Tautology (logic)

For the use tautology in rhetoric, see Tautology (rhetoric).

In propositional logic, a tautology (from the Greek word ταυτολογία) is a propositional formula that is always true, and is sometimes denoted by the symbol ${\displaystyle \top }$ (a symbol also reserved for the truth value 'true').^[1][2] In other words, a tautology cannot be wrong. For example, formulae in maths are tautological, because they always hold true for any values. The philosopher Ludwig Wittgenstein first applied the term to propositional logic in 1921.^[3]

A tautology can also be a figure of speech

In formal terms, a formula ${\displaystyle p}$ is a tautology if it is true under all possible interpretations. In other words, for every interpretation ${\displaystyle M}$, we have ${\displaystyle M\models p}$. An interpretation ${\displaystyle M}$ of a formula ${\displaystyle p}$ is defined as a function that assigns a truth value to ${\displaystyle p}$. That is ${\displaystyle M:p\rightarrow \lbrace {\text{Bool}}\rbrace }$, where ${\displaystyle {\text{Bool}}=\{{\text{true}},{\text{false}}\}}$.^[4]

A statement in logic is considered contingent if it is neither a tautology nor false.

Some examples of tautologies in natural language include:

• "I know that this is Wikipedia because I know that this is Wikipedia."
• "I am president of this club because I am the president of this club."
• "The first rule of the tautology club is the first rule of the tautology club."

These tautologies are sentences of the following form:

• A is true because A is true

Indeed, if A is not true, then A is not true, so the tautology is still true.

Related pages

• Logic