The Loeb space construction in nonstandard analysis is applied to the theory of processes to reveal basic phenomena which cannot be treated using classical methods. An asymptotic interpretation of results established here shows that for a triangular array (or a sequence) of random variables, asymptotic uncorrelatedness or asymptotic pairwise independence is necessary and sufficient for the validity of appropriate versions of the law of large numbers. Our intrinsic characterization of almost sure pairwise independence leads to the equivalence of (...) various multiplicative properties of random variables. (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)

Maurice Merleau-Ponty argued that the seeing of things together (focal figure and background objects) accounted for the sense that things possess unseen depth: they are three-dimensional entities, not facades. I compare this idea to the law of large numbers. In both cases, single entities take on substance, depth, or meaning when assimilated into a large body of comparable instances. Thinking along these lines, Erwin Schrödinger proposed that living processes achieve order by virtue of the multiplicity of their (...) constituent parts, any of which, when considered individually, militate against perception and understanding. He further suggested that those parts take their place as things to be perceived even as they constitute our perceiving faculties, and this is why uncertainty occasions their reality. Thus he offers an instance of Merleau-Ponty’s notion of chiasmic intertwining, which allows for world-body interchange or reversibility beyond the reach of unequivocal apprehension. (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 presents the metaphysics of liberal rights reasoning on one hand and that of demographic reasoning on the other, as exemplifying two worldviews that both compete and complement each other in the contemporary German public debate on demographic decline. First, this essay outlines the way in which liberal theorists of various outlooks, perfectionist and neutralist alike, assume that a wide range of rights serves not only the interests of those individuals who possess them, but that it constitutes the foundations (...) of a just and stable political order in general and therefore is to the advantage of everyone. Second, the essay explains how demographic reasoning questions the assumption of harmony shared by the liberal approaches. Third, it provides an impression of the way in which demographic arguments have been deployed in the public sphere in Germany in the last few years. These arguments associate the autonomy of women with the demise of Germany. They claim that by encouraging women to pursue self-realization as self-interested individuals, the modern secular ethos of Germany as a democratic welfare society may be self-destructive in the long run, since it leads to sub-replacement fertility. Finally, the essay stresses that liberal and demographic perspectives share a “blindness” of historical events. In response, the conclusion brings history back in, by historicizing both demographic reasoning and demographic developments in Germany, with the aim of defusing some of the anxieties that may have been aroused by the current debate. (shrink)

This book details how quantification can serve both as evidence and as an instrument of government, whether when dealing with statistics on employment, occupational health and economic governance, or when developing public management or target-driven policies. In the process, it presents a thought-provoking homage to Alain Desrosières, who pioneered ways to study large numbers and the politics underlying them. It opens with a summary of Desrosières's contributions to the field in which several generations of researchers detail how this (...) statistician and historian profoundly influenced them. This tribute, based on personal testimonies, bears witness to the vitality of the school of thought and analytical framework Desrosières initiated. Next, a collection of essays explores the statistical argument in the neoliberal era, examining issues such as counting the homeless in Europe, measuring the performance of public services, and quantifying the effects of public action on the unemployed in France. The third part details the uses of quantification. It reveals that although statistics are frequently used to the advantage of those in power, they can also play a vital role in challenging and resisting both the conventions underlying the measurements as well as the measurements themselves.Featuring the work of economists, historians, political scientists, sociologists, and statisticians, this title provides readers with a thoughtful look at an influential figure in the history of statistics. It also shows how statistics are used to direct public policy, the degree of conflict that is possible in their production, and the disputes that can develop around their uses. (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)

The main objective of the paper is to propose a frequentist interpretation of probability in the context of model-based induction, anchored on the Strong Law of Large Numbers (SLLN) and justifiable on empirical grounds. It is argued that the prevailing views in philosophy of science concerning induction and the frequentist interpretation of probability are unduly influenced by enumerative induction, and the von Mises rendering, both of which are at odds with frequentist model-based induction that dominates current practice. The (...) differences between the two perspectives are brought out with a view to defend the model-based frequentist interpretation of probability against certain well-known charges, including [i] the circularity of its definition, [ii] its inability to assign ‘single event’ probabilities, and [iii] its reliance on ‘random samples’. It is argued that charges [i]–[ii] stem from misidentifying the frequentist ‘long-run’ with the von Mises collective. In contrast, the defining characteristic of the long-run metaphor associated with model-based induction is neither its temporal nor its physical dimension, but its repeatability (in principle); an attribute that renders it operational in practice. It is also argued that the notion of a statistical model can easily accommodate non-IID samples, rendering charge [iii] simply misinformed. (shrink)

The strong law of large numbers and considerations concerning additional information strongly suggest that Beauty upon awakening has probability 1/3 to be in a heads-awakening but should still believe the probability that the coin landed heads in the Sunday toss to be 1/2. The problem is that she is in a heads-awakening if and only if the coin landed heads. So, how can she rationally assign different probabilities or credences to propositions she knows imply each other? This is (...) the problem I address in this article. I suggest that ‘p whenever q and vice versa’ may be consistent with p and q having different probabilities if one of them refers to a sample space containing ordinary possible worlds and the other to a sample space containing centred possible worlds, because such spaces may fail to combine into one composite probability space and, as a consequence, ‘whenever’ may not be well defined; such is the main contribution of this article. 1The Sleeping Beauty Game2Groisman’s and Peter Lewis’s Approaches3Discussing Beauty’s Credences4The Principle of Equivalence's Failure5Making Sense of the Principle of Equivalence's Failure6Elga’s and Lewis’s Approaches7ConclusionAppendix. (shrink)

Instances of the law of large numbers are used to model many different physical systems. In this paper, I argue for a particular interpretation, of those instances of that law, which appeals to typicality. As I argue, the content of that law, when used to model physical systems, is that the probability of an event typically—rather than probably—approximates the frequency with which that event occurs.

