# False (logic)

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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.^{[1]} Usual notations of the false are 0 (especially in Boolean logic and computer science), O (in prefix notation, O*pq*), and the up tack symbol .^{[2]}^{[3]}^{[4]}

Another approach is used for several formal theories (e.g., 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.^{[5]}^{[6]}^{[7]} It can be treated as an absurd proposition, and is often called absurdity.

## In classical logic and Boolean logicEdit

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 ),^{[2]} while others instead rely upon formulas such as p ∧ ¬p and ¬(p → p).

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.

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 contradictionEdit

In most logical systems, negation, material conditional and false are related as:

- ¬p ⇔ (p → ⊥)

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 ⊢ ⊥). Using the equivalence above, the fact that φ is a contradiction may be derived, for example, from ⊢ ¬φ. A statement that entails false itself is sometimes called a contradiction, and contradictions and false are sometimes not distinguished, especially due to the Latin term *falsum* being used in English to denote either, but false is one specific proposition.

Logical systems may or may not contain the principle of explosion (*ex falso quodlibet in Latin*), ⊥ ⊢ φ for all φ. By that principle, contradictions and false are equivalent, since each entails the other.

## ConsistencyEdit

A formal theory using the " " connective is defined to be consistent, if and only if the false is not among its theorems. In the absence of propositional constants^{[disambiguation needed]}, some substitutes (such as the ones described above) may be used instead to define consistency.

## See alsoEdit

- Contradiction
- Logical truth
- Tautology (logic) (for symbolism of logical truth)
- Truth table

## ReferencesEdit

**^**Jennifer Fisher,*On the Philosophy of Logic*, Thomson Wadsworth, 2007, ISBN 0-495-00888-5, p. 17.- ^
^{a}^{b}"Comprehensive List of Logic Symbols".*Math Vault*. 2020-04-06. Retrieved 2020-08-15. **^**Willard Van Orman Quine,*Methods of Logic*, 4th ed, Harvard University Press, 1982, ISBN 0-674-57176-2, p. 34.**^**"Truth-value | logic".*Encyclopedia Britannica*. Retrieved 2020-08-15.**^**George Edward Hughes and D.E. Londey,*The Elements of Formal Logic*, Methuen, 1965, p. 151.**^**Leon Horsten and Richard Pettigrew,*Continuum Companion to Philosophical Logic*, Continuum International Publishing Group, 2011, ISBN 1-4411-5423-X, p. 199.**^**Graham Priest,*An Introduction to Non-Classical Logic: From If to Is*, 2nd ed, Cambridge University Press, 2008, ISBN 0-521-85433-4, p. 105.**^**Dov M. Gabbay and Franz Guenthner (eds),*Handbook of Philosophical Logic, Volume 6*, 2nd ed, Springer, 2002, ISBN 1-4020-0583-0, p. 12.