Considered by many to be Abraham Robinson's magnum opus, this book offers an explanation of the development and applications of non-standard analysis by the mathematician who founded the subject. Non-standard analysis grew out of Robinson's attempt to resolve the contradictions posed by infinitesimals within calculus. He introduced this new subject in a seminar at Princeton in 1960, and it remains as controversial today as it was then. This paperback reprint of the 1974 revised edition is indispensable (...) reading for anyone interested in non-standard analysis. It treats in rich detail many areas of application, including topology, functions of a real variable, functions of a complex variable, and normed linear spaces, together with problems of boundary layer flow of viscous fluids and rederivations of Saint-Venant's hypothesis concerning the distribution of stresses in an elastic body. (shrink)

This research is motivated by the program of reverse mathematics and non-standard arguments in second-order arithmetic. Within a weak subsystem of second-order arithmetic ACA0, we investigate some aspects of non-standard analysis related to sequential compactness. Then, using arguments of non-standard analysis, we show the equivalence of the Riemann mapping theorem and ACA0 over WKL0. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim).

Within a weak subsystem of second‐order arithmetic WKL0, we develop basic part of non‐standard analysis up to the Peano existence theorem.

We give an elementary introduction to non-standard analysis and its applications to the theory of stochastic processes. This is based on a joint book with J. E. Fenstad, R. Høegh-Krohn and T. Lindstrøm. In particular we give a discussion of an hyperfinite theory of Dirichlet forms with applications to the study of the Hamiltonian for a quantum mechanical particle in the potential created by a polymer. We also discuss new results on the existence of attractive polymer measures in (...) dimension d ≤ 5, with applications to the (φ 1 2 φ 2 2 )d-mode1 of interacting quantum fields. (shrink)

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to (...) teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms. (shrink)

Summary The historical development of the non-standard analysis is sketched. With the help of this mathematical branch infinite and infinitesimal quantities are placed in an extension of the real numbers and so find their justification. In this way an old mathematical and philosophical problem is solved in the 20th century, but not in such a manner, mathematicians with classical „standard methods thought of.

The hope of this work is that these notes will help to convince more mathematicians that non-standard analysis can make a useful addition to the tools of their trade.

Abraham Robinson has twice been the initiator of trends in the foundations of mathematics which have later become recognized as profound and important, although they were generally ignored at first: metamathematical problems of algebra and non-standard analysis. This book considers the second topic; before we begin analysis—especially a treatment of the classical theorems of calculus—we need basic results from logic, model theory in particular. Robinson then sketches non-standard arithmetic and proceeds to develop the usual properties and (...) relations of differentiability and integrability of functions, convergence; he then examines general topology in this new context and fortified with these results develops "modern" functional analysis in some detail: measure theory and functions of a real variable, functions of a complex variable, normed and linear spaces, topological groups and some harmonic analysis. The last two chapters cover selected topics in analysis drawn principally from applied mathematics and the history of the calculus with special reference to infinitesimals. Robinson views the development of non-standard mathematics as the renaissance of ideas about infinitesimals as serious topics of mathematical study. Robinson primarily has confined himself to analysis and not attempted to cover non-standard algebraic systems; several important results of Alling concerning non-standard arithmetic and rings of continuous functions appeared too late to be included. Of course, the real test of non-standard analysis will come in allowing us to prove new theorems about standard analysis, but although Robinson does not attempt to do this here—although he has obtained results in this direction—other mathematicians have already begun to do so: the future of this new branch of mathematics already looks secure.—P. J. M. (shrink)

A popular way to relate probabilistic information to binary rational beliefs is the Lockean Thesis, which is usually formalized in terms of thresholds. This approach seems far from satisfactory: the value of the thresholds is not well-specified and the Lottery Paradox shows that the model violates the Conjunction Principle. We argue that the Lottery Paradox is a symptom of a more fundamental and general problem, shared by all threshold-models that attempt to put an exact border on something that is intrinsically (...) vague. We propose application of the language of relative analysis—a type of non-standard analysis—to formulate a new model for rational belief, called Stratified Belief. This contextualist model seems well-suited to deal with a concept of beliefs based on probabilities ‘sufficiently close to unity’ and satisfies a moderately weakened form of the Conjunction Principle. We also propose an adaptation of the model that is able to deal with beliefs that are less firm than ‘almost certainty’. The adapted version is also of interest for the epistemicist account of vagueness. (shrink)

