You can use the semantics specifications in Herbert Enderton, A Mathematical Introduction to Logic (2nd - 2001), page 83 :
let $\varphi$ a formula in the (first-order) language $\mathcal L$, $\mathcal A$ a structure for the language and $s : Var \mapsto |\mathcal A|$ a function from the set $Var$ of all variables into the universe (or domain) $|\mathcal A|$ of $\mathcal A$.
We may call $s$ a assignment function for the variables.
The "basic" clause of the semantics consists in defining :
what it means for $\mathcal A$ to satisfy $\varphi$ with $s$;
in symbols :
$\mathcal A \vDash \varphi[s]$.
The informal version is :
$\mathcal A \vDash \varphi[s]$ if and only if the translation of $\varphi$ determined by $\mathcal A$, where the variable $x$ is translated as $s(x)$ wherever it occurs free, is true.
In conclusion, the "extension" to predicate logic of the concept of assignment of truth-values to the propositional letters of a propositional formula (which we may call truth assignment or valuation) is that of interpretation, made of a structure for the language and an assignment function for the variables.