The approximate number system (ANS) enables organisms to represent the approximate number of items in an observed collection, quickly and independently of natural language. Recently, it has been proposed that the ANS goes beyond representing natural numbers by extracting and representing rational numbers (Clarke & Beck, 2021a). Prior work demonstrates that adults and children discriminate ratios in an approximate and ratio-dependent manner, consistent with the hallmarks of the ANS. Here, we use a well-known “connectedness illusion” to provide (...) evidence that these ratio-dependent ratio discriminations are (a) based on the perceived number of items in seen displays (and not just non-numerical confounds), (b) are not dependent on verbal working memory, or explicit counting routines, and (c) involve representations with a part-whole (or subset-superset) format, like a fraction, rather than a part-part format, like a ratio. These results vindicate key predictions of the hypothesis that the ANS represents rational numbers. (shrink)

On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system, that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique – the arguments from congruency, confounds, and imprecision – and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds (...) or exotic substitutes for number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind of number being represented. In response, we propose that the ANS represents not only natural numbers, but also non-natural rational numbers. It does not represent irrational numbers, however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research. (shrink)

You ought to save a larger group of people rather than a distinct smaller group of people, all else equal. A consequentialist may say that you ought to do so because this produces the most good. If a non-consequentialist rejects this explanation, what alternative can he or she give? This essay defends the following explanation, as a solution to the so-called numbers problem. Its two parts can be roughly summarised as follows. First, you are morally required to want the survival (...) of each stranger for its own sake. Secondly, you are rationally required to achieve as many of these ends as possible, if you have these ends. (shrink)

We prove that each countable rooted K4 -frame is a d-morphic image of a subspace of the space $\mathbb{Q}$ of rational numbers. From this we derive that each modal logic over K4 axiomatizable by variable-free formulas is the d-logic of a subspace of $\mathbb{Q}$. It follows that subspaces of $\mathbb{Q}$ give rise to continuum many d-logics over K4, continuum many of which are neither finitely axiomatizable nor decidable. In addition, we exhibit several families of modal logics finitely axiomatizable by (...) variable-free formulas over K4 that d-define interesting classes of topological spaces. Each of these logics has the finite model property and is decidable. Finally, we introduce quasi-scattered and semi-scattered spaces as generalizations of scattered spaces, develop their basic properties, axiomatize their corresponding modal logics, and show that they also arise as the d-logics of some subspaces of $\mathbb{Q}$. (shrink)

For a ring R, Hilbert’s Tenth Problem $HTP$ is the set of polynomial equations over R, in several variables, with solutions in R. We view $HTP$ as an enumeration operator, mapping each set W of prime numbers to $HTP$, which is naturally viewed as a set of polynomials in $\mathbb {Z}[X_1,X_2,\ldots ]$. It is known that for almost all W, the jump $W'$ does not $1$ -reduce to $HTP$. In contrast, we show that every Turing degree contains a set W (...) for which such a $1$ -reduction does hold: these W are said to be HTP-complete. Continuing, we derive additional results regarding the impossibility that a decision procedure for $W'$ from $HTP$ can succeed uniformly on a set of measure $1$, and regarding the consequences for the boundary sets of the $HTP$ operator in case $\mathbb {Z}$ has an existential definition in $\mathbb {Q}$. (shrink)

We present a geometric construction that yields completeness results for modal logics including K4, KD4, GL and GL n with respect to certain subspaces of the rational numbers. These completeness results are extended to the bimodal case with the universal modality.

The present study examined whether a dissociation among formats for rational numbers can be obtained in tasks that require comparing a number to a non-symbolic quantity. In Experiment 1, college students saw a discrete or else continuous image followed by a rational number, and had to decide which was numerically larger. In Experiment 2, participants saw the same displays but had to make a judgment about the type of ratio represented by the number. The magnitude (...) task was performed more quickly using decimals, whereas the relation task was performed more accurately with fractions. The pattern observed for percentages was very similar to that for decimals. A dissociation between magnitude comparison and relational processing with rational numbers can be obtained when a symbolic number must be compared to a non-symbolic display. (shrink)

Clarke and Beck suggest that the ratio processing system may be a component of the approximate number system, which they suggest represents rational numbers. We argue that available evidence is inconsistent with their account and advocate for a two-systems view. This implies that there may be many access points for numerical cognition – and that privileging the ANS may be a mistake.

