The barber's paradox can not be expressed in Principia Mathematica (PM). PM defines a theory of sets, classes, 2-classes, 3-classes, etc... so that members of an N-class can only contain elements that are of type K-classes where K < N. The barber's paradox can not be formulated in such a theory.
Consequently PM does not have an issue with the barber's paradox. PM is incomplete in that there are truths that can be expressed but can not be proven, but there are no known paradoxes/inconsistencies.
From a programming point of view PM is like a statically typed programming language. If something type checks in PM then it's correct. But as anyone who programs in Rust can tell you, sometimes you know that something is correct but it's a pain to find a way to express it in the language's rigid type system and sometimes it's simply impossible to express that a program is correct using the language's rigid type system. That's the problem that PM has and the sense in which PM is incomplete.
Consequently PM does not have an issue with the barber's paradox. PM is incomplete in that there are truths that can be expressed but can not be proven, but there are no known paradoxes/inconsistencies.
From a programming point of view PM is like a statically typed programming language. If something type checks in PM then it's correct. But as anyone who programs in Rust can tell you, sometimes you know that something is correct but it's a pain to find a way to express it in the language's rigid type system and sometimes it's simply impossible to express that a program is correct using the language's rigid type system. That's the problem that PM has and the sense in which PM is incomplete.