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Tautology


A tautology is a logical statement in which the conclusion is equivalent to the premise. More colloquially, it is formula in propositional calculus which is always true (Simpson 1992, p. 2015; D'Angelo and West 2000, p. 33; Bronshtein and Semendyayev 2004, p. 288).

If p is a tautology, it is written |=p. A sentence whose truth table contains only 'T' is called a tautology. The following sentences are examples of tautologies:

A ^ B=!(!A v !B)
(1)
A v B=!A=>B
(2)
A ^ B=!(A=>!B)
(3)

(Mendelson 1997, p. 26), where  ^ denotes AND, = denotes "is equivalent to," ! denotes NOT,  v denotes OR, and => denotes implies.


See also

Contingency, Contradiction, Theorem, True

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References

Bronshtein, I. N.; Semendyayev, K. A.; Musiol, G.; and Muehlig, H. Handbook of Mathematics, 4th ed. New York: Springer, 2004.Carnap, R. Introduction to Symbolic Logic and Its Applications. New York: Dover, p. 13, 1958.D'Angelo, J. P. and West, D. B. Mathematical Thinking: Problem-Solving and Proofs, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2000.Mendelson, E. "Tautology." §1.2 in Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, pp. 17-24, 1997.Simpson, J. A. and Weiner, E. S. C. (Preparers). The Compact Oxford English Dictionary, 2nd ed. Oxford, England: Clarendon Press, 1992.

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Tautology

Cite this as:

Weisstein, Eric W. "Tautology." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Tautology.html

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