It is often claimed that irreducibly normative truths would have unacceptable metaphysical implications, and are incompatible with a scientific view of the world. The book argues, on the basis of a general account of the relevance of ontological questions, that this claim is mistaken. It is also a mistake to think that interpreting normative judgments as beliefs would make it impossible to explain their connection with action. An agent’s acceptance of a normative judgment can explain that agent’s subsequent action because (...) it is part of being a rational agent that such an agent’s beliefs about reasons normally, but not invariably, make a difference to the agent’s subsequent behavior. Because facts about reasons are not entities existing apart from us, there is no epistemological problem of how we can “be in touch with” such facts. There are serious worries about normative knowledge, but the problems involved are internal to the normative domain itself. The best solution to these problems would be an overall account of the domain of reasons in normative terms, supported by an argument from reflective equilibrium. But no existing account, constructivist, or based on desires or on an idea of rationality, is plausible, and no alternative is likely to succeed. Conclusions about reasons for action must rest on more piecemeal applications of the method of reflective equilibrium. (shrink)

There is currently intense interest in the question of the source of our presumed knowledge of truths concerning what is, or is not, metaphysically possible or necessary. Some philosophers locate this source in our capacities to conceive or imagine various actual or non-actual states of affairs, but this approach is open to certain familiar and seemingly powerful objections. A different and ostensibly more promising approach has been developed by Timothy Williamson, according to which our capacity for modal knowledge is just (...) an extension, or by-product, of our general capacity to acquire knowledge of true counterfactual conditionals — a capacity that we deploy ubiquitously in everyday life. Williamson’s account crucially involves a thesis to the effect that modal truths can be explained in terms of counterfactual truths. In this paper, I query Williamson’s account on a number of points, including this thesis. My positive proposal, which owes a debt to the work of Kit Fine on modality and essence, appeals instead to our capacity to grasp essences, understood in a neo-Aristotelian fashion, according to which essences are expressed by ‘real definitions’. (shrink)

In "Mathematical Truth", Paul Benacerraf articulated an epistemological problem for mathematical realism. His formulation of the problem relied on a causal theory of knowledge which is now widely rejected. But it is generally agreed that Benacerraf was onto a genuine problem for mathematical realism nevertheless. Hartry Field describes it as the problem of explaining the reliability of our mathematical beliefs, realistically construed. In this paper, I argue that the Benacerraf Problem cannot be made out. There simply is no intelligible problem (...) that satisfies all of the constraints which have been placed on the Benacerraf Problem. The point generalizes to all arguments with the structure of the Benacerraf Problem aimed at realism about a domain meeting certain conditions. Such arguments include so-called "Evolutionary Debunking Arguments" aimed at moral realism. I conclude with some suggestions about the relationship between the Benacerraf Problem and the Gettier Problem. (shrink)

In order to perfectly describe the world, it is not enough to speak truly. One must also use the right concepts - including the right logical concepts. One must use concepts that "carve at the joints", that give the world's "structure". There is an objectively correct way to "write the book of the world". Much of metaphysics, as traditionally conceived, is about the fundamental nature of reality; in the present terms, this is about the world's structure. Metametaphysics - inquiry into (...) the status of metaphysical questions - turns on structure. The question of whether ontological, causal, or modal questions are "substantive" is in large part a question of whether the world has ontological, causal, and modal structure - whether quantifiers, causal relations, and modal operators carve at the joints. (shrink)

This paper explores the prospects for safety-based theories of knowledge in the light of some recent objections.

Modern empiricism has been conditioned in large part by two dogmas. One is a belief in some fundamental cleavage between truths which are analytic, or grounded in meanings independently of matters of fact, and truth which are synthetic, or grounded in fact. The other dogma is reductionism: the belief that each meaningful statement is equivalent to some logical construct upon terms which refer to immediate experience. Both dogmas, I shall argue, are ill founded. One effect of abandoning them is, as (...) we shall see, a blurring of the supposed boundary between speculative metaphysics and natural science. Another effect is a shift toward pragmatism. (shrink)

We think of logic as objective. We also think that we are reliable about logic. These views jointly generate a puzzle: How is it that we are reliable about logic? How is it that our logical beliefs match an objective domain of logical fact? This is an instance of a more general challenge to explain our reliability about a priori domains. In this paper, I argue that the nature of this challenge has not been properly understood. I explicate the challenge (...) both in general and for the particular case of logic. I also argue that two seemingly attractive responses – appealing to a faculty of rational insight or to the nature of concept possession – are incapable of answering the challenge. (shrink)

Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...

