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Summary The chance facts appear to outrun what actually happens, and to involve constraints on what could happen, or on what will probably happen. Humeanism is – roughly – the thought that all such facts can be reduced to facts about what does in fact happen. The most influential characterisation of the view is David Lewis's thesis of Humean supervenience: that all matters of contingent fact supervene on the distribution of qualitative properties in space-time. In the case of chance, Lewis suggested that chanciness reduces to actual, occurrent patterns in the world. So for instance, according to a Humean, a coin's being fair reduces to some sort of fact about the actual ways in which that coin (or similar coins) fall.  Non-Humeans complain that a Humean analysis of chance does not do justice to the modal character of chances. In addition to this central concern, there is a particularly acute objection raised against Humeanism: the so called "Bug". Chances, according to a Humean, are both grounded in patterns of actual events but also provide chances for events which would falsify those same chance facts. Consequently, it appears that we need to believe that such deviant futures have some chance of occurring, but also have no chance of occurring. Working out a bridge between chance-facts and what we should believe that avoids this contradiction is a central feature of the contemporary Humean project. Because non-Humeans are frequently propensity theorists, and because Humeans are frequently frequentists, these PhilPapers categories are also relevant.
Key works The doctrine of Humean supervenience is put forth in Lewis 1986. Lewis's Humean account of chance is first proposed in Lewis 1980, and then refined in Lewis 1994. (See also Oppy 2000 and Loewer 1996.) Important non-Humean contributions are Bigelow et al 1993 and Black 1998. Other recent works of note include Briggs 2009Hall 2004, and Ismael 2008.
Introductions Chapter 7 of Handfield 2012 provides a critical evaluation of Humean accounts of chance, without discussion of the "Bug". Two recommended articles for an introduction to the bug are Bigelow et al 1993 and Briggs 2009.
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  1. Frank Arntzenius & Ned Hall (2003). On What We Know About Chance. British Journal for the Philosophy of Science 54 (2):171-179.
    The ‘Principal Principle’ states, roughly, that one's subjective probability for a proposition should conform to one's beliefs about that proposition's objective chance of coming true. David Lewis has argued (i) that this principle provides the defining role for chance; (ii) that it conflicts with his reductionist thesis of Humean supervenience, and so must be replaced by an amended version that avoids the conflict; hence (iii) that nothing perfectly deserves the name ‘chance’, although something can come close enough by playing the (...)
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  2. John Bigelow, John Collins & Robert Pargetter (1993). The Big Bad Bug: What Are the Humean's Chances? British Journal for the Philosophy of Science 44 (3):443-462.
    Humean supervenience is the doctrine that there are no necessary connections in the world. David Lewis identifies one big bad bug to the programme of providing Humean analyses for apparently non-Humean features of the world. The bug is chance. We put the bug under the microscope, and conclude that chance is no special problem for the Humean.
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  3. Robert Black (1998). Chance, Credence, and the Principal Principle. British Journal for the Philosophy of Science 49 (3):371-385.
    Any adequate theory of chance must accommodate some version of David Lewis's ‘Principal Principle’, and Lewis has argued forcibly that believers in primitive propensities have a problem in explaining what makes the Principle true. But Lewis can only derive (a revised version of) the Principle from his own Humean theory by putting constraints on inductive rationality which cannot be given a Humean rationale.
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  4. Rachael Briggs (2009). The Anatomy of the Big Bad Bug. Noûs 43 (3):428-449.
  5. Adam Elga (2004). Infinitesimal Chances and the Laws of Nature. Australasian Journal of Philosophy 82 (1):67 – 76.
    The 'best-system' analysis of lawhood [Lewis 1994] faces the 'zero-fit problem': that many systems of laws say that the chance of history going actually as it goes--the degree to which the theory 'fits' the actual course of history--is zero. Neither an appeal to infinitesimal probabilities nor a patch using standard measure theory avoids the difficulty. But there is a way to avoid it: replace the notion of 'fit' with the notion of a world being typical with respect to a theory.
