The present study aims at explaining how the Relevance Theory could be a viable approach to weigh up the main functions of some concessive Pragmatic Operators (henceforth POs) in Jordanian Arabic at the production and interpretation levels. A sample of twenty-two speeches delivered by members of the Jordanian Parliament the 16th was randomly selected for scrutiny. Three POs (namely, laakin, bal and wa) detected in their speeches were analyzed at the token level in light of three elementary RT assumptions about discourse connectives in general, namely connectivity (Fraser 1996), the conceptual-procedural distinction (Blakemore 1987, 1988, 1992, 2002; Wilson and Sperber 1993; Grice 1989), and monosemy (Fretheim 2000; cf. Borderia 2008). The major finding of this study was that concessive POs, as a subset of contrastives, are used to optimize relevance: highlight certain dimensions and/or suppress others of the scenario to the background. However, the point of departure from possibly all previous treatments is that the speakers, as politicians, still used them more strategically because total ambiguity resolution should not be a viable alternative in social settings laden with politics.