This paper contains a proposal for a free, nonzero-rest-mass particle’s proper spacetime, determined exclusively by the particle’s rest mass \ and numbers. The approach defines proper time as de Broglie time, which is isomorphic to a sequence of natural numbers \ that count de Broglie time units \\). The approach is based on defining the spatial coordinate as proper following the constructive definition of positive and negative integers as all possible differences of ordered pairs of natural numbers multiplied by the (...) Compton unit \\). The spatial and temporal coordinates that form the particle’s proper spacetime are constructed as Euclidean projections of the de Broglie time. The corresponding expression in the form of an energy-momentum relation reveals the existence, aside from the rest energy term \, of an additional energy term of the same order of magnitude, which is related to large intervals of the \-particle’s proper space. The relation of the numbers-based approach to the foundations of the special theory of relativity and of quantum mechanics is discussed. (shrink)

Recently, in his Rolf Schock Prize Lecture, Derek Parfit has suggested a novel way of avoiding the Repugnant Conclusion by introducing what he calls “imprecision” in value comparisons. He suggests that in a range of important cases, populations of different sizes are only imprecisely comparable. Parfit suggests that this feature of value comparisons opens up a way of avoiding the Repugnant Conclusion without implying other counterintuitive conclusions, and thus solves one of the major challenges in ethics. In this article, I (...) try to clarify Parfit’s proposal and evaluate whether it will help us with the paradoxes in population ethics. (shrink)

Recently, in his Rolf Schock Prize Lecture, Derek Parfit has suggested a novel way of avoiding the Repugnant Conclusion by introducing what he calls “imprecision” in value comparisons. He suggests that in a range of important cases, populations of different sizes are only imprecisely comparable. Parfit suggests that this feature of value comparisons opens up a way of avoiding the Repugnant Conclusion without implying other counterintuitive conclusions, and thus solves one of the major challenges in ethics. In this article, I (...) try to clarify Parfit's proposal and evaluate whether it will help us with the paradoxes in population ethics. (shrink)

This study focuses on implementing electric vehicles (EVs) in developing countries where energy production is mainly based on fossil fuels. Although for these countries the environmental short-run benefits of the EVs cannot offset the short-run costs, it may still be the best option to implement the EVs as soon as possible. Hence, it is necessary to evaluate the alternatives to introducing EVs to the market due to the environmental concerns that created an opportunity for some developing countries to catch up (...) with the international competition. Therefore, we develop a case scenario to explore the decision-making process in implementing the EVs with three alter natives and twelve criteria. We solve the decision-making problem by using Type-2 neutrosophic numbers (T2NNs) based on the RAFSI (Ranking of Alternatives through Functional mapping of criterion sub-intervals into a Single Interval) method. The proposed model combines the advantages of the RAFSI technique, and it applies T2NNs to address the uncertainties. The results show that the alternatives that may suspend the implementation of the EVs are inferior. Direct implementation of EVs is prioritized. The policy implications of the results are discussed in the study. (shrink)

In this paper, we propose a method to design the pseudorandom number generator using three kinds of four-wing memristive hyperchaotic systems with different dimensions as multientropy sources. The principle of this method is to obtain pseudorandom numbers with good randomness by coupling XOR operation on the three kinds of FWMHSs with different dimensions. In order to prove its potential application in secure communication, the security of PRNG based on this scheme is analyzed from the perspective of cryptography. In addition, (...) PRNG has passed the NIST 800.22 and ENT test, which shows that PRNG has good statistical characteristics. Finally, an image encryption algorithm based on PRNG is adopted. In the encryption algorithm, the optimized Arnold matrix scrambling method and the diffusion processing based on XOR are used to obtain the final encrypted image. Through the evaluation of encryption performance, it is concluded that there is no direct relationship between the pristine image and encrypted image. The results show that the proposed image encryption scheme has good statistical output characteristics and security performance in line with cryptography. (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.

