Complex, multigenic biological traits are shaped by the emergent interaction of proteins being the main functional units at the molecular scale. Based on a phenomenological approach, algorithms for quantifying two different aspects of emergence were introduced (Wegner and Hao in Progr Biophys Mol Biol 161:54–61, 2021) describing: (i) pairwise reciprocal interactions of proteins mutually modifying their contribution to a complex trait (denoted as weak emergence), and (ii) formation of a new, complex trait by a set of n ‘constitutive’ proteins at (...) concentrations exceeding individual threshold values (strong emergence). The latter algorithm is modified here to take account of protein redundancy with respect to a complex trait (‘full redundancy’). Irreducibility is considered a necessary and sufficient criterion for strong biological emergence; if one constitutive protein is missing, or its concentration drops below the threshold the trait is lost. A definition based on ‘unpredictability’ is dismissed, because this criterion is irrelevant for the evolution of a complex trait, and apparent unpredictability may rather reflect our basic deficits in understanding unless we can provide an unequivocal proof for it. The phenomenological approach advocated here allows to identify hidden rules according to which strongly emergent traits may be organized. This is of high value for understanding the evolution of complex traits which seems to require the saltational advent of all constitutive proteins ‘in one turn’ to arrive at a functional trait providing for an improved fitness of the organism. Rather than being a purely random process, it may be guided by fundamental structural principles. (shrink)