Solutions to word problems are moderated by the semantic alignment of real-world relations with mathematical operations. Categorical relations between entities are aligned with addition, whereas certain functional relations between entities are aligned with division. Similarly, discreteness vs. continuity of quantities is aligned with different formats for rational numbers. These alignments have been found both in textbooks and in the performance of college students in the USA and in South Korea. The current study examined evidence for alignments in Russia. Textbook (...) analyses revealed semantic alignments for arithmetic word problems, but not for rational numbers. Nonetheless, Russian college students showed semantic alignments both for arithmetic operations and for rational numbers. Since Russian students exhibit semantic alignments for rational numbers in the absence of exposure to examples in school, such alignments likely reflect intuitive understanding of mathematical representations of real-world situations. (shrink)

We agree with Clarke and Beck that the approximate number system represents rational numbers, and we demonstrate our support by highlighting the case of the empty set – the non-symbolic manifestation of zero. It is particularly interesting because of its perceptual and semantic uniqueness, and its exploration reveals fundamental new insights about how numerical information is represented.

The following result is an approximation to the answer of the question of Kokorin about decidability of a quantifier-free theory of field of rational numbers. Let Q0 be a subset of the set of all rational numbers which contains integers 1 and −1. Let be a set containing Q0 and closed by the functions of addition, subtraction and multiplication. For example coincides with Q0 if Q0 is the set of all binary rational numbers or the set of (...) all decimal rational numbers. It is clear that , where Z is the set of all integers and Q is the set of all rational numbers. A negationless theory uses only conjunction and disjunction as logical connectives. Let T be a quantifier-free negationless elementary theory of signature , where Pol is the list of all polynomials with rational coefficients from Q0 in which exponent of the polynomials and rational coefficients are written in binary number system. Theorem 1. Theory T is NP-hard and if is everywhere dense then the theory T is decidable by an algorithm which belongs to EXPTIME. The set of all rational numbers or the set of all binary rational numbers may be taken instead of . The quantifier-free negationless theory of the field of rational numbers may be regarded as a base of constructive mathematics. (shrink)

Let ${{\mathcal L}^{\square\circ}}$ be a propositional language with standard Boolean connectives plus two modalities: an S4-ish topological modality □ and a temporal modality ◦, understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language ${{\mathcal L}^{\square\circ}}$ by interpreting ${{\mathcal L}^{\square\circ}}$ in dynamic topological systems, i.e., ordered pairs 〈X, f〉, where X is a topological space and f is a continuous function on X. Artemov, Davoren and Nerode have axiomatized a logic S4C, and have shown (...) that S4C is sound and complete for this semantics. S4C is also complete for continuous functions on Cantor space (Mints and Zhang, Kremer), and on the real plane (Fernández Duque); but incomplete for continuous functions on the real line (Kremer and Mints, Slavnov). Here we show that S4C is complete for continuous functions on the rational numbers. (shrink)

The relationship between ideals I of Turing degrees and groups of I-recursive automorphisms of the ordering on rationals is studied. We discuss the differences between such groups and the group of all automorphisms, prove that the isomorphism type of such a group completely defines the ideal I, and outline a general correspondence between principal ideals of Turing degrees and the first-order properties of such groups.

Let ƒ be a real number. It is well known [7] that the set of rational numbers which are less than ƒ is a recursive set if and only if ƒ is representable as the limit of a recursive, recursively convergent sequence of rational numbers. In this paper we replace the condition that the set of rational numbers less than ƒ is recursive by the condition that this set is at various points in the Kleene hierarchy, (...) and we replace the recursive, recursively convergent limit by a variety of other recursive limiting processes. (shrink)

There are nine kinds of number: cardinal (measure of class size), ordinal (corresponds to position), generalized ordinal (position in multidimensional discrete manifold), signed (relation between cardinals), rational (different kind of relation between cardinals), real (limit), complex (pair of reals), transfinite (size of reflexive class), and dimension (measure of complexity.