This paper reassesses the criticism of the Lucas-Penrose anti-mechanist argument, based on Gödel’s incompleteness theorems, as formulated by Krajewski : this argument only works with the additional extra-formal assumption that “the human mind is consistent”. Krajewski argues that this assumption cannot be formalized, and therefore that the anti-mechanist argument – which requires the formalization of the whole reasoning process – fails to establish that the human mind is not mechanistic. A similar situation occurs with a corollary to the argument, that (...) the human mind allegedly outperforms machines, because although there is no exhaustive formal definition of natural numbers, mathematicians can successfully work with natural numbers. Again, the corollary requires an extra-formal assumption: “PA is complete” or “the set of all natural numbers exists”. I agree that extra-formal assumptions are necessary in order to validate the anti-mechanist argument and its corollary, and that those assumptions are problematic. However, I argue that formalization is possible and the problem is instead the circularity of reasoning that they cause. The human mind does not prove its own consistency, and outperforms the machine, simply by making the assumption “I am consistent”. Starting from the analysis of circularity, I propose a way of thinking about the interplay between informal and formal in mathematics. (shrink)

Journal Name: Cognitive Linguistics Issue: Ahead of print.

Many applications of intuitionistic fuzzy sets depend on ranking or comparing intuitionistic fuzzy numbers. This paper presents a novel ranking method for intuitionistic fuzzy numbers based on the preference attitudinal accuracy and score functions. The proposed ranking method considers not only the preference attitude of decision maker, but also all the possible values in feasible domain. Some desirable properties of preference attitudinal accuracy and score functions are verified in detail. A total order on the set of intuitionistic fuzzy numbers is (...) established by using the proposed two functions. The proposed ranking method is also applied to select renewable energy. The advantage and validity of the proposed method are shown by comparing with some representative ranking methods. (shrink)

The risk priority number calculation method is one of the critical subjects of failure mode and effects analysis research. Recently, RPN research under a fuzzy uncertainty environment has become a hot topic. Accordingly, increasing studies have ignored the important impact of the random sampling uncertainty in the FMEA assessment. In this study, a fuzzy beta-binomial RPN evaluation method is proposed by integrating fuzzy theory, Bayesian statistical inference, and the beta-binomial distribution. This model can effectively realize real-time, dynamic, and long-term (...) evaluation of RPN under the condition of continuous knowledge accumulation. The major contribution of the proposed model is to use the random uncertainty and fuzzy uncertainty in an integrated model and provide a Markov Chain Monte Carlo method to solve the complex integrated model. The study presented a case study, which presented how to apply this model in practice and indicated the significant influence on the measurement error caused by ignoring the random uncertainty caused by expert evaluation in RPN calculations. (shrink)

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)

The problem of generalizing the power option-pricing model to incorporate more empirical features becomes an urgent and necessary event. A new power option pricing method is designed for the financial market uncertainty that simultaneously involves randomness and fuzziness. The randomness in market uncertainty is modeled by a time-fractional diffusion model, which describes trend memory in underlying asset prices. The fuzziness in market uncertainty is characterized by a triangular interval type-2 fuzzy numbers, which better captures the fuzziness of underlying asset prices. (...) Considering the decision-maker’s subjective judgment, we show the price mean value with the possibility-necessity weight and pessimistic-optimistic index under the type-2 fuzzy environment. We develop the power option pricing model with the time-fractional diffusion model and the triangular interval type-2 fuzzy numbers. Furthermore, the analytic solutions of pricing call power option and put power option are obtained and verified by the variational iterative reconstruction method. Our study shows that power option pricing, which adopts the time-fractional model and the triangular interval type-2 fuzzy numbers, can better capture the trend memory and double fuzziness of the real market. In addition, a numerical example is provided to illustrate that the power option means the value is decreasing with respect to the pessimistic-optimistic index and is fluctuating with respect to the possibility-necessity weight index. (shrink)

Why might it be beneficial for adults to process fractions componentially? Recent research has shown that college-educated adults can capitalize on the bipartite structure of the fraction notation, performing more successfully with fractions than with decimals in relational tasks, notably analogical reasoning. This study examined patterns of relational priming for problems with fractions in a task that required arithmetic computations. College students were asked to judge whether or not multiplication equations involving fractions were correct. Some equations served as structurally inverse (...) primes for the equation that immediately followed it. Students with relatively high math ability showed relational priming both with and without high perceptual similarity. Students with relatively low math ability also showed priming, but only when the structurally inverse equation pairs were supported by high perceptual similarity between numbers. Several additional experiments established boundary conditions on relational priming with fractions. These findings are interpreted in terms of componential processing of fractions in a relational multiplication context that takes advantage of their inherent connections to a multiplicative schema for whole numbers. (shrink)

