In this paper, I highlight the significance of practices of _refutation_ in philosophical inquiry, that is, practices of showing that a claim, person or theory is wrong. I present and contrast two prominent approaches to philosophical refutation: refutation in ancient Greek dialectic (_elenchus_), in its Socratic variant as described in Plato’s dialogues, and as described in Aristotle’s logical texts; and the practice of providing counterexamples to putative definitions familiar from twentieth century analytic philosophy, focusing on the so-called Gettier problem. Moreover, (...) I discuss Lakatos’ method of proofs and refutations, as it offers insightful observations on the dynamics between arguments, refutations, and counterexamples. Overall, I argue that dialectic, in particular in its Socratic variant, is especially suitable for the philosophical purpose of questioning the obvious, as it invites reflection on one’s own doxastic commitments and on the tensions and inconsistencies within one’s set of beliefs. By contrast, the counterexample-based approach to philosophical refutation can give rise to philosophical theorizing that is overly focused on hairsplitting disputes, thus becoming alienated from the relevant human experiences. Insofar as philosophical inquiry treads the fine line between questioning the obvious while still seeking to say something significant about human experiences, perhaps a certain amount of what Lakatos describes as ‘monster-barring’—a rejection of overly fanciful, artificial putative counterexamples—has its place in philosophical argumentation. (shrink)

ABSTRACT Classical and intuitionistic logic both validate Ex Contradictione Quodlibet, according to which any proposition whatsoever follows from a contradiction. Many philosophers have found ECQ counter-intuitive, but criticisms of the principle have almost universally been directed from a position of support for relevance or other orthodox paraconsistent logics, according to which some, but not necessarily all, propositions follow from a contradiction. This paper draws attention to the historically significant view that nothing whatsoever follows from a contradiction – Ex Contradictione Nihil. (...) It addresses the key arguments – model-theoretic, proof-theoretic, meaning theoretic and holist – deployed by proponents of ECQ against their standard opponents, and shows that each can be defeated from a position of support for ECN. In the course of answering these arguments, some understanding is developed of possible motivations for ECN and a solid case is built that ECN deserves further investigation. (shrink)

Minimal adversariality consists in the opposition of contradictory conclusions in argumentation, and its usual metaphorical expression as a game between combating arguers has seen it be criticized from a number of perspectives: the language used, whether cooperation best attains the argumentative telos of epistemic betterment, and the ideal nature of the metaphor itself. This paper explores primarily the idealization of deductive argumentation, which is problematic due to its attenuated applicability to a dialectic involving premises and justificatory biases that are left (...) hidden and unelucidated. To clarify the issue and offer up a solution, we consider minimal adversariality as an involuntary state of affairs before relating this interpretation to a link between rational persuasion and the attainment of epistemic betterment. Through this we see how the idealizing tendencies of minimal adversariality can be reduced even in argumentation involving premises whose justifications for any arguer are inaccessible to any other arguer. (shrink)

I argue against abductivism about logic, which is the view that rational theory choice in logic happens by abduction. Abduction cannot serve as a neutral arbiter in many foundational disputes in logic because, in order to use abduction, one must first identify the relevant data. Which data one deems relevant depends on what I call one's conception of logic. One's conception of logic is, however, not independent of one's views regarding many of the foundational disputes that one may hope to (...) solve by abduction. (shrink)

Epistemic diversity is the ability or possibility of producing diverse and rich epistemic apparati to make sense of the world around us. In this paper we discuss whether, and to what extent, different conceptions of knowledge—notably as ‘justified true belief’ and as ‘distributed and embodied cognition’—hinder or foster epistemic diversity. We then link this discussion to the widespread move in science and philosophy towards monolingual disciplinary environments. We argue that English, despite all appearance, is no Lingua Franca, and we give (...) reasons why epistemic diversity is also deeply hindered is monolingual contexts. Finally, we sketch a proposal for multilingual academia where epistemic diversity is thereby fostered. (shrink)

O livro reúne textos apresentados no II Workshop de Filosofia e Ensino, realizado na UFRGS em 2015, com a temática "Epistemologia e Currículo" -/- .

