Formal logic tends to be concerned with minimal or at least almost-minimal sets of logical connectives. The standard logical connectives are and, or, implies, iff, neg (I couldn't use Latex for their formal symbols) and for quantification forall, exists. In fact, even and, or, neg, forall, exists is a minimal set.
In natural language however, we have all kinds of logical constructions that are syntactically different and allow you to structure sentences in a different ordering than if one had to write them out in the basic logical connectives. For the purpose of this question I'll call this "Extended propositional logic": For example we can say things like:
"Extended propositional logic": A, except when B in which case C. Propositional logic: not B implies A, and B implies C and not A
However, I've never seen such more "exotic" connectives be formalized. In fact, the "except-connective" is not a binary connective, but can be arbitrarily complex:
"Extended propositional logic": A, except when B in which case C, or when D in which case E, or when F in which case G. Propositional logic: not (B or D or F) implies A, and B implies (C and not A), and D implies (E and not A), and F implies (G and not A).
There are certainly others (e.g. even the simple ternary connective "if A then B, else C" isn't used in formal logic, and instead written as A implies B, and notA implies C). It seems like some propositions can be stated more succinctly using appropriate natural language connectives than in terms of the standard formal connectives.
Question: Is there a literature that attempts to formalize of such more "exotic" connectives that people use in natural language, and in particular define symbols for them or at least formal grammars/languages with precise semantics, and with a procedure for turning those "exotic formal symbols" into formulas using only standard connectives? I can imagine that there are also quantifiers that are more exotic than forall, exists but can be restated in terms of those two.
Note: I'm not asking for things like modal logics, which have an entirely different semantics.
Edit: Here is a quantifier-example based on the word "also":
"Extended predicate logic": Only A(a), except when B in which case also A(b) Predicate logic: notB implies (A(a) and forall x not equal to a, notA(x)), and B implies (A(a) and A(b) and forall x not equal to a and not equal to b, notA(x))