The semantical structure of conditionals, and its relation to formal argumentation

Conditionals, i. e. expressions of the logical form “if A, then B”, have been a central topic of study ever since logic was on the academic menu. In contemporary logic, there is a consensus that the semantics of conditionals are best obtained by stipulating a subset of possible worlds in which the antecedent is true, and verifying whether the consequent is true in those worlds. Such a subset of possible worlds can represent, for example, the most typical worlds in which the antecedent is true. This idea has proven a fruitful basis, allowing for many systematic characterisation results as well as for making connections to other topics, such as belief revision and modal logic. In formal argumentation, the potential of these semantical ideas has not gone unnoticed in the last years, and many works have attempted to bridge the worlds of conditionals and arguments on the basis of these ideas. In this article, we give a thorough introduction to the semantics of conditionals and survey the adaptions of these semantics in the literature on computational argumentation, including structured argumentation and generalisations of abstract argumentation such as abstract dialectical frameworks. Furthermore, we highlight opportunities for future research on this topic.