In this paper, we introduce notions of hereditarily weakly selection principles and strongly quasi-selection principles and show that they are different from quasi-selection principles and weakly selection principles which are studied in [1]. By introducing the strongly quasi-separability, the quasi-separability and the weakly separability, we provide relations among these separable properties and weak versions of selection principles. These extend some results of G. Di Maio and Lj.D.R. Kočinac [1].