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Laws of the Roman People: Public Law in the Expansion and Decline of the Roman RepublicDaniel J. GargolaCallie Williamson. The Laws of the Roman People: Public Law in the Expansion and Decline of the Roman Republic. Ann Arbor: University of Michigan Press, 2005. xxviii + 506 pp. 39 tables. 4 maps. Cloth, $75.Laws enacted by citizen assemblies occupy a prominent place in the history of the Roman (...) Republic. In this large book, Williamson aims to examine the culture and practices of lawmaking and to link legislation to Roman expansion over Italy and to the creation of structures of accommodation of sufficient strength to allow the formation of a fairly integrated Italian world. Williamson has set a large task for herself, one that addresses many major themes in Roman history, and she has produced a wide-ranging and complicated argument that should be set out in detail.The book is divided into three broad parts, each made up of three substantial chapters. Part 1 is the core. In the first chapter, Williamson addresses change over time in the frequency of legislative assemblies, the relationship between frequency of lawmaking and periods of crisis, and the functions of public lawmaking. She finds that legislative activity was concentrated in times of crisis, but that the number of laws proposed or enacted in non-crisis years increased markedly from the third to the first centuries and that Roman officeholders proposed laws primarily in instances where the matter was controversial, suggesting that the primary function of lawmaking through popular assemblies was a legitimizing one in just those circumstances where the legitimacy of an action might be questioned.In general, these conclusions are possible and, in some cases, probable, [End Page 469] but they rest on foundations that are not as firm as one would like. Williamson's demonstration centers on a collection of laws and proposed laws from c. 350 to 44 B.C.E., which she has arranged chronologically in twenty-five-year segments that furnish the means to discern changes over time and that, in later chapters, provide the essential temporal structure to integrate lawmaking with a wide range of developments. In appendix B, intended to justify her approach, she acknowledges and dismisses two potential sources of error: the possible use of a statistically invalid sampling procedure and the possible bias introduced by the differing interests of the authors who provide the bulk of the evidence: Cicero, Livy, Cassius Dio, Appian, Plutarch, and Polybius. Yet there is another possibility that Williamson does not consider. Our sources certainly favor some periods over others. Thus, historical accounts become fuller for the Second Punic War and as they approach the first century, just the periods that supposedly exhibit clear signs of an increase in legislative activity. We may know of more laws or proposed laws in periods of war and political conflict than in times of peace, not only because more laws were put forward then but also because such times attracted more interest on the part of historians. Finally, substantial portions of the texts of Livy, Dio Cassius, and Polybius are lost, and the gaps cover much of the third and second centuries. The evidence, then, is extremely uneven over time.One way that Williamson uses the evidence is especially problematic. She notes that three methods of enacting programs or regulations existed simultaneously: magisterial edicts, senatorial decrees, and popular legislation. Since she finds that the kinds of matters addressed in legislation are indistinguishable from the range of issues addressed by the senate, she concludes that, to some extent, it was a matter of choice whether a magistrate went to the senate or to the people for authorization. Thus, when a proposal was moved forward by a law rather than by an edict or senatorial decree, she suggests that the measure was probably controversial and required the legitimizing function of a popular assembly. Unfortunately, our sources often fail to set out clearly just what procedures a magistrate followed. Sometimes, as Williamson notes, a senatorial decree and the proposal of legislation mark successive stages in the legislative process. And certain operations, such as the creation of provincial assignments or promagistracies, could be addressed... (shrink)

Review of Sylla's translation of Jacob Bernoulli's Art of Conjecturing, emphasising Bernoulli's success in understanding multiple quantifiers to formulate and prove a law of large numbers.

During the COVID-19 pandemic, the race to find an effective vaccine or treatment saw an ‘extraordinary number’ of clinical trials being conducted. While there were some key success stories, not all trials produced results that informed patient care. There was a significant amount of waste in clinical research during the pandemic which is said to have hampered an evidence-based response. Conducting trials which could have been predicted to fail to answer the research question (e.g. because they are not large (...) enough to provide a definitive result) is not only a waste of resources but also a breach of research participants’ trust and a violation of research ethics. The issues seen in COVID-19 clinical trials are symptomatic of a wider trial design crisis where many trials do not provide informative results. This paper examines the roles of key stakeholders in delivering ethical and informative trials and whether guidance published by ‘The Good Clinical Trials Collaborative’ could be used to align key stakeholder groups and enable a joined-up approach to improve clinical trial design. (shrink)

. We demonstrate that a comprehensive nonstandard set theory can be developed in the standard $\displaystyle{\in}$ -language. As an illustration, a nonstandard ${\sf Law of Large Numbers}$ is obtained.

In the epigraph, Fisher is blaming two generations of theoretical biologists, from Darwin on, for ignoring Quetelet's statistical techniques and hence harboring confusions about evolution and natural selection. He is right to imply that Darwin and his contemporaries were aware of the core of Quetelet's work. Quetelet's seminal monograph, Sur L'homme, was widely discussed in Darwin's academic circles. We know that Darwin owned a copy (Schweber 1977). More importantly, we have in Darwin's notebooks two entries referring to Quetelet's work on (...) the cause of a large-scale global phenomenon where each year more boys were born than girls. The first entry is written sometime between April and July 1838. Darwin writes "Find out from the Statistical Society—where M. Quetelet has published his laws about sexes relative to age of Marriages" (C 268, Barrett et al., 1987, p. 324). The second is written sometime after October 16, 1838: "In the Atheneum Numbers 406, 407, 409, Quetelet papers are given, & I think facts there mentioned about proportion of sexes, at birth & causes" (Ibid, p. 379). So, even if Darwin did not read Sur L'homme directly it is likely (though not certain) that he read its review in the Atheneum. There is no doubt that Darwin eventually became familiar with Quetelet's work in statistics. The smoking gun is an essay that Darwin writes in 1874, entitled, "On the Males and Complemental Males of Certain Cirripedes, and on Rudimentary Structures" where he discusses Quetelet's laws of variation. (shrink)

