I am working on this problem in which I have a theory $T$ based on language $\mathcal{L}$ and the only information we have is that T is consistent and $\vdash \lnot(A \rightarrow A)$. 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 $\lnot(A \rightarrow A)$ from axioms and inference rules, $\lnot(A \rightarrow A)$ 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 say about this theory?
Thank you!