False (logic)

In logic, false or untrue is the state of possessing negative truth value or a nullary logical connective. In a truth-functional system of propositional logic it is one of two postulated truth values, along with its negation, truth. Usual notations of the false are 0 (especially in Boolean logic and computer science), O (in prefix notation, Opq), and the up tack symbol ⊥.

Another approach is used for several formal theories (for example, intuitionistic propositional calculus) where the false is a propositional constant (i.e. a nullary connective) ⊥, the truth value of this constant being always false in the sense above.

Boolean logic defines the false in both senses mentioned above: "0" is a propositional constant, whose value by definition is 0. In a classical propositional calculus, depending on the chosen set of fundamental connectives, the false may or may not have a dedicated symbol. Such formulas as p ∧ ¬p and ¬(p → p) may be used instead.

In both systems the negation of the truth gives false. The negation of false is equivalent to the truth not only in classical logic and Boolean logic, but also in most other logical systems, as explained below.

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