The probability that a fair coin tossed yesterday landed heads is either 0 or 1, but the probability that it would land heads was 0.5. In order to account for the latter type of probabilities, past probabilities, a temporal restriction operator is introduced and axiomatically characterized. It is used to construct a representation of conditional past probabilities. The logic of past probabilities turns out to be strictly weaker than the logic of standard probabilities.

The Past Hypothesis defended by David Wallace in his 2011 account of macroscopic irreversibility is technically distinct from, but in the same spirit as, that of David Albert in his 2000 book Time and Chance. I am concerned in this essay with the role of objective probability in both accounts, which I find obscure. Most of the analysis will be devoted to the classical treatments by both authors, but a final section will question whether Wallace's quantum version involving (...) unitary dynamics removes this obscurity. (shrink)

If the Past Hypothesis underlies the arrows of time, what is the status of the Past Hypothesis? In this paper, I examine the role of the Past Hypothesis in the Boltzmannian account and defend the view that the Past Hypothesis is a candidate fundamental law of nature. Such a view is known to be compatible with Humeanism about laws, but as I argue it is also supported by a minimal non-Humean "governing'' view. Some worries arise from (...) the non-dynamical and time-dependent character of the Past Hypothesis as a boundary condition, the intrinsic vagueness in its specification, and the nature of the initial probability distribution. I show that these worries do not have much force, and in any case they become less relevant in a new quantum framework for analyzing time's arrows---the Wentaculus. Hence, the view that the Past Hypothesis is a candidate fundamental law should be more widely accepted than it is now. (shrink)

This essay presents and discusses the ways that prior experience constitutes a logical black box in Umberto Eco’s discussion of serial forms in ‘Interpreting Serials’ by using the complex adaptive system model for how complexity arises and is sustained over time, as proposed by John Holland. In exploring how Holland’s model can account for some aspects of Eco’s past experience, it becomes evident that a modification of both theories to accommodate multiple, contradictory potentials simultaneously suggests we consider meaning as (...) a range of potentials rather than as a singular choice. Even though the model proposed here is incomplete and tentative, it is suggestive of possible strategies for justifying interpretations without precluding their rejection or revision at a later time. Individual interpretations are justified not by comparison to an external truth but by the existence of other possible interpretations with shared characteristics that nevertheless contradict each other. (shrink)

We discuss the meaning of probabilities in the many worlds interpretation of quantum mechanics. We start by presenting very briefly the many worlds theory, how the problem of probability arises, and some unsuccessful attempts to solve it in the past. Then we criticize a recent attempt by Deutsch to derive the quantum mechanical probabilities from the nonprobabilistic parts of quantum mechanics and classical decision theory. We further argue that the Born probability does not make sense even as (...) an additional probability rule in the many worlds theory. Our conclusion is that the many worlds theory fails to account for the probabilistic statements of standard (collapse) quantum mechanics. (shrink)

Another title in the reissued Oxford Classic Texts in the Physical Sciences series, Jeffrey's Theory of Probability, first published in 1939, was the first to develop a fundamental theory of scientific inference based on the ideas of Bayesian statistics. His ideas were way ahead of their time and it is only in the past ten years that the subject of Bayes' factors has been significantly developed and extended. Until recently the two schools of statistics were distinctly different and (...) set apart. Recent work has changed all that and today's graduate students and researchers all require an understanding of Bayesian ideas. This book is their starting point. (shrink)

A collection of newly commissioned papers on themes from David Albert's Time and Chance (HUP, 2000), with replies by Albert. Introduction [Barry Loewer, Brad Weslake, and Eric Winsberg] I. Overview of Time and Chance 1. The Mentaculus: A Probability Map of the Universe [Barry Loewer] II. Philosophical Foundations 2. The Metaphysical Foundations of Statistical Mechanics: On the Status of PROB and PH [Eric Winsberg] 3. The Logic of the Past Hypothesis [David Wallace] 4. In What Sense Is the (...) Early Universe Fine-Tuned? [Sean M. Carroll] 5. The Meta-Reversibility Objection [Christopher J. G. Meacham] 6. Typicality versus Humean Probabilities as the Foundation of Statistical Mechanics [Dustin Lazarovici] 7. The Past Hypothesis and the Nature of Physical Laws [Eddy Keming Chen] 8. On the Albertian Demon [Tim Maudlin] III. Underwriting the Asymmetries of Knowledge and Intervention 9. Reading the Past in the Present [Nick Huggett] 10. Causes, Randomness, and the Past Hypothesis [Mathias Frisch] 11. Time, Flies, and Why We Can’t Control the Past [Alison Fernandes] 12. The Concept of Intervention in Time and Chance [Sidney Felder] Conclusion [David Albert]. (shrink)

Gott ( 1993 ) has used the ‘Copernican principle’ to derive a probability distribution for the total longevity of any phenomenon, based solely on the phenomenon’s past longevity. Leslie ( 1996 ) and others have used an apparently similar probabilistic argument, the ‘Doomsday Argument’, to claim that conventional predictions of longevity must be adjusted, based on Bayes’s Theorem, in favor of shorter longevities. Here I show that Gott’s arguments are flawed and contradictory, but that one of his conclusions (...) is plausible and mathematically equivalent to Laplace’s famous—and notorious—‘rule of succession’. On the other hand, the Doomsday Argument, though it appears consistent with some common‐sense grains of truth, is fallacious; the argument’s key error is to conflate future longevity and total longevity. Applying the work of Hill ( 1968 ) and Coolen ( 1998 , 2006 ) in the field of nonparametric predictive inference, I propose an alternative argument for quantifying how past longevity of a phenomenon does provide evidence for future longevity. In so doing, I identify an objective standard by which to choose among counting time intervals, counting population, or counting any other measure of past longevity in predicting future longevity. *Received May 2007; revised October 2008. †To contact the author, please e‐mail: [email protected]. (shrink)

