Definition:Value of Formula under Assignment

Definition
Let $\mathbf A$ be a WFF in the language of predicate logic $\mathcal L_1$.

Let $\mathcal A$ be an $\mathcal L_1$-structure.

Let $\sigma$ be an assignment for $\mathbf A$ in $\mathcal A$.

Then the value of $\mathbf A$ under $\sigma$, denoted $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma}\right]$, is defined recursively by:


 * If $\mathbf A = p \left({\tau_1, \ldots, \tau_n}\right)$ with $\tau_i$ terms and $p \in \mathcal P_n$ an $n$-ary predicate symbol:


 * $\mathop{ \operatorname{val}_{\mathcal A} \left({p \left({\tau_1, \ldots, \tau_n}\right) }\right) } \left[{\sigma}\right] :=  p_{\mathcal A} \left({ \mathop{ \operatorname{val}_{\mathcal A} \left({\tau_1}\right) } \left[{\sigma}\right], \ldots, \mathop{ \operatorname{val}_{\mathcal A} \left({\tau_n}\right) } \left[{\sigma}\right] }\right)$


 * where $p_{\mathcal A}$ denotes the interpretation of $p$ in $\mathcal A$ and $\mathop{ \operatorname{val}_{\mathcal A} \left({\tau_i}\right) } \left[{\sigma}\right]$ is the value of $\tau_i$ under $\sigma$.


 * If $\mathbf A = \neg \mathbf B$ with $\mathbf B$ a WFF:


 * $\mathop{ \operatorname{val}_{\mathcal A} \left({\neg \mathbf B}\right) } \left[{\sigma}\right] :=  f^\neg \left({ \mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B}\right) } \left[{\sigma}\right] }\right)$


 * where $f^\neg$ denotes the truth function of $\neg$.


 * If $\mathbf A = \left({\mathbf B \circ \mathbf B'}\right)$ with $\mathbf B, \mathbf B'$ WFFs and $\circ$ one of $\land, \lor, \implies, \iff$:


 * $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B \circ \mathbf B'}\right) } \left[{\sigma}\right] :=  f^\circ \left({ \mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B}\right) } \left[{\sigma}\right], \mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B'}\right) } \left[{\sigma}\right] }\right)$


 * where $f^\circ$ denotes the respective truth function of $\circ$.


 * If $\mathbf A = \left({ \exists x: \mathbf B}\right)$ with $x \in \mathrm{VAR}$ and $\mathbf B$ a WFF.


 * $\mathop{ \operatorname{val}_{\mathcal A} \left({\exists x: \mathbf B}\right) } \left[{\sigma}\right] :=  \begin{cases}

T & \text{if for some $a \in A$, $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B}\right) } \left[{\sigma + (x / a)}\right] = T$} \\ F & \text{otherwise} \end{cases}$


 * where $\sigma + (x / a)$ is the extension of $\sigma$ by mapping $x$ to $a$.


 * If $\mathbf A = \left({ \forall x: \mathbf B}\right)$ with $x \in \mathrm{VAR}$ and $\mathbf B$ a WFF.


 * $\mathop{ \operatorname{val}_{\mathcal A} \left({\forall x: \mathbf B}\right) } \left[{\sigma}\right] :=  \begin{cases}

T & \text{if for all $a \in A$, $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf B}\right) } \left[{\sigma + (x / a)}\right] = T$} \\ F & \text{otherwise} \end{cases}$


 * where $\sigma + (x / a)$ is the extension of $\sigma$ by mapping $x$ to $a$.

Also see

 * Definition:Model (Predicate Logic)