This paper considers non-standard analysis and a recently introduced computational methodology based on the notion of ①. The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology (...) does not use any of these notions and proposes a more physical treatment of mathematical objects separating the objects from tools used to study them. It both offers a possibility to create new numerical methods using infinities and infinitesimals in floating-point computations and allows one to study certain mathematical objects dealing with infinity more accurately than it is done traditionally. In these notes, we explain that even though both methodologies deal with infinities and infinitesimals, they are independent and represent two different philosophies of Mathematics that are not in a conflict. It is proved that texts :539–555, 2017; Gutman and Kutateladze in Sib Math J 49:835–841, 2008; Kutateladze in J Appl Ind Math 5:73–75, 2011) asserting that the ①-based methodology is a part of non-standard analysis unfortunately contain several logical fallacies. Their attempt to show that the ①-based methodology can be formalized within non-standard analysis is similar to trying to show that constructivism can be reduced to the traditional Mathematics. (shrink)

This paper presents an analysis of the sorites paradox for collective nouns and gradable adjectives within the framework of classical logic. The paradox is explained by distinguishing between qualitative and quantitative representations. This distinction is formally represented by the use of a different mathematical model for each type of representation. Quantitative representations induce Archimedean models, but qualitative representations induce non-Archimedean models. By using a non-standard model of \ called \, which contains infinite and infinitesimal numbers, the two paradoxes (...) are shown to have distinct structures. The sorites paradox for collective nouns arises from the use of infinite numbers, whereas the sorites paradox for gradable adjectives arises from the use of infinitesimal numbers. Each paradox can be traced to a different source of vagueness. The sorites paradox for collective nouns is caused by \, and the sorites paradox for gradable adjectives is caused by \ \. If correct, this analysis implies that infinite and infinitesimal numbers are cognitively real, and that they play a role in the semantic interpretation of natural language. (shrink)

Non-standard models were introduced by Skolem, first for set theory, then for Peano arithmetic. In the former, Skolem found support for an anti-realist view of absolutely uncountable sets. But in the latter he saw evidence for the impossibility of capturing the intended interpretation by purely deductive methods. In the history of mathematics the concept of a nonstandard model is new. An analysis of some major innovations–the discovery of irrationals, the use of negative and complex numbers, the modern concept (...) of function, and non-Euclidean geometry–reveals them as essentially different from the introduction of non-standard models. Yet, non-Euclidean geometry, which is discussed at some length, is relevant to the present concern; for it raises the issue of intended interpretation. The standard model of natural numbers is the best candidate for an intended interpretation that cannot be captured by a deductive system. Next, I suggest, is the concept of a wellordered set, and then, perhaps, the concept of a constructible set. One may have doubts about a realistic conception of the standard natural numbers, but such doubts cannot gain support from non-standard models. Attempts to utilize non-standard models for an anti-realist position in mathematics, which appeal to meaning-as-use, or to arguments of the kind proposed by Putnam, fail through irrelevance, or lead to incoherence. Robinson’s skepticism, on the other hand, is a coherent position, though one that gives up on providing a detailed philosophical account. The last section enumerates various uses of non-standard models. (shrink)

We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions.We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau’s solution.We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn–Banach extension principle which does not hold in Colombeau theory. We establish a connection between (...) our theory with non-standard analysis and thus answer, although indirectly, a question raised by Colombeau. This article provides a bridge between Colombeau theory of generalized functions and non-standard analysis. (shrink)

In this paper I am concerned with an analysis of negative existential sentences that contain proper names only by using negative or neutral free logic. I will compare different versions of neutral free logic with the standard system of negative free logic (Burge, Sainsbury) and aim to defend my version of neutral free logic that I have labeled non-standard neutral free logic.