Stalnaker's Thesis about indicative conditionals is, roughly, that the probability one ought to assign to an indicative conditional equals the probability that one ought to assign to its consequent conditional on its antecedent. The thesis seems right. If you draw a card from a standard 52-card deck, how confident are you that the card is a diamond if it's a red card? To answer this, you calculate the proportion of red cards that are diamonds -- that is, you (...) calculate the probability of drawing a diamond conditional on drawing a red card. Skyrms' Thesis about counterfactual conditionals is, roughly, that the probability that one ought to assign to a counterfactual equals one's rational expectation of the chance, at a relevant past time, of its consequent conditional on its antecedent. This thesis also seems right. If you decide not to enter a 100-ticket lottery, how confident are you that you would have won had you bought a ticket? To answer this, you calculate the prior chance--that is, the chance just before your decision not to buy a ticket---of winning conditional on entering the lottery. The central project of this article is to develop a new uniform theory of conditionals that allows us to derive a version of Skyrms' Thesis from a version of Stalnaker's Thesis, together with a chance-deference norm relating rational credence to beliefs about objective chance. (shrink)

Illuminates legal reasoning -- and its justification At least since plato and Aristotle, thinkers have pondered the relationship between philosophical arguments and the "sophistical" arguments offered by the Sophists -- who were the first professional lawyers. Judges wield substantial political power, and the justifications they offer for their decisions are a vital means by which citizens can assess the legitimacy of how that power is exercised. However, to evaluate judicial justifications requires close attention to the method of reasoning behind decisions. (...) This new collection illuminates and explains the political and moral importance in justifying the exercise of judicial power. Explores enduring questions Focusing on work over the past century, the essays address important recurring questions, such as: When a judge or a lawyer reasons to a conclusion about what is legally required in a given case, must he also ask what is morally required? To what extent do a judge's personal, political, or moral biases affect his legal reasoning? What is the impact of such biases? Can all such biases be avoided? Is legal reasoning similar to reasoning in mathematics, logic, and linguistics, the physical sciences, the social sciences, or literature and history? Do formal logical modes of argument play any roles in legal reasoning? Solid coverage, well organized The articles were chosen to present some of the most influential works on the topic, as well as less familiar works that are thought provoking and informative. Each volume also offers a representative range of theoretical approaches to its topic, contains an introduction that locates the subject within the larger framework oftheories of legal reasoning and jurisprudence, and includes bibliographical notes on further readings. Many 19th century legal theorists argued that deduction is the central mode of legal reasoning, and that legal argument is like a deductive proof in mathematics or logic. They were attracted to this "deductivist" concept because it suggested that legal reasoning could be politically and morally neutral. This volume brings together some of the most thought-provoking articles on both sides of the debate. It also contains several leading articles that explain the role of probabilistic judgments and presumptions in various types of legal arguments, including the laws of evidence and criminal procedure. The collection is a solid introduction to the basic modes of logical argument in legal reasoning. (shrink)

Definitions of time symmetry and examples of time-directed behaviour are discussed in the framework of discrete Markov processes. It is argued that typical examples of time-directed behaviour can be described using time-symmetric transition probabilities. Some current arguments in favour of a distinction between past and future on the basis of probabilistic considerations are thereby judged to be unjustified.

In his recent book, Time and Chance, David Albert claims that by positing that there is a uniform probability distribution defined, on the standard measure, over the space of microscopic states that are compatible with both the current macrocondition of the world, and with what he calls the “past hypothesis”, we can explain the time asymmetry of all of the thermodynamic behavior in the world. The principal purpose of this paper is to dispute this claim. I argue that (...) Albert's proposal fails in his stated goal—to show how to use the time‐reversible dynamics of Newtonian physics to “underwrite the actual content of our thermodynamic experience” (Albert 2000, 159). Albert's proposal can satisfactorily explain why the overall entropy of the universe as a whole is increasing, but it does not and cannot explain the increasing entropy of relatively small, relatively short‐lived systems in energetic isolation without making use of a principle that leads to reversibility objections. (shrink)

Frederick the Great of Prussia was hailed by many as the model of an ‘Enlightened Despot’. Historians continue to debate both the concept of ‘Enlightened Despotism’ and Frederick’s credentials as an enlightened monarch. Should we talk in terms of ‘enlightened absolutism’? Of ‘reform absolutism’? Or simply drop the use of any such terms for a monarch who used his enlightened philosophising and flute playing as window dressing for a system of governance that was essentially conventional absolutism? In light of continuing (...) debate about the nature of Frederick’s reign, it is worth revisiting the views of contemporaries. As a friend of Voltaire, Frederick’s place was well established in traditional depictions of the Enlightenment as centred on the French philosophes. In the past two decades, however, scholars have broadened and deepened our conception of Enlightenment by researching the ‘social history of ideas’ and illuminating Enlightenment in various national and cultural contexts. In this vein, an analysis of perceptions of Frederick the Great can shed light on the nature of Enlightenment in Britain. Frederick’s popularity in Britain reached dizzying heights in the late 1750s as he won spectacular victories in the Continental campaign against Britain’s enemies in the Seven Years War. From the 1760s on, however, British opinion was generally critical of his regulated and militaristic state and his aggrandisement through diplomacy. While it is not within the scope of this essay to explore broader perceptions of Frederick II in late eighteenth-century Britain, it appears that writers who can be located within ‘conservative’ enlightened thought had a more positive perception of Frederick than dissenters. Samuel Johnson declared that the King of Prussia could get away with wearing plain cloths because ‘of the dignity of his character’. On separate occasions in 1780 Edmund Burke praised Frederick for the economy and efficiency of his royal court and for his religious toleration. An article is in preparation that will explore the range of British opinion on Frederick the Great at the end of his long reign. This essay, however, will focus on how some leading Rational Dissenters perceived the self-consciously enlightened King of Prussia, and provide an exposition of the Memoirs of the life and reign of Frederick (1788) by Joseph Towers. (shrink)

This paper addresses the question of how we should regard the probability distributions introduced into statistical mechanics. It will be argued that it is problematic to take them either as purely ontic, or purely epistemic. I will propose a third alternative: they are almost objective probabilities, or epistemic chances. The definition of such probabilities involves an interweaving of epistemic and physical considerations, and thus they cannot be classified as either purely epistemic or purely ontic. This conception, it will be (...) argued, resolves some of the puzzles associated with statistical mechanical probabilities: it explains how probabilistic posits introduced on the basis of incomplete knowledge can yield testable predictions, and it also bypasses the problem of disastrous retrodictions, that is, the fact the standard equilibrium measures yield high probability of the system being in equilibrium in the recent past, even when we know otherwise. As the problem does not arise on the conception of probabilities considered here, there is no need to invoke a Past Hypothesis as a special posit to avoid it. (shrink)

