Help with notation in logic

Question

In propositional logic I know that if I have a set of atomic formulas $S=\{A_1, . . . ,A_n \}$

That an assignment $A$ of $S$, designates truth values to each of the atomic formula in $S$,

But is there some kind of generalization or extension of this word 'assignment' in predicate logic, like say I have the propositional function,

$P:(\text{X likes Y})$

And I wanted to assign values to the propositional variables ${x,y}$.

What would I call this assignment of variables?

Answer 1

One term that is used is valuation. It comes up in one of the standard ways to define what is meant by a sentence of the language $L$ to be true in a particular $L$-structure $M$. Formally, a valuation is a function from the set of variable symbols to the underlying set of $M$.

Answer 2

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.