Geared to preparing students to make the transition from solving problems to proving theorems, this text teachs them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. To help students construct their own proofs, this new edition (...) contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. Previous Edition Hb (1994) 0-521-44116-1 Previous Edition Pb (1994) 0-521-44663-5. (shrink)

We propose the use of variable declarations in natural deduction. A variable declaration is a line in a derivation that introduces a new variable into the derivation. Semantically, it can be regarded as declaring that the variable denotes an element of the universe of discourse. Undeclared variables, in contrast, do not denote anything, and may not occur free in any formula in the derivation. Although most natural deduction systems in use today do not have variable declarations, the idea can be (...) traced back to one of the first papers on natural deduction. We show how the use of variable declarations in natural deduction leads to a formal system that has a number of desirable features: It is simple, easy to use and understand, and corresponds closely to ordinary informal reasoning. Soundness and completeness of the system are easily proven. Furthermore, the system clarifies the role of the existential instantiation rule in natural deduction. (shrink)

This book provides an accessible, critical introduction to the three main approaches that dominated work in the philosophy of mathematics during the twentieth century: logicism, intuitionism and formalism.

Dieses Buch blickt in eine bedeutende Epoche der Philosophie der Mathematik zurück, deren Strömungen die heutige Gestalt der Mathematik prägten. In der Wende vom 19. zum 20. Jahrhundert befand sich die Mathematik in einem fundamentalen Umbruch, der die Mathematiker dieser Zeit herausforderte. Sie mussten Stellung beziehen. Die Grundsätze und Wege der philosophischen Richtungen, die dieses Buch verständlich, kritisch und anerkennend beschreibt, wurden von Mathematikern formuliert. Eine Zeit gravierender Disharmonien begann, die bis in Streit und Feindschaften mündeten und zugleich faszinierende und (...) fruchtbare Ergebnisse hervorbrachten, mathematisch wie philosophisch. Es war ein aufregendes, intellektuelles Abenteuer zu Beginn des 20. Jahrhunderts auf einem außergewöhnlich scharfsinnigen und kreativen Niveau. Die Debatte über die unversöhnlichen Ansichten versiegte allmählich und inzwischen ist wieder relative Ruhe in die Gemeinde der Mathematiker eingekehrt. Zentrale philosophische Fragen aber, die damals die Protagonisten spalteten, sind nach wie vor unbeantwortet. Die Suche nach dem Wesen der Mathematik geht weiter und greift auf die Ideen dieser Kontroversen zurück. (shrink)

I offer an account of how ideals motivate us. My account suggests that although emulating an ideal is often rational, it can lead us to do irrational things. * This is the third in a series of four papers on narrative self-conceptions and their role in moral motivation. In the first paper, “The Self as Narrator” (to appear in Autonomy and the Challenges to Liberalism: New Essays, ed. Joel Anderson and John Christman), I explore the motivational role of narrative self-conceptions, (...) drawing on Daniel Dennett’s notion of the self as a “center of narrative gravity”. In the second paper, “Willing the Law” (in Practical Conflicts, ed. Monika Betzler and Peter Baumann [Cambridge: Cambridge University Press, forthcoming]), I explore the role of selfconceptions in Kantian “conflicts in the will”, drawing on Christine Korsgaard’s notion of “practical identities”. In a fourth paper, “A Sense of Self” (ms), I explore the role of narrative self-con-ceptions in Kantian “conflicts in conception”, drawing on the work of Thomas Nagel and John Perry on the self. (shrink)

In this paper I examine Bernard Williams’ claim that an appealing conception of love can come into conflict with impartial morality. First, I explain how Williams’ claim can survive one strategy to head off the possibility of conflict. I then examine J.D.Velleman’s Kantian conception of love as another possible way to reject Williams’ claim. I argue, however, that Velleman’s attempt to transcend love’s partiality in his account of love produces an unappealing and unconvincing ideal. This is made particularly (...) clear, I suggest, by the analysis that Velleman is forced to give of the kind of case that generated Williams’ observations in the first place. Thus Velleman’s account should be rejected. (shrink)

The potential to collect brain data more directly, with higher resolution, and in greater amounts has heightened worries about mental and brain privacy. In order to manage the risks to individuals posed by these privacy challenges, some have suggested codifying new privacy rights, including a right to “mental privacy.” In this paper, we consider these arguments and conclude that while neurotechnologies do raise significant privacy concerns, such concerns are—at least for now—no different from those raised by other well-understood data collection (...) technologies, such as gene sequencing tools and online surveillance. To better understand the privacy stakes of brain data, we suggest the use of a conceptual framework from information ethics, Helen Nissenbaum’s “contextual integrity” theory. To illustrate the importance of context, we examine neurotechnologies and the information flows they produce in three familiar contexts—healthcare and medical research, criminal justice, and consumer marketing. We argue that by emphasizing what is distinct about brain privacy issues, rather than what they share with other data privacy concerns, risks weakening broader efforts to enact more robust privacy law and policy. (shrink)

