Models based on causal capacities, or independent causal influences/mechanisms, are widespread in the sciences. This paper develops a natural mathematical framework for representing such capacities by extending and generalizing previous results in cognitive psychology and machine learning, based on observations and arguments from prior philosophical debates. In addition to its substantial generality, the resulting framework provides a theoretical unification of the widely-used noisy-OR/AND and linear models, thereby showing how they are complementary rather than competing. This unification helps to explain many (...) of the shared cognitive and mathematical properties of those models. (shrink)