The use of the symbol ∨for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol ∨ in its historical and logical context. Some sources say that disjunction in its use as connecting propositions or formulas was introduced by Peano; others suggest that it originated as an abbreviation of the Latin word for “or,” vel. We show that the origin of the symbol ∨ for disjunction can be traced to Whitehead and (...) Russell’s pre-Principia work in formal logic. Because of Principia’s influence, its notation was widely adopted by philosophers working in logic (the logical empiricists in the 1920s and 1930s, especially Carnap and early Quine). Hilbert’s adoption of ∨ in his Grundzüge der theoretischen Logik guaranteed its widespread use by mathematical logicians. The origins of other logical symbols are also discussed. (shrink)

Interactive theorem provers might seem particularly impractical in the history of philosophy. Journal articles in this discipline are generally not formalized. Interactive theorem provers involve a learning curve for which the payoffs might seem minimal. In this article I argue that interactive theorem provers have already demonstrated their potential as a useful tool for historians of philosophy; I do this by highlighting examples of work where this has already been done. Further, I argue that interactive theorem provers can continue to (...) be useful tools for historians of philosophy in the future; this claim is defended through a more conceptual analysis of what historians of philosophy do that identifies argument reconstruction as a core activity of such practitioners. It is then shown that interactive theorem provers can assist in this core practice by a description of what interactive theorem provers are and can do. If this is right, then computer verification for historians of philosophy is in the offing. (shrink)

In his 1918 logical atomism lectures, Russell argued that there are no molecular facts. But he posed a problem for anyone wanting to avoid molecular facts: we need truth-makers for generalizations of molecular formulas, but such truth-makers seem to be both unavoidable and to have an abominably molecular character. Call this the problem of generalized molecular formulas. I clarify the problem here by distinguishing two kinds of generalized molecular formula: incompletely generalized molecular formulas and completely generalized molecular formulas. I next (...) argue that, if empty worlds are logically possible, then the model-theoretic and truth-functional considerations that are usually given address the problem posed by the first kind of formula, but not the problem posed by the second kind. I then show that Russell’s commitments in 1918 provide an answer to the problem of completely generalized molecular formulas: some truth-makers will be non-atomic facts that have no constituents. This shows that the neo-logical atomist goal of defending the principle of atomicity—the principle that only atomic facts are truth-makers—is not realizable. (shrink)

This book offers a comprehensive critical survey of issues of historical interpretation and evaluation in Bertrand Russell's 1918 logical atomism lectures and logical atomism itself. These lectures record the culmination of Russell's thought in response to discussions with Wittgenstein on the nature of judgement and philosophy of logic and with Moore and other philosophical realists about epistemology and ontological atomism, and to Whitehead and Russell’s novel extension of revolutionary nineteenth-century work in mathematics and logic. Russell's logical atomism lectures have had (...) a lasting impact on analytic philosophy and on Russell's contemporaries including Carnap, Ramsey, Stebbing, and Wittgenstein. Comprised of 14 original essays, this book will demonstrate how the direct and indirect influence of these lectures thus runs deep and wide. (shrink)

Bertrand Russell’s work in philosophy of science has been identified as a progenitor of structuralism in contemporary philosophy. It is often unclear, however, how the philosophical problems facing contemporary structuralist programmes relate to the problems of philosophy as Russell saw them. We contend that Russell has been mistakenly identified as an epistemic structural realist. The goal of this essay is to clarify the relationship between Russell’s programme and contemporary structuralist projects. In doing so, we hope to display the motivation for (...) a broad, truly Russellian structuralist project in the philosophy of science. (shrink)

