While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental (...) mathematics on the one hand and computerized formal proofs on the other hand. Along the way, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed. (shrink)

Numbers and other mathematical objects are exceptional in having no locations in space or time or relations of cause and effect. This makes it difficult to account for the possibility of the knowledge of such objects, leading many philosophers to embrace nominalism, the doctrine that there are no such objects, and to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects. This book cuts through a host of technicalities that have obscured previous (...) discussions of these projects, and presents clear, concise accounts of a dozen strategies for nominalistic interpretation of mathematics, thus equipping the reader to evaluate each and to compare different ones. The authors also offer critical discussion, rare in the literature, of the aims and claims of nominalistic interpretation, suggesting that it is significant in a very different way from that usually assumed. (shrink)

The great logician Gottlob Frege attempted to provide a purely logical foundation for mathematics. His system collapsed when Bertrand Russell discovered a contradiction in it. Thereafter, mathematicians and logicians, beginning with Russell himself, turned in other directions to look for a framework for modern abstract mathematics. Over the past couple of decades, however, logicians and philosophers have discovered that much more is salvageable from the rubble of Frege's system than had previously been assumed. A variety of repaired systems have been (...) proposed, each a consistent theory permitting the development of a significant portion of mathematics. This book surveys the assortment of methods put forth for fixing Frege's system, in an attempt to determine just how much of mathematics can be reconstructed in each. John Burgess considers every proposed fix, each with its distinctive philosophical advantages and drawbacks. These systems range from those barely able to reconstruct the rudiments of arithmetic to those that go well beyond the generally accepted axioms of set theory into the speculative realm of large cardinals. For the most part, Burgess finds that attempts to fix Frege do less than advertised to revive his system. This book will be the benchmark against which future analyses of the revival of Frege will be measured. (shrink)

The most important distinctively American contribution to philosophy is the pragmatist tradition. In this short, lucid, and completely convincing exposition, Professor John P. Murphy begins by exploring the roots of this tradition as found in the work of Peirce, James, and Dewey, demonstrating its power and originality. Historians of philosophy will appreciate the insight Murphy brings to these figures, but the special value of this book lies in his discussion of how the pragmatist spirit has flowered in contemporary philosophy (...) in the work of Quine, Rorty, and Davidson.Throughout, Murphy emphasizes the logic and structure of the views held by these six philosophers and what it is they have in common that makes their work especially “pragmatist.” There is no better introduction to this historical tradition and perhaps no better way into the philosophies of the contemporaries whom Murphy discusses.Interest in pragmatist ideas is undergoing a revival at present, and this book shows us why. It will be of interest to both historians of philosophy and students of contemporary philosophy. (shrink)

Philosophical Logic is a clear and concise critical survey of nonclassical logics of philosophical interest written by one of the world's leading authorities on the subject. After giving an overview of classical logic, John Burgess introduces five central branches of nonclassical logic, focusing on the sometimes problematic relationship between formal apparatus and intuitive motivation. Requiring minimal background and arranged to make the more technical material optional, the book offers a choice between an overview and in-depth study, and it balances (...) the philosophical and technical aspects of the subject.The book emphasizes the relationship between models and the traditional goal of logic, the evaluation of arguments, and critically examines apparatus and assumptions that often are taken for granted. Philosophical Logic provides an unusually thorough treatment of conditional logic, unifying probabilistic and model-theoretic approaches. It underscores the variety of approaches that have been taken to relevantistic and related logics, and it stresses the problem of connecting formal systems to the motivating ideas behind intuitionistic mathematics. Each chapter ends with a brief guide to further reading.Philosophical Logic addresses students new to logic, philosophers working in other areas, and specialists in logic, providing both a sophisticated introduction and a new synthesis. (shrink)

In this paper Sullins argues that in certain circumstances robots can be seen as real moral agents. A distinction is made between persons and moral agents such that, it is not necessary for a robot to have personhood in order to be a moral agent. I detail three requirements for a robot to be seen as a moral agent. The first is achieved when the robot is significantly autonomous from any programmers or operators of the machine. The second is when (...) one can analyze or explain the robot's behavior only by ascribing to it some predisposition or 'intention' to do good or harm. And finally, robot moral agency requires the robot to behave in a way that shows and understanding of responsibility to some other moral agent. Robots with all of these criteria will have moral rights as well as responsibilities regardless of their status as persons. (shrink)

Introduction A brief look at the competing present-day interpretations of Hume's philosophy will leave the uninitiated reader completely baffled. On the one hand , Hume is seen as a philosopher who attempted to analyse concepts with ...