This paper addresses the question of how we should regard the probability distributions introduced into statistical mechanics. It will be argued that it is problematic to take them either as purely subjective credences, or as objective chances. I will propose a third alternative: they are "almost objective" probabilities, or "epistemic chances". The definition of such probabilities involves an interweaving of epistemic and physical considerations, and so cannot be classified as either purely subjective or purely objective. This conception, it will (...) be argued, resolves some of the puzzles associated with statistical mechanical probabilities; it explains how probabilistic posits introduced on the basis of incomplete knowledge can yield testable predictions, and it also bypasses the problem of disastrous retrodictions, that is, the fact the standard equilibrium measures yield high probability of the system being in equilibrium in the recent past, even when we know otherwise. As the problem does not arise on the conception of probabilities considered here, there is no need to invoke a Past Hypothesis as a special posit to avoid it. (shrink)

In a quantum universe with a strong arrow of time, it is standard to postulate that the initial wave function started in a particular macrostate---the special low-entropy macrostate selected by the Past Hypothesis. Moreover, there is an additional postulate about statistical mechanical probabilities according to which the initial wave function is a ''typical'' choice in the macrostate. Together, they support a probabilistic version of the Second Law of Thermodynamics: typical initial wave functions will increase in entropy. Hence, there are (...) two sources of randomness in such a universe: the quantum-mechanical probabilities of the Born rule and the statistical mechanical probabilities of the Statistical Postulate. I propose a new way to understand time's arrow in a quantum universe. It is based on what I call the Thermodynamic Theories of Quantum Mechanics. According to this perspective, there is a natural choice for the initial quantum state of the universe, which is given by not a wave function but by a density matrix. The density matrix plays a microscopic role: it appears in the fundamental dynamical equations of those theories. The density matrix also plays a macroscopic / thermodynamic role: it is exactly the projection operator onto the Past Hypothesis subspace. Thus, given an initial subspace, we obtain a unique choice of the initial density matrix. I call this property "the conditional uniqueness" of the initial quantum state. The conditional uniqueness provides a new and general strategy to eliminate statistical mechanical probabilities in the fundamental physical theories, by which we can reduce the two sources of randomness to only the quantum mechanical one. I also explore the idea of an absolutely unique initial quantum state, in a way that might realize Penrose's idea of a strongly deterministic universe. (shrink)

Thirty years ago, grounded cognition had roots in philosophy, perception, cognitive linguistics, psycholinguistics, cognitive psychology, and cognitive neuropsychology. During the next 20 years, grounded cognition continued developing in these areas, and it also took new forms in robotics, cognitive ecology, cognitive neuroscience, and developmental psychology. In the past 10 years, research on grounded cognition has grown rapidly, especially in cognitive neuroscience, social neuroscience, cognitive psychology, social psychology, and developmental psychology. Currently, grounded cognition appears to be achieving increased acceptance throughout (...) cognitive science, shifting from relatively minor status to increasing importance. Nevertheless, researchers wonder whether grounded mechanisms lie at the heart of the cognitive system or are peripheral to classic symbolic mechanisms. Although grounded cognition is currently dominated by demonstration experiments in the absence of well-developed theories, the area is likely to become increasingly theory driven over the next 30 years. Another likely development is the increased incorporation of grounding mechanisms into cognitive architectures and into accounts of classic cognitive phenomena. As this incorporation occurs, much functionality of these architectures and phenomena is likely to remain, along with many original mechanisms. Future theories of grounded cognition are likely to be heavily influenced by both cognitive neuroscience and social neuroscience, and also by developmental science and robotics. Aspects from the three major perspectives in cognitive science—classic symbolic architectures, statistical/dynamical systems, and grounded cognition—will probably be integrated increasingly in future theories, each capturing indispensable aspects of intelligence. (shrink)

This chapter unfolds a central philosophical problem of statistical mechanics. This problem lies in a clash between the Static Probabilities offered by statistical mechanics and the Dynamic Probabilities provided by classical or quantum mechanics. The chapter looks at the Boltzmann and Gibbs approaches in statistical mechanics and construes some of the great controversies in the field — for instance the Reversibility Paradox — as instances of this conflict. It furthermore argues that a response to this conflict is a critical choice (...) that shapes one's understanding of statistical mechanics itself, namely, whether it is to be conceived as a special or fundamental science. The chapter details some of the pitfalls of the latter ‘globalist’ position and seeks defensible ground for a kind of ‘localist’ alternative. (shrink)

Greek literature embodies a number of instances of the optative in main clauses that are acknowledged either definitely or probably to have a reference to past time. These are mostly well known, but the object of this article is to reconsider them and to attempt an explanation. They are certainly commonest in the Homeric period, but later examples are by no means wanting.