Internal sets and the Boolean algebras of the collection of the internal sets are of central importance in non-standard analysis. Boolean algebras are the algebraization of propositional logic while the logic applied in non-standard analysis (in non-standard stochastics) is the first order or the higher order logic (type theory). We present here a first order logic algebraization for the collection of internal sets rather than the Boolean one. Further, we define an unusual probability on this (...) algebraization. (shrink)

We describe a recent program from the study of definable groups in certain o-minimal structures. A central notion of this program is that of a lattice. We propose a definition of a lattice in an arbitrary first-order structure. We then use it to describe, uniformly, various structure theorems for o-minimal groups, each time recovering a lattice that captures some significant invariant of the group at hand. The analysis first goes through a local level, where a pertinent notion of pregeometry (...) and generic elements is each time introduced. (shrink)

Social media commentary, satellite imagery and GPS data are a part of ‘alternative data’, that is, data that originate outside of the standard repertoire of market data but are considered useful for predicting stock prices, detecting different risk exposures and discovering new price movement indicators. With the availability of sophisticated machine-learning analytics tools, alternative data are gaining traction within the investment management and algorithmic trading industries. Drawing on interviews with people working in investment management and algorithmic trading firms utilizing (...) alternative data, as well as firms providing and sourcing such data, we emphasize social media-based sentiment analytics as one manifestation of how alternative data are deployed for stock price prediction purposes. This demonstrates both how sentiment analytics are developed and subsequently utilized by investment management firms. We argue that ‘alternative data’ are an open-ended placeholder for every data source potentially relevant for investment management purposes and harnessing these disparate data sources requires certain standardization efforts by different market participants. Besides showing how market participants understand and use alternative data, we demonstrate that alternative data often undergo processes of prospecting and assetization. We further contend that the widespread embracement of alternative data in investment management and trading encourages a financialization process at the data level which raises new governance issues. (shrink)

Cylindric set algebras are algebraizations of certain logical semantics. The topic surveyed here, i.e. probabilities defined on cylindric set algebras, is closely related, on the one hand, to probability logic (to probabilities defined on logical formulas), on the other hand, to measure theory. The set algebras occuring here are associated, in particular, with the semantics of first order logic and with non-standard analysis. The probabilities introduced are partially continous, they are continous with respect to so-called cylindric sums.

In this paper I explore Plato's reasons for his rejection of the so-called standard analysis of knowledge as justified true belief. I argue that Plato held that knowledge is an infallible mental state in which (a) the knowable is present in the knower and (b) the knower is aware of this presence. Accordingly, knowledge (epistm) is non-propositional. Since there are no infallible belief states, the standard analysis, which assumes that knowledge is a type of belief, cannot (...) be correct. In addition, I argue that Plato held that belief (doxa) is only possible for the sort of being capable of knowledge. This is because self-reflexivity is necessary for infallible knowledge and self-reflexivity is only possible if the intellect is immaterial. This capacity for self-reflexivity is also essential for belief, since beliefs are, paradigmatically, not dispositions but self-reflexive mental states. (shrink)

In the fifth version of our reply article [26] to Imamura's critique, we recall that Neutrosophic Non-Standard Logic was never used by the neutrosophic community in any application, that the quarter-century old (1995-1998) neutrosophic operators criticized by Imamura were never used as they were improved soon after, but omits to talk about their development, and that in real-world applications we need to convert/approximate the hyperreals, monads and bi-nads of Non-Standard Analysis to tiny intervals with the desired precision; (...) otherwise they would be inapplicable. We pointed out several errors and false statements by Imamura [21] regarding the inf/sup of nonstandard subsets, also Imamura's "rigorous definition of neutrosophic logic" is incorrect, as is his definition of nonstandard unit interval, and we showed that there is no total order in Neutrosophic Computing and Machine Learning , Vol. 23, 2022 Florentin Smarandache, Definición mejorada de la lógica neutrosófica no estándar e introducción a los hiperreales neutrosóficos (Quinta versión) 2 the set of hyperreals (due to the recently introduced Neutrosophic Hyperreals which are indeterminate), so the Transfer Principle from R to R* is questionable. After his critique, several reply posts on non-standard theoretical neutrosophy followed in 2018-2022. As such, I extended the Nonstandard Analysis by adding the right-closed left monad, the left-closed right monad, the punctured binad (which we introduced in 1998), and the nonpunctured binad - all in order to close the newly extended nonstandard space (R*) under nonstandard addition, nonstandard subtraction, nonstandard multiplication, nonstandard division, and nonstandard power operations [23, 24]. Improved definitions of the Nonstandard Unitary Interval and Nonstandard Neutrosophic Logic are presented, along with Nonstandard Neutrosophic Operators. (shrink)

