There are two parts to this question:
How do I determine what is trivial and non-trivial in FOL? I saw several examples in linear algebra, but not in FOL.
What does it mean to be non-trivially consistent? I understand consistency, and am hoping this will be made obvious by answering #1.
Thank you.
$\Sigma$ is an inconsistent set of sentences just in case, for some $\varphi$, $\Sigma$ entails both $\varphi$ and $\neg\varphi$.
Suppose for some $\varphi$, $\Sigma$ already contains both $\varphi$ and $\neg\varphi$ as members. Then, quite trivially, $\Sigma$ is inconsistent -- we need do no real work to show that, we don't need to derive any new consequences from premisses in $\Sigma$. In such a case, it is natural to say that $\Sigma$ is trivially inconsistent. When $\Sigma$ is inconsistent but does not already contain a pair of the form $\varphi$, $\neg\varphi$, then we can say that $\Sigma$ is non-trivially inconsistent.
That isn't standard jargon, but it is entirely natural, and I imagine is all that is intended here.