In this chapter I critically examine the hitherto neglected influence that Dorothy Wrinch had on her teacher, friend, and informal thesis adviser, Bertrand Russell, and the puzzling fact that Russell never cited Wrinch’s mathematical papers on Principia Mathematica. Wrinch never reshaped Russell’s general outlook; indeed, Wrinch adopted as her own many of Russell’s 1911–1919 views about logic, philosophy, science, and their relationships that are characteristic of logic-centered twentieth-century analytic philosophy. Still, the influence was not just in one direction, from teacher (...) to student, but also ran the other way. And while Wrinch’s own life and ideas have enjoyed far more attention in recent literature, most historians of philosophy have, for too long, largely overlooked how Wrinch influenced Russell. Here I critically discuss how Wrinch influenced Russell and tell a more well-rounded story of mutual influence and collaboration. (shrink)

I characterize and argue against the standard interpretation of logical atomism. The argument against this reading is historical: the standard interpretation of logical atomism (1) fails to explain how the view is inspired by nineteenth-century developments in mathematics, (2) fails to explain how logic is central to logical atomism, and (3) fails to explain how logical atomism is a revolutionary and new "scientific philosophy." In short, the standard interpretation is a bad history of logical atomism. A novel interpretation of the (...) view that repairs these difficulties is sketched in the concluding section. (shrink)

I claim that a relatively new position in philosophy of mathematics, pluralism, overlaps in striking ways with the much older Jain doctrine of anekantavada and the associated doctrines of nyayavada and syadvada. I first outline the pluralist position, following this with a sketch of the Jain doctrine of anekantavada. I then note the srrong points of overlaps and the morals of this comparison of pluralism and anekantavada.

Russell derived many of his logical symbols from the pioneering notation of Giuseppe Peano. Principia Mathematica (1910–13) made these “Peanese” symbols (and others) famous. Here I focus on one of the more peculiar notational derivatives from Peano, namely, Principia ’s dual use of a squared dot or dots for both conjunction and scope. As Dirk Schlimm has noted, Peano always had circular dots and only used them to symbolize scope distinctions. In contrast, Principia has squared dots and conventions such that (...) some dots mark scope distinctions while others symbolize conjunction. How did this come to pass? In this paper I trace a genealogy of Principia ’s square dots back to Russell’s appropriation of Peano’s use of circular dots. Russell never explicitly justifies appropriating Peano’s notations to symbolize two distinct notions, but below I explain why Russell deployed Peano’s dot notations in this manner. Further, I argue that it was Cambridge University Press who squared the circular dots. (shrink)

This chapter contextualizes the book and summarizes its subsequent chapters.

Some philosophers, like Kripke, Williamson, Hawthorne, and Turri, have offered examples of claims that are allegedly contingent and a priori justifiable. If any of these examples is genuine, this would upend the traditional epistemological classification on which (a) all and only a priori justifiable claims are necessary and (b) all and only a posteriori ones are contingent. I argue here that these examples are not genuine. This conclusion is not new, but the strategy pursued here is to formalize these muchdiscussed (...) examples in symbolic logics. Once formalized, a perspicuous representation of their logical form will bring into sharp relief that these examples are not both contingent and a priori. Two takeaways are (1) that the traditional epistemological classification remains plausible and (2) that one’s proposed examples of contingent a priori claims should be supported by a formalization in one’s preferred background symbolic logic. (shrink)

Book Review: Samuel Lebens (2017) "Bertrand Russell and the Nature of Propositions: a History and Defense of the Multiple Relation Theory of Judgement".

In this paper I argue that the two-dimensional character of Frege’s Begriffsschrift plays an epistemological role in his argument for the analyticity of arithmetic. First, I motivate the claim that its two-dimensional character needs a historical explanation. Then, to set the stage, I discuss Frege’s notion of a Begriffsschrift and Kant’s epistemology of mathematics as synthetic a priori and partly grounded in intuition, canvassing Frege’s sharp disagreement on these points. Finally, I argue that the two-dimensional character of Frege’s notations play (...) the epistemological role of facilitating our grasp of logical truths independently of intuition. The rest of this paper critically evaluates Frege’s view and discusses Macbeth’s account. (shrink)

