This paper presents an agent-based model of the emergence and transmission of a language system for the expression of logical combinations of propositions. The model assumes the agents have some cognitive capacities for invention, adoption, repair, induction and adaptation, a common vocabulary for basic categories, and the ability to construct complex concepts using recursive combinations of basic categories with logical categories. It also supposes the agents initially do not have a vocabulary for logical categories (i.e. logical connectives), nor grammatical constructions for expressing logical combinations of basic categories through language. The results of the experiments we have performed show that a language system for the expression of logical combinations emerges as a result of a process of self-organisation of the agents' linguistic interactions. Such a language system is concise, because it only uses words and grammatical constructions for three logical categories (i.e. and, or, not). It is also expressive, since it allows the communication of logical combinations of categories of the same complexity as propositional logic formulas, using linguistic devices such as syntactic categories, word order and auxiliary words. Furthermore, it is easy to learn and reliably transmitted across generations, according to the results of our experiments.