Understanding emergence along the lines of self-organization has become so ubiquitous the two terms have just about become synonymous. However, the usual connotations of self-organization result in a misleading account of emergence by downplaying the radical novelty characterizing emergent phenomena. It is this radical novelty which generates the necessary explanatory gap between the antecedent, lower level properties of emergent substrates and the consequent, higher level properties of emergent phenomena. Without this explanatory gap, emergent phenomena are not unpredictable, are not non-deducible, are not irreducible, and thus are not truly emergent. For emergent phenomena to be genuinely emergent, processes of emergence must accomplish the seemingly paradoxical feat of producing an explanatory gap while simultaneously maintaining some degree of continuity with the substrate level.