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  6. Roman Frigg & Carl Hoefer (2010). Determinism and Chance From a Humean Perspective. In Friedrich Stadler, Dennis Dieks, Wenceslao González, Hartmann J., Uebel Stephan, Weber Thomas & Marcel (eds.), The Present Situation in the Philosophy of Science. Springer. 351--72.
    On the face of it ‘deterministic chance’ is an oxymoron: either an event is chancy or deterministic, but not both. Nevertheless, the world is rife with events that seem to be exactly that: chancy and deterministic at once. Simple gambling devices like coins and dice are cases in point. On the one hand they are governed by deterministic laws – the laws of classical mechanics – and hence given the initial condition of, say, a coin toss it is determined whether (...)
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  7. Roman Frigg & Carl Hoefer (2007). Probability in GRW Theory. Studies in History and Philosophy of Science Part B 38 (2):371-389.
    GRW Theory postulates a stochastic mechanism assuring that every so often the wave function of a quantum system is `hit', which leaves it in a localised state. How are we to interpret the probabilities built into this mechanism? GRW theory is a firmly realist proposal and it is therefore clear that these probabilities are objective probabilities (i.e. chances). A discussion of the major theories of chance leads us to the conclusion that GRW probabilities can be understood only as either single (...)
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  8. Luke Glynn (2011). D. H. MELLOR The Matter of Chance. British Journal for the Philosophy of Science 62 (4):899-906.
    Though almost forty years have elapsed since its first publication, it is a testament to the philosophical acumen of its author that 'The Matter of Chance' contains much that is of continued interest to the philosopher of science. Mellor advances a sophisticated propensity theory of chance, arguing that this theory makes better sense than its rivals (in particular subjectivist, frequentist, logical and classical theories) of ‘what professional usage shows to be thought true of chance’ (p. xi) – in particular ‘that (...)
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  9. Joshua Haddock (2011). The Principal Principle and Theories of Chance: Another Bug? Philosophy of Science 78 (5):854-863.
  10. Ned Hall (2004). Two Mistakes About Credence and Chance. Australasian Journal of Philosophy 82 (1):93 – 111.
    David Lewis's influential work on the epistemology and metaphysics of objective chance has convinced many philosophers of the central importance of the following two claims: First, it is a serious cost of reductionist positions about chance (such as that occupied by Lewis) that they are, apparently, forced to modify the Principal Principle--the central principle relating objective chance to rational subjective probability--in order to avoid contradiction. Second, it is a perhaps more serious cost of the rival non-reductionist position that, unlike reductionism, (...)
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  11. Ned Hall (2004). Two Mistakes About Credence and Chance. Australasian Journal of Philosophy 82 (1):93 – 111.
    David Lewis's influential work on the epistemology and metaphysics of objective chance has convinced many philosophers of the central importance of the following two claims: First, it is a serious cost of reductionist positions about chance (such as that occupied by Lewis) that they are, apparently, forced to modify the Principal Principle--the central principle relating objective chance to rational subjective probability--in order to avoid contradiction. Second, it is a perhaps more serious cost of the rival non-reductionist position that, unlike reductionism, (...)
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  12. John F. Halpin, On Chance and the Best-System Account of Law.
    David Lewis[ii] has long defended an account of scientific law acceptable even to an empiricist with significant metaphysical scruples. On this account, the laws are defined to be the consequences of the best system for axiomitizing all occurrent fact. Here "best system" means the set of sentences which yields the best combination of strength of descriptive content[iii] with simplicity of exposition. And occurrent facts, the facts to be systematized, are roughly the particular facts about a localized space-time region that are (...)
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  13. John F. Halpin (1998). Lewis, Thau, and Hall on Chance and the Best-System Account of Law. Philosophy of Science 65 (2):349-360.
    August 16, 1997 David Lewis2 has long defended an account of scientific law acceptable even to an empiricist with significant metaphysical scruples. On this account, the laws are defined to be the consequences of the best system for axiomitizing all occurrent fact. Here "best system" means the set of sentences which yields the best combination of strength of descriptive content 3 with simplicity of exposition. And occurrent facts, the facts to be systematized, are roughly the particular facts about a localized (...)
