  We define a class of singularity on arbitrary pairs of a normal variety and
an effective $\mathbb{R}$-divisor on it, which we call pseudo-lc in this paper.
This is a generalization of the usual lc singularity of pairs and log canonical
singularity of normal varieties introduced by de Fernex and Hacon. By giving
examples of pseudo-lc pairs which are not lc or log canonical in the sense of
de Fernex--Hacon's paper, we show that pseudo-lc singularity is a strictly
extended notion of those singularities. We prove that pseudo-lc pairs admit a
small lc modification. We also discuss a criterion of log canonicity.
