Tautology (logic)  

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Train wreck at Montparnasse (October 22, 1895) by Studio Lévy and Sons.
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Train wreck at Montparnasse (October 22, 1895) by Studio Lévy and Sons.

In propositional logic, a tautology (from the Greek word ταυτολογία) is a propositional formula that is true under any possible valuation (also called a truth assignment or an interpretation) of its propositional variables. The philosopher Ludwig Wittgenstein first applied the term to propositional logic in 1921.

A tautology's negation is a contradiction, a propositional formula that is false regardless of the truth values of its propositional variables. Such propositions are called unsatisfiable. Conversely, a contradiction's negation is a tautology. A formula that is neither a tautology nor a contradiction is said to be logically contingent. Such a formula can be made either true or false based on the values assigned to its propositional variables.

A key property of tautologies is that an effective method exists for testing whether a given formula is always satisfied (or, equivalently, whether its complement is unsatisfiable). One such method uses truth tables. The decision problem of determining whether a formula is satisfiable is the Boolean satisfiability problem, an important example of an NP-complete problem in computational complexity theory.

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Unless indicated otherwise, the text in this article is either based on Wikipedia article "Tautology (logic)" or another language Wikipedia page thereof used under the terms of the GNU Free Documentation License; or on original research by Jahsonic and friends. See Art and Popular Culture's copyright notice.

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