Definition:Relative Semantic Equivalence/Term

Definition
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 $\FF$ :


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

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

 * Definition:Relative Semantic Equivalence of Well-Formed Formulas