# Tautology

From Encyclopedia of Mathematics

A formula of the language of propositional calculus taking the truth value "true" independently of the truth values "true" or "false" taken by its propositional variables. Examples: $A\supset A$, $A\lor\neg A$, $(A\supset B)\supset(\neg B\supset\neg A)$.

In general one can check whether a given propositional formula is a tautology by simply examining its truth table: the finite set of all combinations of values of its propositional variables. It is usual to give a presentation of propositional calculus which is both *sound*: every theorem deducible in the system is a tautology; and *complete*: every tautology is a theorem.

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#### References

[a1] | Yu.I. Manin, "A course in mathematical logic" , Springer (1977) pp. 31, 54 (Translated from Russian) |

[b1] | Peter J. Cameron, "Sets, Logic and Categories" Springer (2012) ISBN 1447105893 |

**How to Cite This Entry:**

Tautology.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Tautology&oldid=39785

This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article