In this essay I take up Plato’s critique of poetry, which has little to do with epistemology and representational imitation, but rather the powerful effects that poeticperformances can have on audiences, enthralling them with vivid image-worlds and blocking the powers of critical reflection. By focusing on the perceived psychological dangers of poetry in performance and reception, I want to suggest that Plato’s critique was caught up in the larger story of momentous shifts in the Greek world, turning on the rise (...) of literacy and its far-reaching effects in modifying the original and persisting oral character of Greek culture. The story of Plato’s Republic in certain ways suggests something essential for comprehending the development of philosophy in Greece (and in any culture, I would add): that philosophy, as we understand it, would not have been possible apart from the skills and mental transformations stemming from education in reading and writing; and that primary features of oral language and practice were a significant barrier to the development of philosophical rationality (and also a worthy competitor for cultural status and authority). Accordingly, I go on to argue that the critique of writing in the Phaedrus is neither a defense or orality per se, nor a dismissal of writing, but rather a defense of a literate soul over against orality and the indiscriminate exposure of written texts to unworthy readers. (shrink)

Aristotle’s Poetics defends the value of tragic poetry, presumably to counter Plato’s critique in the Republic. Can this defense resonate with something larger and rather surprising, that Aristotle’s overall philosophy displays a tragic character? I define the tragic as pertaining to indigenous and inescapable limits on life, knowledge, control, achievement, and agency. I explore how such limits figure in Aristotle’s physics, metaphysics, and biological works. Accordingly I want to disturb the common account of Aristotle’s thought as a neat system of (...) ontological order and metaphysical closure—not to exclude such elements but to place them within a world-view that includes certain limits at the edges of being. (shrink)

Like most philosophers, Heidegger gave little attention to childhood, but his philosophical emphasis on pre-reflective practice and understanding seems uniquely qualified to help make sense of a child’s experience and development. Moreover, it seems to me that many central Heideggerian concepts are best defended, exemplified, and articulated by bringing child development into the discussion. A Heideggerain emphasis on pre-theoretical world-involvement opens up a rich array of phenomena for studying child development, which can improve upon standard theories that have over-emphasized exclusive (...) conditions or criteria. I begin by laying out some basic features of Heidegger’s conception of being-in-the-world as a preparation for understanding the world of the child. Then I will briefly discuss some of Heidegger’s remarks on childhood, followed by some reflections on language acquisition and the correlation of anxiety and meaning. (shrink)

Observers have noted a decline (in the US) in attributions of genetically-based inferiority (e.g. in intelligence) to Blacks, and a rise in attributions of culturally-based inferiority. Is this "culturalism" merely warmed-over racism ("cultural racism") or a genuinely distinct way of thinking about racial groups? The question raises a larger one about the relative place of biology and culture in racist thought. I develop a typology of culturalisms as applied to race: (1) inherentist or essentialist culturalism (inferiorizing cultural characteristics wrongly but (...) intelligibly regarded as inhering in the nature of racial groups). This view has an historical pedigree in the work of J.G. von Herder, and the tradition of "national racism." (2) non-inherentist culturalism (groups regarded as possessing changeable, malleable inferiorizing cultural characteristics). (3) colonalist culturalism (colonial subjects regarded as uncivilized but capable of becoming civilized under European tutelage). (4) neo-racism (cultures of former colonial subjects in or potentially immigrating to European countries are regarded in an inherentist manner and incompatible with but not inferior to European cultures). Non-inherentist culturalism (#2) can be put to either a racist or an anti-racist use. (shrink)