> Future people count. There could be a lot of them. We can make their lives better. > > -- William MacAskill, What We Owe The Future > > [Longtermism is] quite possibly the most dangerous secular belief system in the world today. > > -- Émile P. Torres, Against Longtermism Philosophers,1 2 psychologists,3 4 politicians5 and even some tech billionaires6 have sounded the alarm about artificial intelligence (AI) and the dangers it may pose to the long-term future of humanity. (...) Some believe it poses an existential risk (X-Risk)7 to our species, potentially causing our extinction or bringing about the collapse of human civilisation as we know it. The above quote from philosopher Will MacAskill captures the key tenets of ‘longtermism’, an ethical standpoint that places the onus on current generations to prevent AI-related—and other—X-Risks for the sake of people living in the future.1 2 Developing from an adjacent social movement commonly associated with utilitarian philosophy, ‘effective altruism’, longtermism has amassed following of its own. Its supporters argue that preventing X-Risks is at least as morally significant as addressing current challenges like global poverty. However, critics are concerned that such a distant-future focus will sideline efforts to tackle the many pressing moral issues facing humanity now.8 9 Indeed, according to ‘strong’ longtermism,10 future needs arguably should take precedence over present ones. In essence, the claim is that there is greater expected utility to allocating available resources to prevent human extinction in the future than there is to focusing on present lives, since doing so stands to benefit the incalculably large number of people in later generations who will far outweigh existing populations.1 Taken to the extreme, this view suggests it would be morally permissible, or even required, to actively neglect, harm or destroy large swathes of humanity as …. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:176 Reviews c:\users\ken\documents\type3402\rj 3402 050 red.docx 2015-02-04 9:19 PM THE GRUNDGESETZE Nicholas Griffin Russell Research Centre / McMaster U. Hamilton, on, Canada l8s 4l6 [email protected] Gottlob Frege. Basic Laws of Arithmetic. Derived Using Concept-script. Volumes i and ii. Translated and edited by Philip A. Ebert and Marcus Rossberg with Crispin Wright. Oxford: Oxford U. P., 2013. Pp. xxxix + xxxii + 253 + xv + 285 + A–42 + (...) i–11. isbn 978-0-19-928174-9. £60; us$110. iven the steadily rising interest in Frege’s work since the 1950s, it is surprising that his Grundgesetze der Arithmetik, the work he thought would be the crowning achievement of his career, has not previously been fully translated into English. Other Frege works have long been available in English. The earlier Grundlagen der Arithmetik was nicely, if not always entirely accurately, translated by J. L. Austin and published in 1950.1 Translations of Frege’s other philosophical writings began to appear in influential editions about the same time,2 initially a small corpus that has been gradually added to over the years to include works from his Nachlass and letters from his correspondence.3 Somewhat later, the Begriffsschrift, Frege’s momentous first 1 The Foundations of Arithmetic (Oxford: Blackwell, 1950; 2nd edn. 1953). The later translation under the same title by Dale Jacquette (London: Longman, 2007) is much less satisfactory. 2 The first importance source for these was Peter Geach and Max Black’s Translations from the Philosophical Writing of Gottlob Frege (1952; 2nd edn. 1960; 3rd edn. 1980), though it included reprints of some much earlier translations. 3 The two most comprehensive English sources for the shorter writings are now Brian d= Reviews 177 c:\users\ken\documents\type3402\rj 3402 050 red.docx 2015-02-04 9:19 PM contribution to modern logic, the work which Quine 4 said had made logic a great subject, was translated in full despite the difficulties of Fregean notation.5 But through all this the Grundgesetze remained largely (though not entirely ) untranslated. The view seems, for a long time, to have been that Frege’s contributions to logic were to be found in the Begriffsschrift and his contributions to philosophy in the Grundlagen and a small number of seminal articles. The Grundgesetze, on the other hand, was widely thought not to have added much to these magni ficent achievements, but to have employed them in a project that was doomed by the discovery of Russell’s paradox; namely, the attempt to show that arithmetic could be rigorously derived from purely logical principles. In the bleak aftermath of Russell’s paradox it came to seem as if whatever was of value in the Grundgesetze had already been published elsewhere, while what was new in it was mistaken.6 That assessment of the book’s value, combined with the huge difficulties of typesetting Frege’s idiosyncratic two-dimensional notation, and the sheer scale of the work (even without the anticipated third volume, which was abandoned in the face of Russell’s paradox, it was, by far, Frege’s largest work) made it seem as though a translation of the whole Grundgesetze was not only a prohibitively large undertaking, but also one which was not really necessary. And so, through much of the twentieth century, the Grundgesetze vied with Principia Mathematica as the world’s bestknown but least-read philosophical book. But this understanding of the Grundgesetze’s importance could not withstand the development of neo-logicism, brought to prominence by Crispin Wright’s Frege’s Conception of Numbers as Objects.7 The key insight of neoMcGuinness, ed., Frege, Collected Papers on Mathematics, Logic, and Philosophy (1984); and Michael Beaney, ed., The Frege Reader (1997). 4 At least in early editions of Mathematical Logic. The remark was removed in later ones in deference to Boole. 5 By Stefan Bauer-Mengelberg in Jean van Heijenoort, ed., From Frege to Gödel (1967), pp. 5–82. 6 Not surprisingly the one part of the Grundgesetze which has appeared most frequently in translation is the “Nachwort” responding to Russell’s paradox which Frege added as the book was in press. Geach... (shrink)

Let be n events in a probability space, and suppose that we have only partial information about the distribution: The probabilites of the events themselves, and their pair intersections. With this partial information we cannot, usually, deternine the probability of an event B in the algebra generated by the 's, but we can obtain lower and upper bounds. This is done by a linear program related to the correlation polytope c(n), a structure introduced in [3], [4]. In the first part (...) of the paper I demonstrate how laws of large numbers (for sequences of events which are not necessarily independent) can be proved, using only the duality theorem of linear programming. These include the weak law of large numbers (necessary and sufficient condition) and various sufficient conditions for strong laws. The connection between these laws and the facet structure of the correlation polytope is established. In the second part of the paper I consider a more general case. Assume that our information consists of the values of the probabilities of all intersections of the 's up to size k, k < n. The techniques of linear programming lead naturally to an application of the theory of polynomial approximation in estimating the size of various events. In particular, I prove an approximate version of the central limit theorem. (shrink)

