====== Tautology (Logic) ======

in a logical context: a proposition that is //necessary true//.

For example:

> //Either it rains, or it doesn’t.//

If there is no third state thinkable, outside of “raining” and “not raining”, the statement must be true in all situations: it is //tautological//.

===== Description =====

Any proposition that is //always true//, independent of external circumstances, is called a “tautology”.

The following logical formulae are examples of //tautologies//, as they can never be //false//:

> <span "Tautology">''⊤''</span>: <span "A or not A">''A ∨ ⌐A''</span>   (//A or not A//)
> <span "Tautology">''⊤''</span>: <span "If A, then A">''A → A''</span>   (//If A, then A//)

The same is true for all (valid) mathematical equations or inequations, for example:

> <span "Tautology">''⊤''</span>: <span "Two plus three equals five">''2 + 3 = 5''</span>
> <span "Tautology">''⊤''</span>: <span "X squared is greater or equal to zero">''x² ≥ 0''</span>
> <span "Tautology">''⊤''</span>: <span "One divided by X is unequal zero">''¹⁄<sub>x</sub> ≠ 0''</span>

In all of these, the propositions are //necessary true//.

Tautologies can also be hidden in definitions, e.g. as in the following:

> //If// a number is //even//, //then// it is divisible by 2.

As //divisibility by// 2 is a possible definition of //even// numbers, both [[glossary:antecedent|antecedent]] and [[glossary:consequent|consequent]] extend to the same  this is a tautological statement.

==== Opposite ====

The opposite of a //tautology// is a [[glossary:contradiction|contradiction]]: a proposition that is //necessarily false//.

===== Symbols =====

On this site, the logical symbol <span "Tautology">''⊤''</span> is used for tautologies. As every //tautology// is inherently //true//, it can also be replaced by the term „''true''“ or its equivalent in the various formal systems (e.g. ''1'' in [[wp>Boolean algebra|Boolean logic]]).

Occasionally, also the //double turnstile// symbol (<span "Double turnstile">''⊨''</span>) is used in this meaning. This is not recommended, as this symbol is also used for “implication”.

===== See also =====

  * [[glossary:contradiction|Contradiction]]
  * [[glossary:vacuous_truth|Vacuous truth]]

===== More information =====

  * [[wp>Tautology (logic)]] on //Wikipedia//
  * [[https://mathworld.wolfram.com/Tautology.html|Tautology]] on //WolframMathWorld//

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