The contact argument is widely cited as making a strong case against a gunk-free metaphysics with point-sized simples. It is shown here that the contact argument's reasoning is faulty even if all its background assumptions and desiderata for contact are accepted. Further, the simples theorist can offer both metric and topological accounts of contact that satisfy all the contact argument's desiderata. This indicates that the contact argument's persuasiveness stems from a tacit reliance on the thesis that objects in contact are (...) inseparable: the simples theorist must allow that separated objects might be in contact. The concluding section critically considers this contact-separability thesis and argues that rejecting it is not so terrible. The upshot of all this is that the contact argument is simply unconvincing. (shrink)

This book examines Bertrand Russell’s complicated relationships to the women around him, and to feminism more generally. The essays in this volume offer scholarly reassessments of these relationships and their import for the history of feminism and of analytic philosophy. Russell is a founder of analytic philosophy. He has also been called a feminist due to his public, decades-long advocacy for women’s rights and equality of the sexes. But his private behavior towards wives and sexual partners, and his apparently dismissive (...) (occasionally public) responses to some women philosophers, raises the question of what sort of feminist (or chauvinist) Russell actually was. Focusing on women in Russell’s circle of acquaintance, including feminist activists and his philosophical interlocutors, this book casts new light on a timeless thinker’s feminism and the women who played critical roles in the making of analytic philosophy. (shrink)

I critically discuss a new proposal for a metaphysics of sense-data. This proposal is due to Peter Forrest. Forrest argues that, if we accept Platonism about universals, sense-data are best understood as structured universals–in particular, as structured universals with temporal and spatial properties as components. Against this proposal, I argue sense-data as structured universals are not universals at all.

I here defend logical atomism. This defense rests on reinterpreting logical atomism as a search for logical forms. This reinterpretation has two parts comprising six chapters. In the first part, I do some historically-driven recovery. In the introduction, I review the literature on Russell's logical atomism. In Chapter 1, I argue that the dominant interpretation of logical atomism is wrong on historical grounds: it accounts for neither the history of logical atomism nor for crucial elements of the logical atomist texts. (...) In Chapter 2, I then use Russell's writings to recover what I argue is the core of logical atomism. I explicate the critical notions and essential ingredients of logical atomism using "Principia Mathematica" as the archetype of logical atomism. I argue that logical atomsts are term busters. The essential ingredient of a logical atomist's term busting practice is a higher-order logic with the power of impredicative comprehension. In Chapter 3, I discuss the widespread view that Wittgenstein held a version of logical atomism. Focusing on his pre-"Tractatus" writings and changes in his earlier views, I argue that Wittgenstein embraced a philosophy of logic incompatible with emulating impredicative comprehension in April 1914. As such, Wittgenstein was a logical atomist, if ever, in October 1913, possibly through April 1914. In the second part, having clarified what logical atomism is, I present a modern logical atomism. In Chapter 4, I develop a philosophy of logic for logical atomism based on the notion of a pure logic. I critically discuss normativity in logic, the epistemology of pure logic, and logical pluralism. In Chapter 5, I propose a formal logic for logical atomism. I argue for the logic of logical atomism being an infinitely-descending and infinitely-ascending simple type theory with impredicative comprehension compatible with a domain empty of particulars. In Chapter 6, I critically discuss what the ontology of logical atomism should be, that is, what the ontology of the logical atomist's logic must be. This includes an ontology of logical concepts and of logical forms as completely-general, necessarily-existing logical facts with no constituents. I conclude by indicating avenues for new work on logical atomism. (shrink)

We examine the influence of word choices on mathematical practice, i.e. in developing definitions, theorems, and proofs. As a case study, we consider Euclid’s and Euler’s word choices in their influential developments of geometry and, in particular, their use of the term ‘polyhedron’. Then, jumping to the twentieth century, we look at word choices surrounding the use of the term ‘polyhedron’ in the work of Coxeter and of Grünbaum. We also consider a recent and explicit conflict of approach between Grünbaum (...) and Shephard on the one hand and that of Hilton and Pedersen on the other, elucidating that the conflict was engendered by disagreement over the proper conceptualization, and so also the appropriate word choices, in the study of polyhedra. (shrink)