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  14. Toby Handfield (2012). A Philosophical Guide to Chance: Physical Probability. Cambridge University Press.
    Contents: 1. The concept of chance; 2. The classical picture; 3. Ways the world might be; 4. Possibilities of thought; 5. Chance in phase space; 6. Possibilist theories of chance; 7. Actualist theories of chance; 8. Anti-realist theories of chance; 9. Chance in quantum physics; 10. Chance in branching worlds; 11. Time and evidence; 12. Debunking chance.
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  15. Carl Hoefer (2011). Physics and the Humean Approach to Probability. In Claus Beisbart & Stephan Hartmann (eds.), Probabilities in Physics. Oxford University Press. 321.
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  16. Carl Hoefer (2007). The Third Way on Objective Probability: A Sceptic's Guide to Objective Chance. Mind 116 (463):549-596.
    The goal of this paper is to sketch and defend a new interpretation or 'theory' of objective chance, one that lets us be sure such chances exist and shows how they can play the roles we traditionally grant them. The account is 'Humean' in claiming that objective chances supervene on the totality of actual events, but does not imply or presuppose a Humean approach to other metaphysical issues such as laws or causation. Like Lewis (1994) I take the Principal Principle (...)
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  17. Carl Hoefer (2007). The Third Way on Objective Probability: A Sceptic's Guide to Objective Chance. Mind 116 (463):549-596.
    The goal of this paper is to sketch and defend a new interpretation or theory of objective chance, one that lets us be sure such chances exist and shows how they can play the roles we traditionally grant them. The subtitle obviously emulates the title of Lewis seminal 1980 paper A Subjectivist s Guide to Objective Chance while indicating an important difference in perspective. The view developed below shares two major tenets with Lewis last (1994) account of objective chance: (1) (...)
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  18. Carl Hoefer (1997). On Lewis's Objective Chance: "Humean Supervenience Debugged". Mind 106 (422):321-334.
  19. Jenann Ismael, Raid! The Big, Bad Bug Dissolved.
    There’s a long history of discussion of probability in philosophy, but objective chance separated itself off and came into its own as a topic with the advent of a physical theory - quantum mechanics - in which chances play a central, and apparently ineliminable, role. In 1980 David Lewis wrote a paper pointing out that a very broad class of accounts of the nature of chance apparently lead to a contradiction when combined with a principle that expresses the role of (...)
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  20. David Lewis (1994). Humean Supervenience Debugged. Mind 103 (412):473-490.
    Tn this paper I explore and to an extent defend HS. The main philosophical challenges to HS come from philosophical views that say that nomic concepts-laws, chance, and causation-denote features of the world that fail to supervene on non-nomic features. Lewis rejects these views and has labored mightily to construct HS accounts of nomic concepts. His account of laws is fundamental to his program, since his accounts of the other nomic notions rely on it. Recently, a number of philosophers have (...)
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  21. David Lewis (1994). Humean Supervenience Debugged. Mind 103 (412):473--490.
    Tn this paper I explore and to an extent defend HS. The main philosophical challenges to HS come from philosophical views that say that nomic concepts-laws, chance, and causation-denote features of the world that fail to supervene on non-nomic features. Lewis rejects these views and has labored mightily to construct HS accounts of nomic concepts. His account of laws is fundamental to his program, since his accounts of the other nomic notions rely on it. Recently, a number of philosophers have (...)
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  22. David Lewis (1980). A Subjectivist's Guide to Objective Chance. In Richard C. Jeffrey (ed.), Studies in Inductive Logic and Probability. University of California Press. 83--132.
