A Furstenberg family $$\mathcal{F}$$ is a family, consisting of some subsets of the set of positive integers, which is hereditary upwards, i.e. A ⊂ B and A ∈ $$\mathcal{F}$$ imply B ∈ $$\mathcal{F}$$ . For a given system (i.e., a pair of a complete metric space and a continuous self-map of the space) and for a Furstenberg family $$\mathcal{F}$$ , the definition of $$\mathcal{F}$$ -scrambled pairs of points in the space has been given, which brings the well-known scrambled pairs in Li-Yorke sense and the scrambled pairs in distribution sense to be $$\mathcal{F}$$ -scrambled pairs corresponding respectively to suitable Furstenberg family $$\mathcal{F}$$ . In the present paper we explore the basic properties of the set of $$\mathcal{F}$$ -scrambled pairs of a system. The generically $$\mathcal{F}$$ -chaotic system and the generically strongly $$\mathcal{F}$$ -chaotic system are defined. A criterion for a generically strongly $$\mathcal{F}$$ -chaotic system is showed.