RésuméL'analyse non‐standard fournit une base solide à la théorie des infinitésimaux. L'approche axiomatique qu'en donne Nelson est basée sur un nouveau predicat qui est ajouté au langage de la théorie usuelle des ensembles. Nous interprétons ce prédicat et formulons les axio‐mes de Nelson ?on;une façon qui peut être comparee à la discussion de P. R. Halmos dans son livre Naïve Set Theory .SummaryNon‐standard analysis gives a proper foundation to the theory of infinitesimals. Nelson's axiomatic approach of it (...) uses a new predicate which is added to the classical language of set theory. We interpret this new predicate and formulate Nelson's axioms in a way that can be compared to P. R. Halmos' discussion of set axioms in his book Naïve Set Theory .ZusammenfassungDie nicht‐standard Analyse liefert eine solide Grundlage für die Theorie der infinitesimalen Grössen. Die von Nelson vorgeschlagene Axiomatisierung beruht auf einem neuen Prädikat, das den in der Sprache der Mengentheorie üblichen hinzugefügt wird. Wir interpretieren dieses Prädikat und formulieren Nelsons Axiome auf eine Art, die mit der Diskus‐sion von P.R. Halmos in Naïve Set Theory verglichen werden kann. (shrink)

As political and economic systems of the world become increasingly globalized, education systems in developed nations have established standardized curriculum. Meanwhile, opposing interpretations of globalization, neoliberalism and progressivism, have inspired the growth of different paradigmatic models for global education. Grounded in neoliberalism, the human capital model views education as a means for preparing a skilled workforce; derived from progressivism, the world systems model focuses on educating individuals for global citizenship and civic participation. The purpose of this document analysis is (...) two-fold: (1) to examine the social studies content standards of the U.S. and other developed countries to determine how globalization is presented in state-mandated curricula, and (2) to evaluate whether the depiction of globalization in each set of standards reflects a human capital or world systems model. Our findings indicate that U.S. state content standards heavily emphasize the economics of globalization; non-U.S. standards present a more holistic picture of globalization. Consequently, U.S. standards are more aligned with a human capital model of education whereas non-U.S. standards reflect a world systems interpretation. These findings suggest that students in the U.S. are prepared differently than their counterparts in other developed countries, which has implications for international relations. (shrink)

This study explores the phenomenon of the so-called “third reading” of propositional attitude reports. This reading, which was originally explored in the dissertation of J. Fodor (1973) and has since become one of the significant problems in the formal semantics of natural languages, differs from the more well-known de re and de dicto readings by being an intermediate case. If the de re interpretation can be referred to as transparent specific, and the de dicto interpretation as opaque non-specific, then the (...) third reading is transparent non-specific. Fodor's standard solution has been the subject of much discussion in the literature and has given rise to a series of widely accepted counterexamples that are thought to demonstrate the limitations of Fodor's solution. At the same time, alternative approaches to the explication of these readings also suffer from formal shortcomings (for example, some of them did not satisfy the requirement of meaning compositionality, which is a basic requirement in formal semantic literature). This study points out that the analysis of all complex cases of the third reading did not fully take into account their syntactic structure. This remained unnoticed due to the ellipsis in many of these cases. It is shown that a restoration of the ellided syntactic structure makes it possible to analyze all hard cases as basic ones using the classical standard analysis proposed by Fodor supplemented by the principles of admissibility of substitution of L-equivalent expressions in intensional contexts (known since G. Frege and R. Carnap). In the final part of the work, it is demonstrated how exactly the main complex cases of the “third reading” are explicated in terms of the standard approach. (shrink)

The conditional analysis of ability statements has many versions. In this paper I will deal with the version which claims that ‘x can do y’ is equivalent to ‘if x were to choose to do y, then x would do y.’ However, my comments should be equally applicable to any analysis of ability statements that can properly be called a version of the conditional analysis. The intuition behind the conditional analysis is that what it is for (...) one to be able to do something is for one's choice to be effective. To have an ability to do y is for it to be true that one's choosing to do y would be effective - one's choosing to do y would result in one's doing y. But this intuition is not captured by the conditional analysis in its standard form, and a restriction is needed to mend this defect. This restriction is based on a distinction among counterfactual conditionals. (shrink)