Epistemic freedom is the freedom to affirm anyone of several incompatible propositions without risk of being wrong. We sometimes have this freedom, strange as it seems, and our having it sheds some light on the topic of free will and determinism. This paper sketches a potential explanation for our feeling of freedom. The freedom that I postulate is not causal but epistemic (in a sense that I shall define), and the result is that it is quite compatible with determinism. I (...) therefore claim that insofar as we feel metaphysically free-free in a sense that would be incompatible with determinism-we are mistaking the epistemic freedom that we have for a kind of freedom that we may lack. This claim will lead me, at the end of the paper, to a projectivist account of moral responsibility. Ascriptions of moral responsibility, I shall suggest, should be treated in the same way as ascriptions of color or other secondary qualities. (shrink)

I ask you to X. You now have a reason to X. My request gave you a reason. How? One unpopular theory is the epistemic account, according to which requests do not create any new reasons but instead simply reveal information. For instance, my request that you X reveals that I desire that you X, and my desire gives you a reason to X. Peter Schaber has recently attacked both the epistemic account and other theories of the reason-giving force of (...) requests. Schaber defends a new theory of the reason-giving force of requests according to which request give reasons because it is valuable for requesters and requestees that requests have this power. In this paper I argue that Schaber's attack on the epistemic account fails, and that his own theory ought to be rejected because it faces compelling objections. (shrink)

The Possibility of Practical Reason explores the foundational questions of moral psychology: How can any of our behavior qualify as acting for a reason? How can any considerations qualify as reasons for us to act? David Velleman argues that both possibilities depend on there being a constitutive aim of action―something that makes for success in action as such. These twelve essays―five of which were not included in the previous edition, two of them previously unpublished―discuss topics such as freedom of (...) the will, shared intention, the relation between value and practical reasoning, the foundations of decision theory, and the motivational role of the imagination. (shrink)

“X-Firsters” hold that there is some normative feature that is fundamental to all others (and, often, that there’s some normative feature that is the “mark of the normative”: all other normative properties have it, and are normative in virtue of having it). This view is taken as a starting point in the debate about which X is “on first.” Little has been said about whether or why we should be X-Firsters, or what we should think about normativity if we aren’t (...) X-Firsters. Hence the chapter’s two main goals. First, to provide a simple argument that one shouldn’t be an X-Firster about the normative domain, which starts with the observation that analogous views have dubious merits in analogous domains. Second, to offer an alternative view—taking normativity to be a determinable explained in terms of its determinates—that offers an interesting way to think about the structure and unity of normativity. (shrink)

Plato's dialogues frequently criticize traditional Greek myth, yet Plato also integrates myth with his writing. Daniel S. Werner confronts this paradox through an in-depth analysis of the Phaedrus, Plato's most mythical dialogue. Werner argues that the myths of the Phaedrus serve several complex functions: they bring nonphilosophers into the philosophical life; they offer a starting point for philosophical inquiry; they unify the dialogue as a literary and dramatic whole; they draw attention to the limits of language and the limits (...) of knowledge; and they allow Plato to co-opt cultural authority as a way of defining and legitimating the practice of philosophy. Platonic myth, as a species of traditional tale, is thus both distinct from philosophical dialectic and similar to it. Ultimately, the most powerful effect of Platonic myth is the way in which it leads readers to participate in Plato's dialogues and to engage in a process of self-examination. (shrink)

Supporters of open borders sometimes argue that the state has no pro tanto right to restrict immigration, because such a right would also entail a right to exclude existing citizens for whatever reasons justify excluding immigrants. These arguments can be defeated by suggesting that people have a right to stay put. I present a new form of the exclusion argument against closed borders which escapes this “right to stay put” reply. I do this by describing a kind of exclusion that (...) has not been discussed in depth, which I call “territorial exclusion.” Territorial exclusion is the process according to which the group that wishes to exclude current citizens secedes from the territory in which those citizens reside. I argue that the wrongness of territorial exclusion explains why there is no pro tanto right for a state to exclude immigrants, because otherwise there would be a pro tanto right for the state to kick people out by seceding from the territory they inhabit. Because kicking people out like this is typically wrong, borders cannot be closed. (shrink)

