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