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In logic, false or untrue is a 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.〔Jennifer Fisher, ''On the Philosophy of Logic'', Thomson Wadsworth, 2007, ISBN 0-495-00888-5, (p. 17. )〕 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 (unicode:⊥).〔Willard Van Orman Quine, ''Methods of Logic'', 4th ed, Harvard University Press, 1982, ISBN 0-674-57176-2, (p. 34. )〕 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) (unicode:⊥), the truth value of this constant being always false in the sense above.〔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. )〕 == 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 and 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. == False, negation and contradiction == In most logical systems, negation, material conditional and false are related as: : 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, ISBN 1-4020-0583-0, (p. 12. )〕 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. The contradiction is a statement which entails the false, i.e. . Using the equivalence above, the fact that φ is a contradiction may be derived, for example, from . 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''), . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「False (logic)」の詳細全文を読む スポンサード リンク
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