I develop a theory of counterfactuals about relative computability, i.e. counterfactuals such as 'If the validity problem were algorithmically decidable, then the halting problem would also be algorithmically decidable,' which is true, and 'If the validity problem were algorithmically decidable, then arithmetical truth would also be algorithmically decidable,' which is false. These counterfactuals are counterpossibles, i.e. they have metaphysically impossible antecedents. They thus pose a challenge to the orthodoxy about counterfactuals, which would treat them as uniformly true. What’s more, I (...) argue that these counterpossibles don’t just appear in the periphery of relative computability theory but instead they play an ineliminable role in the development of the theory. Finally, I present and discuss a model theory for these counterfactuals that is a straightforward extension of the familiar comparative similarity models. (shrink)

This book symposium comprises a précis of Nuno Venturinha’s Description of Situations: An Essay in Contextualist Epistemology together with four critical commentaries on different aspects of the book by Marcelo Carvalho, João Vergílio Gallerani Cuter, Marcos Silva and Darlei Dall’Agnol, and the author’s replies.

In recent years, substructural approaches to paradoxes have become quite popular. But whatever restrictions on structural rules we may want to enforce, it is highly desirable that such restrictions be accompanied by independent philosophical motivation, not directly related to paradoxes. Indeed, while these recent developments have shed new light on a number of issues pertaining to paradoxes, it seems that we now have even more open questions than before, in particular two very pressing ones: what (independent) motivations do we have (...) (if any) for restrictions on structural rules, and what to make of the plurality of new logics emerging from these restrictions, i.e. how to ‘choose’ among the different options. In this paper, we address these two questions from the perspective of a dialogical conception of logic that we've been advocating in recent years. We will argue that dialogical interpretations of structural rules, that is, as rules determining specific properties of the dialogues in question, provide a conveniently neutral framework to adjudicate between the different substructural proposals that have been made in the literature on paradoxes. (shrink)

To adequately model mathematical arguments the analyst must be able to represent the mathematical objects under discussion and the relationships between them, as well as inferences drawn about these objects and relationships as the discourse unfolds. We introduce a framework with these properties, which has been used to analyse mathematical dialogues and expository texts. The framework can recover salient elements of discourse at, and within, the sentence level, as well as the way mathematical content connects to form larger argumentative structures. (...) We show how the framework might be used to support computational reasoning, and argue that it provides a more natural way to examine the process of proving theorems than do Lamport’s structured proofs. (shrink)

Mathematicians often intentionally leave gaps in their proofs. Based on interviews with mathematicians about their refereeing practices, this paper examines the character of intentional gaps in published proofs. We observe that mathematicians’ refereeing practices limit the number of certain intentional gaps in published proofs. The results provide some new perspectives on the traditional philosophical questions of the nature of proof and of what grounds mathematical knowledge.

Dialogical egalitarianism is the thesis that any proposition asserted in dialogue, if questioned, must be supported or else retracted. Dialogical foundationalism is the thesis that some propositions are privileged over this burden of proof, standing in no need of support unless and until support for their negation is provided. I first discuss existing arguments for either thesis, dismissing each one of them. Absent a successful principled argument, I then examine which thesis it is pragmatically more advantageous to adopt in analytic (...) philosophical dialogue. This requires identifying the goal of such dialogue, to the attainment of which the thesis would be so advantageous. To identify this goal, I draw on Douglas Walton’s typology of dialogues for an analysis of the types of dialogue of 110 representatively selected journal articles in current analytic philosophy. 95% of articles are found to instantiate persuasion dialogue. In light of the thus prevalent goal of persuading one’s opponent, I argue that the adoption of dialogical egalitarianism in analytic philosophical dialogue is pragmatically inescapable. (shrink)

This paper transforms a development of an argument against pantheism into an objection to the usual account of God within contemporary analytic philosophy. A standard criticism of pantheism has it that pantheists cannot offer a satisfactory account of God as personal. My paper will develop this criticism along two lines: first, that personhood requires contentful mental states, which in turn necessitate the membership of a linguistic community, and second that personhood requires limitation within a wider context constitutive of the ’setting’ (...) of the agent’s life. Pantheism can, I argue, satisfy neither criterion of personhood. At this point the tables are turned on the Swinburnian theist. If the pantheist cannot defend herself against the personhood-based attacks, neither can the Swinburnian, and for instructively parallel reasons: for neither doctrine is God in the material world; in the pantheist case God is identical with the world, in the Swinburnian case God transcends it. Either way both the pantheist and the Swinburnian are left with a dilemma: abandon divine personhood or modify the doctrine of God so as to block the move to personhood. (shrink)

