False (logic) - False, Negation and Contradiction

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.