In lieu of an abstract, here is a brief excerpt of the content:The Logic of Probabilities in Hume's Argument against Miracles Fred Wilson The position is often stated that Hume's discussion of miracles is inconsistent with his views on the logical or ontological status oflaws ofnature and with his more general scepticism. Broad, for one, has so argued.1 Hume's views on induction are assumed to go somethinglike this. Any attempt to demonstrate knowledge ofmatters offact presupposes causal reasoning, but the latter (...) is based not on any perception of necessary connections, but on an unreasoned expectation that because events have been constantly conjoined in the past they will be constantly conjoined in the future. The fact that our past experience gives rise to certain expectations provides absolutely no reason to think that these expectations will be fulfilled. But this is forgotten as soon as he turns to the topic of miracles where the argument requires certain assumptions about laws of nature. Here he claims that the laws of nature arebased upon afirm andunalterable experience and dismisses what he apparently admits to be strong evidence for miracles on the basis of the claim that such events are absolutely impossible. The inconsistency between the discussion of miracles and the earlier discussions of induction and causality is clear. Nonetheless, there is a certain implausibility to this claim that makes it a difficult one to entertain seriously, since it is unlikely that a philosopher as careful as Hume would have failed to recognize the inconsistency if it existed. What I propose here to argue is that there is in fact no inconsistency, and that this becomes clear once one places the discussion ofmiracles in the broader context ofthe overall argument ofthe first Enquiry. In particular, the charge ofinconsistency disappears once one eliminates the caricature ofHume's views on causal reasoning upon which it rests. To come to grips with the discussion of miracles in Hume it is necessary to place that discussion in its historical context. In the generally empiricist atmosphere that developed in Britain following the mid-seventeenth century civil wars, the general defence of the reasonableness of Christianity consisted of two parts. First, it washeldthatrational argument, whether causal or teleological orboth, could yield a reasonable belief that a deity ofa more or less traditional sort existed. Then, second, in order to establish further that Jesus was the Son ofGod it was held that one could rely upon a chain oftestimony Volume XV Number 2 255 FRED WILSON leading back from the present to the past to infer the existence of miracles testifying to the divinity ofJesus. Hume attacks this second inference in the essay on miracles andthe first in the essay that follows in the Enquiries on God's particular providence (the themes of which are developed more fully in the Dialogues on Natural Religion). Given the central place that the appeal to miracles held in Christian apologetics in the 17th and 18th centuries, Hume's systematic attack on the argument from miracles constituted a clear attack on Christianity. To be sure, Hume's argument would never have persuaded JohnWesley, but that would hardly have surprised Hume himself, nor should it surprise us: those in the grips of religious enthusiasm, whether it be John Wesley or Jimmy Swaggart, will hardly be persuaded by rational argument, however much it is still true that they ought to be persuaded by it. The appeal to miracles tojustify specifically Christian beliefs had a long history. One can find it already in the late Roman period in Eusebius. Muslim thinkers were to challenge this traditional defence by arguing that chains oftestimony become more unreliable the longer they become. As for their own Islamic beliefs, these they admitted had also to be justified by an appeal to miracles, but there was, it was claimed, a unique self-justifying miracle in the Koran, the specially revealed Word of God. The critical theme of the Muslim apologists, that the longer a chain of testimony the less assurance it gives of the fact that a miracle had occurred, was developed in Britain in the 17th and 18th centuries.6 It seems to have been introduced by the Oxford orientalist Edward Pocock who had discovered this line of Muslim apologetics while... (shrink)

People are ‘biased toward the future’: all else being equal, we typically prefer to have positive experiences in the future, and negative experiences in the past. Several explanations have been suggested for this pattern of preferences. Adjudicating among these explanations can, among other things, shed light on the rationality of future-bias: For instance, if our preferences are explained by unjustified beliefs or an illusory phenomenology, we might conclude that they are irrational. This paper investigates one hypothesis, according to which (...) future-bias is explained by our having a phenomenology that we describe, or conceive of, as being as of time robustly passing. We empirically tested this hypothesis and found no evidence in its favour. Our results present a puzzle, however, when compared with the results of an earlier study. We conclude that although robust passage phenomenology on its own probably does not explain future-bias, having this phenomenology and taking it to be veridical may contribute to future-bias. (shrink)

In the past twenty years, interest in the epistemic status of religious belief has greatly increased. Leading this revival are the philosophers Basil Mitchell and Richard Swinburne, who believe that {eligious belief can be justified using inductive "best explanation" arguments. However, while Swinburne's approach is formal, using the calculus of Bayes Theorem, Mitchell's is informal, based on his recognition of judgment as central to such an assessment. This book is the first full length comparison of these two men and (...) their styles of justifying religious belief. The first half addresses the issues of rationality, endorsing Mitchell's methodology; the second half explores the concept of theistic explanation and demonstrates that the ontological argument has a place in the comprehensive explanatory power of theism. (shrink)

Studies in social psychology inadvertently call for a subject's reconstruction of past behaviors when using interviews, questionnaires, or personality inventories. Since subjects' past behaviors are unobservables, subjects reconstruct their past retroductively. However, since the behaviors are not perceptually observed, such inquiry is decontextualized and probabilistic. Hence, reconstructions are frequently organized in terms of commonsense plausibility and personal accountability rather than causality. It is proposed that such inquiry may be improved by having subjects not only endorse preformatted material, (...) but also by providing warrants for subjects' endorsements. The provision of warrants has been shown to structure past reality perceptions of laypersons . The provision of warrants not only recontextualizes recall of past behaviors but also discloses historical continuity which, in turn, suggests higher probabilities of future behaviors. (shrink)

Leibniz’s account of probability has come into better focus over the past decades. However, less attention has been paid to a certain domain of application of that account, that is, the application of it to the moral or ethical domain—the sphere of action, choice and practice. This is significant, as Leibniz had some things to say about applying probability theory to the moral domain, and thought the matter quite relevant. Leibniz’s work in this area is conducted at (...) a high level of abstraction. It establishes a proof of concept, rather than concrete guidelines for how to apply calculations to specific cases. Still, this highly abstract material does allow us to begin to construct a framework for thinking about Leibniz’s approach to the ethical side of probability. (shrink)

Reconstructing the Past seeks to clarify and help resolve the vexing methodological issues that arise when biologists try to answer such questions as whether human beings are more closely related to chimps than they are to gorillas. It explores the case for considering the philosophical idea of simplicity/parsimony as a useful principle for evaluating taxonomic theories of evolutionary relationships. For the past two decades, evolutionists have been vigorously debating the appropriate methods that should be used in systematics, the (...) field that aims at reconstructing phylogenetic relationships among species. This debate over phylogenetic inference, Elliott Sober observes, raises broader questions of hypothesis testing and theory evaluation that run head on into long standing issues concerning simplicity/parsimony in the philosophy of science. Sober treats the problem of phylogenetic inference as a detailed case study in which the philosophical idea of simplicity/parsimony can be tested as a principle of theory evaluation. Bringing together philosophy and biology, as well as statistics, Sober builds a general framework for understanding the circumstances in which parsimony makes sense as a tool of phylogenetic inference. Along the way he provides a detailed critique of parsimony in the biological literature, exploring the strengths and limitations of both statistical and nonstatistical cladistic arguments. (shrink)

