Please @user21820, ignore these last comments. My goal is to have a definition of logic (or type of logics) that corresponds to a model for real world deductions, just as Enderton says in his book, and it seems that you have the same goal with your strong and robustness criteria. My feeling was that it had to be done through a classification of logics based on signature and, for each signature, criterias, different from yours, on these structures, but I am not sure it can work.

I did not like the fact that you started with a very flexible notions of logics, because, if every logic or type of logics tries to model the same real world deductions, then any two of them cannot be so different. However, I think you actually try to capture this point with your robustness and strong criteria. I say "or type of logics" to cover the case where one must combine all logics of a given type to model (or hope to model) all possible deductions.

Above, I wrote "criterias, different from yours, on these structures". By "these structures", I meant "the structures of the given signature."

@Dominic108 You are welcome! Have a fruitful time in your other work. =)

Thanks again !