Adults increase the certainty of their inductive inferences by observing more diverse instances. However, most young children fail to do so. The present study tested the hypothesis that children's sensitivity to instance diversity is determined by three variables: ability to discriminate among instances ( Discrimination ); an intuition that large numbers of instances increase the strength of conclusion ( Monotonicity ); ability to detect subcategories and evaluate numerical differences between the subcategories, or Extraction . A total of 219 Chinese children (...) aged 6 to 11 were tested for sensitivity to diversity by means of Discrimination, Monotonicity, and Extraction. The results indicated that children at all ages were able to discriminate instances and attend to set size. However, only 9- and 11-year-olds demonstrated Extraction and sensitivity to diversity. Furthermore, among all children diversity scores increased linearly with the level of Extraction. These results suggest that the law of large numbers plays a role in children's diversity-based reasoning. (shrink)

This chapter examines the pioneering work of Gottfried Wilhelm Leibniz (1646-1716) on various number systems, in particular binary, which he independently invented in the mid-to-late 1670s, and hexadecimal, which he invented in 1679. The chapter begins with the oft-debated question of who may have influenced Leibniz’s invention of binary, though as none of the proposed candidates is plausible I suggest a different hypothesis, that Leibniz initially developed binary notation as a tool to assist his investigations in mathematical problems that (...) were exercising him at the time, namely those concerning the divisibility of composite numbers, primality, and perfect numbers. The chapter then explores Leibniz’s development of binary, his little-known work on binary fractions and expansions, his use of binary as a symbol for creation in his philosophical-theology, and his response to the suggestion that there was a correlation between binary numeration and the hexagrams of the ancient Chinese divinatory text, the Yijing. The chapter then focuses on Leibniz’s work on other number systems, in particular his invention and exploration of hexadecimal as well as his work on duodecimal. The chapter concludes by revealing a hitherto unknown practical application of binary that Leibniz devised in the last year of his life. (shrink)

Synaesthesia is a heterogeneous phenomenon, even when considering one particular sub-type. The purpose of this study was to design a reliable and valid questionnaire for grapheme-colour synaesthesia that captures this heterogeneity. By the means of a large sample of 628 synaesthetes and a factor analysis, we created the Coloured Letters and Numbers questionnaire with 16 items loading on 4 different factors . These factors were externally validated with tests which are widely used in the field of synaesthesia research. The questionnaire (...) showed good test–retest reliability and construct validity . Our findings are discussed in the light of current theories and new ideas in synaesthesia research. More generally, the questionnaire is a useful tool which can be widely used in synaesthesia research to reveal the influence of individual differences on various performance measures and will be useful in generating new hypotheses. (shrink)

Real-world information is imperfect and is usually described in natural language (NL). Moreover, this information is often partially reliable and a degree of reliability is also expressed in NL. In view of this, the concept of a Z-number is a more adequate concept for the description of real-world information. The main critical problem that naturally arises in processing Z-numbers-based information is the computation with Z-numbers. Nowadays, there is no arithmetic of Z-numbers suggested in existing literature. This book is the (...) first to present a comprehensive and self-contained theory of Z-arithmetic and its applications. Many of the concepts and techniques described in the book, with carefully worked-out examples, are original and appear in the literature for the first time. The book will be helpful for professionals, academics, managers and graduate students in fuzzy logic, decision sciences, artificial intelligence, mathematical economics, and computational economics. (shrink)

The productivity of researchers and the impact of the work they do are a preoccupation of universities, research funding agencies, and sometimes even researchers themselves. The h-index is the most popular of different metrics to measure these activities. This research deals with presenting a practical approach to model the h-index based on the total number of citations and the duration from the publishing of the first article. To determine the effect of every factor on h, we applied a set (...) of simple nonlinear regression. The results indicated that both NC and D1 had a significant effect on h. The determination of coefficient for these equations to estimate the h-index was 93.4% and 39.8%, respectively, which verified that the model based on NC had a better fit. Then, to record the simultaneous effects of NC and D1 on h, multiple nonlinear regression was applied. The results indicated that NC and D1 had a significant effect on h. Also, the determination of coefficient for this equation to estimate h was 93.6%. Finally, to model and estimate the h-index, as a function of NC and D1, multiple nonlinear quartile regression was used. The goodness of the fitted model was also assessed. (shrink)