As a concrete basis for thought, we first share the experience of a correspondence analysis applied to a morphosyntactically tagged corpus. The automatic tagging has been partially checked and shows errors. Knowing how factorial analysis works, and having identified the noisiest tags, we adjust the data to be submitted to the analysis so that it is both reliable and clear, making sense for the interpretation of the results to be drawn.This experiment calls for a three-fold discussion. Firstly, about the evaluation (...) of such a process: indeed, by bypassing (rather than correcting) errors (noise), we shift the analysis framework, which no longer allows an evaluation based on the usual comparison of results and improvement measurement. Next, we justify how tweaking the data can still be a sound scientific approach, when the process is methodical and transparent. Then we consider whether corpus size matters, in relationship with the law of large numbers: statistics neutralize the noise of random fluctuations, but not that of repetitive bias.Thus, we argue for “taming” noise: that is, knowing your corpus and your tools, so as to be able to design quality analyses while managing the residual noise, and freeing yourself from a prior and probably illusory requirement for data perfection. (shrink)

The best philosophical account of human rights regards them as protections of the values we attach to human agency. The international law of human rights is embodied in a large number of declarations, conventions, covenants, charters, and judicial decisions. There are many discrepancies between the lists of human rights that emerge from these two authoritative sources. This lecture explores the significance of these discrepancies.

I extract some philosophical morals from some aspects of Lagrangian mechanics. One main moral concerns methodology: Lagrangian mechanics provides a level of description of phenomena which has been largely ignored by philosophers, since it falls between their accustomed levels---``laws of nature'' and ``models''. Another main moral concerns ontology: the ontology of Lagrangian mechanics is both more subtle and more problematic than philosophers often realize. The treatment of Lagrangian mechanics provides an introduction to the subject for philosophers, and is technically elementary. (...) In particular, it is confined to systems with a finite number of degrees of freedom, and for the most part eschews modern geometry. (shrink)

According to E. G. Lorenzen, the international theory of the conflict of laws rests almost wholly on fiction. The discipline of conflict of laws is created by a huge number of national internal laws and quite a large number of international instruments. Concepts used there are interpreted in many ways illustrating the variety of legal solutions and values. Some of the instruments facilitating the correct application of law are unregulated. One of them is qualification, which starts the process of (...) indication and application of the proper law and assists in its completion. (shrink)

According to Hugh Mellor in Real Time II (1998, Ch. 12), assuming the logical independence of causal facts and the 'law of large numbers', causal loops are impossible because if they were possible they would produce inconsistent sets of frequencies. I clarify the argument, and argue that it would be preferable to abandon the relevant independence assumption in the case of causal loops.

A number of high-profile campaigns relating to diversity have focussed on the large law firm sector since the mid-2000s. Reflecting on what has been called the 'diversity approach' to equality management, they have emphasised voluntary action based on business case reasoning. This paper considers the impact of these campaigns in practice, focusing on the dynamics of diversity policy-making within firms. Drawing upon empirical work conducted in large law firms, it explores the perspective of newly appointed diversity staff who (...) have day to day responsibility for diversity policies in the majority of large law firms. It discusses research findings that show while certain policies were being implemented across the large law firm sector, much turns on the position of diversity staff within firms and their ability to conduct in-house negotiations effectively, to compromise and downplay the potential for disruption to the status quo. On the basis of this research I consider the ongoing debates about holding law firms to account with respect to the diversity of their workforce. While the strategy of pressuring firms to disclose 'diversity data' is gathering pace, the paper recommends an open discussion about the goals of diversity policy-making as a pre-condition of greater accountability. (shrink)

This paper addresses and rejects claims that one can demonstrate experimentally that most untutored subjects are systematically and incurably irrational in their probability judgements and in some deductive reasoning tasks. From within a strongly subjectivist theory of probability, it develops the notions of resiliency —a measure of stability of judgements—and robustness —a measure of expected stability. It then becomes possible to understand subjects' behaviour in the Wason selection task, in examples which have been claimed to involve a 'base-rate fallacy', in (...) appearing to ignore laws of large numbers and to overestimate their own success rate as entirely rational: a matter of combining first-order probability assignments with judgements about the robustness of those assignments. (shrink)

We present statistical analyses of the large‐scale structure of 3 types of semantic networks: word associations, WordNet, and Roget's Thesaurus. We show that they have a small‐world structure, characterized by sparse connectivity, short average path lengths between words, and strong local clustering. In addition, the distributions of the number of connections follow power laws that indicate a scale‐free pattern of connectivity, with most nodes having relatively few connections joined together through a small number of hubs with many connections. These (...) regularities have also been found in certain other complex natural networks, such as the World Wide Web, but they are not consistent with many conventional models of semantic organization, based on inheritance hierarchies, arbitrarily structured networks, or high‐dimensional vector spaces. We propose that these structures reflect the mechanisms by which semantic networks grow. We describe a simple model for semantic growth, in which each new word or concept is connected to an existing network by differentiating the connectivity pattern of an existing node. This model generates appropriate small‐world statistics and power‐law connectivity distributions, and it also suggests one possible mechanistic basis for the effects of learning history variables (age of acquisition, usage frequency) on behavioral performance in semantic processing tasks. (shrink)

The present study aims to assess the restriction of private ownership on the cultural-historical property based on the public interest and evaluate how this restriction is explained and what restrictions the cultural heritage rules and regulations impose on the private ownership. The present descriptive and analytical study seeks to examine the above-mentioned questions using the library method. Based on the results, statute law has defined a large number of restrictions for the owner including the owner restriction in the obligation (...) to declare, implement protection plans, as well as sale and transfer of the cultural-historical property. Theories such as public interest are considered as the basis for the restriction of ownership on historical-cultural property. (shrink)

The foundation of the mathematical theory of probabilities is still a controversial subject. There are schools of insufficient reasoning and of cogent reasoning, of a priori determination and of frequency determination, of subjective and of objective probability. Two main difficulties exist. The first is the definition of equally like events. The second difficulty is the relation between the laws of causal natural science and the laws of statistical regularity. Is it really necessary to add to the laws of mechanics one (...) or more independent “laws of large numbers” to explain the regularities in dice throwing? In all these cases different answers exist, and the existence of a mathematical theory of probabilities remains a kind of miracle, as esoteric to the further domain of natural science as the resurrection of the dead. (shrink)

