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A Structured Argumentation Framework for Modeling Debates in the Formal Sciences


Scientific research in the formal sciences comes in multiple degrees of formality: fully formal work; rigorous proofs that practitioners know to be formalizable in principle; and informal work like rough proof sketches and considerations about the advantages and disadvantages of various formal systems. This informal work includes informal and semi-formal debates between formal scientists, e.g. about the acceptability of foundational principles and proposed axiomatizations. In this paper, we propose to use the methodology of structured argumentation theory to produce a formal model of such informal and semi-formal debates in the formal sciences. For this purpose, we propose ASPIC-END, an adaptation of the structured argumentation framework ASPIC+ which can incorporate natural deduction style arguments and explanations. We illustrate the applicability of the framework to debates in the formal sciences by presenting a simple model of some arguments about proposed solutions to the Liar paradox, and by discussing a more extensive—but still preliminary—model of parts of the debate that mathematicians had about the Axiom of Choice in the early twentieth century.

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  1. 1.

    In this paper, we use the word rule in the way in which it is usually used in the structured argumentation literature. There is one important difference between this usage of rule and the way the word is usually used in the logical literature outside of structured argumentation theory: A rule, as the word is used in structured argumentation theory, is what would normally be called an instance of a rule. For this reason, it makes sense to speak of a rule scheme (as we will frequently do in Sect. 5), which is what would normally be just called a rule.

  2. 2.

    The early formalisms of Pollock (1987, 1995) also allowed for arguments involving hypothetical reasoning. Most of the work in structured argumentation theory that built on this early work of Pollock ignored this type of arguments. In a recent paper, Beirlaen et al. (2018) have critically assessed the way hypothetical arguments function in Pollock’s formalisms and have identified three problematic features of the formalism in Pollock (1995). By not allowing defeasible rules within hypothetical reasoning, we avoid these problematic features.

  3. 3.

    Note that our aim here is not to present a detailed case study of how a debate about a semantic paradox can be formalized in ASPIC-END, but only to illustrate the way ASPIC-END works and could be used for such a case study in future work. For this reason, we restrict ourselves to a simple exposition of the Liar paradox and two very simple explanations of it, a truth-value gap explanation and a paracomplete explanation. See Field (2008) for comprehensive presentations of truth-value gap and paracomplete explanations, besides many others. Additionally note that, for the sake of simplicity, we only include in our model those instances of rules that are actually used in the explanations that we formalize, so we leave out other instances of the general rules (rule schemes) that lie behind these instances. A detailed case study would have to consider what happens when all instances of these rules are included; for this purpose, other paradoxes like Curry’s paradox and various revenge versions of the Liar paradox would need to be considered as well, as the instances of these rules applied to the paradoxical sentences from these other paradoxes would be included in the model.


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Correspondence to Marcos Cramer.

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This paper is a significant extension of a workshop paper presented at the 2017 International Workshop on Theory and Applications of Formal Argument (Dauphin and Cramer 2017). Jérémie Dauphin has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 690974 for the Project “MIREL: MIning and REasoning with Legal texts”.

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Cramer, M., Dauphin, J. A Structured Argumentation Framework for Modeling Debates in the Formal Sciences. J Gen Philos Sci 51, 219–241 (2020).

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  • Argumentation theory
  • Formal sciences
  • Natural deduction
  • Hypothetical reasoning
  • Axiom of choice