This paper presents an agent-based model that studies the emergence and evolution of a language system of logical constructions, i.e. a vocabulary and a set of grammatical constructions that allow the expression of logical combinations of categories. The model assumes the agents have a common vocabulary for basic categories, the ability to construct logical combinations of categories using Boolean functions, and some general purpose cognitive capacities for invention, adoption, induction and adaptation. But it does not assume the agents have a vocabulary for Boolean functions nor grammatical constructions for expressing such logical combinations of categories through language. The results of the experiments we have performed show that a language system of logical constructions emerges as a result of a process of self-organisation of the individual agents' interactions when these agents adapt their preferences for vocabulary and grammatical constructions to those they observe are used more often by the rest of the population, and that such a language system is transmitted from one generation to the next.