I know people have used lambda terms which "compile into" first-order logic, but based on what I have seen that is not particularly great for representing complex sentences with causal or temporal relations, or dealing with certain quantifiers. There are so many flavors of logic (modal, epistemic, temporal, higher-order) - has anybody opined about the logic which can represent the maximal subset of English language? I've read about Episodic Logic, which seems promising, but I haven't seen it used by researchers outside of Computer Science.
The wording of your question seems to imply an equivalence between first order predicate logic and higher order logics. They are not equivalent. First order predicate logic was shown to be consistent and complete by Kurt Gödel, but "Stronger logics, such as second-order logic, are not complete." (Consistency.)
Whether first order predicate logic is sufficient to describe the logic of natural languages is an open question.
I'd recommend looking at Jerry Hobbs' way of parsing and representing English, it's one of the most elaborate ones when it comes to logic and commonsense representation: