Jonathan Livengood and Justin Sytsma have published a series of studies on “Actual Causation and Compositionality,” in which they investigate causal attributions of laypeople. We use one of their vignettes to follow up on their research. Our findings cast doubt on their conclusion that ordinary causal attributions tend to violate the compositionality constraint if one looks at cases in which someone is responsible for an effect by way of an intermediary that does not share in the responsibility.

What are the causes and consequences of colonial rule? This introduction to the special issue “Comparative Politics of Colonialism and Its Legacies” surveys recent literature in political science, sociology, and economics that addresses colonial state building and colonial legacies. Past research has made important contributions to our understanding of colonialism’s long-term effects on political, social, and economic development. Existing work emphasizes the role of critical junctures and institutions in understanding the transmission of those effects to present-day outcomes and embraces the (...) idea of design-based inference for empirical analysis. The four articles of this special issue add to existing research but also represent new research trends: increased attention to the internal dynamics of colonial intervention; noninstitutional transmission mechanisms; the role of context conditions at times of colonial intervention; and a finer-grained disaggregation of outcomes, explanatory factors, and units of analysis. (shrink)

Within the literature on emotion and behavioral action, studies on approach-avoidance take up a prominent place. Several experimental paradigms feature successful conceptual replications but many original studies have not yet been replicated directly. We present such a direct replication attempt of two seminal experiments originally conducted by Chen and Bargh (1999). In their first experiment, participants affectively evaluated attitude objects by pulling or pushing a lever. Participants who had to pull the lever with positively valenced attitude objects and push the (...) lever with negatively valenced attitude objects (i.e., congruent instruction) did so faster than participants who had to follow the reverse (i.e., incongruent) instruction. In Chen and Bargh's second experiment, the explicit evaluative instructions were absent and participants merely responded to the attitude objects by either always pushing or always pulling the lever. Similar results were obtained as in Experiment 1. Based on these findings, Chen and Bargh concluded that (1) attitude objects are evaluated automatically; and (2) attitude objects automatically trigger a behavioral tendency to approach or avoid. We attempted to replicate both experiments and failed to find the effects reported by Chen and Bargh as indicated by our pre-registered Bayesian data analyses; nevertheless, the evidence in favor of the null hypotheses was only anecdotal, and definitive conclusions await further study. (shrink)

A frequentist confidence interval can be constructed by inverting a hypothesis test, such that the interval contains only parameter values that would not have been rejected by the test. We show how a similar definition can be employed to construct a Bayesian support interval. Consistent with Carnap’s theory of corroboration, the support interval contains only parameter values that receive at least some minimum amount of support from the data. The support interval is not subject to Lindley’s paradox and provides an (...) evidence-based perspective on inference that differs from the belief-based perspective that forms the basis of the standard Bayesian credible interval. (shrink)

Besides sensory characteristics of food, food-evoked emotion is a crucial factor in predicting consumer’s food preference and therefore in developing new products. Many measures have been developed to assess food-evoked emotions. The aim of this literature review is (i) to give an exhaustive overview of measures used in current research and (ii) to categorize these methods along measurement level (physiological, behavioral, and cognitive) and emotional processing level (unconscious sensory, perceptual / early cognitive, and conscious / decision making) level. This 3x3 (...) categorization may help researchers to compile a set of complementary measures (‘toolbox’) for their studies. We included 101 peer-reviewed articles that evaluate consumer’s emotions and were published between 1997 to 2016, providing us with 59 different measures. More than 60% of these measures are based on self-reported, subjective ratings and questionnaires (cognitive measurement level) and assess the conscious / decision-making level of emotional processing. This multitude of measures and their overrepresentation in a single category hinders the comparison of results across studies and building a complete multi-faceted picture of food-evoked emotions. We recommend (1) to use widely applied, validated measures only, (2) to refrain from using (highly correlated) measures from the same category but use measures from different categories instead, preferably covering all three emotional processing levels, and (3) to acquire and share simultaneously collected physiological, behavioral, and cognitive datasets to improve the predictive power of food choice and other models. (shrink)

