2014 introductory article: https://www.linuxjournal.com/content/promise-theory—what-it
Basic book on the subject: https://www.amazon.com/Thinking-Promises-Designing-Systems-C...
It's not "maths" like arithmentic but it's "maths" like graph theory:
Promise Theory, in the context of information science, is a model of voluntary cooperation between individual, autonomous actors or agents who publish their intentions to one another in the form of promises. It is a form of labelled graph theory, describing discrete networks of agents joined by the unilateral promises they make.
It's less like graph theory and more like inversion of control: an architecture, not a set of theorems and their proofs. Even Burgess' own book you mentioned is nothing like a mathematical handbook.
I'm a great fan of Mark Burgess and his promise theory, but calling it a mathematical theory or a mathematical domain is simply incorrect.
The book I mentioned (Thinking in Promises) is the introductory-level public book.
https://www.amazon.com/Promise-Theory-Principles-Application... is the heavy-duty scientific stuff.
"In Search of Certainty" (https://www.amazon.com/Search-Certainty-Science-Information-...) is somewhere in-between.
I would say promise theory is its own kind of logic and notation. Thanks for the correction @dozzie.
Promise theory is not math, despite its name. It doesn't predict anything, it doesn't explain any phenomena. It's an architectural approach. Brilliant, led to a really great software (CFEngine), but it's not "maths".