We investigate the potential of logic programming (LP) to model morality aspects studied in philosophy and psychology. We do so by identifying three morality aspects that appear, in our view, amenable to computational modeling by appropriately exploiting LP features: the dual-process model (reactive and deliberative) in moral judgments; justification of moral judgments by contractualism; and intention in moral permissibility. The research aims at developing an LP-based system with features needed in modeling moral settings, putting emphasis on modeling these above-mentioned morality aspects. We have currently co-developed two essential ingredients of the LP system, i.e., abduction and logic program updates, by exploiting the benefits of tabling features in logic programs. They serve as the basis for our whole system, into which other reasoning facets will be integrated, to model the surmised morality aspects. Moreover, we touch upon the potential of our ongoing studies of LP-based cognitive features for the emergence of computational morality in populations of agents enabled with the capacity for intention recognition, commitment and apology.