in a logical context: a proposition that is necessary true.
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”.
- Tautology (logic) on Wikipedia
- Tautology on WolframMathWorld