Over the past two decades, gamblers have begun taking mathematics into account more seriously than ever before. While probability theory is the only rigorous theory modeling the uncertainty, even though in idealized conditions, numerical probabilities are viewed not only as mere mathematical information, but also as a decision-making criterion, especially in gambling. This book presents the mathematics underlying the major games of chance and provides a precise account of the odds associated with all gaming events. It begins by (...) explaining in simple terms the meaning of the concept of probability for the layman and goes on to become an enlightening journey through the mathematics of chance, randomness and risk. It then continues with the basics of discrete probability, combinatorics and counting arguments for those interested in the supporting mathematics. These mathematic sections may be skipped by readers who do not have a minimal background in mathematics; these readers can skip directly to the Guide to Numerical Results to pick the odds and recommendations they need for the desired gaming situation. Doing so is possible due to the organization of that chapter, in which the results are listed at the end of each section, mostly in the form of tables. The chapter titled The Mathematics of Games of Chance presents these games not only as a good application field for probability theory, but also in terms of human actions where probability-based strategies can be tried to achieve favorable results. Through suggestive examples, the reader can see what are the experiments, events and probability fields in games of chance and how probability calculus works there. The main portion of this work is a collection of probability results for each type of game. Each game s section is packed with formulas and tables. Each section also contains a description of the game, a classification of the gaming events and the applicable probability calculations. The primary goal of this work is to allow the reader to quickly find the odds for a specific gaming situation, in order to improve his or her betting/gaming decisions. Every type of gaming event is tabulated in a logical, consistent and comprehensive manner. The complete methodology and complete or partial calculations are shown to teach players how to calculate probability for any situation, for every stage of the game for any game. Here, readers can find the real odds, returned by precise mathematical formulas and not by partial simulations that most software uses. Collections of odds are presented, as well as strategic recommendations based on those odds, where necessary, for each type of gaming situation. The book contains much new and original material that has not been published previously and provides great coverage of probabilities for the following games of chance: Dice, Slots, Roulette, Baccarat, Blackjack, Texas Hold em Poker, Lottery and Sport Bets. Most of games of chance are predisposed to probability-based decisions. This is why the approach is not an exclusively statistical one, but analytical: every gaming event is taken as an individual applied probability problem to solve. A special chapter defines the probability-based strategy and mathematically shows why such strategy is theoretically optimal.". (shrink)

Many have interpreted symbolic material culture in the deep past as evidencing the origins sophisticated, modern cognition. Scholars from across the behavioural and cognitive sciences, including linguists, psychologists, philosophers, neuroscientists, primatologists, archaeologists, and palaeoanthropologists have used such artefacts to assess the capacities of extinct human species, and to set benchmarks, milestones, or otherwise chart the course of human cognitive evolution. To better calibrate our expectations, the present paper instead explores the material culture of three contemporary African forager groups. Results (...) show that, although these groups are unequivocally behaviourally modern, they would leave scant long-lasting evidence of symbolic behaviour. Artefact sets are typically small, perhaps as a consequence of residential mobility. When traded materials are excluded, few artefacts have components with moderate–to–strong taphonomic signatures. The present analyses show that artefact function influences preservation probability, such that utilitarian tools for the processing of materials and the preparation of food are disproportionately likely to contain archaeologically traceable components. There are substantial differences in material use among populations, which create important population-level variation in preservation probability, independent of cognitive differences. I discuss the factors – cultural, ecological, and practical – that influence material choice. In so doing, I highlight the difficulties of using past material culture as an evolutionary or cognitive yardstick. (shrink)

I attempt to get as clear as possible on the chain of reasoning by which irreversible macrodynamics is derivable from time-reversible microphysics, and in particular to clarify just what kinds of assumptions about the initial state of the universe, and about the nature of the microdynamics, are needed in these derivations. I conclude that while a “Past Hypothesis” about the early Universe does seem necessary to carry out such derivations, that Hypothesis is not correctly understood as a constraint on (...) the early Universe’s entropy. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Literature 27.2 (2003) 402-414 [Access article in PDF] Past's Weight, Future's Promise:Reading Electra William Junker I SOPHOCLES' Electrapresents as its main character a woman who is tortured by the remembrance of things past: Even my pitiful bed remembers, there in that dreadful house, my long night-watches grieving my unlucky father who found no foreign resting place in war but died when my mother and Aegisthus, (...) her lover, took an axe to his head as a woodman does a tree. (85-95) 1The story she tells is a familiar one. Her father, Agamemnon, having returned from the Trojan War, is murdered by Clytemnestra and her lover Aegisthus. Clytemnestra's motives are complex and unclear, for Agamemnon had previously sacrificed their daughter, Iphigenia, in order to ensure military success, and it is an open question as to what extent maternal vengeance as opposed to romantic intrigue motivated [End Page 402] her action. After the murder, Electra rescues her brother Orestes from probable death by sending him abroad (Aegisthus fearing revenge would have killed the child) while she grows and lives as an outcast in her own home, awaiting the day when the grown Orestes will return and bring their father's murderers to justice.The story, and its ensuing consequences, had already been handled masterfully in Aeschylus' Oresteia; but in Sophocles' play there is considerably more interest shown in Electra herself. 2 This focus results in a magnification of Electra's mourning for her father along with a sustained meditation upon how this mourning results in her inability to live, in any full sense, with the knowledge of his death. She cannot escape the memory of the injustice done him, and this memory fuels her hopes for Orestes' return even as it renders her, for all practical purposes, incapacitated.Her character thus described draws continual attention during the course of the play, and in their first appearance the Chorus admonishes her: "When you go on too far with grief, / you come to endless anguish, and destroy yourself. / There is no salvation from your troubles here, / so why should you persist in suffering" (136-39). Electra herself views her persistent suffering as necessary and noble, a show of allegiance to her murdered father and a refusal to compromise with his killers. The Chorus and Chrysothemis, however, argue that by so carrying on Electra is in reality causing more damage to herself than her situation warrants. Her ragged appearance, quite noticeable stench, and public outcries do nothing to help redress her father's honor; they rather highlight her mania and point toward insanity. At one point Chrysothemis implies as much. To Electra's accusation that her position of political compromise is dishonorable, she rebuts: "There's honor in not letting madness defeat you" (395)—suggesting both a different conception of honor and a veiled appraisal of her sister's mental health. In the service of life and sanity, the Chorus and Chrysothemis seem to say, leave off your suffering. In the service of honor and the dead, Electra rejoins, I cannot. The movement of the play in fact continually juxtaposes the position of Electra with that of her critics, and this juxtaposition is one that calls for interpretive elucidation. 3An especially fertile way of adjudicating these issues in Electra is afforded us by Nietzsche, and, specifically, by his On the Advantage and Disadvantage of History for Life. This is the case because, as I want to suggest, Nietzsche's essay provides a framework which reconstructs, and in a unique way, the central problem of Electra. That problem is [End Page 403] essentially one of reconciling the conflicting claims of familial honor and memory with the claims of life. Through Nietzsche's exposition of history and its relation to life, this problem is both illumined and given greater subtlety—illumined, in that the potential agôn between history and life, while present in Sophocles' Electra, is not just obvious; and made subtle, in that Nietzsche's proposed solution to this agôn allows for a reading of the... (shrink)