The Defining Issues Test 2 (DIT-2) and Engineering Ethical Reasoning Instrument (EERI) are designed to measure ethical reasoning of general (DIT-2) and engineering-student (EERI) populations. These tools—and the DIT-2 especially—have gained wide usage for assessing the ethical reasoning of undergraduate students. This paper reports on a research study in which the ethical reasoning of first-year undergraduate engineering students at multiple universities was assessed with both of these tools. In addition to these two instruments, students were also asked to create personal (...) concept maps of the phrase “ethical decision-making.” It was hypothesized that students whose instrument scores reflected more postconventional levels of moral development and more sophisticated ethical reasoning skills would likewise have richer, more detailed concept maps of ethical decision-making, reflecting their deeper levels of understanding of this topic and the complex of related concepts. In fact, there was no significant correlation between the instrument scores and concept map scoring, suggesting that the way first-year students _conceptualize_ ethical decision making does not predict the way they behave when _performing_ scenario-based ethical reasoning (perhaps more situated). This disparity indicates a need to more precisely quantify engineering ethical reasoning and decision making, if we wish to inform assessment outcomes using the results of such quantitative analyses. (shrink)

This paper argues that the concept and role of ‘institutional racism’ in contemporary discussions of race should be reconsidered. It starts by distinguishing between ‘intrinsic institutional racism’, which holds that institutions are racist in virtue of their constitutive features, and ‘extrinsic institutional racism’, which holds that institutions are racist in virtue of their negative effects. It accepts intrinsic institutional racism, but argues that a ‘disparate impact’ conception of extrinsic conception faces a number of objections, the most serious being that it (...) has no plausible account of what it is that makes extrinsically racist institutions racist. It also argues that claims about the explanatory indispensability of institutional racism are overstated (individual racism is at least as important), critiques structural approaches to racial inequality, and suggests that there is reason to doubt whether institutional reform can provide us with all that morality may require in the racial domain. (shrink)

This article discusses the adequacy of Rawls’ theory of justice as a tool for racial justice. It is argued that critics like Charles W Mills fail to appreciate both the insights and limits of the Rawlsian framework. The article has two main parts spread out over several different sections. The first is concerned with whether the Rawlsian framework suffices to prevent racial injustice. It is argued that there are reasons to doubt whether it does. The second part is concerned with (...) whether a Rawlsian framework has the resources to rectify past racial injustice. It is argued that it has more resources to do this than Mills allows. This second part of the article centers on two Rawlsian ideas: ideal theory and the fair equality of opportunity principle. It is argued that ideal theory is essential for the kind of rectificatory work that Mills wants nonideal theory to do, and that where there is a socioeconomic legacy of past injustice, it is hard to see how FEO could be implemented if it did no rectificatory work, a result which means that there is less need to turn to nonideal theory at all. (shrink)

... the topic of 'meaning' is the one topic discussed in philosophy in which there is literally nothing but 'theory' - literally nothing that can be labelled or even ridiculed as the 'common sense view'. Putnam, 'The Meaning of Meaning' This book explores some truths behind the truism that experimentation is a hallmark of scientific activity. Scientists' descriptions of nature result from two sorts of encounter: they interact with each other and with nature. Philosophy of science has, by and large, (...) failed to give an account of either sort of interaction. Philosophers typically imagine that scientists observe, theorize and experiment in order to produce general knowledge of natural laws, knowledge which can be applied to generate new theories and technologies. This view bifurcates the scientist's world into an empirical world of pre-articulate experience and know how and another world of talk, thought and argument. Most received philosophies of science focus so exclusively on the literary world of representations that they cannot begin to address the philosophical problems arising from the interaction of these worlds: empirical access as a source of knowledge, meaning and reference, and of course, realism. This has placed the epistemological burden entirely on the predictive role of experiment because, it is argued, testing predictions is all that could show that scientists' theorizing is constrained by nature. Here a purely literary approach contributes to its own demise. The epistemological significance of experiment turns out to be a theoretical matter: cruciality depends on argument, not experiment. (shrink)