Highlighting previously neglected democratic strains in Machiavelli's major writings, McCormick excavates institutions through which the common people of ancient, medieval, and Renaissance republics constrained the power of wealthy citizens and public magistrates, and he imagines how such institutions might be revived today.

Darwinism, Democracy, and Race examines the development and defence of an argument that arose at the boundary between anthropology and evolutionary biology in twentieth-century America. In its fully articulated form, this argument simultaneously discredited scientific racism and defended free human agency in Darwinian terms. The volume is timely because it gives readers a key to assessing contemporary debates about the biology of race. By working across disciplinary lines, the book's focal figures--the anthropologist Franz Boas, the cultural anthropologist Alfred Kroeber, the (...) geneticist Theodosius Dobzhansky, and the physical anthropologist Sherwood Washburn--found increasingly persuasive ways of cutting between genetic determinist and social constructionist views of race by grounding Boas's racially egalitarian, culturally relativistic, and democratically pluralistic ethic in a distinctive version of the genetic theory of natural selection. Collaborators in making and defending this argument included Ashley Montagu, Stephen Jay Gould, and Richard Lewontin. Darwinism, Democracy, and Race will appeal to advanced undergraduates, graduate students, and academics interested in subjects including Philosophy, Critical Race Theory, Sociology of Race, History of Biology and Anthropology, and Rhetoric of Science. (shrink)

Used as the basis of the program at the Equinox Holistic Alternative School in Toronto, The Holistic Curriculum advocates for an integrative approach to teaching and learning with a focus on developing a deep connection between mind and body.

This volume reprints a dozen of the author’s papers, most with substantial postscripts, and adds one new one. The bulk of the material is on topics in philosophy of language, but there are also two papers on philosophy of mathematics written after the appearance of the author’s collected papers on that subject, and one on epistemology. As to the substance of Field’s contributions, limitations of space preclude doing much more below than indicating the range of issues addressed, and the general (...) orientation taken towards them. As to the style of his writing, it well exhibits the first of the two virtues, clarity and conciseness, that one looks for in philosophical prose. (shrink)

The complexity of the set of truths of arithmetic is determined for various theories of truth deriving from Kripke and from Gupta and Herzberger.

What does it mean to consider the world of AI through a Christian lens? Rapid developments in AI continue to reshape society, raising new ethical questions and challenging our understanding of the human person. Encountering Artificial Intelligence draws on Pope Francis’ discussion of a culture of encounter and broader themes in Catholic social thought in order to examine how current AI applications affect human relationships in various social spheres and offers concrete recommendations for better implementation. The document also explores questions (...) regarding personhood, consciousness, and the kinds of relationships humans might have with even the most advanced AI. Through these discussions, this book investigates the theoretical and practical challenges to interpersonal encounter raised by the age of AI. (AI Research Group for the Centre for Digital Culture of the Dicastery of Culture and Education of the Holy See). (shrink)

The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.

The source, status, and significance of the derivation of the necessity of identity at the beginning of Kripke’s lecture “Identity and Necessity” is discussed from a logical, philosophical, and historical point of view.

This book will appeal to mathematicians and philosophers interested in the foundations of mathematics.

A new axiomatization of set theory, to be called Bernays-Boolos set theory, is introduced. Its background logic is the plural logic of Boolos, and its only positive set-theoretic existence axiom is a reflection principle of Bernays. It is a very simple system of axioms sufficient to obtain the usual axioms of ZFC, plus some large cardinals, and to reduce every question of plural logic to a question of set theory.