  23. B. Loewer (2001). Determinism and Chance. Studies in History and Philosophy of Science Part B 32 (4):609-620.
    It is generally thought that objective chances for particular events different from 1 and 0 and determinism are incompatible. However, there are important scientific theories whose laws are deterministic but which also assign non-trivial probabilities to events. The most important of these is statistical mechanics whose probabilities are essential to the explanations of thermodynamic phenomena. These probabilities are often construed as 'ignorance' probabilities representing our lack of knowledge concerning the microstate. I argue that this construal is incompatible with the role (...)
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  24. Barry Loewer (2004). David Lewis's Humean Theory of Objective Chance. Philosophy of Science 71 (5):1115--25.
    The most important theories in fundamental physics, quantum mechanics and statistical mechanics, posit objective probabilities or chances. As important as chance is there is little agreement about what it is. The usual “interpretations of probability” give very different accounts of chance and there is disagreement concerning which, if any, is capable of accounting for its role in physics. David Lewis has contributed enormously to improving this situation. In his classic paper “A Subjectivist's Guide to Objective Chance” he described a framework (...)
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  25. Joshua Luczak (2013). On Chance. Metascience 22 (1):85-87.
  26. Christopher J. G. Meacham (forthcoming). Autonomous Chances and the Conflicts Problem. In Toby Handfield & Alastair Wilson (eds.), Asymmetries in Chance and Time. Oxford University Press.
    In recent work, Callender and Cohen (2009) and Hoefer (2007) have proposed variants of the account of chance proposed by Lewis (1994). One of the ways in which these accounts diverge from Lewis’s is that they allow special sciences and the macroscopic realm to have chances that are autonomous from those of physics and the microscopic realm. A worry for these proposals is that autonomous chances may place incompatible constraints on rational belief. I examine this worry, and attempt to determine (...)
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  27. Christopher J. G. Meacham (2010). Contemporary Approaches to Statistical Mechanical Probabilities: A Critical Commentary - Part I: The Indifference Approach. Philosophy Compass 5 (12):1116-1126.
    This pair of articles provides a critical commentary on contemporary approaches to statistical mechanical probabilities. These articles focus on the two ways of understanding these probabilities that have received the most attention in the recent literature: the epistemic indifference approach, and the Lewis-style regularity approach. These articles describe these approaches, highlight the main points of contention, and make some attempts to advance the discussion. The first of these articles provides a brief sketch of statistical mechanics, and discusses the indifference approach (...)
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  28. Christopher J. G. Meacham (2010). Contemporary Approaches to Statistical Mechanical Probabilities: A Critical Commentary - Part II: The Regularity Approach. Philosophy Compass 5 (12):1127-1136.
    This pair of articles provides a critical commentary on contemporary approaches to statistical mechanical probabilities. These articles focus on the two ways of understanding these probabilities that have received the most attention in the recent literature: the epistemic indifference approach, and the Lewis-style regularity approach. These articles describe these approaches, highlight the main points of contention, and make some attempts to advance the discussion. The second of these articles discusses the regularity approach to statistical mechanical probabilities, and describes some areas (...)
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  29. Christopher J. G. Meacham (2005). Three Proposals Regarding a Theory of Chance. Philosophical Perspectives 19 (1):281–307.
    I argue that the theory of chance proposed by David Lewis has three problems: (i) it is time asymmetric in a manner incompatible with some of the chance theories of physics, (ii) it is incompatible with statistical mechanical chances, and (iii) the content of Lewis's Principal Principle depends on how admissibility is cashed out, but there is no agreement as to what admissible evidence should be. I proposes two modifications of Lewis's theory which resolve these difficulties. I conclude by tentatively (...)
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  30. D. H. Mellor (2004). The Matter of Chance. Cambridge University Press.
    This book deals not so much with statistical methods as with the central concept of chance, or statistical probability, which statistical theories apply to nature.
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  31. Richard Pettigrew (2013). What Chance‐Credence Norms Should Not Be. Noûs 47 (3).
    A chance-credence norm states how an agent's credences in propositions concerning objective chances ought to relate to her credences in other propositions. The most famous such norm is the Principal Principle (PP), due to David Lewis. However, Lewis noticed that PP is too strong when combined with many accounts of chance that attempt to reduce chance facts to non-modal facts. Those who defend such accounts of chance have offered two alternative chance-credence norms: the first is Hall's and Thau's New Principle (...)
