1. In logic
In logic, the non-contradiction principle prohibits the affirmation of contradictory propositions. In other words:
— The conjunction “P and not-P” expresses a contradiction, and, as such, is a self-destructing statement, which is necessarily false.
— The disjunction “P or not-P” is necessarily true.
One of the two propositions P and not-P must be true, both cannot be true simultaneously. The same thing cannot be and not be. This principle is considered by classical logic as a law of thought, and as an axiom by contemporary logicians. A logical system respecting the principle of non-contradiction does not contain any antinomies; it is said to be consistent.
Negation — Using the truth-table method, the negation operator is defined as follows:
This table expresses the principle of the excluded middle. It reads:
line 1: “when P is true, then not-P is false”
line 2: “when P is false, then not-P is true”
2. In natural language
The application of the non-contradiction principle to everyday language is complex, because it presupposes that P is plainly true or false, not far from true or practically false, not true or not according to the circumstances.
The non-contradiction principle applies not only to logical, argumentative discourse, but also to any kind of discourse; inconsistent narrations or descriptions for example, are rejected as such.
According to the basic Aristotelian dialectical rule, any discourse resulting in a contradiction is irrational and must be abandoned. Hegelian dialectic sees in the ongoing treatment of contradictions the motor of History. The cynical politician can lay claim to Hegel to hide his opportunism:
Stalin’s speech on the five-year plan serves as an ardent apology for contradiction as a “vital value” and an “instrument of struggle”. One of Lenin’s great strengths was his ability never to feel a prisoner of what he had preached as true the day before […] Mussolini’s famous word “Let us beware of the mortal trap of coherence” could be signed by all those who intend to pursue a work within currents they cannot foresee.
Julien Benda, [The Betrayal of the Intellectuals], 
The affirmation of a paradox as an oxymoron makes it possible to withstand the contradiction: “O wound without scar!”. Such paradoxical assertion is not seen as absurd or fallacious and eliminated as such, but triggers a quest to identify the deeper, symbolic meaning of the words wound and scar used in this context.
 Julien Benda, La Trahison des Clercs, . Excerpt from the Preface to the 1946 edition. Paris: Grasset, 1975, p. 78-79.