A story does more than recount events; it recounts events in a way that renders them intelligible, thus conveying not just information but also understanding. We might therefore be tempted to describe narrative as a genre of explanation. When the police invite a suspect to “tell his story,” they are asking him to explain the blood on his shirt or his absence from home on the night of the murder; and whether he is judged to have a “good story” will (...) depend on its adequacy as an explanation. Can we account for the explanatory force of narrative with the models of explanation available in the philosophy of science? Or does narrative convey a different kind of understanding, which requires a different model and perhaps even a term other than ‘explanation’? (shrink)

Motion—and, in particular, local motion or change in location—plays a central role in Kenelm Digby’s natural philosophy and in his arguments for the immateriality of the soul. Despite this, Digby’s account of what motion consists in has yet to receive much scholarly attention. In this paper, I advance a novel interpretation of Digby on motion. According to it, Digby holds that for a body to move is for it to divide from and unify with other bodies. This is a view (...) of motion—as change in relations of parthood—that Alison Peterman attributes to Digby’s contemporary and acquaintance, Margaret Cavendish. Having shown that Digby’s presentation of the view predates Cavendish’s by more than a decade, I make a case that Digby’s work influenced Cavendish’s on this topic. In developing and defending my reading, I consider to what extent the Digbean account of motion and the arguments for it accord with the ideals of the mechanical philosophy emerging in the early modern period. (shrink)

Abstract This paper offers an appraisal of Phillip Pettit's approach to the problem how a merely finite set of examples can serve to represent a determinate rule, given that indefinitely many rules can be extrapolated from any such set. I argue that Pettit's so-called ethnocentric theory of rule-following fails to deliver the solution to this problem he sets out to provide. More constructively, I consider what further provisions are needed in order to advance Pettit's general approach to the problem. I (...) conclude that what is needed is an account that, whilst it affirms the view that agents' responses are constitutively involved in the exemplification of rules, does not allow such responses the pride of place they have in Pettit's theory. (shrink)

This book is a study of the role of intellect in human action as described by Thomas Aquinas. One of its primary aims is to compare the interpretation of Aristotle by Aquinas with the lines of interpretation offered in contemporary Aristotelian scholarship. The book seeks to clarify the problems involved in the appropriation of Aristotle's theory by a Christian theologian, including such topics as the practical syllogism and the problems of akrasia. Westberg argues that Aquinas was much closer to Aristotle (...) than is often recognized, and he puts forward important new interpretations of the relation of intellect and will in the stages of intention, deliberation, decision, and execution. In the concluding section of the book, he shows how this new interpretation yields fruitful insights on a range of theological topics, including sin, law, love, and the moral virtues. (shrink)

The problem known as Buridan’s Ass says that a hungry donkey equipoised between two identical bales of hay will starve to death. Indecision kills the ass. Some philosophers worry about human analogs. Computer scientists since the 1960s have known about the computer versions of such cases. From what Leslie Lamport calls ‘Buridan’s Principle’—a discrete decision based on a continuous range of input-values cannot be made in a bounded time—it follows that the possibilities for human analogs of Buridan’s Ass are far (...) more wide-ranging and securely provable than has been acknowledged in philosophy. We are never necessarily decisive. This is mathematically provable. I explore four consequences: first, increased interest of the literature’s solutions to Buridan’s Ass; second, a new asymmetry between responsibility for omissions and responsibility for actions; third, clarification of the standard account of akrasia; and, fourth, clarification of the role of credences in normative decision-theory. (shrink)

This paper aims to understand Hegel’s claim in the introduction to his Philosophy of Mind that mind is an actualization of the Idea and argues that this claim provides us with a novel and defensible way of understanding Hegel’s naturalism. I suggest that Hegel’s approach to naturalism should be understood as ‘formal’, and argue that Hegel’s Logic, particularly the section on the ‘Idea’, provides us with a method for this approach. In the first part of the paper, I present an (...) interpretation of Hegel’s method in which life plays a central role. In the second part of the paper, I develop Hegel’s method by providing a reading of Hegel’s Subjective Spirit, focusing on the sections ‘Anthropology’ and ‘Phenomenology’ in particular, arguing that they display the dialectic between life and cognition outlined by Hegel’s Idea. (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)

