•  655
    Five theories of reasoning: Interconnections and applications to mathematics
    with Alison Pease
    Logic and Logical Philosophy 20 (1-2): 7-57. 2011.
    The last century has seen many disciplines place a greater priority on understanding how people reason in a particular domain, and several illuminating theories of informal logic and argumentation have been developed. Perhaps owing to their diverse backgrounds, there are several connections and overlapping ideas between the theories, which appear to have been overlooked. We focus on Peirce’s development of abductive reasoning [39], Toulmin’s argumentation layout [52], Lakatos’s theory of reasoni…Read more
  •  509
    Intellectual humility and argumentation
    In Mark Alfano, Michael Lynch & Alessandra Tanesini (eds.), The Routledge Handbook of the Philosophy of Humility, Routledge. pp. 325-334. 2020.
    In this chapter I argue that intellectual humility is related to argumentation in several distinct but mutually supporting ways. I begin by drawing connections between humility and two topics of long-standing importance to the evaluation of informal arguments: the ad verecundiam fallacy and the principle of charity. I then explore the more explicit role that humility plays in recent work on critical thinking dispositions, deliberative virtues, and virtue theories of argumentation.
  •  265
    The philosophy of alternative logics
    In Leila Haaparanta (ed.), The Development of Modern Logic, Oxford University Press. pp. 613-723. 2009.
    This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of an especial…Read more
  •  262
    Virtue theories have lately enjoyed a modest vogue in the study of argumentation, echoing the success of more far-reaching programmes in ethics and epistemology. Virtue theories of argumentation (VTA) comprise several conceptually distinct projects, including the provision of normative foundations for argument evaluation and a renewed focus on the character of good arguers. Perhaps the boldest of these is the pursuit of the fully satisfying argument, the argument that contributes to human flouri…Read more
  •  249
    Arrogance and deep disagreement
    In Alessandra Tanesini & Michael Lynch (eds.), Polarisation, Arrogance, and Dogmatism: Philosophical Perspectives, Routledge. pp. 39-52. 2020.
    I intend to bring recent work applying virtue theory to the study of argument to bear on a much older problem, that of disagreements that resist rational resolution, sometimes termed "deep disagreements". Just as some virtue epistemologists have lately shifted focus onto epistemic vices, I shall argue that a renewed focus on the vices of argument can help to illuminate deep disagreements. In particular, I address the role of arrogance, both as a factor in the diagnosis of deep disagreements and …Read more
  •  235
    This paper explores some surprising historical connections between philosophy and pornography (including pornography written by or about philosophers, and works that are both philosophical and pornographic). Examples discussed include Diderot's Les Bijoux Indiscrets, Argens's Therésè Philosophe, Aretino's Ragionamenti, Andeli's Lai d'Aristote, and the Gor novels of John Norman. It observes that these works frequently dramatize a tension between reason and emotion, and argues that their existen…Read more
  •  191
    Evidence, Proofs, and Derivations
    ZDM 51 (5): 825-834. 2019.
    The traditional view of evidence in mathematics is that evidence is just proof and proof is just derivation. There are good reasons for thinking that this view should be rejected: it misrepresents both historical and current mathematical practice. Nonetheless, evidence, proof, and derivation are closely intertwined. This paper seeks to tease these concepts apart. It emphasizes the role of argumentation as a context shared by evidence, proofs, and derivations. The utility of argumentation theory,…Read more
  •  183
    Deep disagreements are characteristically resistant to rational resolution. This paper explores the contribution a virtue theoretic approach to argumentation can make towards settling the practical matter of what to do when confronted with apparent deep disagreement, with particular attention to the virtue of courage.
  •  163
    The Uses of Argument in Mathematics
    Argumentation 19 (3): 287-301. 2005.
