Adapated from talks at the UCLA Logic Center and the Pitt Philosophy of Science Series. Exposition of material from Fixing Frege, Chapter 2 (on predicative versions of Frege’s system) and from “Protocol Sentences for Lite Logicism” (on a form of mathematical instrumentalism), suggesting a connection. Provisional version: references remain to be added. To appear in Mathematics, Modality, and Models: Selected Philosophical Papers, coming from Cambridge University Press.
Fixing Frege is one of the most important investigations to date of Fregean approaches to the foundations of mathematics. In addition to providing an unrivalled survey of the technical program to which Frege’s writings have given rise, the book makes a large number of improvements and clarifications. Anyone with an interest in the philosophy of mathematics will enjoy and benefit from the careful and well informed overview provided by the first of its three chapters. Specialists will find the book an (...) indispensable reference and an invaluable source of insights and new results. Although Frege is widely regarded as the father of analytic philosophy, his work on the foundations of mathematics was for a long time rather peripheral to the ongoing research. The main reason for this is no doubt Russell’s discovery in 1901 that the paradox now bearing his name can be derived in Frege’s logical system. But recent decades have seen a huge surge of interest in Fregean approaches to the foundations of mathematics. (The work of George Boolos, Kit Fine, Bob Hale, Richard Heck, Stewart Shapiro, and Crispin Wright is singled out for particular attention in the present monograph.) A variety of consistent theories have been discovered that can be salvaged from Frege’s inconsistent system, and foundational and philosophical claims have been made on behalf of many of these theories. Burgess claims quite plausibly that the significance of any such modified Fregean theory will in large part depend on how much of ordinary mathematics it enables us to develop.1 His.. (shrink)
The discovery of the note cards for Quine’s previously unpublished 1946 lecture on nominalism provides an obvious occasion for commenting on the differences between the issue of nominalism as Quine first publicized it to a wide philosophical audience and the issue of nominalism as debated among Quine’s successors today. Yet as I read and reread the text of Quine’s lecture, I found myself struck less by the differences between Quine’s position there and the positions of present-day writers than by differences (...) between Quine’s position there and the positions of Quine himself in later writings — and not his writings from many years later but his writings from the next few years, and especially one of his writings from the very next year, his notorious joint paper with Goodman. (shrink)
This long-awaited volume is a must-read for anyone with a serious interest in\nphilosophy of mathematics. The book falls into two parts, with the primary focus of\nthe first on ontology and structuralism, and the second on intuition and\nepistemology, though with many links between them. The style throughout involves\nunhurried examination from several points of view of each issue addressed, before\nreaching a guarded conclusion. A wealth of material is set before the reader along\nthe way, but a reviewer wishing to summarize the author’s views (...) crisply will be\nfrustrated. The chapter-by-chapter survey below conveys at best a very incomplete\nand imperfect impression of the work’s virtues, and even of its contents, falling\nshort even of supplying a full menu for the banquet of food for thought that Parsons\nserves up to his readers. (shrink)
A revision of a sermon on the evils of calling model theory “semantics”, preached at Notre Dame on Saint Patrick’s Day, 2005. Provisional version: references remain to be added. To appear in Mathematics, Modality, and Models: Selected Philosophical Papers, coming from Cambridge University Press.
What is the simplest and most natural axiomatic replacement for the set-theoretic definition of the minimal fixed point on the Kleene scheme in Kripke’s theory of truth? What is the simplest and most natural set of axioms and rules for truth whose adoption by a subject who had never heard the word "true" before would give that subject an understanding of truth for which the minimal fixed point on the Kleene scheme would be a good model? Several axiomatic systems, old (...) and new, are examined and evaluated as candidate answers to these questions, with results of Harvey Friedman playing a significant role in the examination. (shrink)
One textbook may introduce the real numbers in Cantor’s way, and another in Dedekind’s, and the mathematical community as a whole will be completely indifferent to the choice between the two. This sort of phenomenon was famously called to the attention of philosophers by Paul Benacerraf. It will be argued that structuralism in philosophy of mathematics is a mistake, a generalization of Benacerraf’s observation in the wrong direction, resulting from philosophers’ preoccupation with ontology.
