Definition:Relative Semantic Equivalence/WFF
< Definition:Relative Semantic Equivalence(Redirected from Definition:Relative Semantic Equivalence of Well-Formed Formulas)
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Definition
Let $\FF$ be a theory in the language of predicate logic.
Let $\mathbf A, \mathbf B$ be WFFs.
Let $\mathbf C$ be the universal closure of $\mathbf A \iff \mathbf B$.
Then $\mathbf A$ and $\mathbf B$ are semantically equivalent with respect to $\FF$ if and only if:
- $\FF \models_{\mathrm{PL}} \mathbf C$
That is, if and only if $\mathbf C$ is a semantic consequence of $\FF$.
Also see
Sources
- 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\mathrm{II}.8$ Further Semantic Notions: Definition $\mathrm{II}.8.4$