Suppose that we want to frame a conception of reasons that isn't relativized to the inclinations of particular agents. That is, we want to identify particular things that count as reasons for acting simpliciter and not merely as reasons for some agents rather than others, depending on their inclinations. One way to frame such a conception is to name some features that an action can have and to say that they count as reasons for someone whether or not he is (...) inclined to care about them. The problem with the resulting conception, as we have seen, is that it entails the normative judgment that one ought to be inclined to care about the specified features, on pain of irrationality, and this normative judgment requires justification. The advantage of internalism is that it avoids these normative commitments. It says that things count as reasons for someone only if he is inclined to care about them, and so it leaves the normative question of whether to care about them entirely open. Yet if we try to leave this question open, by defining things as reasons only for those inclined to care about them, we'll end up with a definition that's relativized to the inclinations of particular agents—won't we? Not necessarily. For suppose that all reasons for acting are features of a single kind, whose influence depends on a single inclination. And suppose that the inclination on which the influence of reasons depends is, not an inclination that distinguishes some agents from others, but rather an inclination that distinguishes agents from nonagents. In that case, to say that these features count as reasons only for those who are inclined to care about them will be to say that they count as reasons only for agents—which will be to say no less than that they are reasons for acting, period, since applying only to agents is already part of the concept of reasons for acting. The restriction on the application of reasons will drop away from our definition, since it restricts their application, not to some proper subset of agents, but rather to the set of all agents, which is simply the universe of application for reasons to act. (shrink)

The terms `endurance' and `perdurance' are commonly thought to denote distinct ways for an object to persist, but it is surprisingly hard to say what these are. The common approach, defining them in terms of temporal parts, is mistaken, because it does not lead to two coherent philosophical alternatives: endurance so understood becomes conceptually incoherent, while perdurance becomes not just true but a conceptual truth. Instead, we propose a different way to articulate the distinction, in terms of identity rather than (...) temporal parts: an object endures if its identity is determined at every moment at which it exists. We make precise what it means for the identity of an object to be determined at a moment. We also discuss what role the endurance / perdurance distinction, so understood, should play in the debates about time, material objects and personal identity. (shrink)

This paper aims to understand Hegel’s claim in the introduction to his Philosophy of Mind that mind is an actualization of the Idea and argues that this claim provides us with a novel and defensible way of understanding Hegel’s naturalism. I suggest that Hegel’s approach to naturalism should be understood as ‘formal’, and argue that Hegel’s Logic, particularly the section on the ‘Idea’, provides us with a method for this approach. In the first part of the paper, I present an (...) interpretation of Hegel’s method in which life plays a central role. In the second part of the paper, I develop Hegel’s method by providing a reading of Hegel’s Subjective Spirit, focusing on the sections ‘Anthropology’ and ‘Phenomenology’ in particular, arguing that they display the dialectic between life and cognition outlined by Hegel’s Idea. (shrink)

A story does more than recount events; it recounts events in a way that renders them intelligible, thus conveying not just information but also understanding. We might therefore be tempted to describe narrative as a genre of explanation. When the police invite a suspect to “tell his story,” they are asking him to explain the blood on his shirt or his absence from home on the night of the murder; and whether he is judged to have a “good story” will (...) depend on its adequacy as an explanation. Can we account for the explanatory force of narrative with the models of explanation available in the philosophy of science? Or does narrative convey a different kind of understanding, which requires a different model and perhaps even a term other than ‘explanation’? (shrink)

This essay considers the critical response to Hegel's view of Socrates we find in Kierkegaard's dissertation, The Concept of Irony. I argue that this dispute turns on the question whether or not the examination of particular thinkers enters into Socrates’ most basic aims and interests. I go on to show how Kierkegaard's account, which relies on an affirmative answer to this question, enables him to provide a cogent defence of Socrates' philosophical practice against Hegel's criticisms.

Abstract: According to a rationalist theory of introspection, rational agents have a capacity to believe they are in conscious states when they are in them, much as they have the capacity, for example, to avoid obvious contradictions in their beliefs. For the agent to know or believe by introspection, on this view, is for them to exercise that capacity. According to an acquaintance theory of introspection, by contrast, whenever an agent is in a conscious state, the agent is aware of (...) or is acquainted with being in the state, where the background notion of acquaintance is understood to be distinct from sense perception, on the one hand, and belief or knowledge, or the other. For the agent to know or believe by introspection, on this view, is for them to take advantage of the epistemic position they occupy in virtue of being acquainted in this way. These theories are not in conflict; it is possible to hold both an acquaintance theory and a rationalist theory. This paper, however, sets out and recommends a different possibility, namely, that of holding a rationalist theory while rejecting any acquaintance theory. (shrink)

“What do you see when you look at your face in the mirror?” asks J. David Velleman in introducing his philosophical theory of action. He takes this simple act of self-scrutiny as a model for the reflective reasoning of rational agents: our efforts to understand our existence and conduct are aided by our efforts to make it intelligible. Reflective reasoning, Velleman argues, constitutes practical reasoning. By applying this conception, _Practical Reflection_ develops philosophical accounts of intention, free will, and (...) the foundation of morals. This new edition of _Practical Reflection_ contains the original 1989 text along with a new introduction and is the latest entry in The David Hume Series of Philosophy and Cognitive Science Reissues, which keeps in print previously published indispensable works in the area of cognitive science. _ _ _ _. (shrink)