# Definition:Tautology

## Definition

A **tautology** is a statement which is *always true*, independently of any relevant circumstances that could theoretically influence its truth value.

It is epitomised by the statement form:

- $p \implies p$

that is:

An example of a "relevant circumstance" here is the truth value of $p$.

The archetypal **tautology** is symbolised by $\top$, and referred to as Top.

### Tautologies in Formal Semantics

Let $\LL$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\LL$.

A logical formula $\phi$ of $\LL$ is a **tautology for $\mathscr M$** if and only if:

That $\phi$ is a **tautology for $\mathscr M$** can be denoted as:

- $\models_{\mathscr M} \phi$

## Also known as

**Tautologies** are also referred to as **logical truths**.

## Also defined as

Some sources define a **tautology** as a statement form which can be epitomised by:

- $p \lor \lnot p$

which, while intuitively obvious, it not a universal definition as it does not apply in contexts in which Law of Excluded Middle does not necessarily hold.

## Also see

- Definition:Top (Logic), a symbol often used to represent
**tautologies**in logical languages. - Definition:Contradiction
- Definition:Contingent Statement

## Sources

- 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.4$: Statement Forms - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{I}: 3$: Logical Constants $(2)$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**tautology** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**tautology** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**tautology** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**tautology**