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  32. Richard Pettigrew (2012). Accuracy, Chance, and the Principal Principle. Philosophical Review 121 (2):241-275.
    In ‘A Non-Pragmatic Vindication of Probabilism’, Jim Joyce attempts to ‘depragmatize’ de Finetti’s prevision argument for the claim that our partial beliefs ought to satisfy the axioms of probability calculus. In this paper, I adapt Joyce’s argument to give a non-pragmatic vindication of various versions of David Lewis’ Principal Principle, such as the version based on Isaac Levi's account of admissibility, Michael Thau and Ned Hall's New Principle, and Jenann Ismael's Generalized Principal Principle. Joyce enumerates properties that must be had (...)
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  33. John T. Roberts (2001). Undermining Undermined: Why Humean Supervenience Never Needed to Be Debugged (Even If It's a Necessary Truth). Proceedings of the Philosophy of Science Association 2001 (3):S98-.
    The existence of "undermining futures" appears to show that a contradiction can be deduced from the conjunction of Humean supervenience (HS) about chance and the Principal Principle. A number of strategies for rescuing HS from this problem have been proposed recently. In this paper, a novel way of defending HS from the threat is presented, and it is argued that this defense has advantages not shared by others. In particular, it requires no revisionism about chance, and it is equally available (...)
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  34. Wolfgang Spohn, Chance and Necessity : From Humean Supervenience to Humean Projection.
    From Humean Supervenience to Humean Projection": This paper attempts to develop a projectivistic understanding of chance or objective probability or partial determination. It does so by critically examining David Lewis’ philosophy of probability and his defense of Humean Supervenience, building thereupon the constructive projectivistic alternative, which will basically be a suitable reinterpretation of de Finetti’s position. Any treatment of the topic must show how it extends to natural necessity or deterministic laws or full determination in perfect parallel. The paper indicates (...)
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  35. S. Sturgeon (1998). Humean Chance: Five Questions for David Lewis. [REVIEW] Erkenntnis 49 (3):321-335.
    David Lewis's approach to objective chance is doubly distinctive. On the one hand, Lewis uses an epistemic principle to disclose the nature of chance. One the other, Lewis conjoins realism about chance with a reductive Humean metaphysics. I aim to undermine both aspects of his view. Specifically, I argue that reductive Humeanism fails across the board, and I use my discussion of chance to explain why. I also argue Lewis's "best-systems" approach to chance fails his own criteria for a metaphysics (...)
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  36. Peter B. M. Vranas (2002). Who's Afraid of Undermining? Why the Principal Principle Might Not Contradict Humean Supervenience. Erkenntnis 57 (2):151 - 174.
    The Principal Principle (PP) says that, for any proposition A, given any admissible evidence and the proposition that the chance of A is x%, one's conditional credence in A should be x%. Humean Supervenience (HS) claims that, among possible worlds like ours, no two differ without differing in the spacetime-point-by-spacetime-point arrangement of local properties. David Lewis (1986b, 1994a) has argued that PP contradicts HS, and the validity of his argument has been endorsed by Bigelow et al. (1993), Thau (1994), Hall (...)
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  37. Barry Ward (2005). Projecting Chances: A Humean Vindication and Justification of the Principal Principle. Philosophy of Science 72 (1):241-261.
    Faced with the paradox of undermining futures, Humeans have resigned themselves to accounts of chance that severely conflict with our intuitions. However, such resignation is premature: The problem is Humean supervenience (HS), not Humeanism. This paper develops a projectivist Humeanism on which chance claims are understood as normative, rather than fact stating. Rationality constraints on the cotenability of norms and factual claims ground a factual-normative worlds semantics that, in addition to solving the Frege-Geach problem, delivers the intuitive set of possibilia (...)
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  38. Alastair Wilson (ed.) (2014). Chance and Temporal Asymmetry. Oxford University Press.