John Norton says that philosophers have been led astray for thousands of years by their attempt to treat induction formally. He is correct that such an attempt has caused no end of trouble, but he is wrong about the history. There is a rich tradition of non-formal induction. In fact, material theories of induction prevailed all through antiquity and from the Renaissance to the mid-1800s. Recovering these past systems would not only fill lacunae in Norton’s own theory but would (...) highlight areas where Norton has not freed himself from the straightjacket of formal induction as much as he might think. This essay begins that recovery. (shrink)

David Hume's A Treatise of Human Nature presents the most important account of skepticism in the history of modern philosophy. In this lucid and thorough introduction to the work, John P. Wright examines the development of Hume's ideas in the Treatise, their relation to eighteenth-century theories of the imagination and passions, and the reception they received when Hume published the Treatise. He explains Hume's arguments concerning the inability of reason to establish the basic beliefs which underlie science and morals, (...) as well as his arguments showing why we are nevertheless psychologically compelled to accept such beliefs. The book will be a valuable guide for those seeking to understand the nature of modern skepticism and its connection with the founding of the human sciences during the Enlightenment. (shrink)

By utilizing new information from both clinical and experimental (lesion, electrophysiological, and gene-activation) studies with animals, the anatomy underlying anterograde amnesia has been reformulated. The distinction between temporal lobe and diencephalic amnesia is of limited value in that a common feature of anterograde amnesia is damage to part of an comprising the hippocampus, the fornix, the mamillary bodies, and the anterior thalamic nuclei. This view, which can be traced back to Delay and Brion (1969), differs from other recent models in (...) placing critical importance on the efferents from the hippocampus via the fornix to the diencephalon. These are necessary for the encoding and, hence, the effective subsequent recall of episodic memory. An additional feature of this hippocampalanterior thalamic and the perirhinal–medial dorsal thalamic systems are compromised, leading to severe deficits in both recall and recognition. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:John P. Anton C. P. CAVAFY'S ARS POETICA ' It is generally recognized that Constantine P. Cavafy (1863-1933) was not born a poet but became one only through persistence and labor, reaching his "first step" sometime after the midpoint of his life. In his effort to assess the quality of his earlier poetic production and sharpen his sensitivity in facing self-criticism, he decided to put in writing his (...) personal thoughts on his work sometime in 1903. The text was written in English in a personal shorthand Cavafy had devised for his own use. It was found among the papers he left to A. Sengopoulos and published for the first time in 1963 by M. Peridis, who also supplied the suitable title Ars Poetica. It falls outside the scope of this essay to discuss the antecedents of this document. Taken as a whole, the text has all the clarity and directness required of a testament designed to serve as a personal guide and handbook. The problems it discusses are limited to the needs of the poet. In this regard, the Ars Poetica has neither the range of topics nor the theoretical scope of, for instance, Aristotle's Poetics. It is quite certain that the ideas that preoccupy Cavafy in the Ars Poetica had been on his mind for a period of years. However, the Ars Poetica is his first prose piece where he gives a systematic treatment to technical and critical issues directly related to the assessment of his work and his role as a poet. In order to project here a clear image of Cavafy's struggle to reach his technical maturity which came after 1910, we need first to discuss the persistent problems he faced during this crucial decade, then to examine the attitudes and values he embodied in certain representative early poems, and finally to relate them to the guiding principles he formulated in his self-addressed poetics of 1903. A number of interrelated problems confronted Cavafy in his late twenties and throughout his thirties. They emerged one by one, to 85 86Philosophy and Literature make more pressing the demand for a consistent solution and a complete recasting of his outlook. Coming to terms with them was extremely difficult. Aside from the emotional and sexual pressures he had to face alone, there was the absence of suitable ideas and conceptual tools to provide him with even the rudiments of an intellectual framework. Whatever he borrowed from Europe and England, or resurrected from his own Hellenic heritage, proved either beyond assimilation or insufficiently digested. The lingering issues fell roughly into four areas. (i) Despite his determination to devote his life to the writing of poetry, the quality of his work was discouraging. He knew this and as a result he often expressed uncertainty about his ability to proceed. Time and again he appeared suffering from a feeling of inadequacy, but nevertheless he sought to engage a reflective mood from which to draw encouragement and hope. (ii) On the theoretical side, he identified with the long tradition which assigned a coveted cultural role to the savant-poet as discloser of truth and spokesman for humanity; this despite the prevailing climate of opinion in his immediate environment which recognized no such high position in his case. Cavafy knew that his standing with the Greek community of Alexandria had gradually dwindled with the loss of family wealth and his lowly employment, and also that his expectations to be respected in his role as a poet and leader in communal affairs were futile. This denial of personal recognition in a society dominated by commerce and finance contributed to his mood of depression. The feeling of being an outcast, despite his associations with the upper social circles of Alexandria, became a constant irritant to his sense of pride. (iii) The course of culture, the destiny of human institutions and the workings of history, became puzzles as he tried to view them through the opaque glass of disillusionment. These were the grand themes actively discussed in the writings of leading European intellectuals, but to the young poet they had a different personal urgency. His vital concern in this connection was not understanding science... (shrink)

