Can morality be so demanding that we have reason not to follow its dictates? According to many, it can, if that morality is a consequentialist one. We take the plausibility and coherence of this objection – the Demandingness Objection – as a given and are also not concerned with finding the best response to the Objection. Instead, our main aim is to explicate the intuitive background of the Objection and to see how this background could be investigated. This double aim leads to different albeit connected threads of inquiry. We first outline the Objection, its different forms and how intuition figures in them. After this, we move on to consider the ongoing debate about the use of intuitions in (moral) philosophy with a focus on two challenges: what intuitions are and how we can detect them. To answer these challenges, we propose an account according to which moral intuitions are seemings that are characterized by being non-inferential, spontaneous, non-doxastic, phenomenologically distinctive, non-sensory, intrinsically motivating, and stable. Armed with these seven “markers” of moral intuition, we put a forward a complex experimental methodology and raise and respond to possible problems with it.