Definition:Relative Semantic Equivalence/Term

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

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 $\sigma$.

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