How do people stretch their understanding of magnitude from the experiential range to the very large quantities and ranges important in science, geopolitics, and mathematics? This paper empirically evaluates how and whether people make use of numerical categories when estimating relative magnitudes of numbers across many orders of magnitude. We hypothesize that people use scale words—thousand, million, billion—to carve the large number line into categories, stretching linear responses across items within each category. If so, discontinuities in position (...) and response time are expected near the boundaries between categories. In contrast to previous work that suggested only that a minority of college undergraduates employed categorical boundaries, we find that discontinuities near category boundaries occur in most or all participants, but that accurate and inaccurate participants respond in opposite ways to category boundaries. Accurate participants highlight contrasts within a category, whereas inaccurate participants adjust their responses toward category centers. (shrink)

The main result of this paper is a formal argument for the principle of maximizing expected utility that does not rely on the law of large numbers. Unlike the well-known arguments by Savage and von Neumann & Morgenstern, this argument does not presuppose the sure-thing principle or the independence axiom. The principal idea is to use the concept of transformative decision rules for decomposing the principle of maximizing expected utility into a sequence of normatively reasonable subrules. It is (...) shown that this procedure provides a resolution of Allais's paradox that cannot be obtained by Savage-style or von Neumann & Morgenstern-style arguments. (shrink)

Deterministic models in population dynamics often are really approximations to stochastic models, justified by an appeal to the law of large numbers. It is proposed to call such models pseudodeterministic. Four questions are discussed in this article: (1) What errors may be made by equating deterministically predicted values to expectations? (2) When, and in what sense, may numbers be assumed to be large? (3) How large are the variances, coefficients of variations, etc., as assigned to (...) the variables in the stochastic versions of the models? (4) What role may pseudodeterministic models play in empirical research, where problems of statistical reliability arise?As an example, a modified Nicholson-Bailey model of the interaction between insect parasitoids and their hosts is discussed; the modification consists of assigning a random (density-independent) mortality to the parasitoid population. A stochastic version of this model is discussed. The expectation of the final host density is compared with the value computed from the deterministic model. The latter value is systematically lower than the former. The magnitude of the difference depends on parameter values. The variability to be expected with the stochastic model is characterized by the coefficient of variation of the final host density; its dependence on parameter values and initial conditions is discussed. It is concluded that it is worthwhile in practical applications to estimate parasitoid mortality, and that the coefficient of variation in real situations may be far from negligible. (shrink)

Australia is second only to Israel in being the world’s most culturally diverse nation, based largely on high levels of immigration in the second part of the 20th century. From the 1970s onwards, Australia formally recognized the massive social changes brought about by postwar immigration, and provided legislation to incorporate cultural diversity into everyday lives. One such ‘legislative’ enactment saw the establishment of multicultural broadcasting in Australia, as arguably a world-first, both in its comprehensiveness and diversity. Today, Australia has a (...) public sector corporation, the Special Broadcasting Service, administering five radio services in 68 languages. Also, the Community Radio sector produces multicultural programming in 100 languages through a number of its 330 broadcast and 207 narrowcast stations. This article examines the relationship between radio and its communities. It argues that despite the ‘profile’ of SBS television, radio is much closer to its constituent communities, and therefore plays a greater role in enabling those communities to speak their own histories, beyond the confines of a consensual Anglophile paradigm. (shrink)

In the TACITUS project for using commonsense knowledge in the understanding of texts about mechanical devices and their failures, we have been developing various commonsense theories that are needed to mediate between the way we talk about the behavior of such devices and causal models of their operation. Of central importance in this effort is the axiomatization of what might be called commonsense metaphysics. This includes a number of areas that figure in virtually every domain of discourse, such as granularity, (...) scales, time, space, material, physical objects, shape, causality, functionality, and force. Our effort has been to construct core theories of each of these areas, and then to define, or at least characterize, a large number of lexical items in terms provided by the core theories. In this paper we discuss our methodological principles and describe the key ideas in the various domains we are investigating. (shrink)

Two studies are reported which demonstrate that analytic responding on everyday reasoning problems can be increased and bias eliminated after training on the law of large numbers. Critical thinking problems involving belief-consistent, neutral, and inconsistent conclusions were presented. Belief bias was eliminated when a written justification of argument strength was elicited. However, belief-based responding was still evident when evaluations of the arguments were elicited using rating scales. This finding demonstrates a dissociation between analytic and belief-based responding as a (...) function of response format. In Experiment 2 an instructional condition designed to foster decontextualised reasoning was included but was ineffective in reducing the degree to which judgements were biased by beliefs. It was concluded that training which makes available the analytic strategies necessary to evaluate a problem has the potential to facilitate performance only if the requested response triggers conscious deliberation of the evidence. (shrink)