This article explicates the views on both race and ethnicity of these three prominent Latinx philosophers, compares them (somewhat), and offers some criticisms. Corlett jettisons race as a categorization of groups, but accepts a form of racialization somewhat at odds with this jettisoning. Gracia adopts as a general principle that an account of both ethnicity and race should help us see aspects of reality that would otherwise be obscured; but this is at odds with his regarding the Latin American view (...) of race as more rational than the U.S. version with its “one-drop rule.” The latter has structured the reality of race in the U.S. for African Americans. Alcoff is much more concerned with the phenomenology of race and ethnicity than the other two, and she clearly adds “pan-ethnicity” to the mix of concepts required to understand Latino/a Americans. I argue that she fails to see the agentic and political aspect of black identity in the U.S., and in a sense shares with Gracia a misplaced sense that the mixedness of Latin American racial identity is somehow to be preferred to the more binary U.S. form. (shrink)

By re-examining Nietzsche's notion of the “eternal-feminine” and his views on women and feminism, this volume offers new perspectives on some of his key ideas. It brings together a diverse group of scholars to critically engage with Nietzsche's use of late-19th-century gender stereotypes and the ways in which they served his critique of values, including his use of “woman” as a trope for truth. -/- Among other subjects, the contributors consider the role of psychology in Nietzsche's thought, his concern with (...) style, self-creation, and advocacy of perfectionism, his views on romantic love and marriage, and his aim of revaluing all values to instigate a distant philosophy of the future. They investigate parallels between Nietzsche's thought and Shaktism, his relation to Goethe and Stendahl, and his influence on Beauvoir, Butler, and Dohm. With the inclusion of two seminal essays on Nietzsche and women by Lawrence J. Hatab and Kelly Oliver, the volume also illustrates some of the ways in which scholarship on these subjects has evolved over the last four decades. -/- Providing fresh insights into these inter-related subjects, Nietzsche on Women and the Eternal-Feminine highlights the enduring relevance of his thought and its still-underappreciated potential for re-thinking both the bases for and aims of feminism and other emancipatory movements. (shrink)

There can be no doubt that discrimination based on sex, race, ethnicity, religion or beliefs should not be tolerated in academia. Surprisingly, however, in recent years, policies of Diversity, Inclusion and Equity(DIE), officially introduced to counteract discrimination, have increasingly led to quite the opposite result: the exclusion of individuals who do not share a radical 'woke' ideology on identity politics (feminism, other gender activisms, critical race theory, etc.), and to the suppression of the academic freedom to discuss such dogmas. This (...) subversion of academia ― disguised Trojan-horse style as universal human rights advocacy ― is unacceptable because academia must be politically neutral and protect freedom of speech, a cornerstone of professional scholarly activity without which universities as we know them will slowly but surely suffocate. -/- Our purpose here is to put together some particularly illustrative cases of such repression in a single book, testifying to a ubiquitous trend within western culture, irreducible to a few isolated complaints. The essays contained here illustrate the abuse of power, censorship and witch-hunts at many universities and research centres in the name of DIE. -/- List of coauthors in alphabetical order: Dorian Abbot, Tomonori Agoh, Gerhard Amendt, Ivar Arpi, David Benatar, Peter Boghossian, Civitas Research Team, David Díaz Pardo de Vera, Pedro Domingos, Janice Fiamengo, Étienne Forest, Jorge Gibert Galassi, Norman Goldstuck, José L. González Quirós, Lawrence M. Krauss, Patrick LaBelle, Martín López Corredoira, Heather Mac Donald, Martin Malmgren, Erik. J. Olsson, Jordan Peterson, Constantin Polychronakos, Philip C. Salzman, Alessandro Strumia, Tom Todd, Andrei Yafaev. (shrink)

