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 truthfunctional 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, Opq), and the up tack symbol ⊥.^{[2]}
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.^{[3]}^{[4]}^{[5]}
Contents
In classical logic and Boolean logic
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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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,^{[6]} 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.
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
 Contradiction
 Logical truth
 Tautology (logic) (for symbolism of logical truth)
References
 ^ Jennifer Fisher, On the Philosophy of Logic, Thomson Wadsworth, 2007, ISBN 0495008885, p. 17.
 ^ Willard Van Orman Quine, Methods of Logic, 4th ed, Harvard University Press, 1982, ISBN 0674571762, p. 34.
 ^ 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 144115423X, p. 199.
 ^ Graham Priest, An Introduction to NonClassical Logic: From If to Is, 2nd ed, Cambridge University Press, 2008, ISBN 0521854334, p. 105.
 ^ Dov M. Gabbay and Franz Guenthner (eds), Handbook of Philosophical Logic, Volume 6, 2nd ed, Springer, 2002, ISBN 1402005830, p. 12.