This article discusses agent-based simulation (ABS) as a tool of sociological understanding. I argue that agent-based simulations can play an important role in the expansion of explanatory understanding in the social sciences. The argument is based on an inferential account of understanding (Ylikoski 2009, Ylikoski & Kuorikoski 2010), according to which computer simulations increase our explanatory understanding by expanding our ability to make what-if inferences about social processes and by making these inferences more reliable. The inferential account also suggests a (...) number of ways in which the use of simulation methodology might give rise to illusory understanding. The structure of the article is as .. (shrink)

Agent-based modeling has become a common and well-established tool in the social sciences and certain of the humanities. Here, we aim to provide an overview of the different modeling approaches in current use. Our discussion unfolds in two parts: we first classify different aspects of the model-building process and identify a number of characteristics shared by most agent-based models in the humanities and social sciences; then we map relevant differences between the various modeling approaches. We classify these into different (...) dimensions including the type of target systems addressed, the intended modeling goals, and the models’ degree of abstraction. Along the way, we provide reference to related debates in contemporary philosophy of science. (shrink)

In the last decade, the notion of triad was reintroduced by Eric Scerri, who suggested it as a possible categorical criterion to represent chemical periodicity. In particular, he reformulated the notion of triad in terms of atomic number instead of atomic weights; in this way, the value of the intermediate term of the triad became the exact average of the values of the two extremes. Following the inspiration of Scerri’s work, the main purpose of this article is to obtain (...) a representation where all the relations among the chemical elements that form the groups of the periodic system can be reconstructed on the basis of triads, without using electronic configurations. (shrink)

In his [1937, 1938], Paul Dirac proposed his “Large Number Hypothesis” (LNH), as a speculative law, based upon what we will call the “Large Number Coincidences” (LNC’s), which are essentially “coincidences” in the ratios of about six large dimensionless numbers in physics. Dirac’s LNH postulates that these numerical coincidences reflect a deeper set of law-like relations, pointing to a revolutionary theory of cosmology. This led to substantial work, including the development of Dirac’s later [1969/74] cosmology, and other alternative (...) cosmologies, such as the Brans-Dicke modification of GTR, and to extensive empirical tests. We may refer to the generic hypothesis of “Large Number Relations” (LNR’s), as the proposal that there are lawlike relations of some kind between the dimensionless numbers, not necessarily those proposed in Dirac’s early LNH. Such relations would have a profound effect on our concepts of physics, but they remain shrouded in mystery. Although Dirac’s specific proposals for LNR theories have been largely rejected, the subject retains interest, especially among cosmologists seeking to test possible variations in fundamental constants, and to explain dark energy or the cosmological constant. In the first two sections here we briefly summarize the basic concepts of LNR’s. We then introduce an alternative LNR theory, using a systematic formalism to express variable transformations between conventional measurement variables and the true variables of the theory. We demonstrate some consistency results and review the evidence for changes in the gravitational constant G. The theory adopted in the strongest tests of Ġ/G, by the Lunar Laser Ranging (LLR) experiments, assumes: Ġ/G = 3(dr/dt)/r – 2(dP/dt)/P – (dm/dt)/m, as a fundamental relationship. Experimental measurements show the RHS to be close to zero, so it is inferred that significant changes in G are ruled out. However when the relation is derived in our alternative theory it gives: Ġ/G = 3(dr/dt)/r – 2(dP/dt)/P – (dm/dt)/m – (dR/dt)/R. The extra final term (which is the Hubble constant) is not taken into account in conventional derivations. This means the LLR experiments are consistent with our LNR theory (and others), and they do not really test for a changing value of G at all. This failure to transform predictions of LNR theories correctly is a serious conceptual flaw in current experiment and theory. (shrink)

Math anxiety results in a drop in performance on various math-related tasks, including the symbolic number ordering task in which participants decide whether a triplet of digits is presented in order (e.g. 3-5-7) or not (e.g. 3-7-5). We investigated whether the strategy repertoire and reaction times during a symbolic ordering task were affected by math anxiety. In study 1, participants performed an untimed symbolic number ordering task and indicated the strategy they used on a trial-by-trial basis. The use (...) of the memory retrieval strategy, based on the immediate recognition of the triplet, decreased with high math anxiety, but disappeared when controlling for general anxiety. In the study 2, participants completed a timed version of the number order task. High math-anxious participants used the decomposition strategy (e.g. 5 is larger than 3 and 7 is larger than 5 to decide whether 3-5-7 is in the correct order) more often, and were slower in responding when both memory- and other decomposition strategies were used. Altogether, both studies demonstrate that high-math anxious participants are not only slower to decide whether a number triplet is in the correct order, but also rely more on procedural strategies. (shrink)