My dissertation focuses on mathematical explanation found in proofs looked at from a historical point of view, while stressing the importance of mathematical practices. Current philosophical theories on explanatory proofs emphasize the structure and content of proofs without any regard to external factors that influence a proof’s explanatory power. As a result, the major philosophical views have been shown to be inadequate in capturing general aspects of explanation. I argue that, in addition to form and content, a proof’s explanatory power (...) depends on its targeted audience. History is useful here, because from it, we are able to follow the transition from a first-generation proof, which is usually non-explanatory, into its explanatory version. By tracking the similarities and differences between these proofs, we are able to gain a better understanding of what makes a proof explanatory according to mathematicians who have the relevant background to evaluate it as so. My first chapter discusses why history is important for understanding mathematical practices. I describe two kinds of history: one that presents a narrative of events, which influenced developments in mathematics both directly and indirectly, and another, typically used in mathematical research, which concentrates only on technical developments. I contend that both versions of the past benefit the philosopher. History used in research gives us an idea of what mathematicians desire or find to be important, while history written by historians shows us what effects these have on mathematical practices. The next two chapters are about explanatory proofs. My second chapter examines the main theories of mathematical explanation. I argue that these theories are short-sighted as they only consider what appears in a proof without considering the proof’s purported audience or background knowledge necessary to understand the proof. In the third chapter, I propose an alternative way of analyzing explanatory proofs. Here, I suggest looking at a theorem’s history, which includes its successive proofs, as well as the mathematicians who wrote them. From this, we can better understand how and why mathematicians prove theorems in multiple ways, which depends on the purposes of these theorems. The last chapter is a case study on the computer proof of the Four Color Theorem by Appel and Haken. Here, I compare and contrast what philosophers and mathematicians have had to say about the proof. I argue that the main philosophical worry regarding the theorem—its unsurveyability—did not make a strong impact on the mathematical community and would have hindered mathematical development in computer-assisted proofs. By studying the history of the theorem, we learn that Appel and Haken relied on the strategy of Kempe’s flawed proof from the 1800s (which, obviously, did not involve a computer). Two later proofs, also aided by computer, were developed using similar methods. None of these proofs are explanatory, but not because of their massive lengths. Rather, the methods used in these proofs are a series of calculations that exhaust all possible configurations of maps. (shrink)

In this paper we argue that questions about which mathematical ideas mathematicians are exposed to and choose to pay attention to are epistemologically relevant and entangled with power dynamics and social justice concerns. There is a considerable body of literature that discusses the dissemination and uptake of ideas as social justice issues. We argue that these insights are also relevant for the epistemology of mathematics. We make this visible by a journalistic exploration of relevant cases and embed our insights into (...) the larger question how mathematical ideas are taken up in mathematical practices. We argue that epistemologies of mathematics ought to account for questions of exposure to and choice of attention to mathematical ideas, and remark on the political relevance of such epistemologies. (shrink)

Argument maps represent some arguments more effectively than others. The goal of this article is to account for that variability, so that those who wish to use argument maps can do so with more foresight. I begin by identifying four properties of argument maps that make them useful tools for evaluating arguments. Then, I discuss four types of argument that are difficult to map well: reductio ad absurdum arguments, charges of equivocation, logical analogies, and mathematical arguments. The difficulties presented by (...) these four types appear unrelated to one another, but I show that, in each case, the difficulty can be traced back to the use of metalinguistic reasoning. The need to represent a transition between object language and metalanguage can undermine one or more of the benefits that argument map representation would otherwise confer. (shrink)

This paper argues that adopting a particular dialogical account of logical consequence quite directly gives rise to an interesting form of logical pluralism, the form of pluralism in question arising out of the requirement that deductive proofs be explanatory.

In The Enigma of Reason, Mercier and Sperber (M&S) present and defend their interactionist account of reason. In this piece, I discuss briefly the points of agreement between M&S and myself and, more extensively, the points of disagreement, most of which pertain to details of the evolutionary components of their account. I discuss in particular the purported modular nature of reason; their account of myside bias as an optimum/adaptation; and the claim that reason thus construed must be an individual‐level and (...) not a group‐level adaptation. In the final section, I offer brief considerations on an alternative account of reasoning, where the focus is on how sociocultural environments may tune the social production and evaluation of arguments. (shrink)