Ang Noli Me Tangere at El Filibusterismo ay mga nobelang isinulat ni Dr. Jose P. Rizal na sumasalamin sa kalagayan ng mga Pilipino noong panahon ng mga Español. Ikinubli ni Rizal ang mga di-makatáong pagtrato ng mga Español sa mga itinuring na Indio sa pamamagitan ng mga pangyayari at mga karakter sa kaniyang dalawang akda. Mula sa Teorya ng Moral na Pag-unlad (Theory of Moral Development) ng isang Amerikanong Sikolohistang si Lawrence Kohlberg ay sinuri ng papel na ito ang (...) 10 tagpo na nagpapakita ng katungkulang moral sa dalawang nobela gámit ang Antas at Yugto ng Moral na Pag-unlad ni Kohlberg (Kohlberg’s Levels and Stages of Moral Development). Mula sa ginawang pagsusuri, napagtanto ng papel na ito na gumamit si Rizal ng estratehiya upang ipakita na ang aksiyong isinagawa ng ahente sa bawat tagpo ay taliwas sa kaniyang kabuoang moralidad, bagaman ang pamamaraang ito ay nagdulot sa mga mambabása ng Noli Me Tangere at El Filibusterismo ng iba’t ibang interpretasyon at pagkaunawa sa mga isinulat ng may-akda. Mapanghámon ang mga nobela ni Rizal na sumusubok sa tradisyonal na kaugalian at panuntunan ng Pilipinas maging sa pagkukunwari ng lipunang Pilipino. Malaki ang naging gampanin ng nabuong tema na sumisimbolo sa mga isyung panlipunang nais palutangin ni Rizal. Gayunpaman, dahil sa mataas na pangangatwirang moral na nakapaloob sa kaniyang mga gawa, nararapat na gamítin ito hindi lámang bílang materyal sa pagaaral ng Filipino at sa mga kaugnay na larang, kundi pati rin sa pag-aaral ng kritisismo sa politika at lipunan, maging sa pag-aaral ng etika at moralidad, partikular na sa konteksto ng Pilipinas. (shrink)

In this paper the author deals with the new development of Metaphysics among American Thomists. In contrast to Gilson, there is revaluation of 'essence' among some authors, insofar form has an instrumental role for the existence of things (see e.g. Lawrence Dewan). The example of Stephen L. Brock is presented as an alternative to the excessive Apophaticism of some interpretations of Aquinas such as the one of J.-L. Marion.

Are science and religion compatible when it comes to understanding cosmology (the origin of the universe), biology (the origin of life and of the human species), ethics, and the human mind (minds, brains, souls, and free will)? Do science and religion occupy non-overlapping magisteria? Is Intelligent Design a scientific theory? How do the various faith traditions view the relationship between science and religion? What, if any, are the limits of scientific explanation? What are the most important open questions, problems, or (...) challenges confronting the relationship between science and religion, and what are the prospects for progress? These and other questions are explored in Science and Religion: 5 Questions--a collection of thirty-three interviews based on 5 questions presented to some of the world's most influential and prominent philosophers, scientists, theologians, apologists, and atheists. Contributions by Simon Blackburn, Susan Blackmore, Sean Carroll, William Lane Craig, William Dembski, Daniel C. Dennett, George F.R. Ellis, Owen Flanagan, Owen Gingerich, Rebecca Newberger Goldstein, John F. Haught, Muzaffar Iqbal, Lawrence Krauss, Colin McGinn, Alister McGrath, Mary Midgley, Seyyed Hossein Nasr, Timothy O'Connor, Massimo Pigliucci, John Polkinghorne, James Randi, Alex Rosenberg, Michael Ruse, Robert John Russell, John Searle, Michael Shermer, Victor J. Stenger, Robert Thurman, Michael Tooley, Charles Townes, Peter van Inwagen, Keith Ward, Rabbi David Wolpe. (shrink)

Does consciousness collapse the quantum wave function? This idea was taken seriously by John von Neumann and Eugene Wigner but is now widely dismissed. We develop the idea by combining a mathematical theory of consciousness (integrated information theory) with an account of quantum collapse dynamics (continuous spontaneous localization). Simple versions of the theory are falsified by the quantum Zeno effect, but more complex versions remain compatible with empirical evidence. In principle, versions of the theory can be tested by experiments with (...) quantum computers. The upshot is not that consciousness-collapse interpretations are clearly correct, but that there is a research program here worth exploring. (shrink)

