Definition:Relative Semantic Equivalence/Term

Definition
Let $\mathcal F$ 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 $\mathcal F$ :


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

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

 * Definition:Relative Semantic Equivalence of Well-Formed Formulas