While interest in the emotion of awe has surged in psychology, philosophers have yet to devote a single self-standing article to awe’s conceptual contours and moral standing. The present article aims to rectify this imbalance and begin to make up for the unwarranted philosophical neglect. In order to do so, awe is given the standard Aristotelian treatment to uncover its conceptual contours and moral relevance. Aristotelianism typically provides the most useful entry point to ‘size up’ any emotion – more (...) problematically here, however, as Aristotle did not himself explicitly identify awe. The article critiques and proposes to improve upon existing psychological conceptual analyses of awe, probes the question why Aristotle ignored it and addresses an often-presumed link between awe and humility which bears on its moral status. (shrink)

In this paper, we discuss the notion of inevitable ignorance that the Italian Constitutional Court has introduced in justifying a restriction of the legal maxim Ignorantia legis non excusat. In particular, we argue that the epistemic flavor of the notion extends to the notion of inevitability beside that of ignorance, and we offer an epistemic analysis of the notion. This analysis is based both on the legal-theoretical framework defined by the justification of the restriction of the maxim, and (...) on a discussion of some paradigmatic Italian cases where the standard of excusability involving inevitable ignorance is applied. The analysis reveals that the notion of inevitable ignorance is closely connected to a number of notions also used in formal epistemology, such as belief, evidence, rationality, and trust. (shrink)

In this contribution, we focus on probabilistic problems with a denumerably or non-denumerably infinite number of possible outcomes. Kolmogorov (1933) provided an axiomatic basis for probability theory, presented as a part of measure theory, which is a branch of standard analysis or calculus. Since standard analysis does not allow for non-Archimedean quantities (i.e. infinitesimals), we may call Kolmogorov's approach "Archimedean probability theory". We show that allowing non-Archimedean probability values may have considerable epistemological advantages in the infinite (...) case. The current paper focuses on the motivation for our new axiomatization. (shrink)

The Received View on quantum non-individuality is, roughly speaking, the view according to which quantum objects are not individuals. It seems clear that the RV finds its standard expression nowadays through the use of the formal apparatuses of non-reflexive logics, mainly quasi-set theory. In such logics, the relation of identity is restricted, so that it does not apply for terms denoting quantum particles; this “lack of identity” formally characterizes their non-individuality. We face then a dilemma: on the one hand, (...) identity seems too important to be given up, on the other hand the RV seems to require that identity be given up. In this paper we shall discuss how the specific characterization of the RV through non-reflexive logics came to be framed. We examine some of the main objections to this version of the RV and argue that they are misguided under this specific “non-reflexive” understanding of the RV. Finally, we shall also argue that this non-reflexive view is not the only option for a metaphysical articulation of the RV: less radical approaches to identity and logic are open. In particular, some of these alternative approaches to the RV we present may be immune to most of the criticisms presented against the non-reflexive approach. (shrink)

This thesis studies collider phenomenology of physics beyond the Standard Model at the Large Hadron Collider (LHC). It also explores in detail advanced topics related to Higgs boson and supersymmetry - one of the most exciting and well-motivated streams in particle physics. In particular, it finds a very large enhancement of multiple Higgs boson production in vector-boson scattering when Higgs couplings to gauge bosons differ from those predicted by the Standard Model. The thesis demonstrates that due to the (...) loss of unitarity, the very large enhancement for triple Higgs boson production takes place. This is a truly novel finding. The thesis also studies the effects of supersymmetric partners of top and bottom quarks on the Higgs production and decay at the LHC, pointing for the first time to non-universal alterations for two main production processes of the Higgs boson at the LHC-vector boson fusion and gluon-gluon fusion. Continuing the exploration of Higgs boson and supersymmetry at the LHC, the thesis extends existing experimental analysis and shows that for a single decay channel the mass of the top quark superpartner below 175 GeV can be completely excluded, which in turn excludes electroweak baryogenesis in the Minimal Supersymmetric Model. This is a major new finding for the HEP community. This thesis is very clearly written and the introduction and conclusions are accessible to a wide spectrum of readers. (shrink)