In this era when results of empirical scientific research are being appealed to all across philosophy, when we even find moral philosophers invoking the results of brain scans, many profess to practice "naturalized epistemology," or to be "epistemological naturalists." Such phrases derive from the title of a well-known essay by Quine,[1] but Paul Gregory's thesis in the work under review is that there is less connection than is usually assumed between Quine's variety of naturalized epistemology and what is today taken, (...) by opponents and proponents alike, to constitute epistemological naturalism. To put it bluntly, as Gregory does in the opening sentence of his introduction, Quine "has not been well understood." If there is less connection between the Quinian and other epistemological naturalisms than there has often been taken to be, on Gregory's account there is also much more connection between Quine's position on epistemology and his positions on other contentious issues. (shrink)
It is shown that for invariance under the action of special groups the statements "Every invariant PCA is decomposable into (1 invariant Borel sets" and "Every pair of invariant PCA is reducible by a pair of invariant PCA sets" are independent of the axioms of set theory.
Metalinguistic descriptivism is the view that proper names are semantically equivalent to descriptions featuring their own quotations (e.g., ?Socrates? means ?the bearer of ?Socrates??). The present paper shows that Millians can actually accept an inferential version of this equivalence thesis without running afoul of the modal argument. Indeed, they should: for it preserves the explanatory virtues of more familiar forms of descriptivism while avoiding objections (old and new) to Kent Bach's nominal description theory. We can make significant progress on Frege's (...) puzzle and Plato's beard without committing ourselves one way or the other on the semantic values of proper names. The view on offer can also be motivated by analogy with Tarski's schema T, inviting the idea that the equivalence between a name and the associated nominal description has more to do with the semantics of representational locutions than it does with names per se. My response to the modal argument exploits the Kripkean distinction between reference at a world and reference in a world, and can be accepted by metalinguistic descriptivists and Millians alike. (shrink)
There are statements of the form “There are no Fs” that we would like to count as true, yet it is hard to see how they could be true (at least, operating within the semantic framework of structured propositions). The relevant Fs are general terms that we take to be semantically fundamental or primitive, especially those native to metaphysical discourse. A case can be made the problem is no less difficult than the corresponding problem for singular terms.
This paper aims to make three contributions to decision theory. First there is the hope that it will help to re-establish the legitimacy of the problem, pace various recent analyses provided by Maitzen and Wilson, Slezak and Priest. Second, after pointing out that analyses of the problem have generally relied upon evidence that is conditional on the taking of one particular option, this paper argues that certain assumptions implicit in those analyses are subtly flawed. As a third contribution, the piece (...) aims to draw attention to an important similarity between Newcomb’s problem and the toxin puzzle. In short, both problems illustrate the fact that you can have a reason to intend to φ without having a reason to actually φ. (shrink)
Recent philosophy of language has been profoundly impacted by the idea that mainstream, model-theoretic semantics is somehow incompatible with deflationary accounts of truth and reference. The present article systematizes the case for incompatibilism, debunks circularity and “modal confusion” arguments familiar in the literature, and reconstructs the popular thought that truth-conditional semantics somehow “presupposes” a correspondence theory of truth as an inference to the best explanation. The case for compatibilism is closed by showing that this IBE argument fails to rule out (...) two kinds of deflationism: the position Field famously accused Tarski of having; and a less familiar version of the view that defines reference in terms of a deflated notion of truth. Finally, the distinction between unifying and constitutive explanation is used to forestall the response that correspondence theory is literally part of mainstream semantics. (shrink)
