# 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*:

`⊤`

:`A ∨ ⌐A`

(A or not A)

`⊤`

:`A → A`

(If A, then A)

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

`⊤`

:`2 + 3 = 5`

`⊤`

:`x² ≥ 0`

`⊤`

:`¹⁄`

_{x}≠ 0

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

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

Ifa number iseven,thenit is divisible by 2.

As *divisibility by* 2 is a possible definition of *even* numbers, both antecedent and consequent extend to the same this is a tautological statement.

### Opposite

The opposite of a *tautology* is a contradiction: a proposition that is *necessarily false*.

## Symbols

On this site, the logical symbol `⊤`

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 Boolean logic).

Occasionally, also the *double turnstile* symbol (`⊨`

) is used in this meaning. This is not recommended, as this symbol is also used for “implication”.

## See also

## More information

- Tautology (logic) on
*Wikipedia* - Tautology on
*WolframMathWorld*