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logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, false or untrue is the state of possessing negative truth value or a nullary
logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...
. In a truth-functional system of propositional logic, it is one of two postulated truth values, along with its negation,
truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as belief ...
. Usual notations of the false are 0 (especially in Boolean logic and
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
), O (in prefix notation, O''pq''), and the up tack symbol \bot. Another approach is used for several
formal theories Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements (forms, in Ancient Greek). They may refer to: Dress code and events * Formal wear, attire for formal events * Semi-formal atti ...
(e.g., intuitionistic propositional calculus), where a propositional constant (i.e. a nullary connective), \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 logic

In Boolean logic, each variable denotes a truth value which can be either true (1), or false (0). In a classical
propositional calculus Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
, 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 \bot), while others instead rely upon formulas such as and . 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. 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: : In fact, this is the definition of negation in some systems,Dov M. Gabbay and Franz Guenthner (eds), ''Handbook of Philosophical Logic, Volume 6'', 2nd ed, Springer, 2002,
p. 12.
/ref> such as intuitionistic logic, and can be proven in propositional calculi where negation is a fundamental connective. Because 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., ). Using the equivalence above, the fact that φ is a contradiction may be derived, for example, from . A statement that entails false itself is sometimes called a contradiction, and contradictions and false are sometimes not distinguished, especially due to the
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
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 Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
), for all . By that principle, contradictions and false are equivalent, since each entails the other.


Consistency

A formal theory using the "\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 also

* Contradiction * Logical truth * Tautology (logic) (for symbolism of logical truth) *
Truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arg ...


References

{{Common logical symbols Logical connectives