    Stephen Toulmin once observed that ”it has never been customary for philosophers to pay much attention to the rhetoric of mathematical debate’ [Toulmin et al., 1979, An Introduction to Reasoning, Macmillan, London, p. 89]. Might the application of Toulmin’s layout of arguments to mathematics remedy this oversight? Toulmin’s critics fault the layout as requiring so much abstraction as to permit incompatible reconstructions. Mathematical proofs may indeed be represented by fundamentally distinct l…Read more
  •  159
    In Andrew Aberdein & Matthew Inglis (eds.), Advances in Experimental Philosophy of Logic and Mathematics, Bloomsbury Academic. pp. 1-13. 2019.
    There has been little overt discussion of the experimental philosophy of logic or mathematics. So it may be tempting to assume that application of the methods of experimental philosophy to these areas is impractical or unavailing. This assumption is undercut by three trends in recent research: a renewed interest in historical antecedents of experimental philosophy in philosophical logic; a “practice turn” in the philosophies of mathematics and logic; and philosophical interest in a substantial b…Read more
  •  154
    Virtue in argument
    Argumentation 24 (2): 165-179. 2010.
    Virtue theories have become influential in ethics and epistemology. This paper argues for a similar approach to argumentation. Several potential obstacles to virtue theories in general, and to this new application in particular, are considered and rejected. A first attempt is made at a survey of argumentational virtues, and finally it is argued that the dialectical nature of argumentation makes it particularly suited for virtue theoretic analysis.
  •  148
    Managing Informal Mathematical Knowledge: Techniques from Informal Logic
    Lecture Notes in Artificial Intelligence 4108 208--221. 2006.
    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 inf…Read more
  •  140
    Mathematics and argumentation
    Foundations of Science 14 (1-2): 1-8. 2009.
    Some authors have begun to appeal directly to studies of argumentation in their analyses of mathematical practice. These include researchers from an impressively diverse range of disciplines: not only philosophy of mathematics and argumentation theory, but also psychology, education, and computer science. This introduction provides some background to their work.
  •  130
    Raising the tone: Definition, bullshit, and the definition of bullshit
    In G. Reisch & G. Hardcastle (eds.), Bullshit and Philosophy, Open Court. pp. 151-169. 2006.
    Bullshit is not the only sort of deceptive talk. Spurious definitions are another important variety of bad reasoning. This paper will describe some of these problematic tactics, and show how Harry Frankfurt’s treatment of bullshit may be extended to analyze their underlying causes. Finally, I will deploy this new account of definition to assess whether Frankfurt’s definition of bullshit is itself legitimate.
  •  119
    Logic for dogs
    In Steven D. Hales (ed.), What Philosophy Can Tell You About Your Dog, Open Court. pp. 167-181. 2008.
    Imagine a dog tracing a scent to a crossroads, sniffing all but one of the exits, and then proceeding down the last without further examination. According to Sextus Empiricus, Chrysippus argued that the dog effectively employs disjunctive syllogism, concluding that since the quarry left no trace on the other paths, it must have taken the last. The story has been retold many times, with at least four different morals: (1) dogs use logic, so they are as clever as humans; (2) dogs use logic, so usi…Read more
  •  116
    The Vices of Argument
    Topoi 35 (2): 413-422. 2016.
    What should a virtue theory of argumentation say about fallacious reasoning? If good arguments are virtuous, then fallacies are vicious. Yet fallacies cannot just be identified with vices, since vices are dispositional properties of agents whereas fallacies are types of argument. Rather, if the normativity of good argumentation is explicable in terms of virtues, we should expect the wrongness of bad argumentation to be explicable in terms of vices. This approach is defended through analysis of s…Read more
  •  114
    What do mathematicians mean when they use terms such as ‘deep’, ‘elegant’, and ‘beautiful’? By applying empirical methods developed by social psychologists, we demonstrate that mathematicians' appraisals of proofs vary on four dimensions: aesthetics, intricacy, utility, and precision. We pay particular attention to mathematical beauty and show that, contrary to the classical view, beauty and simplicity are almost entirely unrelated in mathematics.
  •  112
    Arguments with losers
    Florida Philosophical Review 16 (1): 1-11. 2016.
