Definition:Relative Semantic Equivalence

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Definition

Let $\FF$ be a theory in the language of predicate logic.


Relative Semantic Equivalence of WFFs

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 $\FF$.


Relative Semantic Equivalence of Terms

Let $\tau_1, \tau_2$ be terms.


Then $\tau_1$ and $\tau_2$ are semantically equivalent with respect to $\FF$ if and only if:

$\map {\operatorname{val}_\AA} {\tau_1} \sqbrk \sigma = \map {\operatorname{val}_\AA} {\tau_2} \sqbrk \sigma$

for all models $\AA$ of $\FF$ and assignments $\sigma$ for $\tau_1,\tau_2$ in $\AA$.

Here $\map {\operatorname{val}_\AA} {\tau_1} \sqbrk \sigma$ denotes the value of $\tau_1$ under $\sigma$.


Also see