The patient preference predictor is a proposed computer-based algorithm that would predict the treatment preferences of decisionally incapacitated patients. Incorporation of a PPP into the decision-making process has the potential to improve implementation of the substituted judgement standard by providing more accurate predictions of patients’ treatment preferences than reliance on surrogates alone. Yet, critics argue that methods for making treatment decisions for incapacitated patients should be judged on a number of factors beyond simply providing them with the treatments they (...) would have chosen for themselves. These factors include the extent to which the decision-making process recognises patients’ freedom to choose and relies on evidence the patient themselves would take into account when making treatment decisions. These critics conclude that use of a PPP should be rejected on the grounds that it is inconsistent with these factors, especially as they relate to proper respect for patient autonomy. In this paper, we review and evaluate these criticisms. We argue that they do not provide reason to reject use of a PPP, thus supporting efforts to develop a full-scale PPP and to evaluate it in practice. There are no data in this work. (shrink)

We show how trajectories can be reintroduced in quantum mechanics provided that its spatial continuum is modelled by a variable real number (qr-number) continuum. Such a continuum can be constructed using only standard Hilbert space entities. In this approach, the geometry of atoms and subatomic objects differs from that of classical objects. The systems that are non-local when measured in the classical space-time continuum may be localized in the quantum continuum. We compare trajectories in this new description of (...) space-time with the corresponding Bohmian picture. (shrink)

Clarke and Beck's view that numbers are both second-order and sensible is based on an empirically dubious claim, which is required to show that what they call the “weak sensitivity principle” is satisfied. The explanatory benefits that they say are gained by positing a sensory relation to numbers are also gained by positing such a relation to multitudes of objects.

This article considers what the implications of decertification would be for single-sex services such as domestic and sexual violence support. Some reform options attached to decertification could (re)allocate authority away from the state to organisations or individuals to determine gender criteria. What would the consequences of such re-allocation be in determining eligibility to receive or access services or excluding people on the basis of a characteristic protected under equality law? Engaging with this in the context of domestic and sexual violence (...) support service provision raises a number of questions. Firstly, does the existence of gender-based violence and/or of the effects it produces require a stable category in order to address them? What benefits may emerge from providing single-sex spaces that could not be replicated in other settings? And finally, what criteria of exclusion and inclusion are currently used to determine access to spaces beyond legal gender status? (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public (...) domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

This paper argues that neurolinguistics has the potential to yield insights that can feed back into corpus-based Cognitive Linguistics. It starts by discussing how far the cognitive realism of probabilistic statements derived from corpus data currently goes. Against this background, it argues that the cognitive realism of usage-based models could be further enhanced through deeper engagement with neurolinguistics, but also highlights a number of common misconceptions about what neurolinguistics can and cannot do for linguistic theorizing.

Failed expectations about “bringing in the Kingdom” may lie at the root of the decline in the missionary force of mainline denominations. Is a missiological motivation based on “evangelising the world to bring back the King” giving rise to similar unrealistic expectations? Optimistic statistics on aspects of church growth are vulnerable to question. Reports of the numbers of missionaries forget the aspect of proportionality to size of population, reports of church growth include the effect of babies born to Christians of (...) all denominational groups, and increases in the numbers of missionaries include short-termers. The growth of the new category of Great Commission Christians and the reduction in number of unreached people groups are based on faulty descriptive bases. If the world remains unevangelised by 2001, these attempts to motivate Christians to evangelism on the basis of current optimistic statistical improvements will undermine a proper basis for motivating people for evangelism which is based on faith and not on sight. (shrink)

