The main ingredient of this paper is the derivation of the generalized version of Hamilton's rule. This version is derived with the Generalized Price equation. The generalized version of Hamilton's rule generalizes the original rule, in the sense that it produces a set of rules; one rule for every different model of how social interactions affect fitnesses. Every such Hamilton-like rule is generally valid; they all correctly determine when altruism, or costly cooperation, will be selected for, whatever model they are combined with. Every such rule, however, only has a meaningful interpretation in combination with the model it belongs to. The classic Hamilton's rule is the generalized Hamilton's rule that goes with the linear model. The insight that there are many Hamilton-like rules, all of which are generally valid, but none of which is generally meaningful, helps understand the controversy surrounding Hamilton's rule, and provides a constructive way to always find a rule that both gets the direction of selection right, and has a meaningful interpretation.