We give a new generalization of Strict-tolerant logic. The main idea is that strict truth is the standard of truth accepted by everyone, and tolerant truth the standard accepted by someone. To formalize the idea, we use generalized matrices in abstract algebraic logic, defining strict truth as the intersection of the closure system in the given generalized matrix, and tolerant truth the union of it. We connect our generalized strict-tolerant logics with p-consequences and propose conditions for such logics to be classical. Compared to existing generalizations of Strict-tolerant logic, our generalization is both intuitive and general enough.