Research on the cognitive abilities involved in decision making has shown that, under objective risk conditions (i.e., when explicit information about possible outcomes and risks is available), superior decisions are especially predicted by executive functions and exact number processing skills, also referred to as objective numeracy. So far, decision-making research has mainly focused on exact number processing skills, such as performing calculations or transformations of symbolic numbers. There is evidence that such exact numeric skills are based on approximate (...) number processing (ANP) skills, which enable quick and accurate processing of non-symbolic numbers (e.g. Chen and Li, 2014). Very few studies, however, have investigated ANP skills in the context of risky decision making and have analyzed direct associations among the aforementioned sub functions. Possible interactions between the closely related skills have not been considered. The current study (N = 128) examines interactions of ANP skills with executive functions and objective numeracy, in predicting risky choice behavior. ANP skills are represented by the accuracy in a dot-comparison task. Decision making is measured by two versions of the Game of Dice Task (GDT), which place different emphases on the reflection of potential risks. The results show two-way as well as three-way interactions between the measures of ANP skills, executive functions, and objective numeracy in predicting risky decisions in both GDT versions. The riskiest decisions were most frequently made in case of low scores in all of the three competencies, while good performance in any one of them resulted in significant reductions of disadvantageous decisions. The findings indicate that high ANP skills can positively affect choice behavior in individuals who have weaknesses in reflectively attributed skills, namely executive functions and objective numeracy. Potential compensatory effects and mechanisms of ANP in decision making are discussed. (shrink)

Begins with a review of Benacerraf's metaphysical challenge to mathematical realism based on sets: how, for example, can number theory be the study of particular sets when other sets with the same structural relations would seem to do just as well? The set theoretic realist gives the straightforward response that numbers are not particular sets, but properties of sets. I close with a digression on the prospects for ‘Frege numbers’—i.e. numbers construed as proper classes.

Conscience-based refusals by health care professionals to provide care to eligible patients are problematic, given the monopoly such professionals hold on the provision of such services. This article reviews standard ethical arguments in support of conscientious refuser accommodation and finds them wanting. It discusses proposed compromise solutions involving efforts aimed at testing the genuineness and reasonability of refusals and rejects those solutions too. A number of jurisdictions have introduced policies requiring conscientious refusers to provide effective referrals. These policies have (...) turned out to be unworkable. They subject patients to a health care delivery lottery, which is incompatible with the fundamental values of medical professionalism. This paper sheds light on transnational efforts aimed at undermining progress made in reproductive health by means of conscientious refusal accommodation claims. The view that the accommodation of conscientious refusers is indefensible on consequentialist ethical grounds, as well as on grounds related to medical professionalism itself, is defended. (shrink)

Von Mises defined a "physical" probability as a strict limit of the relative frequency of occurrence of an event in repeated trials. As a result of a number of criticisms of von Mises's approach, the more favored approach became the "propensity" interpretation. It is argued here that this interpretation is not compelling and that the only problem in von Mises's approach is the assumption that the relative frequency converges in a strict sense. This problem is then remedied by deducing (...) the axioms of probability theory from the assumption that the relative frequency converges only in the sense of the weak law of large numbers. (shrink)

This paper has two aims. The first is to present and defend a new argument for rights contributionism – the view that the notion of a moral claim-right is a contributory (or pro tanto) rather than overall normative notion. The argument is an inference to the best explanation: it is argued that (i) there are contributory moral factors that contrast with standard moral reasons by way of having a number of formal properties that are characteristic of rights, even though (...) they can be overridden, and (ii) that this is best explained by a view that takes these factors to correlate with rights. The second aim is to show that the truth of rights contributionism matters for normative ethics. More specifically, it is argued that rights contributionism clears the way for deontologists to justify the pre-theoretically plausible verdict that we have a duty to save the greater number in so-called Taurek scenarios – scenarios in which we have to choose between saving either a greater or a smaller number of different people. The chapter offers a novel and distinctively deontological explanation of this verdict that is based on the assumption that everyone has a pro tanto right to be saved in a Taurek scenario. (shrink)

The acceptability of _any_-DPs in existential modal sentences presents a challenge for theories of NPI licensing: existential modal sentences appear to differ substantially from other environments in which _any_-DPs are acceptable (in particular, they lack a downward-entailing operator). One approach to this challenge has been to, first, take _any_-DPs to be subject to an environment-based downward-entailingness condition—they have to occur in an environment that is Strawson downward-entailing with respect to their domain (cf. Kadmon and Landman 1993 )—and, second, to derive (...) such an environment in existential modal sentences by means of exhaustification (e.g., Fox 2007 ). This note presents new evidence for such a two-layered approach (cf. Crnič 2017, 2019 ). The evidence comes from a striking contrast in the behavior of singular vs. plural _any_-DPs in existential modal sentences. The paper concludes by charting some relations between _any_-DPs and other polarity items. (shrink)