Saul Kripke has made fundamental contributions to a variety of areas of logic, and his name is attached to a corresponding variety of objects and results. 1 For philosophers, by far the most important examples are ‘Kripke models’, which have been adopted as the standard type of models for modal and related non-classical logics. What follows is an elementary introduction to Kripke’s contributions in this area, intended to prepare the reader to tackle more formal treatments elsewhere.2 2. WHAT IS A (...) MODEL THEORY? Traditionally, a statement is regarded as logically valid if it is an instance of a logically valid form, where a form is regarded as logically valid if every instance is true. In modern logic, forms are represented by formulas involving letters and special symbols, and logicians seek therefore to define a notion of model and a notion of a formula’s truth in a model in such a way that every instance of a form will be true if and only if a formula representing that form is true in every model. Thus the unsurveyably vast range of instances can be replaced for purposes of logical evaluation by the range of models, which may be more tractable theoretically and perhaps practically. Consideration of the familiar case of classical sentential logic should make these ideas clear. Here a formula, say (p & q) ∨ ¬p ∨ ¬q, will be valid if for all statements P.. (shrink)
The argument from potential has been hard to assess because the versions presented by friends and those presented by enemies have born very little resemblance to each other. I here try to improve this situation by attempting to bring both versions into enforced contact. To this end, I sketch a more detailed analysis of the modern concept of potential than any hitherto attempted. As one would expect, arguments from potential couched in terms of that notion are evident non-starters. I then (...) ask how the modern notion of potential needs to be supplemented in order to produce a more convincing argument. I then enquire whether the supplementations utilised in the most distinguished recent presentations of the argument have anything better than an ad hoc role to play in contemporary metaphysics. I conclude that the rehabilitation of the argument is unlikely; in any event, the onus of proof seems to be on the friend of that argument to show that it is uncontrived. Finally, I argue that the (modern) notion of potential has an important role to play in any plausible account of foetal value. (shrink)
This paper re-examines the question of whether quirks of early human foetal development tell against the view (conceptionism) that we are human beings at conception. A zygote is capable of splitting to give rise to identical twins. Since the zygote cannot be identical with either human being it will become, it cannot already be a human being. Parallel concerns can be raised about chimeras in which two embryos fuse. I argue first that there are just two ways of dealing with (...) cases of fission and fusion and both seem to be available to the conceptionist. One is the Replacement View according to which objects cease to exist when they fission or fuse. The other is the Multiple Occupancy View – both twins may be present already in the zygote and both persist in a chimera. So, is the conceptionist position tenable after all? I argue that it is not. A zygote gives rise not only to a human being but also to a placenta – it cannot already be both a human being and a placenta. Neither approach to fission and fusion can help the conceptionist with this problem. But worse is in store. Both fission and fusion can occur before and after the development of the inner cell mass of the blastocyst – the entity which becomes the embryo proper. The idea that we become human beings with the arrival of the inner cell mass leads to bizarre results however we choose to accommodate fission and fusion. (shrink)
My contribution to the symposium on Goedel’s philosophy of mathematics at the spring 2006 Association for Symbolic Logic meeting in Montreal. Provisional version: references remain to be added. To appear in an ASL volume of proceedings of the Goedel sessions at that meeting.
