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 specific mathematical fallacies may be characterized in terms of argumentation (...) schemes. The third section considers the phenomenon of correct answers which result from incorrect methods. This turns out to pose some deep questions concerning the nature of mathematical knowledge. In particular, it is argued that a satisfactory epistemology for mathematical practice must address the role of luck. (shrink)

Is it possible to distinguish communities of arguers by tracking the argumentation schemes they employ? There are many ways of relating schemes to communities, but not all are productive. Attention must be paid not only to the admissibility of schemes within a community of argumentational practice, but also to their comparative frequency. Two examples are discussed: informal mathematics, a convenient source of well-documented argumentational practice, and anthropological evidence of nonstandard reasoning.

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.

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.

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.

This paper considers the application to mathematical fallacies of techniques drawn from informal logic, specifically the use of ”argument schemes’. One such scheme, for Appeal to Expert Opinion, is considered in some detail.

Informal logic is a method of argument analysis which is complementary to that of formal logic, providing for the pragmatic treatment of features of argumentation which cannot be reduced to logical form. The central claim of this paper is that a more nuanced understanding of mathematical proof and discovery may be achieved by paying attention to the aspects of mathematical argumentation which can be captured by informal, rather than formal, logic. Two accounts of argumentation are considered: the pioneering work of (...) Stephen Toulmin [The uses of argument, Cambridge University Press, 1958] and the more recent studies of Douglas Walton, [e.g. The new dialectic: Conversational contexts of argument, University of Toronto Press, 1998]. The focus of both of these approaches has largely been restricted to natural language argumentation. However, Walton’s method in particular provides a fruitful analysis of mathematical proof. He offers a contextual account of argumentational strategies, distinguishing a variety of different types of dialogue in which arguments may occur. This analysis represents many different fallacious or otherwise illicit arguments as the deployment of strategies which are sometimes admissible in contexts in which they are inadmissible. I argue that mathematical proofs are deployed in a greater variety of types of dialogue than has commonly been assumed. I proceed to show that many of the important philosophical and pedagogical problems of mathematical proof arise from a failure to make explicit the type of dialogue in which the proof is introduced. (shrink)

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)

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 layouts. However, cases of genuine conflict characteristically reflect an underlying (...) disagreement about the nature of the proof in question. (shrink)

Charles Stevenson introduced the term 'persuasive definition’ to describe a suspect form of moral argument 'which gives a new conceptual meaning to a familiar word without substantially changing its emotive meaning’. However, as Stevenson acknowledges, such a move can be employed legitimately. If persuasive definition is to be a useful notion, we shall need a criterion for identifying specifically illegitimate usage. I criticize a recent proposed criterion from Keith Burgess-Jackson and offer an alternative.

This paper rejects the claim that, rationally, we should follow the argument wherever it leads.

Informal logic is the attempt to develop a logic to assess, analyse and improve ordinary language (or "everyday") reasoning. It intersects with attempts to understand such reasoning from the point of view of philosophy, formal logic, cognitive psychology, and a range of other disciplines. Most of the work in informal logic focuses on the reasoning and argument (in the premise-conclusion sense) one finds in personal exchange, advertising, political debate, legal argument, and the social commentary that characterizes newspapers, television, the World (...) Wide Web and other forms of mass media. (shrink)

The issue of the relationship between formal and informal logic depends strongly on how one understands these two designations. While there is very little disagreement about the nature of formal logic, the same is not true regarding informal logic, which is understood in various (often incompatible) ways by various thinkers. After reviewing some of the more prominent conceptions of informal logic, I will present my own, defend it and then show how informal logic, so understood, is complementary to formal logic.

Our article identifies and describes the metaphoric fallacy to a deductive inference (MFDI) that is an example of incorrect reasoning along the lines of the false analogy fallacy. The MFDI proceeds from informal semantical (metaphorical) claims to a supposedly formally deductive and necessary inference. We charge that such an inference is invalid. We provide three examples of the MFDI to demonstrate the structure of this invalid form of reasoning. Our goal is to contribute to the set of known informal fallacies.

The informal fallacies can be conceived as enthymemes that are formally valid. But, then, what accounts for our sense of their fallaciousness? I explain this in terms of the notion of a warrant.

Argument and explanation are distinct forms of reasoning with an underappreciated complementary relationship. In this essay I define these terms precisely, identify the mischief that results from conflating them, elucidate their complementary relationship and employ this relationship to provide a fruitful approach to analyzing the logical structure of the common editorial.