A number of authors have defended permissivism by appealing to rational supererogation, the thought that some doxastic states might be rationally permissible even though there are other, more rational beliefs available. If this is correct, then there are situations that allow for multiple rational doxastic responses, even if some of those responses are rationally suboptimal. In this paper, I will argue that this is the wrong approach to defending permissivism—there are no doxastic states that are rationally (...) supererogatory. By the lights of contemporary linguistics, ‘rational’ is an absolute gradable adjective, and as such, can only be applied to things that satisfy the top of the scale of rationality. For this reason, it is not possible to believe what is rational while also failing to believe what is rationally optimal. (shrink)

Number concepts must support arithmetic inference. Using this principle, it can be argued that the integer concept of exactly ONE is a necessary part of the psychological foundations of number, as is the notion of the exact equality - that is, perfect substitutability. The inability to support reasoning involving exact equality is a shortcoming in current theories about the development of numerical reasoning. A simple innate basis for the natural number concepts can be proposed that embodies the (...) arithmetic principle, supports exact equality and also enables computational compatibility with real- or rational-valued mental magnitudes. (shrink)

In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system represents numbers or numerosities, and why the ANS represents rational numbers.

Different people reason differently, which means that sometimes they reach different conclusions from the same evidence. We maintain that this is not only natural, but rational. In this essay we explore the epistemology of that state of affairs. First we will canvass arguments for and against the claim that rational methods of reasoning must always reach the same conclusions from the same evidence. Then we will consider whether the acknowledgment that people have divergent rational reasoning methods should (...) undermine one’s confidence in one’s own reasoning. Finally we will explore how agents who employ distinct yet equally rational methods of reasoning should respond to interactions with the products of each others’ reasoning. We find that the epistemology of multiple reasoning methods has been misunderstood by a number of authors writing on epistemic permissiveness and peer disagreement. (shrink)

This article exposes the sophistry inherent in the construction of the "number line," as this continuum is named by mathematicians, and shows how another continuum, one which preserves the properties of the old "number line" but which is based on rational foundations, namely the relations to one another of the ratios that continuous magnitudes have to one another, can be generated to replace it.

The arithmetic R♯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {R}}^{\boldsymbol {\sharp }}$$\end{document} is obtained by postulating the Peano axioms on the basis of the relevant logic R. R♯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {R}}^{\boldsymbol {\sharp }}$$\end{document} is a remarkable arithmetic, not least in that it has finite models. In this paper we examine the options for extending R♯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {R}}^{\boldsymbol {\sharp }}$$\end{document} from natural numbers to (...) rational numbers, as this is the essential next step towards providing a relevant basis for mathematics and for applications. Relevant rational number theory is problematic in that the most obvious approaches lead to non-conservative extensions of R♯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {R}}^{\boldsymbol {\sharp }}$$\end{document}. We consider three ways in which relevant theories of rational arithmetic can be formulated, and note in particular how these fare in the finite models of R♯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {R}}^{\boldsymbol {\sharp }}$$\end{document}. (shrink)

Different people reason differently, which means that sometimes they reach different conclusions from the same evidence. We maintain that this is not only natural, but rational. This chapter explores the epistemology of that state of affairs. First it canvasses arguments for and against the claim that rational methods of reasoning must always reach the same conclusions from the same evidence. Then it considers whether the acknowledgment that people have divergent rational reasoning methods should undermine one’s confidence in (...) one’s own reasoning. Finally it explores how agents who employ distinct yet equally rational methods of reasoning should respond to interactions with the products of each others’ reasoning. The chapter finds that the epistemology of multiple reasoning methods has been misunderstood by a number of authors writing on epistemic permissiveness and peer disagreement. (shrink)