Existing accounts of shared intention (by Bratman, Searle, and others) do not claim that a single token of intention can be jointly framed and executed by multiple agents; rather, they claim that multiple agents can frame distinct, individual intentions in such a way as to qualify as jointly intending something. In this respect, the existing accounts do not show that intentions can be shared in any literal sense. This article argues that, in failing to show how intentions can be literally (...) shared, these accounts fail to resolve what seems problematic in the notion of shared intention. It then offers an account in which the problem of shared intention is resolved, because intention can indeed be literally shared. This account is derived from Margaret Gilbert’s notion of a “pool of wills,” to which it applies Searle’s definition of intention. (shrink)

Reductive intellectualists hold that knowledge-how is a kind of knowledge-that. For this thesis to hold water, it is obviously important that knowledge-how and knowledge-that have the same epistemic properties. In particular, knowledge-how ought to be compatible with epistemic luck to the same extent as knowledge-that. It is argued, contra reductive intellectualism, that knowledge-how is compatible with a species of epistemic luck which is not compatible with knowledge-that, and thus it is claimed that knowledge-how and knowledge-that come apart.

Empirical work has lately confirmed what many philosophers have taken to be true: people are ‘biased toward the future’. All else being equal, we usually prefer to have positive experiences in the future, and negative experiences in the past. According to one hypothesis, the temporal metaphysics hypothesis, future-bias is explained either by our beliefs about temporal metaphysics—the temporal belief hypothesis—or alternatively by our temporal phenomenology—the temporal phenomenology hypothesis. We empirically investigate a particular version of the temporal belief hypothesis according to (...) which future-bias is explained by the belief that time robustly passes. Our results do not match the apparent predictions of this hypothesis, and so provide evidence against it. But we also find that people give more future-biased responses when asked to simulate a belief in robust passage. We take this to suggest that the phenomenology that attends simulation of that belief may be partially responsible for future-bias, and we examine the implications of these results for debates about the rationality of future-bias. (shrink)

We argue that any strong version of a knowledge condition on intentional action, the practical knowledge principle, on which knowledge of what I am doing (under some description: call it A-ing) is necessary for that A-ing to qualify as an intentional action, is false. Our argument involves a new kind of case, one that centers the agent’s control appropriately and thus improves upon Davidson’s well-known carbon copier case. After discussing this case, offering an initial argument against the knowledge condition, and (...) discussing recent treatments that cover nearby ground, we consider several objections. One we consider at some length maintains that although contemplative knowledge may be disconnected from intentional action, specifically practical knowledge of the sort Anscombe elucidated escapes our argument. We demonstrate that this is not so. Our argument illuminates an important truth, often overlooked in discussions of the knowledge-intentional action relationship: intentional action and knowledge have different levels of permissiveness regarding failure in similar circumstances. (shrink)

A handful of well-known arguments (the 'diachronic Dutch book arguments') rely upon theorems establishing that, in certain circumstances, you are immune from sure monetary loss (you are not 'diachronically Dutch bookable') if and only if you adopt the strategy of conditionalizing (or Jeffrey conditionalizing) on whatever evidence you happen to receive. These theorems require non-trivial assumptions about which evidence you might acquire---in the case of conditionalization, the assumption is that, if you might learn that e, then it is not the (...) case that you might learn something else that is consistent with e. These assumptions may not be relaxed. When they are, not only will non-(Jeffrey) conditionalizers be immune from diachronic Dutch bookability, but (Jeffrey) conditionalizers will themselves be diachronically Dutch bookable. I argue: 1) that there are epistemic situations in which these assumptions are violated; 2) that this reveals a conflict between the premise that susceptibility to sure monetary loss is irrational, on the one hand, and the view that rational belief revision is a function of your prior beliefs and the acquired evidence alone, on the other; and 3) that this inconsistency demonstrates that diachronic Dutch book arguments for (Jeffrey) conditionalization are invalid. (shrink)

