In Boolean logic, each variable denotes a truth value which can be either true (1), or false (0).
Some systems of classical logic include dedicated symbols for false (0 or
In most logical systems, negation, material conditional and false are related as: In fact, this is the definition of negation in some systems,[8] 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.
A contradiction is the situation that arises when a statement that is assumed to be true is shown to entail false (i.e., φ ⊢ ⊥).
In the absence of propositional constants, some substitutes (such as the ones described above) may be used instead to define consistency.