Family firms are ubiquitous and play a crucial role across all world economies, but how they differ in the disclosure of social and environmental actions from non-family firms has been largely overlooked in the literature. Advancing the discourse on corporate social responsibility reporting, we examine how family influence on a business organization affects CSR reporting. The arguments developed here draw on institutional theory, using a rich body of empirical evidence gathered through a content analysis of the CSR reports of (...) 98 large- and medium-sized Italian firms. The grounded theory analysis informs and contextualizes several differences in the type and content of corporate social responsibility reports of family and non-family firms. Our findings show that in comparison to non-family firms, family firms disseminate a greater variety of CSR reports, are less compliant with CSR standards and place emphasis on different CSR topics. We, thus, contribute to the family business and corporate social responsibility reporting literatures in several ways, offering implications for practice and outlining promising avenues for future research. (shrink)

Aim: Understanding the effects of psychotherapy is a crucial concern for both research and clinical practice, especially when outcome tends to be negative. Yet, while outcome is predominantly evaluated by means of quantitative pre-post outcome questionnaires, it remains unclear what this actually means for patients in their daily lives. To explore this meaning, it is imperative to combine treatment evaluation with quantitative and qualitative outcome measures. This study investigates the phenomenon of non-improvement in psychotherapy, by complementing quantitative pre-post outcome scores (...) that indicate no reliable change in depression symptoms with a qualitative inquiry of patients’ perspectives. Methods: The study took place in the context of a Randomized Controlled Trial evaluating time-limited psychodynamic and cognitive behavioural therapy for major depression. A mixed methods study was conducted including patients’ pre-post outcome scores on the BDI-II-NL and post treatment Client Change Interviews. Nineteen patients whose data showed no reliable change in depression symptoms were selected. A grounded theory analysis was conducted on the transcripts of patients’ interviews. Findings: From the patients’ perspective, non-improvement can be understood as being stuck between knowing versus doing, resulting in a stalemate. Positive changes (mental stability, personal strength and insight) were stimulated by therapy offering moments of self-reflection and guidance, the benevolent therapist approach and the context as important motivations. Remaining issues (ambition to change but inability to do so) were attributed to the therapy hitting its limits, patients’ resistance and impossibility and the context as a source of distress. “No change” in outcome scores therefore seems to involve a “partial change” when considering the patients’ perspectives. Conclusion: The study shows the value of integrating qualitative first-person analyses into standard quantitative outcome evaluation and particularly for understanding the phenomenon of non-improvement. It argues for more multi-method and multi-perspective research to gain a better understanding of (negative) outcome and treatment effects. Implications for both research and practice are discussed. (shrink)

Objective:This study aimed to investigate the effect of robot-assisted gait training therapy combined with non-invasive brain stimulation on lower limb function in patients with stroke and spinal cord injury.Data sourcesPubMed, Cochrane Central Register of Controlled Trials, Ovid MEDLINE, and Web of Science were searched.Study selectionRandomized controlled trials published as of 3 March 2021. RCTs evaluating RAGT combined with NIBS, such as transcranial direct current stimulation and repetitive transcranial magnetic stimulation, for lower limb function and activities in patients with stroke and (...) SCI were included.Data extractionTwo reviewers independently screened the records, extracted the data, and assessed the risk of bias.Data synthesisA meta-analysis of five studies and risk of bias were conducted. Pooled estimates demonstrated that RAGT combined with NIBS significantly improved lower limb function [standardized mean difference = 0.52; 95% confidence interval = 0.06–0.99] but not lower limb activities. Subgroup analyses also failed to find a greater improvement in lower limb function of RAGT with tDCS compared to sham stimulation. No significant differences between participant characteristics or types of NIBS were observed.ConclusionThis meta-analysis demonstrated that RAGT therapy in combination with NIBS was effective in patients with stroke and SCI. However, a greater improvement in lower limb function and activities were not observed using RAGT with tDCS compared to sham stimulation. (shrink)

