For more than thirty years, semantics and pragmatics have benefitted from descriptive studies on discourse connectives, the argumentative function of which has been recognized in different theoretical frameworks: theory of linguistic argumentation (Ducrot et al. in Les mots du discours. Minuit, Paris, 1980 ), Relevance Theory (Blakemore in Semantic constraints on relevance. Blackwell, Oxford, 1987 ; Carston in Thoughts and utterances: the pragmatics of explicit communication. Blackwell, Oxford, 2002 ), cognitive semantics (Sanders and Nordmann in Discourse Processes 29(1):37–60, 2000 ), Rhetorical Structure Theory (Taboada in Journal of Pragmatics 38(4):567–592, 2006 ). However, an issue has not been deeply investigated: What is the difference between discourses with and without connectives? In this chapter, I raise the issue of the function of connectives in discourse, the different ways of defining connectives, the paradox that cognitive approaches to connectives give rise to, the role of mais in argumentation, and the issue of the contribution of connectives to argumentation. In the analysis I show how the contrast between sequences with and without connectives can be thought in a more general framework: argumentation sequences with connectives are more efficient because they make argumentation relations explicit, and have as side effects the minimization of processing costs and the maximizing of relevance; moreover, they are stronger because connectives introduce new focal information, which has more contextual implications than a non-focal one.