Definition:Tautology/Formal Semantics/Predicate Logic
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Definition
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 $\models_{\mathrm{PL_A} } \phi$.
Sources
- 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\text{II}.8$ Further Semantic Notions: Definition $\text{II.8.1}$