Interpretation

Variable Assignment

A variable assignment $\sigma$ from the set of variables $\mathcal{V}=\{v_0,v_1,\dots ,v_n,\dots \}$ into the structure $\mathfrak{A}$ is a function $\sigma :\mathcal{V}\to A$ mapping $\mathcal{V}$ into the universe $A$ of $\mathfrak{A}$. The value/interpretation of a term $t$ under an assignment $\sigma$ is defined in the natural way using the complexity of $t$.

Interpretation of a Term

Let $t$ be a term and let $\sigma$ be a variable assignment into $\mathfrak{A}$. Then the interpretation $t^{\mathfrak{A}}[\sigma ]$ of $t$ in $\mathfrak{A}$ under the assignment $\sigma$ is defined by induction on the complexity of $t$ as follows:

1. if $t$ is a variable $x$, then $t^{\mathfrak{A}}[\sigma ]=\sigma (x)$,
2. if $t$ is a constant symbol $c$, then $t^{\mathfrak{A}}[\sigma ]=c^{\mathfrak{A}}$,
3. if $t$ is a term $f(t_1,\dots ,t_m)$, then $t^{\mathfrak{A}}[\sigma ]=f^{\mathfrak{A}}(t_1^{\mathfrak{A}}[\sigma ],\dots ,t_m^{\mathfrak{A}}[\sigma ]).$