ABSTRACT:Alasdair MacIntyre’sAfter Virtuehas made a significant impact within business ethics. This impact has centered upon applications of the virtues-goods-practices-institutions schema (Moore & Beadle, 2006). In this article, I develop an extension of the practices-institutions schema (Moore, 2017), drawing upon MacIntyre’s later text,Dependent Rational Animals. Two key concepts drawn from this text are “networks of giving and receiving” and “the virtues of acknowledged dependence.” Networks of giving and receiving are non-calculative relationships that enable participants to cope with vulnerability. These relationships (...) are sustained by the virtues of acknowledged dependence, including just generosity,misericordia,and beneficence, virtues that direct participants to treat the needs of others as reasons for action. Drawing upon research in social network theory, I develop an example illustrating the application of these concepts within an organizational and interorganizational context. I then suggest a number of applications and research questions related to this extension of the practices-institutions schema. (shrink)

Rationing in healthcare remains very much a taboo topic. Before COVID-19, it rarely received public attention, even when it occurred in everyday practices, mainly in the form of implicit rationing, as it continues to do today. There are different definitions, types and levels of healthcare rationing, according to different perspectives. With the aim of contributing to a more coherent debate on such a highly emotional healthcare issue as rationing, here are provided a number of reflections from a patient advocate (...) perspective which are specifically focused on bedside rationing, the most troublesome level, both for patients and clinicians, particularly in regard to cancer care. Oncology, with its numerous expensive therapies and increasing number of patients, is undeniably one of the main areas contributing to the increase in healthcare costs. However, the fixed budgets of today's publicly financed health systems cannot allow unlimited access to the potentially beneficial treatments to all patients. Bedside rationing constitutes the last phase of many decision-making processes occurring at different interrelated levels (macro-levels), both inside and outside healthcare systems, which implicitly and inevitably result in a bottleneck determined by the upstream decisions themselves. Shifting from implicit to explicit bedside rationing essentially means moving from a paternalistic to a citizen-before-patient approach; this implies, first of all, a cultural change. Practical bedside rationing is an ethically complex topic, but one that needs to be urgently addressed in a transparent and open debate. In this scenario, the oncological community – patients, patient advocates and clinicians – can and should play an important role. (shrink)

In this paper, I explore the question of whether the expected consequences of holding a belief can affect the rationality of doing so. Special attention is given to various ways in which one might attempt to exert some measure of control over what one believes and the normative status of the beliefs that result from the successful execution of such projects. I argue that the lessons which emerge from thinking about the case ofbelief have important implications for the way we (...) should think about the rationality of a number of other propositional attitudes,such as regret, desire, and fear. Finally,I suggest that a lack of clarity with respect to the relevant issues has given rise to a number of rather serious philosophical mistakes. (shrink)

This paper addresses the so-called Taurek debate on whether or not to save the greater number of victims in rescue conflicts. In this controversy we take a consequentialist and aggregationist pro-numbers position and defend it against various objections. In particular, we consider it to be compatible with the principle of equal concern. Given that the pro-numbers position hinges to a significant extent on the acceptance of the person-neutral value of personal well-being, we offer a coherentist justification for this value (...) claim. Systematically though, we argue that neither pure welfare aggregationism nor its strict rejection are convincing. Rather, rejoining the two ethical dimensions of individual claims and collective well-being requires a hybrid position. Identifying three possible ways of constructing such a position, we briefly defend them against a recent version of the charge that they cannot be coherently expressed in a rational choice framework. ----- Im Rahmen der sogenannten Taurek-Debatte wird die Frage diskutiert, ob in Rettungskonflikten die größere Anzahl an Opfern gerettet werden sollte oder nicht. Wir beziehen in dieser Kontroverse eine konsequentialistische und aggregationistische Pro-Anzahl-Position und verteidigen diese gegen verschiedene Einwände. Insbesondere halten wir die Position für kompatibel mit dem Prinzip gleicher Achtung. Angesichts des Umstandes, dass die Pro-Anzahl-Position in maßgeblicher Weise von der Annahme eines personenneutralen Wertes persönlichen Wohlergehens abhängt, bieten wir eine kohärentistische Rechtfertigung für diese Wertbehauptung. Systematisch argumentieren wir, dass weder ein reiner Wohlergehensaggregationismus noch seine komplette Zurückweisung überzeugend sein können. Die Vereinigung der beiden ethischen Dimensionen individueller Ansprüche und kollektiven Wohlergehens erfordert stattdessen eine hybride Position. Wir identifizieren drei mögliche Wege für die Konstruktion einer solchen Hybridposition und verteidigen diese abschließend kurz gegen einen aktuellen Vorwurf, demzufolge diese nicht auf kohärente Weise im Rahmen der rationalen Entscheidungstheorie abgebildet werden können. (shrink)