This paper sets out an account of trust in AI as a relationship between clinicians, AI applications, and AI practitioners in which AI is given discretionary authority over medical questions by clinicians. Compared to other accounts in recent literature, this account more adequately explains the normative commitments created by practitioners when inviting clinicians’ trust in AI. To avoid committing to an account of trust in AI applications themselves, I sketch a reductive view on which discretionary authority is exercised by AI (...) practitioners through the vehicle of an AI application. I conclude with four critical questions based on the discretionary account to determine if trust in particular AI applications is sound, and a brief discussion of the possibility that the main roles of the physician could be replaced by AI. (shrink)

According to the canonical formulation of the modal account of luck [e.g. Pritchard (2005)], an event is lucky just when that event occurs in the actual world but not in a wide class of the nearest possible worlds where the relevant conditions for that event are the same as in the actual world. This paper argues, with reference to a novel variety of counterexample, that it is a mistake to focus, when assessing a given event for luckiness, on events distributed (...) over just the nearest possible worlds. More specifically, our objection to the canonical formulation of the modal account of luck reveals that whether an event is lucky depends crucially on events distributed over all possible worlds–viz., across the modal universe. It is shown that an amended modal account of luck which respects this point has the additional virtue of avoiding a notable kind of counterexample to modal accounts of luck proposed by Lackey (2008). (shrink)

Until recently, philosophers debating the rationality of time-biases have supposed that people exhibit a first-person hedonic bias toward the future, but that their non-hedonic and third-person preferences are time-neutral. Recent empirical work, however, suggests that our preferences are more nuanced. First, there is evidence that our third-person preferences exhibit time-neutrality only when the individual with respect to whom we have preferences—the preference target—is a random stranger about whom we know nothing; given access to some information about the preference target, third-person (...) preferences mirror first-person preferences. As a result, the simulation hypothesis has been proposed, according to which third-person preferences will mirror first-person preferences when we can simulate the mental states of the preference target. Second, there is evidence that we prefer negative hedonic events to be in our past (we are first-person negatively hedonically future-biased) only when we view future events as fixed and in no way under our control. By contrast, when we perceive it to be within our power to mitigate the badness of future events, we are first-person negatively hedonically past-biased. This is the mitigation hypothesis. We distinguish two versions of the mitigation hypothesis, the squirrelling version and the heuristic version. We ran a study which tested the simulation hypothesis, and which aimed to determine whether the squirrelling or the heuristic version of the mitigation hypothesis enjoys more empirical support. We found support for the heuristic version of the hypothesis, but no support for the squirrelling version. (shrink)

Future-biased individuals systematically prefer pleasures to be in the future and pains to be in the past. Empirical research shows that negative future-bias is robust: people prefer more past pain to less future pain. Is positive future-bias robust or fragile? Do people only prefer pleasures to be located in the future, compared to the past, when those pleasures are of equal value, or do they continue to prefer that pleasures be located in the future even when past pleasures outweigh future (...) ones? Some arguments against the rationality of future-bias require positive future-bias to be robust, while others require it to be fragile. We empirically investigate and show that positive future-bias is robust. (shrink)

We explain and explore class-theoretic potentialism—the view that one can always individuate more classes over a set-theoretic universe. We examine some motivations for class-theoretic potentialism, before proving some results concerning the relevant potentialist systems (in particular exhibiting failures of the $\mathsf {.2}$ and $\mathsf {.3}$ axioms). We then discuss the significance of these results for the different kinds of class-theoretic potentialists.

