# 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:

If a number is even, then it 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”. 