There is a tension between normative and descriptive elements in the theory of rational belief. This tension has been reflected in work in psychology and decision theory as well as in philosophy. Canons of rationality should be tailored to what is humanly feasible. But rationality has normative content as well as descriptive content.A number of issues related to both deductive and inductive logic can be raised. Are there full beliefs – statements that are categorically accepted? Should statements be (...) accepted when they become overwhelmingly probable? What is the structure imposed on these beliefs by rationality? Are they consistent? Are they deductively closed? What parameters, if any, does rational acceptance depend on? How can accepted statements come to be rejected on new evidenceShould degrees of belief satisfy the probability calculus? Does conformity to the probability calculus exhaust the rational constraints that can be imposed on partial beliefs? With the acquisition of new evidence, should beliefs change in accord with Bayes' theorem? Are decisions made in accord with the principle of maximizing expected utility? Should they be?A systematic set of answers to these questions is developed on the basis of a probabilistic rule of acceptance and a conception of interval-valued logical probability according to which probabilities are based onknownfrequencies. This leads to limited deductive closure, a demand for only limited consistency, and the rejection of Bayes' theorem as universally applicable to changes of belief. It also becomes possible, given new evidence, to reject previously accepted statements. (shrink)

Additive theories of rationality, as I use the term, are theories that hold that an account of our capacity to reflect on perceptually-given reasons for belief and desire-based reasons for action can begin with an account of what it is to perceive and desire, in terms that do not presuppose any connection to the capacity to reflect on reasons, and then can add an account of the capacity for rational reflection, conceived as an independent capacity to ‘monitor’ and ‘regulate’ (...) our believing-on-the-basis-of-perception and our acting-on-the-basis-of-desire. I show that a number of recent discussions of human rationality are committed to an additive approach, and I raise two difficulties for this approach, each analogous to a classic problem for Cartesian dualism. The interaction problem concerns how capacities conceived as intrinsically independent of the power of reason can interact with this power in what is intuitively the right way. The unity problem concerns how an additive theorist can explain a rational subject's entitlement to conceive of the animal whose perceptual and desiderative life he or she oversees as ‘I’ rather than ‘it’. I argue that these difficulties motivate a general skepticism about the additive approach, and I sketch an alternative, ‘transformative’ framework in which to think about the cognitive and practical capacities of a rational animal. (shrink)

In the United Kingdom, a number of National Health Service Clinical Commissioning Groups have proposed controversial measures to restrict elective surgery for patients who either smoke or are obese. Whilst the nature of these measures varies between NHS authorities, typically, patients above a certain Body Mass Index and smokers are required to lose weight and quit smoking prior to being considered eligible for elective surgery. Patients will be supported and monitored throughout this mandatory period to ensure their clinical needs (...) are appropriately met. Controversy regarding such measures has primarily centred on the perceived unfairness of targeting certain health states and lifestyle choices to save public money. Concerns have also been raised in response to rhetoric from certain NHS authorities, which may be taken to imply that such measures punitively hold people responsible for behaviours affecting their health states, or simply for being in a particular health state. In this paper, we examine the various elective surgery rationing measures presented by NHS authorities. We argue that, where obesity and smoking have significant implications for elective surgical outcomes, bearing on effectiveness, the rationing of this surgery can be justified on prognostic grounds. It is permissible to aim to maximise the benefit provided by limited resources, especially for interventions that are not urgently required. However, we identify gaps in the empirical evidence needed to conclusively demonstrate these prognostic grounds, particularly for obese patients. Furthermore, we argue that appeals to personal responsibility, both in the prospective and retrospective sense, are insufficient in justifying this particular policy. Given the strength of an alternative justification grounded in clinical effectiveness, rhetoric from NHS authorities should avoid explicit statements, which suggest that personal responsibility is the key justificatory basis of proposed rationing measures. (shrink)