Several areas of welfare economics seek to evaluate states of affairs as a function of interpersonally comparable individual utilities. The aim is to map each state of affairs onto a vector of individual utilities, and then to produce an ordering of these vectors that can be represented by a mathematical function assigning a real number to each. When this approach is used in intertemporal contexts, a central theoretical question concerns the evaluative weight to be applied to utility coming at different (...) times. This question concerns the rate of pure time preference, which is one key determinant of the social discount rate. This article argues that the standard philosophical account of pure time preference is mistaken, because it ascribes to economists a methodological commitment they need not, and often do not, accept. This in turn undercuts the most common philosophical objection to pure time preference, which traces at least to Rawls’s A Theory of Justice. The article then evaluates three further objections to pure time preference, concluding that it might still be defensible under certain circumstances. The article closes by articulating a final argument that is suggested by the “Social, Economic and Ethical Concepts and Methods” chapter of the most recent Intergovernmental Panel on Climate Change report. If this further argument is sound, it would constitute a decisive objection to pure time preference as it currently figures in much intertemporal welfare economics. (shrink)

If individual knowledge and justification can be vanquished by epistemic defeaters, then the same should go for group knowledge. Lackey (2014) has recently argued that one especially strong conception of group knowledge defended by Bird (2010) is incapable of preserving how it is that (group) knowledge is ever subject to ordinary mechanisms of epistemic defeat. Lackey takes it that her objections do not also apply to a more moderate articulation of group knowledge--one that is embraced widely in collective epistemology--and which (...) she does not challenge. This paper argues that given certain background premises that are embraced by orthodox thinking in collectivist epistemology, the more moderate account of group knowledge cannot make sense of either psychological or normative epistemic defeaters. I conclude by offering some suggestions for how the more moderate proposal might avoid this result. (shrink)

Recently, mainstream philosophy journals have tended to implement more and more stringent forms of peer review, probably in an attempt to prevent editorial decisions that are based on factors other than quality. Against this trend, we propose that journals should relax their standards of acceptance, as well as be less restrictive about whom is to decide what is admitted into the debate. We start by arguing, partly on the basis of the history of peer review in the journal Mind, that (...) past and current peer review practices attest to partisanship with respect to philosophical approach. Then, we explain that such partisanship conflicts with the standard aims of peer review, and that it is both epistemically and morally problematic. This assessment suggests that, if feasible, journals should treat all available and proposed standards of acceptance in philosophy as epistemically equal, and that philosophical work should be evaluated in terms of the novelty and significance of its contribution to developing thought in ways that are of value. Finally, we show, in a programmatic way, that improving the current situation is feasible, and can be done fairly easily. (shrink)

An adequate account of laws should satisfy at least five desiderata: it should provide a unified account of laws and chances, it should yield plausible relations between laws and chances, it should vindicate numerical chance assignments, it should accommodate dynamical and non-dynamical chances, and it should accommodate a plausible range of nomic possibilities. No extant account of laws satisfies these desiderata. This paper presents a non-Humean account of laws, the Nomic Likelihood Account, that does.

In a series of works Sosa (in: Knowledge in perspective, Cambridge University Press, Cambridge, 1991; A virtue epistemology: apt belief and reflective knowledge, Oxford University Press, Oxford, 2007; Reflective knowledge: apt belief and reflective knowledge, Oxford University Press, Oxford, 2009; ‘How Competence Matters in Epistemology’, Philos Perspect 24(1):465–475, 2010; Knowing full well, Princeton University Press, Princeton, 2011; Judgment and agency, Oxford University Press, Oxford, 2015; Epistemology, Princeton University Press, Princeton, 2017) has defended the view that there are two kinds or (...) ‘grades’ of knowledge, animal and reflective. One of the most persistent critics of Sosa’s attempts to bifurcate knowledge is Kornblith (in: Greco (ed) Ernest sosa and his critics, Wiley, Hoboken, 2004; ‘Sosa in Perspective’, Philos Stud 144(1):127–136, 2009; On reflection, Oxford University Press, Oxford, 2012). Our aim in this paper is to outline and evaluate Kornblith’s criticisms. We will argue that, while they raise a range of difficult (exegetical and substantive) questions about Sosa’s ‘bi-level’ epistemology, Sosa has the resources to adequately respond to all of them. Thus, this paper is a (qualified) defence of Sosa’s bi-level epistemology. (shrink)