The question, "Which modal logic is the right one for logical necessity?," divides into two questions, one about model-theoretic validity, the other about proof-theoretic demonstrability. The arguments of Halldén and others that the right validity argument is S5, and the right demonstrability logic includes S4, are reviewed, and certain common objections are argued to be fallacious. A new argument, based on work of Supecki and Bryll, is presented for the claim that the right demonstrability logic must be contained in S5, (...) and a more speculative argument for the claim that it does not include S4.2 is also presented. (shrink)

Critics of organ donation after circulatory death (DCD) argue that, even if donors are past the point of autoresuscitation, they have not satisfied the “irreversibility” requirement in the circulatory and respiratory criteria for determining death, since their circulation and respiration could be artificially restored. Thus, removing their vital organs violates the “dead-donor” rule. I defend DCD donation against this criticism. I argue that practical medical-ethical considerations, including respect for do-not-resuscitate orders, support interpreting “irreversibility” to mean permanent cessation of circulation and (...) respiration. Assuming a consciousness-related formulation of human death, I then argue that the loss of circulation and respiration is significant, because it leads to the permanent loss of consciousness and thus to the death of the human person. The DNR request by an organ donor should thus be interpreted to mean “do not restore to consciousness.” Finally, I respond to an objection that if “irreversibility” has a medical-ethical meaning, it would entail the absurd possibility that one of two individuals in the same physical state could be alive and the other dead—an implication that some think is inconsistent with understanding death as an objective biological state of the organism. I argue that advances in medical technology have created phenomena that challenge the assumption that human death can be understood in strictly biological terms. I argue that ethical and ontological considerations about our nature bear on the definition and determination of death and thus on the permissibility of DCD. (shrink)

Much of the debate over the definition and criteria for determining our death has focused on disagreement over the correct biological account of death, i.e., what it means for any organism to die. In this paper, we argue that this exclusive focus on the biology of death is misguided, because it ignores ethical and social factors that bear on the acceptability of criteria for determining our death. We propose that attention shift from strictly biological considerations to ethical and social considerations (...) that bear on the determination of what we call “civil death.” We argue for acceptance of a neurological criterion for determining death on grounds that it is the most reasonable way to synthesize biological, ethical, and social considerations about our death.. (shrink)