This article argues humanities scholarship is often dismissive of the quantitative, and that there is scope for worthwhile interdisciplinary research into the way everyday life is given tone and texture by experiences and cultures of number. Following the work of Mary Poovey (2008) and Steven Connor (2016), it challenges the view, particularly influential in the humanities, that number and associated ideas to do with data, objectivity, mathematics, and the rational, are parasitic upon life. In contrast to this (...) view, this article suggests that even if the idea of ‘the human’ is defined in opposition to number, the relation between the two is more usefully understood as an interweaving of differential tensions, rather than two poles separated by an uncrossable distance. Examples from literary fiction and two smartphone apps are analysed with the intent of initiating a dialogue between different cultural objects that share a concern with number and human experience. (shrink)

Georg Cantor argued that pure mathematics would be better-designated “free mathematics” since mathematical inquiry need not justify its discoveries through some extra-mental standard. Even so, he spent much of his later life addressing ancient and scholastic objections to his own transfinite number theory. Some philosophers have argued that Cantor need not have bothered. Thomas Aquinas at least, and perhaps Aristotle, would have consistently embraced developments in number theory, including the transfinite numbers. The author of this paper asks whether (...) the restriction of arithmetic to the natural numbers that is apparently assumed by Aristotle and Aquinas is necessary in the light of their stated principles. The author concludes that, while some texts from Aristotle and Thomas suggest that such discoveries as zero, rational, and real numbers, and even Cantor’s own transfinite numbers, are legitimate objects of scientific knowledge, a careful analysis shows that they are incompatible with the ultimate arithmetical principle, the unit. (shrink)

Critical Rationalism has become an influential philosophy in many areas including a great number of scientific disciplines. Yet only few studies have been devoted to the role of the philosophy of Sir Karl Popper in the vast field of education. This volume undertakes to fill this gap. Leading scholars in the educational science and in the philosophy of education have critically written for this volume in an attempt to elaborate Popper's methodological and socio-political views and confront them with a (...) globally relevant spectrum of scientific objectives and cultural values. Among the topics discussed are moral values, education for freedom and its consequences for the student, and the critical attitude in political education. Attention is also paid to the historiography of this significant philosophical movement. Regarding pedagogical research, the empirical paradigm, the falsificatory approach to educational research, the complex relationship between educational theory and practice as well as the problem of value-neutrality in educational science are objects of critical analysis. (shrink)

In _Realistic Rationalism_, Jerrold J. Katz develops a new philosophical position integrating realism and rationalism. Realism here means that the objects of study in mathematics and other formal sciences are abstract; rationalism means that our knowledge of them is not empirical. Katz uses this position to meet the principal challenges to realism. In exposing the flaws in criticisms of the antirealists, he shows that realists can explain knowledge of abstract objects without supposing we have causal contact with them, that numbers (...) are determinate objects, and that the standard counterexamples to the abstract/concrete distinction have no force. Generalizing the account of knowledge used to meet the challenges to realism, he develops a rationalist and non-naturalist account of philosophical knowledge and argues that it is preferable to contemporary naturalist and empiricist accounts. The book illuminates a wide range of philosophical issues, including the nature of necessity, the distinction between the formal and natural sciences, empiricist holism, the structure of ontology, and philosophical skepticism. Philosophers will use this fresh treatment of realism and rationalism as a starting point for new directions in their own research. (shrink)

There are decision problems where the preferences that seem rational to many people cannot be accommodated within orthodox decision theory in the natural way. In response, a number of alternatives to the orthodoxy have been proposed. In this paper, I offer an argument against those alternatives and in favour of the orthodoxy. I focus on preferences that seem to encode sensitivity to risk. And I focus on the alternative to the orthodoxy proposed by Lara Buchak’s risk-weighted expected utility (...) theory. I will show that the orthodoxy can be made to accommodate all of the preferences that Buchak’s theory can accommodate. (shrink)

