‘Tipping points’ (TPs) are thresholds of potentially disproportionate changes in the Earth’s climate system associated with future global warming and are considered today as a ‘hot’ topic in environmental sciences. In this study, TP interactions are analysed from an integrated and conceptual point of view using two qualitative Boolean models built on graph grammars. They allow an accurate study of the node TP interactions previously identified by expert elicitation and take into account a range of various large-scale climate processes potentially able to trigger, alone or jointly, instability in the global climate. Our findings show that, contrary to commonly held beliefs, far from causing runaway changes in the Earth’s climate, such as self-acceleration due to additive positive feedbacks, successive perturbations might actually lead to its stabilization. A more comprehensive model defined TPs as interactions between nine (non-exhaustive) large-scale subsystems of the Earth’s climate, highlighting the enhanced sensitivity to the triggering of the disintegration of the west Antarctic ice sheet. We are claiming that today, it is extremely difficult to guess the fate of the global climate system as TP sensitivity depends strongly on the definition of the model. Finally, we demonstrate the stronger effect of decreasing rules (i.e. mitigating connected TPs) over other rule types, thus suggesting the critical role of possible ‘stabilizing points’ that are yet to be identified and studied.