The field of Pragmatics was born of the recognition that certain elements of the propositional form of a sentence (e.g. pronouns, tense), as well as many other conveyed meanings, depend crucially on contextual information for their determination. These context-sensitive aspects of meaning are not easily accommodated in a theory of semantics, for which the primary object under investigation is the context-free sentence, an abstract, formal entity defined within a theory of grammar, and having only logical and causal properties. One linguistic construction which poses a prima facie problem for a truth-conditional semantics is the emphatic "do", taken in its "implicitly contrastive" use. An example of this use is: "Criminals aren't too bothered about the problems they cause victims in the commission for their crime, but as a general rule they do draw a line somewhere," where the contrast is, very roughly, between the first disjunct of the"but" construction and the second disjunct. Beginning from the observation that implicitly contrastive emphatic "do" does not contribute to the truth-conditional content of the utterance that contains it, I will offer a hypothetical analysis based on the classical Gricean theory of implicature, before settling on a thesis that stretches the uses of the cognitive apparatus of Relevance Theory. Following Relevance Theoretic principles, I will argue that emphatic "do" is a linguistic device that does not contribute to truth conditional content but generates an implicature to the effect that the proposition introduced by the "do" contradicts one of more propositions inferentially related to utterances made earlier in the discourse. The Relevance Theoretic solution, while superficially similar to the hypothetical Gricean one, employs a mode of explanation which allows for a more precise statement of the hearer's inferential strategy and the potential for integration into other scientifically-minded research programs.