The Jeffreys–Lindley paradox exposes a rift between Bayesian and frequentist hypothesis testing that strikes at the heart of statistical inference. Contrary to what most current literature suggests, the paradox was central to the Bayesian testing methodology developed by Sir Harold Jeffreys in the late 1930s. Jeffreys showed that the evidence for a point-null hypothesis $${\mathcal {H}}_0$$ H 0 scales with $$\sqrt{n}$$ n and repeatedly argued that it would, therefore, be mistaken to set a threshold for rejecting $${\mathcal {H}}_0$$ H 0 (...) at a constant multiple of the standard error. Here, we summarize Jeffreys’s early work on the paradox and clarify his reasons for including the $$\sqrt{n}$$ n term. The prior distribution is seen to play a crucial role; by implicitly correcting for selection, small parameter values are identified as relatively surprising under $${\mathcal {H}}_1$$ H 1. We highlight the general nature of the paradox by presenting both a fully frequentist and a fully Bayesian version. We also demonstrate that the paradox does not depend on assigning prior mass to a point hypothesis, as is commonly believed. (shrink)

Writing a captivating book about a bureaucrat and his statistical modelling techniques is impossible? Erwin Dekker’s biography of Jan Tinbergen proves otherwise. As he has done before, Dekker tells the history of economic thought and methodology as part and parcel of general intellectual and cultural history. Nevertheless, he never downplays or neglects the analysis of inner-scientific problem situations. Drawing on rich archival material and conversations with Tinbergen’s family, students, and colleagues, Dekker vividly introduces us to an extraordinary personality and career. (...) Tinbergen of course was so much more than a Dutch bureaucrat and Nobel Prize winning econometrician. He became a key figure in transforming the discipline of economics. Perhaps even more sweeping – and reaching far beyond academia – is Tinbergen’s re-invention of the role played by economists in the modern nation state and in the international community. Ultimately, he spearheaded a shift in the public notion of how malleable “the economy” is by conscious policy efforts – or so Dekker convincingly exhibits and critically discusses. Anybody reflecting on the proper role of scientific experts in democratic societies, may want to pick up Dekker’s entertaining and thought-provoking study. (shrink)

The treatise De intellectu attributed to Alexander of Aphrodisias can be divided into four sections. The first is an interpretation of the Aristotelian theory of intellect, and especially of the active intellect referred to in Aristotle, De anima 3.5, which differs from the interpretation in Alexander's own De anima, and whose relation to Alexander's De anima, attribution to Alexander, and date are all disputed. The second is an account of the intellect which is broadly similar to (...) A though differing on certain points. The third is an account of someone's response to the problem of how intellect can enter the human being ‘from outside’ if it is incorporeal and hence cannot move at all; in the fourth the writer who reported Cl criticizes that solution and gives his own alternative one. (shrink)

This article argues that the first-person narrator and antihero of Anthony Burgess's famous dystopia is far from being the symbol for human freedom he has traditionally been taken to be. Quite the opposite, he is to be seen as a symbol for human “self-imposed nonage” at every point of the novel: from his alleged rebellion to his farewell to rape and aggression in the final chapter. All of his apparent acts of freedom are determined by the dynamic interplay of biological (...) disposition and political exploitation. Burgess's theory of freedom, however, is more sophisticated than Alex's “turnspit freedom.” It is displayed in two minor characters, F. Alexander and the prison chaplain, modelled on Burgess and his cousin the Catholic Archbishop George Dwyer, respectively, who display “political awareness” that leads to decisions based on informed judgment. (shrink)

Writing a captivating book about a bureaucrat and his statistical modelling techniques is impossible? Erwin Dekker’s biography of Jan Tinbergen proves otherwise. As he has done before, Dekker tells the history of economic thought and methodology as part and parcel of general intellectual and cultural history. Nevertheless, he never downplays or neglects the analysis of inner-scientific problem situations. Drawing on rich archival material and conversations with Tinbergen’s family, students, and colleagues, Dekker vividly introduces us to an extraordinary personality and career. (...) Tinbergen of course was so much more than a Dutch bureaucrat and Nobel Prize winning econometrician. He became a key figure in transforming the discipline of economics. Perhaps even more sweeping – and reaching far beyond academia – is Tinbergen’s re-invention of the role played by economists in the modern nation state and in the international community. Ultimately, he spearheaded a shift in the public notion of how malleable “the economy” is by conscious policy efforts – or so Dekker convincingly exhibits and critically discusses. Anybody reflecting on the proper role of scientific experts in democratic societies, may want to pick up Dekker’s entertaining and thought-provoking study. (shrink)

Against the common historiographic narratives of evolutionary biology, the first decades of the 20th century were theoretically far richer than usually assumed. This especially refers to the hitherto neglected role that early theoretical biologists played in introducing visionary research perspectives and concepts before the institutionalization of the Modern Synthesis. Here, we present one of these scholars, the German theoretical biologist and ecomorphologist Hans Böker, by reviewing his 1935 paper “Artumwandlung durch Umkonstruktion, Umkonstruktion durch aktives Reagieren der Organismen”, published in the (...) inaugural volume of the journal Acta Biotheoretica. While largely forgotten today, this work represents a melting pot of ideas that adumbrate some of today’s most lively debated empirical and conceptual topics in evolutionary biology: the active role of organisms as actors of their own evolution, environmental induction and phenotypic plasticity, genetic assimilation, as well as developmental bias. We discuss Böker’s views on how species change through "Umkonstruktion," and how such reconstruction is exerted through active reactions of organisms to environmental perturbations. In addition, we outline the aims and wider context of his "biological comparative anatomy," including Boker’s reprehensible political affiliation with the Nazi Party. Finally, we highlight some of the historical reasons for why Böker’s views did not have a larger impact in evolutionary biology, but we also recount some of the direct and indirect legacies of his approach in research areas such as ecomorphology and EvoDevo. Böker’s paper is available as supplementary material in the online version of this article, as part of the journal's "Classics in Biological Theory" collection; the first translation of the paper into English, by Alexander Böhm and Jan Baedke, is also being published in this volume. (shrink)

Around 1900 the relation between religion and sexuality became a favorite subject of art. Munich was the place where some of the most relevant works of art where produced at this time. In 1905 the opera “Salome” was composed by the Munichborn Richard Strauß, based on Oscar Wilde’s drama and dealing with the relation between sexual love, religion and death. In quite a different way Oscar Panizza’s “Liebeskonzil” is concerned with the same topic. Here it is not the decadent court (...) of Herodes with John the Baptist as religious prophet and Salome falling in deadly love with him but the oversexualized Vatican at the time of Alexander Borgia punished by God with the syphilis. Religion and sexuality also form the background of the Wedekind’s unfinished “Ella”, an early stage of the development of his famous “Lulu”, and they play a central role in Thomas Mann’s novel “Gladius Dei”, a byproduct of his drama “Fiorenza”. Here sexuality and art are represented by the Florentine court of Lorenzo de’ Medici whereas the friar Savonarola is the representative of the religious opposition. (shrink)

This paper tries to summarize the main lines of discussion at the conference "Compatibilist Libertarianism: Advantages and Challenges" (October 29, 2021). This conference, organised by Alexander Gebharter and Maria Sekatskaya, served the discussion of Christian List's account of compatibilist libertarianism. Speakers were Taylor W. Cyr, Nadine Elzein, Alexander Gebharter, Christian List, Alfred R. Mele, Leonhard Menges, Tuomas K. Pernu, and Maria Sekatskaya.

Does willful ignorance mitigate blameworthiness? In many legal systems, willfully ignorant wrongdoers are considered as blameworthy as knowing wrongdoers. This is called the ‘equal culpability thesis’. Given that legal practice depends on it, the issue has obvious importance. Interestingly enough, however, there exists hardly any philosophical reflection on ECT. A recent exception is Alexander Sarch, who defends a restricted version of ECT. On Sarch’s view, ECT is true whenever willfully ignorant agents incur additional blameworthiness for their ignorance. In this (...) paper, I defend an alternative view, according to which ECT is true whenever the motives of willfully ignorant and knowing wrongdoers are equally bad. (shrink)

Long ago, when Alexander the Great asked the mathematician Menaechmus for a crash course in geometry, he got the famous reply ``There is no royal road to mathematics.'' Where there was no shortcut for Alexander, there is no shortcut for us. Still, the fact that we have access to computers and mature programming languages means that there are avenues for us that were denied to the kings and emperors of yore. The purpose of this book is to teach (...) logic and mathematical reasoning in practice, and to connect logical reasoning with computer programming in Haskell. Haskell emerged in the 1990s as a standard for lazy functional programming, a programming style where arguments are evaluated only when the value is actually needed. Haskell is a marvelous demonstration tool for logic and maths because its functional character allows implementations to remain very close to the concepts that get implemented, while the laziness permits smooth handling of infinite data structures. This book does not assume the reader to have previous experience with either programming or construction of formal proofs, but acquaintance with mathematical notation, at the level of secondary school mathematics is presumed. Everything one needs to know about mathematical reasoning or programming is explained as we go along. After proper digestion of the material in this book, the reader will be able to write interesting programs, reason about their correctness, and document them in a clear fashion. The reader will also have learned how to set up mathematical proofs in a structured way, and how to read and digest mathematical proofs written by others. This is the updated, expanded, and corrected second edition of a much-acclaimed textbook. Praise for the first edition: 'Doets and van Eijck's ``The Haskell Road to Logic, Maths and Programming'' is an astonishingly extensive and accessible textbook on logic, maths, and Haskell.' Ralf Laemmel, Professor of Computer Science, University of Koblenz-Landau. (shrink)