Consider two epistemic experts—for concreteness, let them be two weather forecasters. Suppose that you aren’t certain that they will issue identical forecasts, and you would like to proportion your degrees of belief to theirs in the following way: first, conditional on either’s forecast of rain being x, you’d like your own degree of belief in rain to be x. Secondly, conditional on them issuing different forecasts of rain, you’d like your own degree of belief in rain to be some weighted (...) average of the forecast of each. Finally, you’d like your degrees of belief to be given by an orthodox probability measure. Moderate ambitions, all. But you can’t always get what you want. (shrink)

The two purposes of this essay. The general philosophical problem with most versions of social libertarianism and how this essay will proceed. The specific problem with liberty explained by a thought-experiment. The positive and abstract theory of interpersonal liberty-in-itself as ‘the absence of interpersonal initiated constraints on want-satisfaction’, for short ‘no initiated impositions’. The individualistic liberty-maximisation theory solves the problems of clashes, defences, and rectifications without entailing interpersonal utility comparisons or libertarian consequentialism. The practical implications of instantiating liberty: three rules (...) of liberty-in-practice, 1) initial ultimate control of one’s body, 2) initial ultimate control of one’s used resources, 3) consensual interpersonal interactions and resource transfers. These rules are economically efficient and maximise general want-satisfaction. Private property and legal remedies are additional practical institutional aspects, but to which ‘initiated impositions’ then apply prima facie. Libertarian law is often mistaken for complete libertarianism. Moral explanations are a separate issue. The three main moral theories imply libertarianism, but it can be morally posited independently of them. Critical rationalism and its application. No empirical or argumentative support for theories. An important ambiguity with ‘justification’. How the epistemology applies to the theory of liberty and its application but remains separable in principle. Conclusion: there are further published explanations but this should be enough to generate useful criticism. Appendix replying to some typical comments. (shrink)

Supervaluationism is often described as the most popular semantic treatment of indeterminacy. There’s little consensus, however, about how to fill out the bare-bones idea to include a characterization of logical consequence. The paper explores one methodology for choosing between the logics: pick a logic thatnorms beliefas classical consequence is standardly thought to do. The main focus of the paper considers a variant of standard supervaluational, on which we can characterizedegrees of determinacy. It applies the methodology above to focus ondegree logic. (...) This is developed first in a basic, single-premise case; and then extended to the multipremise case, and to allowdegreesof consequence. The metatheoretic properties of degree logic are set out. On the positive side, the logic is supraclassical—all classical valid sequents are degree logic valid. Strikingly, metarules such as cut and conjunction introduction fail. (shrink)

In this paper I argue for an association between impurity and explanatory power in contemporary mathematics. This proposal is defended against the ancient and influential idea that purity and explanation go hand-in-hand (Aristotle, Bolzano) and recent suggestions that purity/impurity ascriptions and explanatory power are more or less distinct (Section 1). This is done by analyzing a central and deep result of additive number theory, Szemerédi’s theorem, and various of its proofs (Section 2). In particular, I focus upon the radically impure (...) (ergodic) proof due to Furstenberg (Section 3). Furstenberg’s ergodic proof is striking because it utilizes intuitively foreign and infinitary resources to prove a finitary combinatorial result and does so in a perspicuous fashion. I claim that Furstenberg’s proof is explanatory in light of its clear expression of a crucial structural result, which provides the “reason why” Szemerédi’s theorem is true. This is, however, rather surprising: how can such intuitively different conceptual resources “get a grip on” the theorem to be proved? I account for this phenomenon by articulating a new construal of the content of a mathematical statement, which I call structural content (Section 4). I argue that the availability of structural content saves intuitive epistemic distinctions made in mathematical practice and simultaneously explicates the intervention of surprising and explanatorily rich conceptual resources. Structural content also disarms general arguments for thinking that impurity and explanatory power might come apart. Finally, I sketch a proposal that, once structural content is in hand, impure resources lead to explanatory proofs via suitably understood varieties of simplification and unification (Section 5). (shrink)