The Paradox of Confirmation: New Challenges for Standard Bayesian Solutions by Justin M. Dallmann A Thesis submittedEffi to the Faculry of Graduate Studies of l* çl Uxi\,'rrlìsITT +9.{. ûr,.Nl¡\Nil'ûn in partial fulfilment of the requirements of the degree of MASTER OF ARTS Department of Philosophy University of Manitoba Winnipeg Copyright @2009 by Justin M. Dallmann THE T]NTVERSITY OF MANÍTOBA FACI]LTY OF GRAD.I]ATE STUDIES COPYRIGHT PERMISSION The Paradox of Confirmation: New Challenges for Standard Bayesian Solutions BY Justin M. Dallmann A ThesisÆracticum submitted to the Faculty of Graduate Studies of The University of Manitoba in partial fulfillment of the requirement of the degree MASTER OF ARTS Justin M. Dallmann @ 2009 Permission has been granted to the University of Manitoba Libraries to lend a copy of this thesis/practicum, to Library and Archives Canada (LAC) to lend a copy of this thesis/practicum, and to LAC's agent (UMlÆroQuest) to microfilm, sell copies and to publish an abstract of this thesis/practicum. This reproduction or copy of this thesis has been made available by authority of the copyright owner solely for the purpose of private study and research, and may only be reproduced and copied as permitted by copyright laws or with express written authorization from the copyright own 1 Front Matter/Prefatory Pages 1.1 Abstract In this paper I demonstrate one way that Bayesian confirmation theory can con- tribute to an epistemology of several key metaphysical concepts found at rhe core of the natural sciences. I then use the results to generalize the Paradox of Confir- mation in a way that, I contend, undermines the standard Bayesian solution to the Paradox as well as the more recent refinements proposed in (Howson & Urbach, 2006), (Fitelson, 2006), (Fitelson & Hawthorne, Forth.), and (Vranas, 2004). The formal results presented serve as yet another illustration of the inadequacy of the standard Bayesian solutions to the Paradox of Confirmation, prompting Bayesians of all stripes to reject the standard solution. As an upshot, this thesis also demon- strates that it is imprudent to ignore metaphysical phenomena when constructing a theory of confirmation. t.2 Acknowledgements I would like to thank the Social Sciences and Humanities Research Council of Canada, the Donald Vernon Snider Memorial Foundation and the Province of Man- itoba for their generous support in the form of a Canadian Graduate Scholarship, Donald Vernon Snider Memorial Award and Manitoba Graduate Scholarship, re- spectively. 1.3 Dedication To all those from whom I've learned, my wife Amanda, and my rabbit Bunny. 111 Contents Front Matter/Prefatory Pages I 1.1 Abstract i 7.2 Acknowledgements ii 1.3 Dedication iii 2 Introduction 1 Confirming the Metaphysical 3 3.1 Bayesian Accounts of Confirmation J 3.2 The Argument from Entailment 4 3.3 A Case Study: Causatíon 6 4 A Generalization of Hempel's Paradox 15 4.1 Non-Standard Solutions 76 4.2'The' Standard Solution 79 4.3 The Generalization 24 4.4 A Counterargument to the Generalized Paradox 26 4.5 Weakenings of the Standard Solution 39 4.6 The Obvious'Solution' . 44 5 Conclusion 47 6 Appendix 49 References 54 IV Introduction Bayesianism, as it will be discussed within the context of this thesis, is the view that: (i) our credences come in degrees, (ii) conformiry to the probabiliry calculus is a rationality constraint on our credences, (iii) we are to update our credences by conditionalizing on our available evidence, and (iv) the support a theory garners from a given piece of evidence is a function of our subjective credence in the theory before we have learned that the evidence obtains and our credence in the theory after the evidence has been learned. On most (subjective) Bayesian views, one's level of initial credence in a hypoth- esis is relatively unconstrained. All that is demanded is that one's initial credences are probabilistically consistent. As a trivial example, no agent who assigns a prob- abiliry or credence of 1 to some hypothesis /z could also assign to its negation -/z a degree of credence of .5 without being considered irrational. Such an assignment violates the edicts of the probabiliry calculus. The norms of Bayesian rationality also constrain how one updates one's initial probabilities, or priors, by the process of Bayesian conditionalization: After learning some statement e, one's credence in any statement /z should be updated to Plhlel, one's posterior credence in /z given ¿. In this way, if one's initial assignments were off the mark the evidence will push back. Updating by conditionalization will, "in the long run," force one's credences to converge on the truth . or so the story goes. As far as lhis overly simplified picture is concerned, the Bayesian framework is amenable to metaphysical enquiry. Nothing in it bars the assignment of positive