THE ART OF CAUSAL CONJECTURE Glenn Shafer Table of Contents Chapter 1. Introduction........................................................................................ ...........1 1.1. Probability Trees..........................................................................................3 1.2. Many Observers, Many Stances, Many Natures..........................................8 1.3. Causal Relations as Relations in Nature’s Tree...........................................9 1.4. Evidence............................................................................................ ...........13 1.5. Measuring the Average Effect of a Cause....................................................17 1.6. Causal Diagrams..........................................................................................20 1.7. Humean Events............................................................................................23 1.8. Three Levels of Causal Language................................................................27 1.9. An Outline of the Book................................................................................27 Chapter 2. Event Trees............................................................................................... .....31 2.1. Situations and Events...................................................................................32 2.2. The Ordering of Situations and Moivrean Events.......................................35 2.3. Cuts................................................................................................ ..............39 2.4. Humean Events............................................................................................43 2.5. (...) Moivrean Variables......................................................................................49 2.6. Humean Variables........................................................................................53 2.7. Event Trees for Stochastic Processes...........................................................54 2.8. Timing in Event Trees.................................................................................56 2.9. Intersecting Event Trees...............................................................................60 2.10. Notes on the Literature...............................................................................61 Chapter 3. Probability Trees...........................................................................................63 3.1. Some Types of Probability Trees.................................................................64 ii 3.2. Axioms for the Probabilities of Moivrean Events.......................................68 3.3. Zero Probabilities....................................................................................... ..70 3.4. A Sample-Space Analysis of the Event-Tree Axioms.................................72 3.5. Probabilities and Expected Values for Variables.........................................74 3.6. Martingales......................................................................................... .........79 3.7. The Expectation of a Variable in a Cut........................................................83 3.8. Conditional Expected Value and Conditional Expectation.........................87 Chapter 4. The Meaning of Probability...........................................................................91 4.1. The Interpretation of Expected Value..........................................................92 4.2. The Interpretation of Expectation................................................................95 4.3. The Long Run..............................................................................................98 4.4. Changes in Belief.........................................................................................101 4.5. The Empirical Validation of Probability......................................................106 4.6. The Diversity of Uses of Probability...........................................................108 4.7. Notes on the Literature.................................................................................110 Chapter 5. Independent Events.......................................................................................113 5.1. Independence........................................................................................ .......114 5.2. Weak Independence.....................................................................................118 5.3. The Principle of the Common Cause...........................................................121 5.4. Conditional Independence............................................................................128 5.5. Notes on the Literature.................................................................................133 Chapter 6. Events Tracking Events.................................................................................135 6.1. Tracking............................................................................................ ...........137 6.2. Tracking and Conditional Independence.....................................................142 6.3. Stochastic Subsequence...............................................................................143 6.4. Singular Diagrams for Stochastic Subsequence...........................................147 6.5. Conjunctive and Interactive Forks...............................................................149 Chapter 7. Events as Signs of Events..............................................................................153 iii 7.1. Sign................................................................................................ ..............154 7.2. Weak Sign................................................................................................ ....159 7.3. The Ethics of Causal Talk............................................................................160 7.4. Screening Off...............................................................................................16 2 Chapter 8. Independent Variables...................................................................................167 8.1. Unconditional Independence........................................................................170 8.2. Conditional Independence............................................................................175 8.3. Independence for Partitions.........................................................................177 8.4. Independence for Families of Variables......................................................182 8.5. Individual Properties of the Independence Relations...................................186 Chapter 9. Variables Tracking Variables........................................................................189 9.1. Tracking and Conditional Independence: A Summary...............................190 9.2. Strong Tracking............................................................................................ 192 9.3. Strong Tracking and Conditional Independence..........................................198 9.4. Stochastic Subsequence...............................................................................201 9.5. Functional Dependence................................................................................203 9.6. Tracking in Mean.........................................................................................204 9.7. Linear Tracking............................................................................................ 207 9.8. Tracking by Partitions..................................................................................210 9.9. Tracking by Families of Variables...............................................................212 Chapter 10. Variables as Signs of Variables...................................................................215 10.1. Sign................................................................................................ ............219 10.2. Linear Sign................................................................................................ .222 10.3. Scored Sign................................................................................................ 225 10.4. Families of Variables.................................................................................227 Chapter 11. AnTheory of Event Trees.............................................................229 11.1. Event Trees as Sets of Sets........................................................................230 11.2. Event Trees as Partially Ordered Sets........................................................232 iv 11.3. Regular Event Trees...................................................................................240 11.4. The Resolution of Moivrean Variables......................................................244 11.5. Humean Events and Variables...................................................................246 Chapter 12. Martingale Trees..........................................................................................247 12.1. Examples of Decision Trees......................................................................249 12.2. The Meaning of Probability in a Decision Tree.........................................253 12.3. Martingales......................................................................................... .......257 12.4. The Structure of Martingale Trees.............................................................261 12.5. Probability and Causality...........................................................................265 12.6. Lower and Upper Probability.....................................................................269 12.7. The Law of Large Numbers.......................................................................272 12.8. Notes on the Literature...............................................................................274 Chapter 13. Refining............................................................................................ ...........275 13.1. Examples of Refinement............................................................................277 13.2. A Constructive Definition of Finite Refinement.......................................281 13.3. Axioms for Refinement..............................................................................282 13.4. Lifting Moivrean Events and Variables.....................................................288 13.5. Refining Martingale Trees.........................................................................288 13.6. Grounding........................................................................................... .......294 Chapter 14. Principles of Causal Conjecture..................................................................299 14.1. The Diversity of Causal Explanation.........................................................302 14.2. The Mean Effect of the Happening of a Moivrean Event..........................305 14.3. The Effect of a Humean Variable..............................................................311 14.4. Attribution and Generality.........................................................................316 14.5. The Statistical Measurement of the Effect of a Cause...............................319 14.6. Measurement by Experiment.....................................................................320 14.7. Using Our Knowledge of How Things Work............................................322 v 14.8. Sampling Error...........................................................................................329 14.9. The Sampling Frame..................................................................................329 14.10. Notes on the Literature.............................................................................330 Chapter 15. Causal Models.............................................................................................3 31 15.1. The Causal Interpretation of Statistical Prediction....................................333 15.2. Generalizing to a Family of Exogenous Variables....................................337 15.3 Some Joint Causal Diagrams......................................................................339 15.4. Causal Path Diagrams................................................................................342 15.5. Causal Relevance Diagrams.......................................................................346 15.6. The Meaning of Latent Variables..............................................................352 15.7. Notes on the Literature...............................................................................357 Chapter 16. Representing Probability Trees...................................................................359 16.1. Three Graphical Representations...............................................................361 16.2. Skeletal Simplifications.............................................................................368 16.3. Martingale Trees in Type Theory...............................................................371 Appendix A. Huygens’s Probability Trees.....................................................................379 Huygens’s Manuscript in Translation..................................................................380 Appendix B. Some Elements of Graph Theory..............................................................385 B1. Undirected Graphs........................................................................................385 B2. Directed Graphs............................................................................................38 6 Appendix C. Some Elements of Order Theory...............................................................393 C1. Partial and Quasi Orderings.........................................................................393 C2. Singular and Joint Diagrams for Binary Relations.......................................394 C3. Lattices............................................................................................ .............395 C4. The Lattice of Partitions of a Set..................................................................396 Appendix D. The Sample-Space Framework for Probability.........................................399 D1. Probability Measures....................................................................................399 D2. Variables........................................................................................... ...........400 vi D3. Families of Variables...................................................................................401 D4. Expected Value............................................................................................402 D5. The Law of Large Numbers.........................................................................405 D6. Conditional Probability................................................................................406 D7. Conditional Expected Value........................................................................407 Appendix E. Prediction in Probability Spaces................................................................409 E1. Conditional Distribution...............................................................................411 E2. Regression on a Single Variable...................................................................412 E3. Regression on a Partition or a Family of Variables......................................415 E4. Linear Regression on a Single Variable.......................................................418 E5. Linear Regression on a Family of Variables................................................422 Appendix F. Sample-Space Concepts of Independence.................................................425 F1. Overview............................................................................................ ...........426 F2. Independence Proper.....................................................................................432 F3. Unpredictability in Mean..............................................................................434 F4. Simple Uncorrelatedness..............................................................................437 F5. Mixed Uncorrelatedness...............................................................................438 F6. Partial Uncorrelatedness...............................................................................440 F7. Independence for Partitions..........................................................................442 F8. Independence for Families of Variables.......................................................445 F9. The Basic Role of Uncorrelatedness.............................................................448 F10. Dawid’s Axioms.........................................................................................449 Appendix G. Prediction Diagrams..................................................................................453 G1. Path Diagrams............................................................................................ ..454 G2. Generalized Path Diagrams..........................................................................462 G3. Relevance Diagrams.....................................................................................466 G4. Bubbled Relevance Diagrams......................................................................475 Appendix H. Abstract Stochastic Processes...................................................................479 vii H1. Probability Conditionals and Probability Distributions...............................477 H2. Abstract Stochastic Processes......................................................................479 H3. Embedding Variables and Processes in a Sample Space.............................480 References.......................................................................................... ..............................491. (shrink)