John Mackie's stimulating book is a complete and clear treatise on moral theory. His writings on normative ethics-the moral principles he recommends-offer a fresh approach on a much neglected subject, and the work as a whole is undoubtedly a major contribution to modern philosophy.The author deals first with the status of ethics, arguing that there are not objective values, that morality cannot be discovered but must be made. He examines next the content of ethics, seeing morality as a functional device, (...) basically the same at all times but changing significantly in response to changes in the human condition. He sketches a practical moral system, criticizing but also borrowing from both utilitarian and absolutist views. Thirdly, the frontiers of ethics, areas of contact with psychology, metaphysics, theology, law and politcs, are explored.Throughout, his aim is to discuss a wide range of questions that are both philosophical and practical, working within a distinctive version of subjectivism-an "error" theory of the apparent objectivity of values. John Mackie has drawn on the contributions of such classic thinkers as Plato, Aristotle, Hobbes, Hume, Kant and Sidgwick, and on more recent discussions, to produce a thought-provoking account that will inspire both the general reader and the student of philosophy. (shrink)

All contentious moral issues--from gay marriage to abortion and affirmative action--raise difficult questions about the justification of moral beliefs. How can we be justified in holding on to our own moral beliefs while recognizing that other intelligent people feel quite differently and that many moral beliefs are distorted by self-interest and by corrupt cultures? Even when almost everyone agrees--e.g. that experimental surgery without consent is immoral--can we know that such beliefs are true? If so, how? These profound questions lead to (...) fundamental issues about the nature of morality, language, metaphysics, justification, and knowledge. They also have tremendous practical importance in handling controversial moral questions in health care ethics, politics, law, and education. Sinnott-Armstrong here provides an extensive overview of these difficult subjects, looking at a wide variety of questions, including: Are any moral beliefs true? Are any justified? What is justified belief? The second half of the book explores various moral theories that have grappled with these issues, such as naturalism, normativism, intuitionism, and coherentism, all of which are attempts to answer moral skepticism. Sinnott-Armstrong argues that all these approaches fail to rule out moral nihilism--the view that nothing is really morally wrong or right, bad or good. Then he develops his own novel theory,--"moderate Pyrrhonian moral skepticism"--which concludes that some moral beliefs can be justified out of a modest contrast class but no moral beliefs can be justified out of an extreme contrast class. While explaining this original position and criticizing alternatives, Sinnott-Armstrong provides a wide-ranging survey of the epistemology of moral beliefs. (shrink)

Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set (...) theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science. (shrink)

This book is concerned with `the problem of existence in mathematics'. It develops a mathematical system in which there are no existence assertions but only assertions of the constructibility of certain sorts of things. It explores the philosophical implications of such an approach through an examination of the writings of Field, Burgess, Maddy, Kitcher, and others.

A response is given here to Benacerraf's 1973 argument that mathematical platonism is incompatible with a naturalistic epistemology. Unlike almost all previous platonist responses to Benacerraf, the response given here is positive rather than negative; that is, rather than trying to find a problem with Benacerraf's argument, I accept his challenge and meet it head on by constructing an epistemology of abstract (i.e., aspatial and atemporal) mathematical objects. Thus, I show that spatio-temporal creatures like ourselves can attain knowledge about mathematical (...) objects by simply explaininghow they can do this. My argument is based upon the adoption of a particular version of platonism — full-blooded platonism — which asserts that any mathematical object which possiblycould exist actuallydoes exist. (shrink)

Since Edmund L. Gettier reminded us recently of a certain important inadequacy of the traditional analysis of "S knows that p," several attempts have been made to correct that analysis. In this paper I shall offer still another analysis (or a sketch of an analysis) of "S knows that p," one which will avert Gettier's problem. My concern will be with knowledge of empirical propositions only, since I think that the traditional analysis is adequate for knowledge of nonempirical truths.