Die Erforschung der nordgermanischen Mythologie und Religion hängt wesentlich ab von den Schriften des isländischen Gelehrten und Politikers Snorri Sturluson (1178/79-1241). In über 200 Jahren hat sich diese Forschung disparat entfaltet. Der Autor bietet eine konstruktive Neuorientierung für die Debatten. Er diskutiert kritisch die hochmittelalterlichen Diskurse und liefert auf Basis einer lexematischen Analyse eine Auseinandersetzung mit den überlieferten Fassungen von Snorris Werk.

This chapter focuses on the ontological nature of the past and our experience of the present in relation to the actual. It is argued that philosophers mistakenly equate the real with the actual. The actual is what is presented to us in experience, but what is presented to us in experience is confined to the present; so one might think that it follows that only the present is real. This presentist theory is subsequently rejected. Past, present and future are real (...) and certain mental acts such as memories of the past and expectations of the future have as their content objects at those times; these objects are just as real as the objects of present experiences. This paves the way for a distinctive brand of eternalism that is defended in the remainder of this chapter. (shrink)

§1. I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually called Polish spaces. Typical examples are ℝn, ℂn, Hilbert space and more generally all separable Banach spaces, the Cantor space 2ℕ, the Baire space ℕℕ, the infinite symmetric group S∞, the unitary group, the group of measure preserving transformations of the unit interval, etc.In this theory sets are (...) classified in hierarchies according to the complexity of their definitions and the structure of sets in each level of these hierarchies is systematically analyzed. In the beginning we have the Borel sets in Polish spaces, obtained by starting with the open sets and closing under the operations of complementation and countable unions, and the corresponding Borel hierarchy. After this come the projective sets, obtained by starting with the Borel sets and closing under the operations of complementation and projection, and the corresponding projective hierarchy.There are also transfinite extensions of the projective hierarchy and even much more complex definable sets studied in descriptive set theory, but I will restrict myself here to Borel and projective sets, in fact just those at the first level of the projective hierarchy, i.e., the Borel (), analytic () and coanalytic () sets. (shrink)

