middle or excluded third, for every proposition, either its positive or negative form is true: A∨¬A.
The law of non-contradiction (which states that contradictory statements cannot both be true at the same time) is still valid. For instance, in Heyting arithmetic , one can prove that for any proposition p that does not contain quantifiers , ∀ x , y , z , … ∈ N : p ∨ ¬ p {\displaystyle \forall x,y,z,\ldots \in \mathbb {N} :p\vee \neg p ...
law of non-contradiction A fundamental principle of classical logic stating that contradictory statements cannot both be true in the same sense at the same time. left field See domain. Leibniz's Law The principle of the identity of indiscernibles, stating that if two entities share all the same properties, then they are identical. lemma
Several key mistakes could throw off the accuracy of blood pressure readings for people who take them at home. The average "normal" blood pressure is 120/80, according to the American Heart ...
Weak law of the excluded middle (WLEM) is axiomatised and yields a system where disjunction behaves more like in classical logic than intuitionistic logic, i.e. the disjunction and existence properties don't hold, but where use of non-intuitionistic reasoning is marked by occurrences of double-negation in the conclusion. LEM entails but is not ...