I am working on this problem in which I have a theory T based on L language and the only information we have is that T is consistent and |- not(A -> A). Given this information, how can I know if this theory is sound, complete and/or decidable?
My only guess is that I can say that T is sound because since T is consistent and we can derive not(A->A) from axioms and inference rules, not(A->A) is a theorem and because of that we can assume that T is sound (because the premises and conclusions are true).
Am I correct? What else can I tell about this theory?