# Semantics of brackets

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.

• Brackets are for human readability; as you said, we can do without them. If so, what is the "semantic value" of something that can be avoided without changing the meaning of the resulting expression? Commented May 26, 2021 at 6:44
• Having said that, brackets are punctuation marks; maybe useful The Syntax and Semantics of Punctuation Commented May 26, 2021 at 7:11
• 'Semantic value' is another term for meaning. Following your argument, we could also say that the question of the semantics of variables is quite pointless, since in combinatory logic we can do without them. But of course the semantics of variables is not an idle question. So your argument goes wrong here. Commented May 26, 2021 at 13:45