How cooperation evolves and persists widely remains an open problem for improving humanity across domains ranging from climate change to pandemic response. To shed light on how behavioural norms can resolve social dilemmas around cooperation, we present a formal mathematical model of individuals’ decision making under general social norms, encompassing a variety of concerns and motivations an individual may have beyond simply maximizing their own payoff. Using the canonical Prisoner’s dilemma, we compare four norms: compassion, universalizability, reciprocity and equity, to determine which, if any, social forces can facilitate the evolution of cooperation. We analyse our model through a variety of limiting cases, including weak selection, low mutation and large population sizes. This is complemented by computer simulations of population dynamics via a Fisher process, which confirm our theoretical results. We find that the first two norms lead to the emergence of cooperation in a wide range of games, but the latter two cannot. Owing to our framework’s generality, it can be used to investigate many other norms, and how norms themselves evolve. Our work complements recent work on fair-minded learning dynamics and provides a useful bottom-up perspective into understanding the impact of top-down social norms on collective cooperative intelligence.