Definition:Relative Semantic Equivalence

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Definition

Let $\mathcal F$ 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 $\mathcal F$ if and only if:

$\mathcal F \models_{\mathrm{PL}} \mathbf C$

That is, iff $\mathbf C$ is a semantic consequence of $\mathcal F$.


Relative Semantic Equivalence of Terms

Let $\tau_1, \tau_2$ be terms.


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

$\mathop{ \operatorname{val}_{\mathcal A} \left({\tau_1}\right) } \left[{\sigma}\right] = \mathop{ \operatorname{val}_{\mathcal A} \left({\tau_2}\right) } \left[{\sigma}\right]$

for all models $\mathcal A$ of $\mathcal F$ and assignments $\sigma$ for $\tau_1,\tau_2$ in $\mathcal A$.

Here $\mathop{ \operatorname{val}_{\mathcal A} \left({\tau_1}\right) } \left[{\sigma}\right]$ denotes the value of $\tau_1$ under $\sigma$.


Also see