Definition:Value of Term under Assignment
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Definition
Let $\tau$ be a term in the language of predicate logic $\LL_1$.
Let $\AA$ be an $\LL_1$-structure.
Let $\sigma$ be an assignment for $\tau$ in $\AA$.
Then the value of $\tau$ under $\sigma$, denoted $\map {\operatorname{val}_\AA} \tau \sqbrk \sigma$, is defined recursively by:
- $\map {\operatorname{val}_\AA} x \sqbrk \sigma := \map \sigma x$
- If $\tau = \map f {\tau_1, \ldots, \tau_n}$ with $\tau_i$ terms and $f \in \FF_n$ an $n$-ary function symbol:
- $\map {\operatorname{val}_\AA} {\map f {\tau_1, \ldots, \tau_n} } \sqbrk \sigma := \map {f_\AA} {\map {\operatorname{val}_\AA} {\tau_1} \sqbrk \sigma, \ldots, \map {\operatorname{val}_\AA} {\tau_n} \sqbrk \sigma}$
- where $f_\AA$ denotes the interpretation of $f$ in $\AA$.
If $\tau$ contains no variables, one writes $\map {\operatorname{val}_\AA} \tau$ instead of $\map {\operatorname{val}_\AA} \tau \sqbrk \sigma$.
Sources
- 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\text{II}.7$ First-Order Logic Semantics: Definition $\text{II}.7.4$