The generalized maximum likelihood estimator (GMLE) of a survival function $S_o(t)$ based on truncated interval-censored (IC) data has been studied since 1990's. In this literature there are several issues that have not been properly settled in how to compute the GMLE and in theoretical part: innermost intervals based on these data are not correctly formulated and they lead to inconsistent GMLE, the self-consistent algorithm for the GMLE is wrong, the truncated interval censorship models are not realistic, and the consistency of the GMLE has not been established. These issues are settled in this paper.