ObjectivesTo identify and analyse ethical considerations raised when individuals with body dysmorphic disorder consult for non-surgical cosmetic procedures.MethodsEthical analysis was conducted addressing the issues of best interests and capacity to consent for non-surgical cosmetic procedures in individuals with BDD. Analysis was informed by the findings of semistructured interviews with non-surgical cosmetic practitioners and mental health professionals.FindingsNon-surgical cosmetic interventions were viewed not to be in the best interests of individuals with BDD, as they fail to address core psychological issues, (...) result in dissatisfaction post-procedure, and risk harm. Referral to mental health services was advocated, however numerous obstacles to this were perceived. The issue of capacity to consent to non-surgical cosmetic procedures raised questions regarding whether standard capacity assessment is sensitive to the manner in which BDD may influence decision-making processes. In addition, concerns were voiced that decisions made by individuals with BDD in this context may be judged foolish, and thus wrongly equated with lack of capacity.Discussion/conclusionsEthical analysis, informed by the available evidence base, suggests that it is generally not in the best interests of individuals with BDD to undergo non-surgical cosmetic intervention, and referral to mental health services is indicated. Analysis of capacity draws parallels between BDD and anorexia nervosa, as decision-making capacity in both conditions can be impaired by pathological values derived from the disorder. Means of differentiating clinical assessment of pathological values from inappropriate value judgements are advocated, in order to safeguard against the latter encroaching into capacity assessment. (shrink)

The distribution of the focus particle even is constrained: if it is adjoined at surface structure to an expression that is entailed by its focus alternatives, as in even once, it must be appropriately embedded to be acceptable. This paper focuses on the context-dependent distribution of such occurrences of even in the scope of non-monotone quantifiers. We show that it is explained on the assumption that even can move at LF Syntax and semantics, 1979). The analysis is subsequently extended (...) to occurrences of negative polarity items in these environments, which mirror the abovementioned distribution of even and which invalidate standard characterizations of NPI licensing conditions in terms of downward-entailingness. The idea behind the extension is that NPIs denote weak elements that are associates of covert even. The paper concludes by discussing two comprehensive theories of NPI licensing and how our proposal relates to them. (shrink)

In this paper we provide a detailed proof-theoretical analysis of a natural deduction system for classical propositional logic that (i) represents classical proofs in a more natural way than standard Gentzen-style natural deduction, (ii) admits of a simple normalization procedure such that normal proofs enjoy the Weak Subformula Property, (iii) provides the means to prove a Non-contamination Property of normal proofs that is not satisfied by normal proofs in the Gentzen tradition and is useful for applications, especially in (...) formal argumentation, (iv) naturally leads to defining a notion of depth of a proof, to the effect that, for every fixed natural k, normal k-depth deducibility is a tractable problem and converges to classical deducibility as k tends to infinity. (shrink)

The article examines the effects of non-financial disclosure on corporate social responsibility. We conceptualise trade-offs between two ideal types in relation to CSR. Whereas self-regulation is associated with greater flexibility for businesses to develop best practices, it can also lead to complacency if firms feel no external pressure to engage with CSR. In contrast, government regulation is associated with greater stringency around minimum standards, but can also result in rigidity owing to a ‘one-size-fits-all’ approach. Given these potential trade-offs, we ask (...) how mandatory non-financial disclosure has been shaping CSR practices and examine its potential effectiveness as a regulatory instrument. Our analysis of 24 OECD countries using the Asset4 database shows that firms in countries that require non-financial disclosure adopt significantly more CSR activities. However, we also find that NFD regulation does not lead to lower levels of corporate irresponsibility. Furthermore, our analysis demonstrates that, over time, the variation in CSR activities declines as firms adopt increasingly similar practices. Our study thereby contributes to understanding the impact of government regulation on CSR at firm level. We also discuss the limits of mandatory NFD in addressing regulatory trade-offs between stringency and flexibility in the field of corporate social responsibility. (shrink)

This article discusses how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. Techniques and ideas from non-standard analysis are brought to bear on the problem.

In this chapter, I evaluate various conceptions of distance. Of the two most prominent, one takes distance relations to be intrinsic, the other extrinsic. I recommend pluralism: different conceptions can peacefully coexist as long as each holds sway over a distinct region of logical space. But when one asks which conception holds sway at the actual world, one conception stands out. It is the conception of distance embodied in differential geometry, what I call the Gaussian conception. On this conception, all (...) fundamental facts about distance are “local” facts.” But there is a problem: the Gaussian conception, notwithstanding its mathematical and physical credentials, appears metaphysically suspicious on Humean grounds. In the final section, I suggest that the Gaussian conception can be given a sound metaphysical footing in terms of non-standard analysis. (shrink)