Arguably, Hume's greatest single contribution to contemporary philosophy of science has been the problem of induction (1739). Before attempting its statement, we need to spend a few words identifying the subject matter of this corner of epistemology. At a first pass, induction concerns ampliative inferences drawn on the basis of evidence (presumably, evidence acquired more or less directly from experience)—that is, inferences whose conclusions are not (validly) entailed by the premises. Philosophers have historically drawn further distinctions, often appropriating the term (...) “induction” to mark them; since we will not be concerned with the philosophical issues for which these distinctions are relevant, we will use the word “inductive” in a catch-all sense synonymous with “ampliative”. But we will follow the usual practice of choosing, as our paradigm example of inductive inferences, inferences about the future based on evidence drawn from the past and present. A further refinement is more important. Opinion typically comes in degrees, and this fact makes a great deal of difference to how we understand inductive inferences. For while it is often harmless to talk about the conclusions that can be rationally believed on the basis of some.. (shrink)

This book explores important topics in Darwin’s theory of evolution and in the twentieth- and twenty-first-century theories that grew out of it. These topics include fitness, adaptation, altruism, intragenomic conflict, units of selection, genetic drift, the randomness of mutation, gradualism, common ancestry, taxonomy, race, phylogenetic inference, and optimality models. The book brings these and other biological topics into contact with numerous philosophical ideas − operationalism, reductionism, conventionalism, null hypotheses and default reasoning, instrumentalism versus realism, likelihoods versus probabilities, model selection, essentialism, (...) falsifiability, parsimony, the principle of the common cause, comparisons of causal power, interpretations of probaiblity, indeterminism, sensitivity to initial conditions, and the knowability of the past. (shrink)

The prediction of future events from our knowledge of past events is one of the main functions of Science. Such predictions are made possible by inferring causal relations between events from observed regularities. These relations are then codified into “laws of nature,” and it is through knowledge of these laws that prediction becomes possible. The concept of “causal relation” is thus a fundamental one in the structure of science. Now recent advances in physics have led scientists to modify considerably (...) their views of this concept. It has been found necessary to abandon the idea of rigidly deterministic causal relations, and substitute in its stead relations that are only determined with a certain degree of probability. The main purpose of this paper is to analyse the meaning of these changes in our concepts of causality. (shrink)

There are two main classes of interpretations of probability. The first are those that rely on a measure of frequency. The other is those that take a logical or subjective view of a unique event, independent of past or future events. The interpretation of probability which is used in the book is then defined as evidential probability, a function based on a set of known statements based on frequency or measure. The properties of probability are (...) then enumerated and explained. Probabilities can also be based on statistical inference. Statistical inferences are statistical properties with high probability based on statistical evidence. (shrink)

The problem of the nature of probability laws in physics is a central one from the epistemologic point of view : perhaps the most central one for twentieth century physics, if we consider, on the one hand, the harsh debates to which it has given rise in the recent past (left unconcluded by the protagonists and most often by their successors) and the vagueness of the answers which are generally proposed by contemporary physicists when asked about this problem (...) ; and, on the other hand, the perspectives of theoretical physics as it is developing under our eyes in the various domains - and, perhaps above all, in the "frontier" domains such as particle symmetry fields and cosmology, or, differently, dynamics of continuous media, turbulences and chaos, where probability is a - if not the - fundamental concept. (shrink)

In May of 1973 we organized an international research colloquium on foundations of probability, statistics, and statistical theories of science at the University of Western Ontario. During the past four decades there have been striking formal advances in our understanding of logic, semantics and algebraic structure in probabilistic and statistical theories. These advances, which include the development of the relations between semantics and metamathematics, between logics and algebras and the algebraic-geometrical foundations of statistical theories (especially in the sciences), (...) have led to striking new insights into the formal and conceptual structure of probability and statistical theory and their scientific applications in the form of scientific theory. The foundations of statistics are in a state of profound conflict. Fisher's objections to some aspects of Neyman-Pearson statistics have long been well known. More recently the emergence of Baysian statistics as a radical alternative to standard views has made the conflict especially acute. In recent years the response of many practising statisticians to the conflict has been an eclectic approach to statistical inference. Many good statisticians have developed a kind of wisdom which enables them to know which problems are most appropriately handled by each of the methods available. The search for principles which would explain why each of the methods works where it does and fails where it does offers a fruitful approach to the controversy over foundations. (shrink)

The argument set out here provides a new understanding of the methods followed by the Renaissance architect Andrea Palladio for depicting the monuments of classical antiquity—in both his drawings and the plates of his seminal architectural treatises, the Quattro libri dell’architettura. It accepts the now-established view that Palladio’s early studies were frequently copied from the drawings of previous practitioners, but it reasons that his later output was also heavily dependent on past achievements. This is contrary to the claims Palladio (...) made in the Quattro libri, which contend that his drawings were based on surveys he had carried out himself, and to the confirmatory presumptions of modern-day scholars. The article first examines the ancient temples Palladio illustrated in the treatise, and observes that they have numerous features that are unsupported by the archaeological record but are also seen either in Palladio’s earlier copy drawings or in the graphic works of his predecessors or contemporaries. Secondly, it considers Palladio’s drawings of bath complexes, which remained unpublished only because of his death, noting that some of them are still reckoned to be invaluable and reliable records of then-accessible remains that Palladio was able to inspect but no longer exist. The conclusion here, however, is that his reconstructions were again based on earlier surveys that he judged dependable, but which he often then adapted and adjusted in line with his personal preferences. The article then continues with a discussion of his probable debts to older architects such as Michele Sanmicheli and Antonio da Sangallo the Younger, and his relationship with his contemporary Pirro Ligorio, whose drawings are often markedly similar to Palladio’s. It concludes by considering whether Palladio’s coverage of ancient architecture reveals the extent of such knowledge at the midpoint of the sixteenth century. (shrink)