In this paper, I argue that ad hominem arguments are not always fallacious. More explicitly, in certain cases of practical reasoning, the circumstances of a person are relevant to whether or not the conclusion should be accepted. This occurs, I suggest, when a person gives advice to others or prescribes certain courses of action but fails to follow her own advice or act in accordance with her own prescriptions. This is not an instance of a fallacious tu quoque provided that (...) such circumstantial ad hominem arguments are construed as rebuttals to appeals (administrative) authority (of expertise), or so I argue. (shrink)

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 reasoning in mathematics [23], Pollock’s notions of counterexample [44], and argumentation (...) schemes constructed by Walton et al. [54], and explore some connections between, as well as within, the theories. For instance, we investigate Peirce’s abduction to deal with surprising situations in mathematics, represent Pollock’s examples in terms of Toulmin’s layout, discuss connections between Toulmin’s layout and Walton’s argumentation schemes, and suggest 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 [13], which help to show ways in which the work of Lakatos fits into the informal reasoning community. (shrink)

This article answers John Biro's "Knowability, Believability, and Begging the Question: a Reply to Sanford" in "Metaphilosophy" 15 (1984). Biro and I agree that of two argument instances with the same form and content, one but not the other can beg the question, depending on other factors. These factors include actual beliefs, or so I maintain (against Biro) with the help of some analysed examples. Brief selections from Archbishop Whatley and J S Mill suggest that they also regard reference to (...) actual beliefs as essential to explaining begging the question. (shrink)

According to John A Barker, whether an argument begs the question is purely a matter of logical form (Dialogue, 1976). According to me, it is also a matter of epistemic conditions; some arguments which beg the question in some contexts need not beg the question in every context (Analysis, 1972). I point out difficulties in Barker's treatment and defend my own views against some of his criticisms. In the concluding section, "Alleged difficulties with disjunctive syllogism," I defend the validity of (...) disjunctive syllogism against the views of Alan Ross Anderson and Nuel D Belnap, Jr., ask pointed questions about the notion of intensional disjunction, and suggest how my treatment of begging the question can be extended to deal with the so-called paradoxes of strict implication. (shrink)

A primary purpose of argument is to increase the degree of reasonable confidence that one has in the truth of the conclusion. A question begging argument fails this purpose because it violates what W. E. Johnson called an epistemic condition of inference. Although an argument of the sort characterized by Robert Hoffman in his response (Analysis 32.2, Dec 71) to Richard Robinson (Analysis 31.4, March 71) begs the question in all circumstances, we usually understand the charge that an argument is (...) question begging with reference to the beliefs of the person, or the sort of person, to whom the argument is directed. (shrink)

The paper’s purpose is to get clearer on what it is to express a proposition. A proposition is understood as anything that can be asserted, assumed, conjectured, stated, believed, and so on. It is not something that can be asked, ordered, requested, and so on. The paper tries to provide groundwork for a successful analysis by making distinctions and clarifying problems.

No topic in informal logic is more important than begging the question. Also, none is more subtle or complex. We cannot even begin to understand the fallacy of begging the question without getting clear about arguments, their purposes, and circularity. So I will discuss these preliminary topics first. This will clear the path to my own account of begging the question. Then I will anticipate some objections. Finally, I will apply my account to a well-known and popular response to scepticism (...) by G. E. Moore. (shrink)

This is an introductory guide to the basic principles of constructing good arguments and criticizing bad ones. It is nontechnical in its approach, and is based on 150 key examples, each discussed and evaluated in clear, illustrative detail. The author explains how errors, fallacies, and other key failures of argument occur. He shows how correct uses of argument are based on sound argument strategies for reasoned persuasion and critical questions for responding. Among the many subjects covered are: techniques of posing, (...) replying to, and criticizing questions, forms of valid argument, relevance, appeals to emotion, personal attack, uses and abuses of expert opinion, problems in deploying statistics, loaded terms, equivocation, and arguments from analogy. (shrink)

Although debate is a richly structured and prevalent form of discourse, it has received little scholarly attention. Logicians have focused on the structure of individual arguments—how they divide into premises and conclusions, which in turn divide into various constituents. In contrast, I focus on the structure of sets of arguments, showing how arguments are themselves constituents in high-level dialectical structures. I represent debates and positions by graphs whose vertices correspond to arguments and whose edges correspond to two inter-argument relations: “dispute” (...) and “support,” respectively. On this basis I develop a theory of the structure of debate. (shrink)