This paper investigates the historical origin and ancestors of typicality, which is now a central concept in Boltzmannian Statistical Mechanics and Bohmian Mechanics. Although Ludwig Boltzmann did not use the word typicality, its main idea, namely, that something happens almost always or is valid for almost all cases, plays a crucial role for his explanation of how thermodynamic systems approach equilibrium. At the beginning of the 20th century, the focus on almost always or almost everywhere was fruitful for developing measure (...) theory and probability theory. It was apparently Hugh Everett III who first mentioned typicality in physics in 1957 while searching for a justification of the Born rule in his interpretation of quantum mechanics. The historically closest concept before these developments is moral certainty, which was invented by the medieval French theologian Jean Gerson, and it became a standard concept at least until the Age of Enlightenment, when Jakob Bernoulli proved the Law of Large numbers. (shrink)

In the Medieval period, Roman law and canon law formed ius commune or the common European law. The similarity between Roman and canon law was that they used the same methods and the difference was that they relied on different authoritative texts. In their works canonists and civilists combined the ancient Greek achievements in philosophy with the Roman achievements in the field of law. Canonists were the first who carried out research on the distinctions between various legal sources and systematized (...) them according to a hierarchical order. The Medieval civilists sought solutions in canon law for a large number of problems that Justinian’s Codification did not hinge on or did it only superficially. Solutions offered by canon law were accepted not only in the civil law of Continental Europe, but also in the English law. (shrink)

In this paper, we focus on the launch of the Great Soviet Encyclopedia, which was first published in 1925. We present the context of the launch and explain why it was closely connected to the period of the New Economic Policy. In the last section, we examine four articles about randomness and probability included in the first volumes of the encyclopedia in order to illustrate some debates from within the scientific scene in the USSR during the 1920s.

Rosenberg applies current thinking in philosophy of science to neoclassical economics in order to assess its claims to scientific standing. Although philosophers have used history and psychology as paradigms for the examination of social science, there is good reason to believe that economics is a more appropriate subject for analysis: it is the most systematized and quantified of the social sciences; its practitioners have reached a measure of consensus on important aspects of their subject; and it encompasses a large (...) number of apparently law-like propositions. (shrink)

A number of theoretical positions in psychology—including variants of case‐based reasoning, instance‐based analogy, and connectionist models—maintain that abstract rules are not involved in human reasoning, or at best play a minor role. Other views hold that the use of abstract rules is a core aspect of human reasoning. We propose eight criteria for determining whether or not people use abstract rules in reasoning, and examine evidence relevant to each criterion for several rule systems. We argue that there is substantial evidence (...) that several different inferential rules, including modus ponens, contractual rules, causal rules, and the law of large numbers, are used in solving everyday problems. We discuss the implications for various theoretical positions and consider hybrid mechanisms that combine aspects of instance and rule models. (shrink)