A common way of characterizing Boltzmann’s explanation of thermodynamics in term of statistical mechanics is with reference to three ingredients: the dynamics, the past hypothesis, and the statistical postulate. In this paper I focus on the statistical postulate, and I have three aims. First, I wish to argue that regarding the statistical postulate as a probability postulate may be too strong: a postulate about typicality would be enough. Second, I wish to show that there is no need to (...) postulate anything, for the typicality postulate can be suitably derived from the dynamics. Finally, I discuss how the attempts to give preference to certain stochastic quantum theories (such as the spontaneous collapse theory) over deterministic alternatives on the basis that they do not need the statistical postulate fail. (shrink)

IBE ('Inference to the best explanation' or abduction) is a popular and highly plausible theory of how we should judge the evidence for claims of past events based on present evidence. It has been notably developed and supported recently by Meyer following Lipton. I believe this theory is essentially correct. This paper supports IBE from a probability perspective, and argues that the retrodictive probabilities involved in such inferences should be analysed in terms of predictive probabilities and a priori (...) probability ratios of initial events. The key point is to separate these two features. Disagreements over evidence can be traced to disagreements over either the a priori probability ratios or predictive conditional ratios. In many cases, in real science, judgements of the former are necessarily subjective. The principles of iterated evidence are also discussed. The Sceptic's position is criticised as ignoring iteration of evidence, and characteristically failing to adjust a priori probability ratios in response to empirical evidence. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:David Hume and the Probability of Miracles Barry Gower 1. Introduction Oflate there have been published several discussions ofDavid Hume's famous essay "Of Miracles" which attempt to make precise the reasoning it contains. This, it turns out, requires the use of certain mathematical rules and theorems of the probability calculus which were unknown to Hume or, indeed, to anyone else when the essay was first published. It (...) is suggested, in particular, that the claims he made can best be understood in the light of a theorem of the probability calculus which we now name after a celebrated eighteenth century probabilist, Thomas Bayes. However, the so-called "Bayesian" interpretation of the argument against miracles misrepresents Hume's reasoning in that the manner in which he expressed his thinking does not fit such an interpretation. Moreover, this error needs to be rectified, not just because it involves a mistaken view about the past but, more importantly, because it obscures the legacy of a different mode of thinking evident in Hume's writing about probabilistic inference which deserves to be recovered. Our thinkingisimpoverishedifcertain presumptions aboutprobability become so entrenched that we have great difficulty in seeing them as anything other than obvious. By imputing them to Hume we overlook their contestability and so lose an opportunity to recapture an alternative way of understanding probabilistic reasoning. As is well known, the argument against miracles, though not published until 1748, had been worked out some twelve or thirteen years earlier during Hume's stay at the Jesuit college of La Fleche, where he wrote most of A Treatise of Human Nature. A chapter entitled "Reasonings concerning Miracles" had been written but, at the last moment, it was suppressed for fear that such an explicit attack on religion would give offence and thus deflect attention from the book's main philosophical aim which was to put an end to controversy. On the other hand, Thomas Bayes's famous memoir on probability was not published until 1764, andit is unlikely that Hume knew ofits existence until Richard Price mentionedit, though without identifyingits author, in an elaborate mathematical footnote to a dissertation containing remarks on the probability of miracles, published in 1767. Had Hume taken the trouble to follow up this reference, he would almost certainly Volume XVI Number 1 17 BARRY GOWER have been either baffled by its mathematics or discouraged by its lack of practical advice. That, at least, seems to have been the reaction of Hume's contemporaries, for apart from Price none of them seems to have paid any attention to the memoir. Even today, when anyone with the slightest acquaintance with probability can recite a version of Bayes's theorem, it is hard to follow the reasoning of the memoir and to appreciate the significance of the ideas it contains. As an historian of statistics has recently said, "Bayes's contemporaries were stymied by the work, and modern readers unwilling to devote many hours to it are likely to gain only a superficial understanding ofit." One paradoxical feature of attributing Bayesianism to Hume concerns the role of Richard Price. His dissertation, which had, it seems, been written some years before it was published — perhaps in the early 1750's — is sharply critical of Hume's argument against miracles. Yet it was Price himself who was responsible for the posthumous publication of Bayes's memoir, and his introduction and commentary clearly express approval of its methods and conclusions. We are, then, faced with the suggestion that Hume, not having heard of Bayes, was applying Bayesian ideas about "inverse" inference to miraculous events, whereas Price, who was not only familiar with but also endorsed Bayes's ideas, preferred a quite different approach to reasoning about miracles and about the credibility of testimony generally. Can Hume really be described as a "better intuitive Bayesian" than Price? Before embracing such aparadoxical conclusion we should consider whether it is really justified by what Hume said about the probability ofmiracles. First, though, some attention needs to be paid to a way ofthinking about probable arguments which does not conform to convention, and which implies a non-Bayesian interpretation of the question about miracles. 2. Jakob Bernoulli and the Combination... (shrink)

We illustrate how a variety of logical methods and techniques provide useful, though currently underappreciated, tools in the foundations and applications of reasoning under uncertainty. The field is vast spanning logic, artificial intelligence, statistics, and decision theory. Rather than (hopelessly) attempting a comprehensive survey, we focus on a handful of telling examples. While most of our attention will be devoted to frameworks in which uncertainty is quantified probabilistically, we will also touch upon generalisations of probability measures of uncertainty, which (...) have attracted a significant interest in the past few decades. (shrink)

Evolutionary theory is awash with probabilities. For example, natural selection is said to occur when there is variation in fitness, and fitness is standardly decomposed into two components, viability and fertility, each of which is understood probabilistically. With respect to viability, a fertilized egg is said to have a certain chance of surviving to reproductive age; with respect to fertility, an adult is said to have an expected number of offspring.1 There is more to evolutionary theory than the theory of (...) natural selection, and here too one finds probabilistic concepts aplenty. When there is no selection, the theory of neutral evolution says that a gene’s chance of eventually reaching fixation is 1/(2N), where N is the number of organisms in the generation of the diploid population to which the gene belongs. The evolutionary consequences of mutation are likewise conceptualized in terms of the probability per unit time a gene has of changing from one state to another. The examples just mentioned are all “forwarddirected” probabilities; they describe the probability of later events, conditional on earlier events. However, evolutionary theory also uses “backwards probabilities” that describe the probability of a cause conditional on its effects; for example, coalescence theory allows one to calculate the expected number of generations in the past that the genes in the present generation find their most recent common ancestor. (shrink)