1 Choice conjecture In axiomatizing nonclassical extensions of classical sentential logic one tries to make do, if one can, with adding to classical sentential logic a finite number of axiom schemes of the simplest kind and a finite number of inference rules of the simplest kind. The simplest kind of axiom scheme in effect states of a particular formula P that for any substitution of formulas for atoms the result of its application to P is to count (...) as an axiom. The simplest kind of onepremise inference rule in effect states of a particular pair of formulas P and Q that for any substitution of formulas for atoms, if the result of its application to P is a theorem, then the result of its application to Q is to count as a theorem; similarly for many-premise rules. Such are the schemes and rules of all the best-known modal and tense logics, for instance. Sometimes it is difficult to find such simple schemes and rules (though it is usually even more difficult to prove that none exist). In that case one may resort to less simple schemes or less simple rules. There is no generally recognized rigorous definition of "next simplest kind" of scheme. (In the case of schemes, one fact that makes a rigorous definition difficult is that, if the logic in question is axiomatizable at all, which is to say, if the set of formulas wanted as theorems is recursively enumerable, then by Craig’s trick one can always get a primitive recursive set of schemes of the simplest kind, even if one cannot get a finite set. Intuitively, some primitive recursive sets are much simpler than others, but it is difficult to reduce this intuition to a rigorous definition.) Neither is there any generally recognized definition of "next simplest kind" of rule, and hence there is no fully rigorous enunciation of the choice conjecture, the conjecture that schemes of the next simplest kind can always be avoided in favor of rules of the next simplest kind and vice versa. Nonetheless, there are cases where intuitively one does recognize that the schemes or rules in a given axiomatization are only slightly more complex than the simplest kind, including cases where one does have a choice between adopting slightly-more-complex-than-simplest schemes and adopting slightly-more-complex-than-simplest rules. In tense logic early examples of slightly more complex rules are found in [2] and [3]: there is one example of the embarrassed use of such rules in the former, and many examples of the enthusiastic use of such rules in the latter and its sequels. Accordingly the rules in question have come to be called "Gabbay-style" rules.. (shrink)
This paper addresses a recent wave of criticisms of liberal peacebuilding operations. We decompose the critics’ argument into two steps, one which offers a diagnosis of what goes wrong when things go wrong in peacebuilding operations, and a second, which argues on the basis of the first step that there is some deep principled flaw in the very idea of liberal peacebuilding. We show that the criticism launched in the argument’s first step is valid and important, but that the second (...) step by no means follows. Drawing a connection between liberal peacebuilding and humanitarian intervention, we argue that the problems that the critics point to are in fact best addressed within the framework of liberal internationalism itself. Further, we argue that the development of the notion of human security marks a dawning awareness within liberal internationalism of the kinds of problems that the critics point to, however difficult it may still be to embody these ideas in practice. (shrink)
I aim to show how and why some definitions can be benignly circular. According to Lloyd Humberstone, a definition that is analytically circular need not be inferentially circular and so might serve to illuminate the application-conditions for a concept. I begin by tidying up some problems with Humberstone's account. I then show that circular definitions of a kind commonly thought to be benign have inferentially circular truth-conditions and so are malign by Humberstone's test. But his test is too demanding. The (...) inferences we actually use to establish the applicability of, e.g., colour concepts are designed to establish warranted assertability and not truth. Understood thus, dispositional analyses are not inferentially circular. (shrink)
Religion, spirituality, and contemporary shamanic practice in Scotland : exploring the relationships -- The impacts of transformational cultural change on religion and spirituality -- Seeking a new definition of religion -- What is shamanism? -- A case study of three shamanic practice groups in Scotland -- Exploring connections between cross-cultural shamanic elements and neo-shamanic expressions in Scotland : interviews, participant observation, and analysis -- Applying Hervieu-Lger's analytical model of religion to reveal a lineage of spirituality, not belief, in the shamanic (...) chain of memory -- Patterns, conclusions, and resulting issues -- Seeing differently : a new paradigm of spirituality and religion. (shrink)
Computability and Logic has become a classic because of its accessibility to students without a mathematical background and because it covers not simply the staple topics of an intermediate logic course, such as Godel’s incompleteness theorems, but also a large number of optional topics, from Turing’s theory of computability to Ramsey’s theorem. Including a selection of exercises, adjusted for this edition, at the end of each chapter, it offers a new and simpler treatment of the representability of recursive functions, a (...) traditional stumbling block for students on the way to the Godel incompleteness theorems. (shrink)
This is the verbatim manuscript of a paper which has circulated underground for close to thirty years, reaching a metethical conclusion close to J. L. Mackie’s by a somewhat different route.
