In this paper, we use concepts, structure and tools from argumentation theory to show how conversational implicatures are triggered by conflicts of presumptions. Presumptive implicatures are shown to be based on defeasible forms of inference used in conditions of lack of knowledge, including analogical reasoning, inference to the best explanation, practical reasoning, appeal to pity, and argument from cause. Such inferences are modelled as communicative strategies used to fill knowledge gaps by shifting the burden of proof to provide the missing contrary evidence to the other party in a dialogue. Through a series of illustrative examples, we show how such principles of inference are based on common knowledge about the ordinary course of events shared by participants in a structured dialogue setting in which they take turns putting forward and responding to speech acts.