Definition:Relative Semantic Equivalence/Term
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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 $\FF$ 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
Sources
- 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\text{II}.8$ Further Semantic Notions: Definition $\text{II.8.4}$