One of the issues with introducing animal rights law is whether the problem is quantitative or qualitative, whether it can be achieved by working within existing legal paradigms or whether it requires a new set of paradigms. The answer is fundamental: a quantitative problem can be solved by applying more of the same solutions, while a qualitative problem requires completely different solutions. The qualitative camp can be represented by, say, Professor Gary Francione, demanding not only rights for animals but that (...) they have personhood and are no longer considered “property.” The quantitative approach can be finely represented by Professor David Favre, who has argued for many years that we can make progress by developing existing legal principles to give some rights to some animals, and he does so eloquently in his new book. He is clear in his view that progress is, and can continue to be, made with a series of steps, as he puts it, “climbing the slippery slope toward personhood” (p. viii).His case is supported by looking backward—progress made in establishing the rules governing humans in their relationships with animals—as well as forward to developments that will support continuing improvement in the lives of animals, not least the potential for technology (for example, through cell- and plant-based meat). At the same time, he also rightly rejects the concepts of animal welfare and animal rights being at odds, let alone either of them being a panacea for better lives for animals.One aspect of animal law that Favre believes does need to change is that of property status. As he says, the status of animals as personal property is “critical to the future of animal law” (p. 25). In general, the solutions have been largely binary—that animals simply cannot be property for them to have meaningful rights or that animals can be property and still have rights. The former makes intuitive good sense (and is consistent with the objectives of the human abolitionist movements) but would have enormous effects on economic activity throughout the world. The latter is appealing because it does not change the current paradigm, so can be developed more easily.Favre's premise is that we are moving toward rights for animals, so we need to find a way of integrating animals into the current legal system rather than having to destroy and reconstruct the legal system. His ingenious solution is to propose a fourth category of property, alongside real, personal, and intellectual property: “living property.”Within this category, he envisages using the common law concept of the trust to provide equitable “self-ownership,” where an animal has equitable “ownership” while a human retains legal “ownership.” He then connects this concept to adaptations in other aspects of law that start to recognize the selfhood of animals, creating a platform onto which broader and deeper rights for animals can develop, such as the status of “animalhood” (p. 72) and the corresponding legal right of consideration before decisions are made by the state affecting an animal's fundamental interests (such as continuing to be alive). By putting animals in a separate class—rather than uncomfortably leaving them in personal property—he creates a possible route for animal rights that seems less paradigm-busting than removing them from being property altogether.Drawing on evidence from criminal law, tort law, equity (trusts), property law, and family law, he shows how in many places—with a focus on states in the United States—existing laws are being adapted (“green shoots” p. 81) to recognize that animals are more than personal property, in ways that cannot be explained simply through the growth of regulations. For example, family law is starting to recognize that in the division of matrimonial assets, companion animals cannot be treated simply like any other personal property such as a table or a car based on who paid for them. Because animals are “living property,” the courts should take account of other aspects, such as who cared for the animals and who looked after them. Similarly, the concept of extending trust law so that an animal can be the beneficiary of a trust—in the same way as a human can be but inanimate personal property cannot—shows that the law can understand that companion animals can play a large emotional role in our lives (as well as we in theirs), such that we want to ensure proper care for them after we die.The book raises important issues. First, it is intentionally about domestic animals, with their close links with humans, so the potential for laws to engage is much easier than with, say, farmed animals. The danger with solutions that focus on domestic or charismatic animals (e.g., dogs, chimpanzees, elephants) is that the arguments may be pertinent essentially only to those species and therefore hard to apply to animals of less charismatic species, say in a factory farm (but who are present in much greater numbers and generally badly treated in comparison). He “leaves for another day” (p. 32) the discussion about other animals. Hopefully that day will come, as it would be interesting to see how Favre could extend his concept to animals for whom the public has less affection and less appreciation of their capacities (and for whom the prospect of death is much more real and imminent).Secondly, public support for these ideas about living property may simply not be present. If there is not a sufficient groundswell of support for changing the status of animals, then whether one proposes changing it to not-property or “self-owned” property doesn't matter (if someone doesn't like needles, it doesn't matter whether the needle you propose to use is big or small). This may be too pessimistic, of course (people can be helped to overcome a fear of needles, and one way to start is with small needles). Perhaps the law should start with small changes to recognize at least a few animals—their sentience, their capacities for pleasure and pain, their emotions and preferences, their potential for autonomy—and build public support from there.There is also the underlying and more radical point about whether these ideas—the trust for a beloved companion set up by the human, the divorce settlement that doesn't just give the companion to the party who paid for them—are actually bringing about changes for animals as such or are simply adapting laws to help humans, recognizing human needs, and do not actually change the status of animals. The ends are still humans, and the means are still animals.This is not intended to paint a bleak picture: The more bluntand uncompromising part of the animal rights movement that has been focused on the qualitative problem has hardly been successful at getting animal rights laws passed anywhere in the world, so everything that moves us closer to a better, richer, more thoughtful view of animals will help. Favre's approach has ingenuity, shows good sense, and has the potential to bring about real animal rights. The themes and cases in this book that show how courts and legislatures are becoming more aware of the sentience of at least some animals, as well as proposing a middle path in the debate about animals as property, are important in the journey toward recognition of animal rights. Even more important, they may actually work. (shrink)

We propose an interpretation of physics named potentiality realism. This view, which can be applied to classical as well as to quantum physics, regards potentialities (i.e. intrinsic, objective propensities for individual events to obtain) as elements of reality, thereby complementing the actual properties taken by physical variables. This allows one to naturally reconcile realism and fundamental indeterminism in any theoretical framework. We discuss our specific interpretation of propensities, that require them to depart from being probabilities at the formal level, though (...) allowing for statistics and the law of large numbers. This view helps reconcile classical and quantum physics by showing that most of the conceptual problems that are customarily taken to be unique issues of the latter -- such as the measurement problem -- are actually in common to all indeterministic physical theories. (shrink)

It is argued that existing axiomatic theories of utility do not provide the utility principle or the principle of maximising expected utility with a formal justification. It is also argued that these theories only put mild constraints on a decision-maker in a decision-context. Finally, it is argued that the prospects are not particularly bright for finding formal non-circular arguments for the utility principle that do not rely on the law of large numbers.

A new axiomatic characterization with a minimum of conditions for entropy as a function on the set of states in quantum mechanics is presented. Traditionally unspoken assumptions are unveiled and replaced by proven consequences of the axioms. First the Boltzmann–Planck formula is derived. Building on this formula, using the Law of Large Numbers—a basic theorem of probability theory—the von Neumann formula is deduced. Axioms used in older theories on the foundations are now derived facts.

There are cases in which the literal interpretation of statutes may lead to counterintuitive consequences. When such cases go to high courts, judges may handle these counterintuitive consequences by identifying problematic rule conditions. Given that the law consists of a large number of rule conditions, it is demanding and exhaustive to figure out which condition is problematic. For solving this problem, our work aims to assist judges in civil law systems to resolve counterintuitive consequences using logic program representation of (...) statutes and Legal Debugging. The core principle of Legal Debugging is to cooperate with a user to find a culprit, a root cause of counterintuitive consequences. This article proposes an algorithm to resolve a culprit. Since the statutes are represented by logic rules but changes in law are initiated by cases, we adopt a prototypical case with judgement specified by a set of rules. Then, to resolve a culprit, we reconstruct a program so that it provides reasons as if we applied case-based reasoning to a new set of prototypical cases with judgement, which include a new set of facts relevant to a considering case. (shrink)