  •  104
    We investigated whether mathematicians typically agree about the qualities of mathematical proofs. Between-mathematician consensus in proof appraisals is an implicit assumption of many arguments made by philosophers of mathematics, but to our knowledge the issue has not previously been empirically investigated. We asked a group of mathematicians to assess a specific proof on four dimensions, using the framework identified by Inglis and Aberdein (2015). We found widespread disagreement between ou…Read more
  •  102
    Observations on Sick Mathematics
    In Bart van Kerkhove, Jean Paul van Bendegem & Jonas de Vuyst (eds.), Philosophical Perspectives on Mathematical Practice, College Publications. pp. 269--300. 2010.
    This paper argues that new light may be shed on mathematical reasoning in its non-pathological forms by careful observation of its pathologies. The first section explores the application to mathematics of recent work on fallacy theory, specifically the concept of an ‘argumentation scheme’: a characteristic pattern under which many similar inferential steps may be subsumed. Fallacies may then be understood as argumentation schemes used inappropriately. The next section demonstrates how some speci…Read more
  •  101
    Truth in Fiction: Rethinking its Logic, by John Woods, Springer, 2018 (review)
    Philosophia 49 (2): 873-881. 2021.
    A review of John Woods, Truth in Fiction: Rethinking its Logic. Cham: Springer, 2018.
  •  98
    Mathematical Wit and Mathematical Cognition
    Topics in Cognitive Science 5 (2): 231-250. 2013.
    The published works of scientists often conceal the cognitive processes that led to their results. Scholars of mathematical practice must therefore seek out less obvious sources. This article analyzes a widely circulated mathematical joke, comprising a list of spurious proof types. An account is proposed in terms of argumentation schemes: stereotypical patterns of reasoning, which may be accompanied by critical questions itemizing possible lines of defeat. It is argued that humor is associated w…Read more
  •  94
    Introduction: Virtues and Arguments
    Topoi 35 (2): 339-343. 2016.
    It has been a decade since the phrase virtue argumentation was introduced, and while it would be an exaggeration to say that it burst onto the scene, it would be just as much of an understatement to say that it has gone unnoticed. Trying to strike the virtuous mean between the extremes of hyperbole and litotes, then, we can fairly characterize it as a way of thinking about arguments and argumentation that has steadily attracted more and more attention from argumentation theorists. We hope it is …Read more
  •  78
    Until recently, discussion of virtues in the philosophy of mathematics has been fleeting and fragmentary at best. But in the last few years this has begun to change. As virtue theory has grown ever more influential, not just in ethics where virtues may seem most at home, but particularly in epistemology and the philosophy of science, some philosophers have sought to push virtues out into unexpected areas, including mathematics and its philosophy. But there are some mathematicians already there, …Read more
  •  77
    Mathematical Monsters
    In Diego Compagna & Stefanie Steinhart (eds.), Monsters, Monstrosities, and the Monstrous in Culture and Society, . pp. 391-412. 2019.
    Monsters lurk within mathematical as well as literary haunts. I propose to trace some pathways between these two monstrous habitats. I start from Jeffrey Jerome Cohen’s influential account of monster culture and explore how well mathematical monsters fit each of his seven theses. The mathematical monsters I discuss are drawn primarily from three distinct but overlapping domains. Firstly, late nineteenth-century mathematicians made numerous unsettling discoveries that threatened their understandi…Read more
  •  74
    Rationale of the Mathematical Joke
    In Alison Pease, Markus Guhe & Alan Smaill (eds.), Proceedings of AISB 2010 Symposium on Mathematical Practice and Cognition, Aisb. pp. 1-6. 2010.
    A widely circulated list of spurious proof types may help to clarify our understanding of informal mathematical reasoning. An account in terms of argumentation schemes is proposed.
  •  61
    Douglas Walton’s multitudinous contributions to the study of argumentation seldom, if ever, directly engage with argumentation in mathematics. Nonetheless, several of the innovations with which he is most closely associated lend themselves to improving our understanding of mathematical arguments. I concentrate on two such innovations: dialogue types (§1) and argumentation schemes (§2). I argue that both devices are much more applicable to mathematical reasoning than may be commonly supposed.