On Grice's original conception, conversational implicatures are inferences that listeners make in order to reconcile the speaker's behavior with the assumption that the speaker is cooperative. This social, interactional characterization goes hand-in-hand with complex reasoning about nested beliefs: roughly, for p to count as an implicature, the speaker must believe that the listener will infer that the speaker believes p. In this talk, I'll study a range of related computational models that seek to make good on this characterization. Broadly speaking, these models claim that implicatures follow from decision making in communicatively cooperative contexts. However, I will look equally closely at grammar-driven accounts of implicature that de-emphasize the central aspects of Grice's definition. Broadly speaking, these models claim that implicatures follow from the nature of logical form and independently motivated semantic operators. I'll review the evidence for and against these positions, pitting them against each other in a grand debate. My focus here will be on (i) embedding under attitude predicates, (ii) evidence from quasi-downward-monotone environments, (iii) the status of Hurford's constraint, and (iv) recent experimental evidence concerning non-monotone quantifiers. My primary finding is that there is no real tension between interactional and grammar-driven accounts. The grammar-driven accounts give us precise ways of generating alternatives, and the interactional accounts explain how those alternatives interact with the context and general pragmatic pressures to yield conversational implicatures. Professor Potts' current projects include: Grounded language understanding as social cognition, bringing sentiment analysis and social network analysis together, and pragmatic enrichment and contextual inference. He is currently teaching a course entitled "Programming for Linguists."