One of the most difficult problems in the foundations of physics is what gives rise to the arrow of time. Since the fundamental dynamical laws of physics are (essentially) symmetric in time, the explanation for time's arrow must come from elsewhere. A promising explanation introduces a special cosmological initial condition, now called the Past Hypothesis: the universe started in a low-entropy state. Unfortunately, in a universe where there are many copies of us (in the distant ''past'' or the (...) distant ''future''), the Past Hypothesis is not enough; we also need to postulate self-locating (de se) probabilities. However, I show that we can similarly use self-locating probabilities to strengthen its rival---the Fluctuation Hypothesis, leading to in-principle empirical underdetermination and radical epistemological skepticism. The underdetermination is robust in the sense that it is not resolved by the usual appeal to 'empirical coherence' or 'simplicity.' That is a serious problem for the vision of providing a completely scientific explanation of time's arrow. (shrink)

Recent findings suggest that our capacity to imagine the future depends on our capacity to remember the past. However, the extent to which episodic memory is involved in our capacity to think about what could have happened in our past, yet did not occur , remains largely unexplored. The current experiments investigate the phenomenological characteristics and the influence of outcome valence on the experience of past, future and counterfactual thoughts. Participants were asked to mentally simulate past, (...) future, and counterfactual events with positive or negative outcomes. Features of their subjective experiences during each type of simulation were measured using questionnaires and autobiographical interviews. The results suggest that clarity and vividness were higher for past than future and counterfactual simulations. Additionally, emotional intensity was lower for counterfactual simulations than past and future simulations. Finally, outcome valence influenced participants’ judgment of probability for future and counterfactual simulations. (shrink)

This paper presents a paradox based on the following assumptions: that emotions are warranted when you are justified in thinking that the emotion is fitting, that there are warranted cases of past-directed fear, that fear is fitting in the face of its formal object: dangerousness, and that this formal object consists in a probability of damage or harm to something of value. The paper then discusses three likely solutions: (1) denying that past-directed fear can be warranted, (2) (...) using an alternative formulation of fear's formal object, and (3) giving up warrant as justified as fittingness. Finally, it provides a case for the third solution and presents a warrant-based approach to fittingness and the appropriateness of emotions in general. (shrink)

This book is designed to explain the technical ideas that are taken for granted in much contemporary philosophical writing. Notions like "denumerability," "modal scope distinction," "Bayesian conditionalization," and "logical completeness" are usually only elucidated deep within difficult specialist texts. By offering simple explanations that by-pass much irrelevant and boring detail, Philosophical Devices is able to cover a wealth of material that is normally only available to specialists. The book contains four sections, each of three chapters. The first section is about (...) sets and numbers, starting with the membership relation and ending with the generalized continuum hypothesis. The second is about analyticity, a prioricity, and necessity. The third is about probability, outlining the difference between objective and subjective probability and exploring aspects of conditionalization and correlation. The fourth deals with metalogic, focusing on the contrast between syntax and semantics, and finishing with a sketch of Godel's theorem. Philosophical Devices will be useful for university students who have got past the foothills of philosophy and are starting to read more widely, but it does not assume any prior expertise. All the issues discussed are intrinsically interesting, and often downright fascinating. It can be read with pleasure and profit by anybody who is curious about the technical infrastructure of contemporary philosophy."--p. 4 of cover. (shrink)

Over the past two decades, work relations have changed dramatically. New phenomena like “gig-economy” or “crowd work” not only constitute precarious working conditions but also contradict with our social esteem of work resulting from the social theories of the classical economy of the eighteenth and nineteenth centuries. The central focus of classical economists on building an educated and disciplined workforce provided not only the base for the upcoming industrial society but also resulted in a work-based society where “being employed” (...) became the precondition for social security and social participation. It is the aim of this contribution to show how our positive attitudes towards work, established by the political economic theories of the eighteenth and nineteenth centuries, are jeopardized by the social changes in post-industrialized societies, due to the effects of globalized economies, digitalization and changed industrial relations. This has also far-reaching consequences for managerial theories based on conceptions like meaningful work or discussions about social responsibilities vis-à-vis employees as primary stakeholder groups. (shrink)

This paper examines the logic and morality of the German Stem Cell Act of 2002. After a brief description of the law’s scope and intent, its ethical dimensions are analysed in terms of symbolic threats, indirect consequences, and the encouragement of immorality. The conclusions are twofold. For those who want to accept the law, the arguments for its rationality and morality can be sound. For others, the emphasis on the uniqueness of the German experience, the combination of absolute and qualified (...) value judgments, and the lingering questions of indirect encouragement of immoral activities will probably be too much. (shrink)

We human beings have evolved with brains that have amazing capabilities for rational thought, pattern recognition, judgment, creativity, and imagination, none of which can be readily duplicated by the best computer simulations. However, there is one area, in which the human brain is sadly lacking: the ability to accurately assess probabilities and act on these assessments. The successes of lotteries, Las Vegas, and tribal casinos provide ample evidence that when it comes to estimating the odds and acting accordingly, we humans (...) as a species are really deficient. We think that “winning streaks†are real, that slot machines are “overdue†for a jackpot, that the past pattern of random events somehow influences the odds for the next event. (shrink)

It was Hume's concept of knowledge that marked Masaryk's philosophy more than influential Kant. Masaryk devoted Hume his inaugural lecture The Probability Calculus and Hume's Scepticism at Prague University in 1882. He tried to face his scepticism concerning causation and induction by a formula P = n: or P = : where P means the increasing probability that an event that had happened n times in the past will happen again. Hume stresses that there is an essential (...) difference between probability and certainty, and we have no logical reason to presuppose that the past experience is legitimate for expectation that the future will resemble to the past. Only our belief, not knowledge, leads us to that. Although Masaryk did not accept Hume's scepticism, he had deep understanding for his way of thinking. Reflection of Hume's influence on Masaryk in Czech philosophy is unsatisfactory, which is stated in this article. (shrink)