False, Negation and Contradiction
In most logical systems, negation, material conditional and false are related as:
- ¬p ⇔ (p → ⊥)
This is the definition of negation in some systems, such as intuitionistic logic, and can be proven in propositional calculi where negation is a fundamental connective. Because p → p is usually a theorem or axiom, a consequence is that the negation of false (¬ ⊥) is true.
The contradiction is a statement which entails the false, i.e. φ ⊢ ⊥. Using the equivalence above, the fact that φ is a contradiction may be derived, for example, from ⊢ ¬φ. Contradiction and the false are sometimes not distinguished, especially due to Latin term falsum denoting both. Contradiction means a statement is proven to be false, but the false itself is a proposition which is defined to be opposite to the truth.
Logical systems may or may not contain the principle of explosion (in Latin, ex falso quodlibet), ⊥ ⊢ φ.
Read more about this topic: False (logic)
Famous quotes containing the word negation:
“Michelangelo said to Pope Julius II, “Self negation is noble, self-culture is beneficent, self-possession is manly, but to the truly great and inspiring soul they are poor and tame compared to self-abuse.” Mr. Brown, here, in one of his latest and most graceful poems refers to it in an eloquent line which is destined to live to the end of time—”None know it but to love it, None name it but to praise.””
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)