credence to hypotheses involving cause and effect, dispositions, or claims couched in counterfactuals. However, as a purely sociological observation, the majority of practising Bayesians for the most part stear clear of such matters. Discussions of Bayesian confirmation theory are no exception to this rule. One of the main strands of this thesis will be to examine the Bayesian confirmation project in light of the metaphysics of science. To this end, the thesis commences with a brief preliminary exposition of Bayesian confirmation theory in $3.1. In $3.2, I concern myself with ways that Bayesian con- firmation theory can contribute to an epistemology of several key metaphysical con- cepts found at the core of the natural sciences. I then use the results to generalize the Paradox of Confirmation in $4. After an exposition of the standard Bayesian response to the Paradox of Confirmation in 54.2,I contend that the generalization of the Paradox undermines this solution to the Paradox as well as the recent re- finements of the solution proposed in (Howson & Urbach, 2006), (Fitelson, 2006), (Fitelson & Hawthorne, Forth.), and (Vranas, 2004). Several possible responses are canvassed and rejected in $4.4 (especially subsection 4.4.I). In $4.4.3, I present the upshot of the novel problems introduced by the preceding sections and try to draw out some general lessons for confirmation theory. Sections 4.5 looks at, and rejects, recent weakenings of the standard solution and $4.6 contains the presenta- tion and refutation of a novel solution to the Paradox. Finally, I conclude in $5 that the formal results presented in this thesis serve as yet another illustration of the in- adequacy of the standard Bayesian solution to the Paradox of Confirmation. Hence, Bayesians of all stripes should reject the standard solution. The thesis demonstrates the imprudence of ignoring metaphysical phenomena when constructing a theory of confirmation. 3 Confirming the Metaphysical 3.1 Bayesian Accounts of Confirmation Several Bayesian measures of confirmation, cfh,el, have been proposed and de- fended in the literature. Where /z denotes some hypothesis and e the evidence, a few of the most popular such measures include:1 o The Difference: d[h, e) = ]F Uz I el -PUtl; o The Log-Ratio: r fh, e) = /r (lP lh I e) lP lhl); o The Log-Likelihood-Ratio: I lh, e) = ln (JP te I h) lP fe | -tl).z Despite the diversiry of measures, it is worth pointing out that if dlh, el > 0, then rfh,e) > 0 and llh,el > 0 (as is shown in the Appendix $6, Theorem 1). Hence, to show that some evidence ¿ lends a positive degree of support to /z it is sufficient to show that dlh,el > 0. The proofs in this thesis will thus content themselves with demonstrations that hold for d [, ] whenever this sufficiency condition will deliver the requisite results for any of the above noted confirmation measures. Moreover, anyfuturereferencetoaconfirmationmeasure,c[,],willassumethatc[,]=d[,] unless otherwise stated. r See (Fitelson, 1999) for a more comprehensive list of measures as well as their historical back- ground. Notable proponents of each measure are easy to come by. Proponents of d include Earman, Eells, Gillies, Jeffrey, and Rosenk¡antz. Horwich, Mackie and Milne endorse r. Finally, a short list of advocates of / includes the likes of Fitelson, Good and Pearl. The nomenclature and presentation of the confirmation measures in this section are essentially that of Fiteison and Hawthorne (Forth.). 2Most authors usually assume that confirmation, as weil as subjective probabiliry assignments, are relative to an agent's background knowledge k. I will also make this standard assumption since I think that it is standard for good reason. Howeve6 for ease of presentation, k will be suppressed in future premises except those cited from other texts. Hence, any assignment of the form F [å] is best regarded as shorthand for F[À ll], FUr I el as shorthand for IF[hl e-kf, c[h,e] as shorthand for c fh, e I k), and similarly for the rest. 3.2 The Argument from Entailment It is part of a venerable tradition in the philosophy of science that analyses in their nascent stages are foremost informed by the sciences themselves. In confirmation theory the rough template adduced from scientific practice is also sharpened by ex- amining simple probabiliry models involving urns, cards, dice and coins; each im- posing constraints on the confirmation function. Certainly, the clarity and precision of these austere cases have done much to promote understanding. Nevertheless, analyses that content themselves with the examination of famous episodes from the history of science and simplistic probability models unduly restrict the range of evidence available to them. In this section, I examine a structural constraint on the confirmation function that stems from the often overlooked metaphysics of science.