In his influential book, The Nature of Morality, Gilbert Harman writes: “In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the other hand, one never seems to need to appeal in this way to moral principles.” What is the epistemological relevance of this contrast, if genuine? This chapter argues that ethicists and philosophers of mathematics have misunderstood it. They have confused what the chapter calls the justificatory challenge for realism about an area, D—the (...) challenge to justify our D-beliefs—with the reliability challenge for D-realism—the challenge to explain the reliability of our D-beliefs. Harman’s contrast is relevant to the first, but not, evidently, to the second. One upshot of the discussion is that genealogical debunking arguments are fallacious. Another is that indispensability considerations cannot answer the Benacerraf–Field challenge for mathematical realism. (shrink)

This paper critically examines coincidence arguments and evolutionary debunking arguments against non-naturalist realism in metaethics. It advances a version of these arguments that goes roughly like this: Given a non-naturalist, realist metaethic, it would be cosmically coincidental if our first order normative beliefs were true. This coincidence undermines any prima facie justification enjoyed by those beliefs.

It is widely agreed that the intelligibility of modal metaphysics has been vindicated. Quine's arguments to the contrary supposedly confused analyticity with metaphysical necessity, and rigid with non-rigid designators.2 But even if modal metaphysics is intelligible, it could be misconceived. It could be that metaphysical necessity is not absolute necessity – the strictest real notion of necessity – and that no proposition of traditional metaphysical interest is necessary in every real sense. If there were nothing otherwise “uniquely metaphysically significant” about (...) metaphysical necessity, then paradigmatic metaphysical necessities would be necessary in one sense of “necessary”, not necessary in another, and that would be it. The question of whether they were necessary simpliciter would be like the question of whether the Parallel Postulate is true simpliciter – understood as a pure mathematical conjecture, rather than as a hypothesis about physical spacetime. In a sense, the latter question has no objective answer. In this article, I argue that paradigmatic questions of modal metaphysics are like the Parallel Postulate question. I then discuss the deflationary ramifications of this argument. I conclude with an alternative conception of the space of possibility. According to this conception, there is no objective boundary between possibility and impossibility. Along the way, I sketch an analogy between modal metaphysics and set theory. (shrink)

It is widely agreed that the intelligibility of modal metaphysics has been vindicated. Quine's arguments to the contrary supposedly confused analyticity with metaphysical necessity, and rigid with non-rigid designators.2 But even if modal metaphysics is intelligible, it could be misconceived. It could be that metaphysical necessity is not absolute necessity – the strictest real notion of necessity – and that no proposition of traditional metaphysical interest is necessary in every real sense. If there were nothing otherwise “uniquely metaphysically significant” about (...) metaphysical necessity, then paradigmatic metaphysical necessities would be necessary in one sense of “necessary”, not necessary in another, and that would be it. The question of whether they were necessary simpliciter would be like the question of whether the Parallel Postulate is true simpliciter – understood as a pure mathematical conjecture, rather than as a hypothesis about physical spacetime. In a sense, the latter question has no objective answer. In this article, I argue that paradigmatic questions of modal metaphysics are like the Parallel Postulate question. I then discuss the deflationary ramifications of this argument. I conclude with an alternative conception of the space of possibility. According to this conception, there is no objective boundary between possibility and impossibility. Along the way, I sketch an analogy between modal metaphysics and set theory. (shrink)

This paper explores the prospects for safety-based theories of knowledge in the light of some recent objections.

In The Myth of Morality, Richard Joyce argues that moral discourse is hopelessly flawed. At the heart of ordinary moral judgements is a notion of moral inescapability, or practical authority, which, upon investigation, cannot be reasonably defended. Joyce argues that natural selection is to blame, in that it has provided us with a tendency to invest the world with values that it does not contain, and demands that it does not make. Should we therefore do away with morality, as we (...) did away with other faulty notions such as witches? Possibly not. We may be able to carry on with morality as a 'useful fiction' - allowing it to have a regulative influence on our lives and decisions, perhaps even playing a central role - while not committing ourselves to believing or asserting falsehoods, and thus not being subject to accusations of 'error'. (shrink)

In his précis of a recent book, Richard Joyce writes, “My contention…is that…any epistemological benefit-of-the-doubt that might have been extended to moral beliefs…will be neutralized by the availability of an empirically confirmed moral genealogy that nowhere…presupposes their truth.” Such reasoning – falling under the heading “Genealogical Debunking Arguments” – is now commonplace. But how might “the availability of an empirically confirmed moral genealogy that nowhere… presupposes” the truth of our moral beliefs “neutralize” whatever “epistemological benefit-of-the-doubt that might have been extended (...) to” them? In this article, I argue that there appears to be no satisfactory answer to this question. The problem is quite general, applying to all arguments with the structure of Genealogical Debunking Arguments aimed at realism about a domain meeting two conditions. The Benacerraf-Field Challenge for mathematical realism affords an important special case. (shrink)

