Definition:Relative Semantic Equivalence
Jump to navigation
Jump to search
Definition
Let $\FF$ 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 $\FF$ if and only if:
- $\FF \models_{\mathrm{PL}} \mathbf C$
That is, if and only if $\mathbf C$ is a semantic consequence of $\FF$.
Relative Semantic Equivalence of Terms
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 $\sigma$.