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.

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.

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

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”.

• Contradiction
• Vacuous truth

• Tautology (logic) on Wikipedia
• Tautology on WolframMathWorld