Given a function free and quantifier-free First-Order Logic Formula $\phi(X)$. I wish to define the notion of a truth table over its atoms, and I call it "First Order Truth Table". My attempt is as follows:
First Order Truth Table of a formula $\Phi(X)$ is a truth table $T_{\Phi}(X)$ with columns defined over atoms of $\Phi(X)$. The rows of $T_{\Phi}(X)$ are truth values assigned to atoms of the formula such that $\Phi(X)$ is satisfied.
I will add an example to make clear, what I mean.
$\Phi(x,y) = F(x) \lor F(y)$
The First Order Truth Table is defined as follows,
\begin{array}{|c|c|} \hline F(x)& F(y) \\ \hline T& T \\ \hline T& F \\ \hline F& T \\ \hline \end{array}
I wish to verify if the statement I wrote really conveys the idea even without the example. Any help in reformulating it better or feedback on the statement will be much appreciated. $\Phi(X)$ is quantifier-free is apparent from the context of the article I am writing, hence no need to add it in the definition.