Mencius, who lived in the 4th century B.C., is second only to Confucius in importance in the Confucian tradition. The _Mencius_ consists of sayings of Mencius and conversations he had with his contemporaries. When read side by side with the _Analects_, the _Mencius_ throws a great deal of light on the teachings of ConfuciusMencius developed many of the ideas of Confucius and at the same time discussed problems not touched upon by Confucius. He drew out the implications of Confucius' moral (...) principles and reinterpreted them for the conditions of his time. As the fullest of the four great Confucian texts, the _Mencius_ has been the required reading amongst Chinese scholars for two thousand years, and it still throws considerable light on the character of the Chinese people. (shrink)

_Rhetoric_ is the sixth volume in The New Hackett Aristotle series, a series featuring translations, with Introductions and Notes, by C. D. C. Reeve, Delta Kappa Epsilon Distinguished Professor of Philosophy at The University of North Carolina at Chapel Hill. The series will eventually include all of Aristotle's works.

A record of the words and teachings of Confucius, _The Analects_ is considered the most reliable expression of Confucian thought. However, the original meaning of Confucius's teachings have been filtered and interpreted by the commentaries of Confucianists of later ages, particularly the Neo-Confucianists of the Song dynasty, not altogether without distortion.In this monumental translation by Professor D. C. Lau, an attempt has been made to interpret the sayings as they stand. The corpus of the sayings is taken as an organic (...) whole and the final test of the interpretation rests on the internal consistency it exhibits. In other words, _The Analects_ is read in the light of _The Analects._This results in a truer understanding of Confucius' thought than the traditional interpretation and paves the way for a re-assessment of its importance in the history of Chinese thought and its relevance to the present day world.This volume also contains an introduction to the life and teachings of Confucius, and three appendices on the events in the life of Confucius, on his disciples, and on the composition of _The Analects._. (shrink)

This paper suggests that empirical educational research has not, on the whole, been treated well by philosophers of education. A variety of criticisms have been offered, ranging from triviality, conceptual confusion and the impossibility of empirically studying normative processes. Furthermore, many of those who criticise, or dismiss, empirical research do so without subjecting any specific examples to careful scholarly scrutiny. It is suggested that both philosophy of education, and the empirical research enterprise, stand to profit if philosophers pay more attention (...) to real cases—and this attention is especially important at present, when research funding is being based on spurious scientistic criteria such as the use of ‘gold standard’ randomised experimental research designs. (shrink)

This work provides an interesting, though sometimes rather sweeping, demonstration that the metaphysical problem of the same and the other is also the central problem of literature and literary criticism. The author defends the analogical imagination as the symbolic counterpart of participation in Platonic metaphysics.--D. C. B.

After an exceedingly short treatment of six theories of tragedy, the author concludes that while each has emphasized a necessary component of the tragic, none has really come to grips with its basic "paradox": the fact that while the art of tragedy attempts to explain the mystery of human suffering, such an attempt is doomed to failure.--D. C. B.

This paper suggests that empirical educational research has not, on the whole, been treated well by philosophers of education. A variety of criticisms have been offered, ranging from triviality, conceptual confusion and the impossibility of empirically studying normative processes. Furthermore, many of those who criticise, or dismiss, empirical research do so without subjecting any specific examples to careful scholarly scrutiny. It is suggested that both philosophy of education, and the empirical research enterprise, stand to profit if philosophers pay more attention (...) to real cases—and this attention is especially important at present, when research funding is being based on spurious scientistic criteria such as the use of ‘gold standard’ randomised experimental research designs. (shrink)