Saul Kripke has been a major influence on analytic philosophy and allied fields for a half-century and more. His early masterpiece, _Naming and Necessity_, reversed the pattern of two centuries of philosophizing about the necessary and the contingent. Although much of his work remains unpublished, several major essays have now appeared in print, most recently in his long-awaited collection _Philosophical Troubles_. In this book Kripke’s long-time colleague, the logician and philosopher John P. Burgess, offers a thorough and self-contained guide (...) to all of Kripke’s published books and his most important philosophical papers, old and new. It also provides an authoritative but non-technical account of Kripke’s influential contributions to the study of modal logic and logical paradoxes. Although Kripke has been anything but a system-builder, Burgess expertly uncovers the connections between different parts of his oeuvre. Kripke is shown grappling, often in opposition to existing traditions, with mysteries surrounding the nature of necessity, rule-following, and the conscious mind, as well as with intricate and intriguing puzzles about identity, belief and self-reference. Clearly contextualizing the full range of Kripke’s work, Burgess outlines, summarizes and surveys the issues raised by each of the philosopher’s major publications. _Kripke_ will be essential reading for anyone interested in the work of one of analytic philosophy’s greatest living thinkers. (shrink)

The species of indeterminist tense logic called Peircean by A. N. Prior is proved to be recursively decidable.

Scholars loosely affiliated with the “Cambridge School” (e.g., Pocock, Skinner, Viroli, and Pettit) accentuate rule of law, common good, class equilibrium, and non-domination in Machiavelli's political thought and republicanism generally but underestimate the Florentine's preference for class conflict and ignore his insistence on elite accountability. The author argues that they obscure the extent to which Machiavelli is an anti-elitist critic of the republican tradition, which they fail to disclose was predominantly oligarchic. The prescriptive lessons these scholars draw from republicanism for (...) contemporary politics reinforce rather than reform the “senatorial,” electorally based, and socioeconomically agnostic republican model (devised by Machiavelli's aristocratic interlocutor, Guicciardini, and refined by Montesquieu and Madison) that permits common citizens to acclaim but not determine government policies. Cambridge School textual interpretations and practical proposals have little connection with Machiavelli's “tribunate,” class-specific model of popular government elaborated in The Discourses, one that relies on extra-electoral accountability techniques and embraces deliberative popular assemblies. (shrink)

Nominalism is the view that mathematical objects do not exist. This chapter delimits several types of nominalistic projects: revolutionary programs that attempt to change mathematics and hermeneutic programs that attempt to interpret mathematics. Some programs accord with naturalism, and some oppose naturalism. Steven Yablo’s fictionalism is brought into the fold and discussed at some length.

This is the first in-depth critical appraisal in English of the political, legal, and cultural writings of Carl Schmitt, perhaps this century's most brilliant critic of liberalism. It offers an assessment of this most sophisticated of fascist theorists without attempting either to apologise for or demonise him. Schmitt's Weimar writings confront the role of technology as it finds expression through the principles and practices of liberalism. Contemporary political conditions such as disaffection with liberalism and the rise of extremist political organizations (...) have rendered Schmitt's work both relevant and insightful. John McCormick examines why technology becomes a rallying cry for both right- and left-wing intellectuals at times when liberalism appears anachronistic, and shows the continuities between Weimar's ideological debates and those of our own age. (shrink)

Perhaps if the future existed, concretely and individually, as something that could be discerned by a better brain, the past would not be so seductive: its demands would he balanced by those of the future. Persons might then straddle the middle stretch of the seesaw when considering this or that object. It might be fun. But the future has no such reality (as the pictured past and the perceived present possess); the future is but a figure of speech, a specter (...) of thought. (shrink)

(2013). Informed Consent, Big Data, and the Oxymoron of Research That Is Not Research. The American Journal of Bioethics: Vol. 13, No. 4, pp. 40-42. doi: 10.1080/15265161.2013.768864.

Today there appears to be a widespread impression that W. V. Quine's notorious critique of modal logic, based on certain ideas about reference, has been successfully answered. As one writer put it some years ago: “His objections have been dead for a while, even though they have not yet been completely buried.” What is supposed to have killed off the critique? Some would cite the development of a new ‘possible-worlds’ model theory for modal logics in the 1960s; others, the development (...) of new ‘direct’ theories of reference for names in the 1970s.These developments do suggest that Quine's unfriendliness towards any formal logics but the classical and indifference towards theories of reference for any singular terms but variables were unfortunate. (shrink)