Gintis's article is an example of growing awareness by social scientists of the significance of evolutionary theory for understanding human nature. Although we share its main point of view, we comment on some disagreements related to levels of behavioral analysis, the explanation of social cooperation, and the ubiquity of inter-individual differences in human decision-making. (Published Online April 27 2007).
Blair proposes that fluid intelligence, working memory, and executive function form a unitary construct: fluid cognition. Recently, our group has utilized a combined correlational–experimental cognitive neuroscience approach, which we argue is beneficial for investigating relationships among these individual differences in terms of neural mechanisms underlying them. Our data do not completely support Blair's strong position. (Published Online April 5 2006).
Philosophical Analysis in the Twentieth Century by Scott Soames reminds me of nothing so much as Lectures on Literature by Vladimir Nabokov. Both are works that arose immediately out of the needs of undergraduate teaching, yet each manages to say much of significance to knowledgeable professionals. Each indirectly provides an outline of the history of its field, through a presentation of selected major works, taken in chronological order and including items that are generally recognized as marking decisive turning points. Yet (...) neither Soames’s work nor Nabokov’s is a history in any conventional sense, both being immediately disqualified from that category by the general absence of coverage of minor and middling works and writers. The emphasis is pedagogical rather than historiographical: the emphasis is on introducing the student to the field through very close examination of the limited number of key texts selected for inclusion. The author’s distinctive personality is also apparent in both works. Each writer has a favorite theme he repeatedly sounds: for Soames, the danger of conflating the analytic, the a priori, and the necessary; for Nabokov, the philistinism of expecting an uplifting “message” from works of literary art. Each also includes some quirky, individual selections: The Right and the Good, The Strange Case of Dr. Jekyll and Mr. Hyde. Few others would have taken R. L. Stevenson to be up there with Dickens, Flaubert, and Proust, or W. D. Ross with Russell, Wittgenstein, and Quine. Each also sets aside for separate treatment elsewhere a major body of work one might have expected to be covered. Nabokov reserves Russian literature for a companion volume, while Soames gives only slight coverage to what he describes as “work in logic, the foundations of logic, and the application of logical techniques to the study of language” — a category that in practice turns out to include the bulk of the relevant material (by such writers as Frege, Carnap, and Tarski) that published originally in German without simultaneous English translation.. (shrink)
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 ...
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.
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.
Quine correctly argues that Carnap's distinction between internal and external questions rests on a distinction between analytic and synthetic, which Quine rejects. I argue that Quine needs something like Carnap's distinction to enable him to explain the obviousness of elementary mathematics, while at the same time continuing to maintain as he does that the ultimate ground for holding mathematics to be a body of truths lies in the contribution that mathematics makes to our overall scientific theory of the world. Quine's (...) arguments against the analytic/synthetic distinction, even if fully accepted, still leave room for a notion of pragmatic analyticity sufficient for the indicated purpose. (shrink)
Genome Canada has funded a research project to evaluate the usefulness of different forms of ethical analysis for assessing the moral weight of public opinion in the governance of genomics. This paper will describe a role of public consultation for ethical analysis and a contribution of ethical analysis to public consultation and the governance of genomics/biotechnology. Public consultation increases the robustness of ethical analysis with a more diverse and rich accounts experiences. Consultation must be carefully and respectfully designed to generate (...) sufficiently diverse and rich accounts of moral experiences. Since dominant groupstend to define ethical or policy issues in a manner that excludes some interests or perspectives, it is important to identify the range of interests that diverse publics hold before defining the issue and scope of a consultation. Similarly, a heavy policy focus and pressures to commercialize products risk oversimplification of the discussion and the premature foreclosure of ethical dialogue. Consequently, a significant contribution of ethical dialogue strengthened by social analysis is to consider the context and non-policy use of power to govern genomics and to sustain social debate on enduring ethical issues. (shrink)
