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.