# 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 a propositional constant (i.e. a nullary connective) ⊥, is introduced, the truth value of which being always false in the sense above. It can be treated as an absurd proposition and is often called absurdity.

## In classical logic and Boolean logic

In Boolean logic each variable denotes a Truth value which can be either true (1), or false (0). In a classical propositional calculus each proposition will be assigned a truth value of either true or false. Some systems of classical logic include dedicated symbols for false (0 or ⊥), others instead rely upon formulas such as p ∧ ¬p and ¬(pp).

In both Boolean logic and Classical logic systems, true and false are opposite with respect to negation; The negation of false gives true, and the negation of true gives false.

$x$ $\neg x$ true false
false true

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.

## 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 pp 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.

## Consistency

A formal theory using "⊥" connective is defined to be consistent if and only if the false is not among its theorems. In the absence of propositional constants, some substitutes such as mentioned above may be used instead to define consistency.

## See also

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