I'm interested in finding a categorematic semantics for brackets as used in formal languages, i.e. for indicating derivation histories of expressions. A categorematic semantics for brackets assigns them an appropriate object from a meaning algebra, that is a set of meanings paired with operations from meanings to meanings. I'm curious whether there is some literature on this, since I have some ideas on this topic, but am unsure whether they can put to work.
For instance, take a standard first-order language, where connectives are presented via infix notation (Polish notation would make brackets redundant of course). Syntactically, we can take the left-right-brackets of the language as a 3-place syntactic operation, that acts the identity function on strings of the form A, #, B where A, B are formulas and # one of the 2-place connectives of the language.
Semantically we can consider the left-right-brackets (LRB) to be a sort of generalized functional application on assigment sets and set operations (assignments are of course functions from variables to domain elements): The idea is that LRB are associated with a 3-place operation, O, on the field of sets we get by considering the set of all sets of assignments. O takes as input a pair of assignment sets and one of the 2-place boolean operations (intersection or union) and returns the set we get by applying the boolean operation to the assignment sets.
Consequently, using O seems to yield a fully categorematic semantics for first-order formulas, where every expression is assigned either an assignment set or an operation on assignment sets. E.g. in a first-order model M the assignment set of the formula (A & B) is the assignment set O(A', ∩, B'), where ∩ is of course restricted to the domain of the model.