The Newcomb problem is analysed here as a type of common cause problem. In relation to such problems, if you take the dominated option your expected outcome will be good and if you take the dominant option your expected outcome will be not so good. As is explained, however, these arenot conventional conditional expected outcomes but `conditional evidence expected outcomes' and while in the deliberation process, the evidence on which they are based is only hypothetical evidence.Conventional conditional expected outcomes are (...) more sensitive to your current epistemic state in that they are based purely on actual evidence which is available to you during the deliberation process. So although they are conditional on a certain act being performed, they are not based on evidence that you would have only if that act is performed. Moreover, for any given epistemic state during the deliberation process, your conventional conditional expected outcome for the dominant option will be better than that for the dominated option. The principle of dominance is thus in perfect harmony with the conventional conditional expected outcomes. In relation to the Newcomb problem then, the evidence unequivocally supports two-boxing as the rational option. Yet what is advanced here is not simply a two-boxing strategy. To see why, two stages to the problem need to be recognised. The first stage is that which occurs before the information used by the predictor in making his predictions has been gained. The second stage is after this point. Provided that you are still in the first stage, you have an opportunity to influence whether or not the predictor places the $1m in the opaque box. To maximise the probability that it is, you need to commit yourself to one-boxing. (shrink)
: Decisions about funding health services are crucial to controlling costs in health care insurance plans, yet they encounter serious challenges from intellectual property protection—e.g., patents—of health care services. Using Myriad Genetics' commercial genetic susceptibility test for hereditary breast cancer (BRCA testing) in the context of the Canadian health insurance system as a case study, this paper applies concepts from social contract theory to help develop more just and rational approaches to health care decision making. Specifically, Daniels's and Sabin's "accountability (...) for reasonableness" is compared to broader notions of public consultation, demonstrating that expert assessments in specific decisions must be transparent and accountable and supplemented by public consultation. (shrink)
Recently it has become almost the received wisdom in certain quarters that Kripke models are appropriate only for something like metaphysical modalities, and not for logical modalities. Here the line of thought leading to Kripke models, and reasons why they are no less appropriate for logical than for other modalities, are explained. It is also indicated where the fallacy in the argument leading to the contrary conclusion lies. The lessons learned are then applied to the question of the status of (...) the formula. (shrink)
Hemispatial neglect in imagery implies a spatially organised representation. Reaction times in memory for arrays of locations from shifted viewpoints indicate processes analogous to actual bodily movement through space. Behavioral data indicate a privileged role for this process in memory. A proposed spatial mechanism makes contact with direct recordings of the representations of location and orientation in the mammalian brain.
Appointment as a director of a company board often represents the pinnacle of a management career. Worldwide, it has been noted that very few women are appointed to the boards of directors of companies. Blame for the low numbers of women of company boards can be partly attributed to the widely publicized "glass ceiling". However, the very low representation of women on company boards requires further examination. This article reviews the current state of women's representation on boards of directors and (...) summarizes the reasons as to why women are needed on company boards. Given that more women on boards are desirable, the article then describes how more women could be appointed to boards, and the actions that organizations and women could take to help increase the representation of women. Finally, the characteristics of those women that have succeeded in becoming members of company boards are described from an international perspective. Unfortunately, answers to the vexing question of whether these women have gained board directorships in their own right as extremely competent managers, or whether they are mere token female appointments in a traditional male dominated culture, remains elusive. (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)
For the sentences of languages that contain operators that express the concepts of definiteness and indefiniteness, there is an unavoidable tension between a truth-theoretic semantics that delivers truth conditions for those sentences that capture their propositional contents and any model-theoretic semantics that has a story to tell about how indetifiniteness in a constituent affects the semantic value of sentences which imbed it. But semantic theories of both kinds play essential roles, so the tension needs to be resolved. I argue that (...) it is the truth theory which correctly characterises the notion of truth, per se. When we take into account the considerations required to bring model theory into harmony with truth theory, those considerations undermine the arguments standardly used to motivate supervaluational model theories designed to validate classical logic. But those considerations also show that celebration would be premature for advocates of the most frequently encountered rival approach – many-valued model theory. (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)
