Five theories of reasoning: Interconnections and applications to mathematics
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The last century has seen many disciplines place a greater prior-ity on understanding how people reason in a particular domain, and several illuminating theories of informal logic and argumentation have been devel-oped. Perhaps owing to their diverse backgrounds, there are several con-nections and overlapping ideas between the theories, which appear to have been overlooked. We focus on Peirce's development of abductive reasoning , Toulmin's argumentation layout , Lakatos's theory of reasoning in mathematics , Pollock's notions of counterexample , and argumen-tation schemes constructed by Walton et al. , and explore some connec-tions between, as well as within, the theories. For instance, we investigate Peirce's abduction to deal with surprising situations in mathematics, rep- resent Pollock's examples in terms of Toulmin's layout, discuss connections between Toulmin's layout and Walton's argumentation schemes, and sug-gest new argumentation schemes to cover the sort of reasoning that Lakatos describes, in which arguments may be accepted as faulty, but revised, rather than being accepted or rejected. We also consider how such theories may apply to reasoning in mathematics: in particular, we aim to build on ideas such as Dove's , which help to show ways in which the work of Lakatos fits into the informal reasoning community.