In this fresh interpretation of Heidegger, Alexander S. Duff explains Heidegger's perplexing and highly varied political influence. Heidegger and Politics argues that Heidegger's political import is forecast by fundamental ambiguities about the status of politics in his thought. Duff explores how, in Being and Time as well as earlier and later works, Heidegger analyzes 'everyday' human existence as both irretrievably banal but also supplying our only tenuous path to the deepest questions about human life. Heidegger thus points to two (...) irreconcilable attitudes toward politics: either a total and purifying revolution must usher in an authentic communal existence, or else we must await a future deliverance from the present dispensation of Being. Neither attitude is conducive to moderate politics, and so Heidegger's influence tends towards extremism of one form or another, modified only by explicit departures from his thought. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Kennedy Institute of Ethics Journal 13.1 (2003) 51-52 [Access article in PDF] Congress Considers Incentives for Organ Procurement Alexander S. Curtis [Tables]During the 108th Congressional session, several bills pertaining to ethical incentives for organ donation likely will be introduced. In some cases, they will be similar to bills before the 107th Congress (see Table 1). Bills in both the House of Representatives and the Senate address the establishment (...) and maintenance of donor registries, financial reimbursements for medical and nonmedical expenses incurred by qualifying donors, the presentation of commemorative medals, and tax credits.A bill entitled "Gift of Life Congressional Medal Act" (introduced last year by Senate majority leader Bill Frist (R-TN) as Senate Bill 325), and its corresponding bill of the same title (passed in the House of Representatives last year as H.R. 708), would establish a congressional commemorative medal honoring "any organ donor, or the family or family member of any organ donor." The medal is to be designed and struck with "suitable emblems, devices, and inscriptions" by the Secretary of the Treasury. The eligible donor or donor's family member would be presented with the medal by the Secretary of Health and Human Services. This bill would establish qualifying donations as those of human kidneys, livers, hearts, lungs, pancreases, or any other organ, but excludes corneas and eyes.Another bill, introduced by Representative Michael Bilirakis (R-FL) entitled "Organ Donation Improvement Act" (H.R. 624), would award grants or contracts to states, transplant centers, and qualified organ procurement organizations for the purpose of providing travel, subsistence, and incidental nonmedical expenses to individuals who make living donations. In order to receive reimbursement, the donating individuals must reside in states other than the states in which the intended recipients reside and the annual income of those receiving the organs cannot exceed $35,000. The bill also would appropriate $15,000,000 to make grants to states and public and nonprofit entities for the purpose of carrying out studies, public education demonstrations, and outreach activities "designed to increase the number of organ donors within the state." The bill would be effective from time of enactment through 2005. [End Page 51]Two other bills proposed within the House of Representatives in 2002 called for amendment to the Internal Revenue Code. The "Gift of Life Tax Credit Act (introduced as H.R. 1872 by Representative James Hansen (R-UT)) and the "Help Organ Procurement Expand Act" (introduced as H.R. 2090 by Representative Chris Smith (R-NJ)) present a refundable credit to individuals or to the estates of those who agree either to be living donors or to donate their organs upon death. The "Gift of Life Tax Credit Act" refunds $10,000 to the estates of eligible individuals "for the taxable year which includes the date of the individual's death." The "Help Organ Procurement Expand Act" offers a refund of $2,500 to qualified persons. Donations of kidneys, livers, pancreases, islet cells, lungs, and intestines are accepted, however several conditions restrict permissible donors. The "Gift of Life Tax Credit Act" requires that the legally competent individual's death not be due to a suicide and that the intention to donate was clearly declared. The "Help Organ Procurement Expand Act" prohibits donations from individuals who were killed with physician assistance, died as a result of a suicide, have been indicted or convicted of a felony or a misdemeanor offense, or are actively under criminal investigation for a possible felony offense. The "Help Organ Procurement Expand Act" also restricts procuring organs from unborn children or human fetuses and under circumstances in which the living donor would be subject to "unacceptable levels of medical risk of death or permanent debilitation." Alexander S. Curtis is a research assistant at the Kennedy Institute of Ethics, Georgetown University, Washington, DC. He received his bachelor's degree from Miami University.... (shrink)

The article examines the processes taking place in contemporary musical culture, and the emphasis is placed on the situation that has developed today in the musical life of Russia. It is noted that many contemporary composers in Russia are breaking with the domestic artistic traditions, focusing on the creative initiatives of composers living in the West. Among the Russian composers, the SoMa group composers are distinguished by a special intransigence to the principles of artistic creativity that tradionally developed in Russia, (...) they loudly declared: “Contemporary Russian music is us.” At the same time, it should be taken into account that Russian music is inextricably linked with the artistic traditions of Russia, which, first of all, is manifested in the reliance of the musical material on the songfulness that originates in the liturgical Orthodox singing. It is emphasized that today in Russia there are composers who maintain contact with the tradition, thereby preserving the achievements of Russian music. Among them, first of all, are the composers who are part of the MOST group. Moreover, the composers work in all cities of Russia – from Kaliningrad to Vladivostok – and increase the glory of Russia by their creativity. An interview is given with one of them – the Perm composer Nikita Shirokov. In conclusion, the author discusses the prospects for the development of the journal’s section – it is planned to publish materials and interviews related to the contemporary musical life in other countries. (shrink)

This book presents the first detailed account of Werner Heisenberg’s failed attempt to find a theory of everything in the autumn of his career. It further investigates what we can learn from his failure in relation to the search for a final theory of physics, an endeavour that continues to define research in fundamental physics to this day. Thereby it provides the first historically informed contribution to the current debate on post-empirical physics and the state of particle physics.

This essay argues that the thinker whose teaching played a key role in the formation of the Russian political and legal paradigm was not Marx but Hegel. It analyzes the impact of the Hegelian philosophy on the development of the Russian constitutional tradition, and examines its political implications.

A survey is given of three-valued paraconsistent propositionallogics connected with Jaśkowski’s criterion for constructing paraconsistentlogics. Several problems are raised and four new matrix three-valued paraconsistent logics are suggested.

