This article needs additional citations for verification. (March 2024) |
In logic, false or untrue is the state of possessing negative truth value and is 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 (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 (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. 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 
), 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.
This section needs expansion. You can help by adding to it. (February 2012) |
False, negation and contradiction
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, 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.
Consistency
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, some substitutes (such as the ones described above) may be used instead to define consistency.
See also
- Contradiction
- Logical truth
- Tautology (logic) (for symbolism of logical truth)
- Truth table
References
- Its noun form is falsity.
- Jennifer Fisher, On the Philosophy of Logic, Thomson Wadsworth, 2007, ISBN 0-495-00888-5, p. 17.
- 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.
This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources False logic news newspapers books scholar JSTOR March 2024 Learn how and when to remove this message In logic false or untrue is the state of possessing negative truth value and is 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 displaystyle bot Another approach is used for several formal theories e g intuitionistic propositional calculus where a propositional constant i e a nullary connective displaystyle bot 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 logicIn 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 displaystyle bot 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 x displaystyle x x displaystyle neg x true falsefalse 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 This section needs expansion You can help by adding to it February 2012 False negation and contradictionIn most logical systems negation material conditional and false are related as p p In fact 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 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 f Using the equivalence above the fact that f is a contradiction may be derived for example from f 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 f for all f By that principle contradictions and false are equivalent since each entails the other ConsistencyA formal theory using the displaystyle bot 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 the ones described above may be used instead to define consistency See alsoWikiquote has quotations related to Falsehood Contradiction Logical truth Tautology logic for symbolism of logical truth Truth tableReferencesIts noun form is falsity Jennifer Fisher On the Philosophy of Logic Thomson Wadsworth 2007 ISBN 0 495 00888 5 p 17 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