This is a major contribution to the theory of rational choice which will be of particular interest to philosophers and economists. The author sets out the foundations of rational choice, and then sketches a dynamic choice framework in which principles of ordering and independence follow from a number of apparently plausible conditions. However, there is potential conflict among these conditions, and when they are weakened to avoid it the usual foundations of rational choice no longer prevail. (...) The thrust of the argument is to suggest that the theory of rational choice is less deterministic than many suppose. (shrink)

A number of philosophers argue that law enforcement officers may have good reasons to racially profile suspects under certain conditions. Their conclusions rest on a claim of epistemic rationality: if members of some races are at an increased risk of criminality, then it may be rational for law enforcement officers to subject them to increased scrutiny. In this paper I contest the epistemic rationality of racial profiling by appealing to recent work in criminology and the sociology of race (...) and crime. I argue that recent studies indicate that race is a comparatively poor baseline for judging criminality. Law enforcement officers are therefore not making a cognitive error by ignoring race to focus on other correlates of crime but are keeping up with our best social science. (shrink)

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than the (...) ‘size’ ofB(part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher). (shrink)

A number of authors, Including Thomas Schelling and David Lewis, have envisaged a model of the generation of action in coordination problems in which salience plays a crucial role. Empirical studies suggest that human subjects are likely to try for the salient combination of actions, a tendency leading to fortunate results. Does rationality dictate that one aim at the salient combination? Some have thought so, Thus proclaiming that salience is all that is needed to resolve coordination problems for agents (...) who are rational in the sense of game theory. I argue against this position; rational agents will not necessarily aim for the salient. It remains to explain how the salient comes to be chosen by human beings. Various possibilities are noted. One involves a mechanism invoked by Hume and Wittgenstein in other contexts: we may project an unreasoned compulsion onto reason, falsely believing that rationality dictates our choice of the salient. (shrink)

A number of formal epistemologists have argued that perfect rationality requires probabilistic coherence, a requirement that they often claim applies only to ideal agents. However, in “Rationality as an Absolute Concept,” Roy Sorensen contends that ‘rational’ is an absolute term. Just as Peter Unger argued that being flat requires that a surface be completely free of bumps and blemishes, Sorensen claims that being rational requires being perfectly rational. When we combine these two views, though, they lead (...) to counterintuitive results. If being rational requires being perfectly rational, and only the probabilistically coherent are perfectly rational, then this indicts all ordinary agents as irrational. In this paper, I will attempt to resolve this conflict by arguing that Sorensen is only partly correct. One important sense of ‘rational’, the sanctioning sense of ‘rational’, is an absolute term, but another important sense of ‘rational’, the sense in which someone can have rational capacities, is not. I will, then, show that this distinction has important consequences for theorizing about ideal rationality, developing an account of the relationship between ordinary and ideal rationality. Because the sanctioning sense of ‘rational’ is absolute, it is rationally required to adopt the most rational attitude available, but which attitude is most rational can change depending on whether we are dealing with ideal agents or people more like ourselves. (shrink)

There are not enough solid organs available to meet the needs of patients with organ failure. Thousands of patients every year die on the waiting lists for transplantation. Yet there is one currently available, underutilized, potential source of organs. Many patients die in intensive care following withdrawal of life-sustaining treatment whose organs could be used to save the lives of others. At present the majority of these organs go to waste.In this paper we consider and evaluate a range of ways (...) to improve the number and quality of organs available from this group of patients. Changes to consent arrangements (for example conscription of organs after death) or changes to organ donation practice could dramatically increase the numbers of organs available, though they would conflict with currently accepted norms governing transplantation.We argue that one alternative, Organ Donation Euthanasia, would be a rational improvement over current practice regarding withdrawal of life support. It would give individuals the greatest chance of being able to help others with their organs after death. It would increase patient autonomy. It would reduce the chance of suffering during the dying process. We argue that patients should be given the choice of whether and how they would like to donate their organs in the event of withdrawal of life support in intensive care.Continuing current transplantation practice comes at the cost of death and prolonged organ failure. We should seriously consider all of the alternatives. (shrink)

Surprise—operationalized as looking time—has a long history in developmental research, providing a window into the perception and cognition of infants. Recently, however, a number of developmental researchers have considered infants’ and children's surprise in its own right. This article reviews empirical evidence and computational models of complex statistical inferences underlying surprise, and discusses how these findings relate to the role that surprise appears to play as a catalyst for learning.

