This is primarily a question about related literature. I am looking for specific references, or terminology that I can use to search for references.
Let A a well defined mathematical structure of some sort: string, graph, etc. Let P be the set of all properties of A. By this I really mean to include all true statements about A. (For example, P could contain, among other things, any Turing machine that outputs a 1 on some representation of A.)
What area, if any, has even attempted to tackle such a formalization? Logic? Formal language theory? Model theory? Set theory?
I know my question is slightly vague, but I am hoping to get some related work which will help me cast my question it in the right framework.
For some family of structures A, I would like to obtain a formal way of talking about all possible properties of A. I need the language for describing these properties to be ``computable:'' I would like to design algorithms that examine these properties and compute using them. I am looking for any terminology or references that will help me achieve this goal. Answers would be more appreciated if they also provide examples of how properties may be discussed in the framework being suggested.