The complex organization of syntax in hierarchical structures is one of the core design features of human language. Duality of patterning refers, for instance, to the organization of the meaningful elements in a language at two distinct levels: a combinatorial level, where meaningless forms are combined into meaningful forms; and a compositional level, where meaningful forms are composed into larger lexical units. The question remains wide open regarding how such structures could have emerged. The aim of this paper is to address these two aspects in a self‐consistent way. First, we introduce suitable measures to quantify the level of combinatoriality and compositionality in a language, and we present a framework to estimate these observables in human natural languages. Second, we show that a recently introduced multi‐agent modeling scheme, namely the Blending Game, provides a mathematical framework to address the problem of how a population of individuals can bootstrap combinatoriality and compositionality. Theoretical predictions based on this model turn out to be in good agreement with empirical data. It is remarkable that the two sides of duality of patterning emerge simultaneously as a consequence of a pure cultural dynamics in a simulated environment that contains meaningful relations, provided a simple constraint on message transmission fidelity is also considered.