Language is a powerful coordination device. We generalize the cheap-talk approach to pre-play communication by way of introducing a meaning correspondence between messages and actions, and by postulating two axioms met by natural languages. Players have a lexicographic preference, second to material payoffs, against deviating from the meaning correspondence. Under two-sided communication in generic and symmetric n × n-coordination games, a Nash equilibrium component in such a lexicographic communication game is evolutionarily stable if and only if it results in the unique Pareto efficient outcome of the underlying game. We extend the analysis to one-sided communication in arbitrary finite two-player games.