A Subjectivist’s Guide to Objective Chance

“D. Lewis (1980), ‘A Subjectivist’s Guide to Objective Chance’, in R.C. Jeffrey ed.”

“I am grateful for helpful conversations about this material with Anthony Aguirre, Adam Bales, Sam Baron, Yann Benétreau-Dupin, Rachael Briggs, Adam Caulton, David Deutsch, Rohan French, Toby Handfield, Nicholas Jones, Chris Meacham, Dave Ripley, Miranda Rose, Simon Saunders, David Wallace, and Robbie Williams, and to…”

“6For a classical discussion, see Lewis (1980). 7For extensive recent discussions about the importance of the”

“Using the same example, we can also show that GC conflicts with (an imprecise version of) the Principal Principle (Lewis 1980): (PP) For any X ∈ and any interval I ⊆ [0, 1], Ci ( X/Pj (X) ∈ I ) ⊆ I if the given agent does not have any inadmissible evidence, where Ci is her credal state at ti and Pj…”

“…could be fully based on objective probabilities, and nowadays it is difficult to find any Bayesians who do not adhere to some version of the so-called Principal Principle, according to which credences should be based on objective probabilities, whenever these are available; see Lewis (1980).”

“Since the first two claims are deeply controversial (see, e.g., Lewis 1980, Hoefer 2005, Nina 2017), the force of this worry is unclear at best and I henceforth bracket it.”

“Revisionists include, for example, Lewis [32,33], Swineburne [47], Holder [24], Herzberg [23], and Norton [34], in addition to Carnap [8] and Skyrms [42,43].”

“…that rational credences must conform to other norms as well, e.g. norms connecting credences to physical chances such as the Principal Principle (Lewis, 1980), norms connecting credences to symmetry considerations such as the Principle of Indifference (Laplace, 1951; Keynes, 1921) and the…”

“The Principal Principle: Objective chances should satisfy the Principal Principle.”

“See Lewis (1980).”

“The leading idea behind Lewis’s (1980) formulation of this principle was that a rational agent should conform her personal degrees of belief (credences) to the (objective) chances.”

“This interpretation of rationality of pH is essentially the content of the Principle Principle [14] and, while it is very natural, one should realize that pH features in it in two roles: (i) standing for the degree of belief and (ii) representing some extra-mental, non-degree-of-belief-type quantities (for instance fr…”

“That is, CrtðA j chtðAÞ ¼ x&EÞ ¼ x: ð45Þ for any admissible E, where ‘‘[a]dmissible propositions are the sort of information whose impact on credence about outcomes comes entirely by way of credence about the chances of those outcomes’’ (Lewis 1980, p. 272).”

“Lewis (1980), by the way, reverses the argument, arguing that chances must follow those axioms, because credences must rationally do so and because credences and chances are related by his Principal Principle.”

“…simulation that constructs a tree (or graph) of lineages adds branches in reverse order, from the present to the past, by estimating probabilities that two or more organisms will have a common parent, based on relatively simple but reasonable modeling assumptions concerning evolutionary processes.”

“21 For related views, see Gluer 2009, Quilty-Dunn 2015, and Lewis 1980b: 239. different orientations of lines, or different shapes.”

“So if you know that a coin is fair, the Principal Principle says that your credence in heads for a future toss of the coin should be 1/2.”

“…famous such norm is the Principal Principle (henceforth (PP)), due to David Lewis, which says (roughly) that our credence in a given proposition conditional on the chances being given by a particular probability function ch should be whatever value ch assigns to that proposition ([Lewis, 1980]).”

“A more remote analogy may be drawn from Calibration (C) to Principal principle of Lewis (1980) .”

“The present logic can assign any c hance value to unique events in the future, while events in the past are assigned only chance values 0 or 1, as required by our deenition of objective c hance.”

“We associate with any physical system a physical state space, or ontic state space Λ, and a set L of subsets of Λ that will be taken to be the measurable sets, that is, the ones that are candidates for ascribing a probability to.9 With any preparation procedure ψ is associated 8This is a principal often tacitly assume…”

“given E. In principle, the probability structure admits two interpretations. Under an objectivist interpretation, it is a feature of the system itself and thus represents objective chance (see, e.g., Lewis 1986, Schaffer 2007, and List and Pivato 2015). Of course, objective chance could be degenerate, i.e., restricted…”

“The driver seems to violate a form of the “Principal Principle” according to which one’s rational credence should match the known objective chances (compare [Lewis 1980]).”

“The second equation results from an application of the Principal Principle.”

“And though David Lewis’s Principal Principle (Lewis, 1986a, 1986b) attempts to bridge the gap between degrees of belief and objective chances, the Principle is off the mark at least in the case of supposedly deterministic examples such as coin tosses.”

“previous work has formalized fairness notions of machine learning using counterfactual epistemic operators. 16 Y. Kawamoto Formalization of statistical properties. Instudies ofphilosophicallogic,Lewis[29] shows the idea that when a random value has various possible probability distributions, then those distributions s…”

“The second way to ensure that each door gets 1/3 credence at the beginning consists in endorsing the Principal Principle (PP, Lewis 1981).”

“Two candidate norms of this sort: the principal principle, which says how an agent’s credences in propositions concerning the objective chances should relate to her credences in other propositions (Lewis [1980]); and the principle of indifference, which says, roughly, that an agent with no evidence should divide her c…”

“…protagonists in this debate are: the subjective Bayesians, who can be further subdivided into the radical (de Finetti 1989) and empirical (Lewis 1980) varieties; the objective Bayesians (Jaynes 2003; Williamson 2010) and those who favour imprecise probabilities (Joyce 2010; Gaifman 1986).”

“Traditionally, following David Lewis (1983) and (1994), the summary is taken to be the set of axioms of the system that best balances simplicity, informativeness, and fit, when written in terms that reference only the perfectly natural, categorical properties.13 Laws are simpler when the sentences are fewer and shorte…”

“53 A similar point is made by Lewis (1987, 84).”

“Recall the following formulation of the Principal Principle given by Lewis (1980): Principal Principle: Let C be any reasonable initial credence function.”

“1Note that this theory of objective chance differs in important respects from Lewis’s own account in [29], while following Lewis in broad outline.”

“He starts out with the principle that the subjective probability of a future chance event should be equal to the known chances (Mellor, 1971; Lewis, 1980).”

“On this probabilistic interpretation the royal rule becomes Lewis’ (1980) principal principle, and the probabilistically re-interpreted set-up allows one to study which ends the principal principle furthers (Pettigrew 2013 is an excellent study of this kind).”

“Note, however, that one can countenance epistemic probabilities even if one rejects principles of indifference (Greaves, 2016, fn9, §IV; Isaacs et al. 2022, p. 247, 269; Lewis 1980, pp. 289-90).”

“But one might think that she can’t if we suppose that, like philosophers such as Lewis (1980) and Schaffer (2007), she thinks that the objective chance of a past event having happened is either 1 or 0, depending on whether it happened or not.”