It is commonly suggested that evolutionary considerations generate an epistemological challenge for moral realism. At first approximation, the challenge for the moral realist is to explain our having many true moral beliefs, given that those beliefs are the products of evolutionary forces that would be indifferent to the moral truth. An important question surrounding this challenge is the extent to which it generalizes. In particular, it is of interest whether the Evolutionary Challenge for moral realism is equally a challenge for (...) mathematical realism. It is widely thought not to be. In this paper, I argue that the Evolutionary Challenge for moral realism is equally a challenge for mathematical realism. Along the way, I substantially clarify the Evolutionary Challenge, discuss its relation to more familiar epistemological challenges, and broach the problem of moral disagreement. The paper should be of interest to ethicists because it places pressure on anyone who rejects moral realism on the basis of the Evolutionary Challenge to reject mathematical realism as well. And the paper should be of interest to philosophers of mathematics because it presents a new epistemological challenge for mathematical realism that bears, I argue, no simple relation to Paul Benacerraf's familiar challenge. (shrink)

A plurality of approaches to foundational aspects of mathematics is a fact of life. Two loci of this are discussed here, the classicism/constructivism controversy over standards of proof, and the plurality of universes of discourse for mathematics arising in set theory and in category theory, whose problematic relationship is discussed. The first case illustrates the hypothesis that a sufficiently rich subject matter may require a multiplicity of approaches. The second case, while in some respects special to mathematics, raises issues of (...) ontological multiplicity and relativity encountered in the natural sciences as well. (shrink)

Part of the ambiguity lies in the various points of view from which this question might be considered. The crudest di erence lies between the point of view of the working mathematician and that of the logician concerned with the foundations of mathematics. Now some of my fellow mathematical logicians might protest this distinction, since they consider themselves to be just more of those \working mathematicians". Certainly, modern logic has established itself as a very respectable branch of mathematics, and there (...) are quite a few highly technical journals in logic, such as The Journal of Sym-. (shrink)

Mark Balaguer argues for full blooded platonism (FBP), and argues that FBP alone can solve Benacerraf's familiar epistemic challenge. I note that if FBP really can solve Benacerraf's epistemic challenge, then FBP is not alone in its capacity so to solve; RFBP—really full blooded platonism—can do the trick just as well, where RFBP differs from FBP by allowing entities from inconsistent mathematics. I also argue briefly that there is positive reason for endorsing RFBP.

The paper clarifies and defends the orthodox view that counterfactual conditionals with impossible antecedents are vacuously true against recent criticisms. It argues that apparent counterexamples to orthodoxy result from uncritical reliance on a fallible heuristic used in the processing of conditionals. A comparison is developed between such counterpossibles and vacuously true universal generalizations.

The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set-theoretic possibilities, a phenomenon that challenges the universe (...) view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for. (shrink)

Having no recourse to ways of knowing about the natural world, ethical non-naturalists are in need of an epistemology that might apply to a normative breed of facts or properties, and intuitionism seems well suited to fill that bill. Here I argue that the metaphysical inspiration for ethical intuitionism undermines that very epistemology, for this pair of views generates what I call the defeater from cosmic coincidence. Unfortunately, we face not a happy union, but a difficult choice: either ethical intuitionism (...) or ethical non-naturalism, but not both. (shrink)

The Evolution of Morality attempts to accomplish two tasks. The first is to clarify and provisionally advocate the thesis that human morality is a distinct adaptation wrought by biological natural selection. The second is to inquire whether this empirical thesis would, if true, have any metaethical implications.

In this paper, we develop an alternative strategy, Platonized Naturalism, for reconciling naturalism and Platonism and to account for our knowledge of mathematical objects and properties. A systematic (Principled) Platonism based on a comprehension principle that asserts the existence of a plenitude of abstract objects is not just consistent with, but required (on transcendental grounds) for naturalism. Such a comprehension principle is synthetic, and it is known a priori. Its synthetic a priori character is grounded in the fact that it (...) is an essential part of the logic in which any scientific theory will be formulated and so underlies (our understanding of) the meaningfulness of any such theory (this is why it is required for naturalism). Moreover, the comprehension principle satisfies naturalist standards of reference, knowledge, and ontological parsimony! As part of our argument, we identify mathematical objects as abstract individuals in the domain governed by the comprehension principle, and we show that our knowledge of mathematical truths is linked to our knowledge of that principle. (shrink)