Building on the correspondence between finitely axiomatised theories in Łukasiewicz logic and rational polyhedra, we prove that the unification type of the fragment of Łukasiewicz logic with $n\geqslant 2$ variables is nullary. This solves a problem left open by V. Marra and L. Spada [Ann. Pure Appl. Logic 164 (2013), pp. 192–210]. Furthermore, we refine the study of unification with bounds on the number of variables. Our proposal distinguishes the number m of variables allowed in the problem (...) and the number n in the solution. We prove that the unification type of Łukasiewicz logic for all $m,n \geqslant 2$ is nullary. (shrink)

Background Rationing of care in nursing is nurses' inability to complete all care activities for patients because of scarcity in time and resource. Literature suggests that rationing of care is closely related to patient safety and quality of care. The phenomena have been defined and studied from varied perspectives and contexts. A systematic review of studies related to the concept was aimed at identifying and synthesizing the finding. Methods The review followed Preferred Reporting Items for Systematic Reviews and Meta‐analysis guidelines, (...) and literature searches were conducted in MEDLINE, CINAHL, PsycInfo, Web of Science and EMBASE databases. Fifty‐seven quantitative studies were included in the review. Findings The review observed that nursing activities addressing the emotional, educational, mobility and hygiene needs of the patients were commonly rationed. Antecedents of rationing included resource inadequacy and organizational work environment. Rationing influenced patient satisfaction, mortality and a number of adverse events and was associated with decreased job satisfaction, increased intention to leave and high turnover among nurses. Discussions This review concludes that rationing in nursing is ubiquitous, embedded in the work environment and poses a threat to the professional health and philosophical base of nursing in addition to having serious implications on patients' safety. Strategies to reframe and reconsider organizational traits, and open discussion with other healthcare stakeholders can reduce rationing of nursing care. The review suggests future researchers adopt different methodological layout to study rationing. (shrink)

In _Realistic Rationalism_, Jerrold J. Katz develops a new philosophical position integrating realism and rationalism. Realism here means that the objects of study in mathematics and other formal sciences are abstract; rationalism means that our knowledge of them is not empirical. Katz uses this position to meet the principal challenges to realism. In exposing the flaws in criticisms of the antirealists, he shows that realists can explain knowledge of abstract objects without supposing we have causal contact with them, that numbers (...) are determinate objects, and that the standard counterexamples to the abstract/concrete distinction have no force. Generalizing the account of knowledge used to meet the challenges to realism, he develops a rationalist and non-naturalist account of philosophical knowledge and argues that it is preferable to contemporary naturalist and empiricist accounts. The book illuminates a wide range of philosophical issues, including the nature of necessity, the distinction between the formal and natural sciences, empiricist holism, the structure of ontology, and philosophical skepticism. Philosophers will use this fresh treatment of realism and rationalism as a starting point for new directions in their own research. (shrink)

The independence of irrelevant alternatives is a popular and important axiom of decision theory. It states, roughly, that one’s choice from a set of options should not be influenced by the addition or removal of further, unchosen options. In recent debates, a number of authors have given putative counterexamples to it, involving intuitively rational agents who violate IIA. Generally speaking, however, these counterexamples do not tend to move IIA’s proponents. Their strategy tends to be to individuate the options (...) that the agent faces differently, so that the case no longer counts as a violation of IIA. In this paper, we examine whether this strategy succeeds. We argue that the ways of individuating options required to save IIA from the most problematic counterexamples—in particular, cases where agents violate IIA due to nonconsequentialist moral beliefs—do so only at the expense of severely compromising its central function within decision theory. (shrink)