So far I have taken prose and poetry where admittedly they exist, in literature, and attempted to discover the difference between them by taking pieces of prose or poetry which have the same or much the same subjects and comparing them with one another. In all these pairs of passages I thought I could detect this difference: that in the poem the subject as rendered in words acquires a life of its own, is a living thing, as it were, living (...) its own life like an animal or plant, is organic, and, in a word, concrete. While the prose, for all its constructive unity, is not self-subsistent, but is descriptive of a given subject, even though that subject is itself created by the writer, as in a novel, and, in a word, is analytic. It is not the greater passion or vividness of the poem, for prose may be vivid, as in Carlyle, and passionate, as in Burke. It is that the poet places himself and places his hearer within the subject itself, and works from within outwards, while the prosaist describes relatively from without. Both of them describe, the poet as well as the prosaist. But the prosaist builds up his subject so as to bring it before your mind. The poet starts with his subject in its integrity, places the hearer's mind within it, and his exposition is the unfolding life of the subject itself. (shrink)

My object in these lectures is to show that Science is a form of Art, though not of fine art; in other words, that it is one example of a process of which fine art is the most obvious example, the process of making out of certain materials a result into which the mind itself enters. Clearly enough the material of the artist, whatever it be, marble or paints or tones or words, is moulded by the artist into a shape (...) which it would not possess unless something in the artist’s mind found expression in it. Science also is a product of the mind’s interference and is artificial. But it differs from fine art, I shall try to show, in this respect: in fine art the material is controlled from the mind, or at least it is principally controlled from the mind. In science the material is the facts of nature ; but while science would not be except for the mind of the scientist, the work is controlled from the side of the material itself, and his mind is instrumental to the product rather than in some sense intrinsic to it as in fine art. (shrink)

Objectivité veut dire coercition. Où réside ce caractère dans le cas des plus hautes valeurs? Elles sont toutes des artifices, et une satisfaction donnée à certaines tendances des hommes. C’est pourquoi leur objectivité consiste non dans la contrainte de la nature sur l’esprit, mais dans celle de l’esprit sur l’esprit. De là naissent des étalons de valeur. Objections : 1° tirée de la valeur absolue : en quel sens elle existe, 2° de la valeur « subhumaine ». La valeur comme (...) forme universelle, son existence comme « élection naturelle ». (shrink)

LetXbe a standard Borel space, and letEbe acountableBorel equivalence relation onX, i.e., a Borel equivalence relationEfor which every equivalence class [x]Eis countable. By a result of Feldman-Moore [FM],Eis induced by the orbits of a Borel action of a countable groupGonX.The structure of general countable Borel equivalence relations is very little understood. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. A countable Borel equivalence relation is calledhyperfiniteif it is induced by a Borel ℤ-action, i.e., by (...) the orbits of a single Borel automorphism. Such relations are studied and classified in [DJK]. It is shown in Ornstein-Weiss [OW] and Connes-Feldman-Weiss [CFW] that for every Borel equivalence relationEinduced by a Borel action of a countable amenable groupGonXand for every probability measure μ onX, there is a Borel invariant setY⊆Xwith μ = 1 such thatE↾Y is hyperfinite. Motivated by this result, Weiss [W2] raised the question of whether everyEinduced by a Borel action of a countable amenable group is hyperfinite. Later on Weiss showed that this is true forG= ℤn. However, the problem is still open even for abelianG. Our main purpose here is to provide a weaker affirmative answer for general amenableG. We need a definition first. Given two standard Borel spacesX, Y, auniversally measurableisomorphism betweenXandYis a bijection ƒ:X→Ysuch that both ƒ, ƒ-1are universally measurable. Notice now that to assert that a countable Borel equivalence relation onXis hyperfinite is trivially equivalent to saying that there is a standard Borel spaceYand a hyperfinite Borel equivalence relationFonY, which isBorelisomorphic toE, i.e., there is a Borel bijection ƒ:X→YwithxEy⇔ ƒFƒ. We have the following theorem. (shrink)

This paper studies the notion of virtuality in the Bohr-Kramers-Slater theory of 1924. We situate the virtual entities of BKS within the tradition of the correspondence principle and the radiation theory of the Bohr model. We show how, in this context, virtual oscillators emerged as classical substitute radiators and were used to describe the otherwise elusive quantum transitions. They played an effective role in the quantum theory of radiation while remaining categorically distinct and ontologically separated from the quantum world of (...) the Bohr model. The notion of virtuality thus differs markedly from its counterpart in quantum mechanics or QFT. (shrink)