Definition:Tautology/Formal Semantics
Definition
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:
- $\models_{\mathscr M} \phi$
Examples
Tautology for Boolean Interpretations
Let $\mathbf A$ be a WFF of propositional logic.
Then $\mathbf A$ is called a tautology (for boolean interpretations) if and only if:
- $\map v {\mathbf A} = \T$
for every boolean interpretation $v$ of $\mathbf A$.
That $\mathbf A$ is a tautology may be denoted as:
- $\models_{\mathrm {BI} } \mathbf A$
Tautology for Predicate Logic
Let $\mathbf A$ be a WFF of predicate logic.
Then $\mathbf A$ is a tautology if and only if, for every structure $\AA$ and assignment $\sigma$:
- $\AA, \sigma \models_{\mathrm{PL_A} } \mathbf A$
that is, if $\mathbf A$ is valid in every structure $\AA$ and assignment $\sigma$.
That $\mathbf A$ is a tautology can be denoted as:
- $\models_{\mathrm{PL_A} } \mathbf A$
Also known as
In this context, tautologies are also referred to as (logically) valid formulas.
However, on $\mathsf{Pr} \infty \mathsf{fWiki}$, this can easily be confused with a formula that is valid in a single structure, and is therefore discouraged.
Also denoted as
When the formal semantics under discussion is clear from the context, $\models \phi$ is a common shorthand for $\models_{\mathscr M} \phi$.
Also see
- Definition:Top (Logic), a symbol often used to represent tautologies in logical languages.
- Definition:Contradiction
- Definition:Contingent Statement
Sources
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.6$: Truth Tables and Tautologies