User Tools

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”.

See also

More information

About this site

Ad Hominem Info is a project to explain and categorize the most common systematic fallacies and fallacies. On this page, you will find a background article that briefly explains an important logical concept, which may be needed to better understand another article in this area.
For more information, please see the main category “logic

This website uses cookies. By using the website, you agree with storing cookies on your computer. Also, you acknowledge that you have read and understand our Privacy Policy. If you do not agree, please leave the website.

More information