Much work in MKM depends on the application of formal logic to mathematics. However, much mathematical knowledge is informal. Luckily, formal logic only represents one tradition in logic, specifically the modeling of inference in terms of logical form. Many inferences cannot be captured in this manner. The study of such inferences is still within the domain of logic, and is sometimes called informal logic. This paper explores some of the benefits informal logic may have for the management of informal mathematical (...) knowledge. (shrink)

In the seventeenth century, a vision arose which was to captivate the Western imagination for the next three hundred years: the vision of Cosmopolis, a society as rationally ordered as the Newtonian view of nature. While fueling extraordinary advances in all fields of human endeavor, this vision perpetuated a hidden yet persistent agenda: the delusion that human nature and society could be fitted into precise and manageable rational categories. Stephen Toulmin confronts that agenda—its illusions and its consequences for our present (...) and future world. "By showing how different the last three centuries would have been if Montaigne, rather than Descartes, had been taken as a starting point, Toulmin helps destroy the illusion that the Cartesian quest for certainty is intrinsic to the nature of science or philosophy."—Richard M. Rorty, University of Virginia "[Toulmin] has now tackled perhaps his most ambitious theme of all.... His aim is nothing less than to lay before us an account of both the origins and the prospects of our distinctively modern world. By charting the evolution of modernity, he hopes to show us what intellectual posture we ought to adopt as we confront the coming millennium."—Quentin Skinner, New York Review of Books. (shrink)

Toulmin’s scheme for the layout of arguments (1958, The Uses of Argument, Cambridge University Press, Cambridge) represents an influential tool for the analysis of arguments. The scheme enriches the traditional premises-conclusion model of arguments by distinguishing additional elements, like warrant, backing and rebuttal. The present paper contains a formal elaboration of Toulmin’s scheme, and extends it with a treatment of the formal evaluation of Toulmin-style arguments, which Toulmin did not discuss at all. Arguments are evaluated in terms of a so-called (...) dialectical interpretation of their assumptions. In such an interpretation, an argument’s assumptions can be evaluated as defeated, e.g., when there is a defeating reason against the assumption. The present work builds on recent research on defeasible argumentation (cf. e.g. the work of Pollock, Reiter, Loui, Vreeswijk, Prakken, Hage and Dung). More specifically, the author’s work on the dialectical logic DEFLOG and the argumentation tool ARGUMED serve as starting points. (shrink)