The Open Handbook of Formal Epistemology
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Presidential Address
Empowering Philosophy Christia Mercer COLUMBIA UNIVERSITY Presidential Address delivered at the one hundred sixteenth Eastern Division meeting of the American Philosophical Association in Philadelphia, PA, on January 10, 2020. The main goal of my presidential address in January 2020 was to show that philosophy’s past offers a means to empower its present. I hoped to encourage colleagues to make the philosophy we teach and practice more inclusive (both textually and topically) and to adopt a more public- facing engagement with our discipline. As I add these introductory remarks to my January lecture, it is June 2020 and the need to empower philosophy has never seemed more urgent. We’ve witnessed both the tragic death of George Floyd and the popular uprising of a diverse group of Americans in response to the ongoing violence against Black lives. Many white Americans—and many philosophers—have begun to realize that their inattentiveness to matters of diversity and inclusivity must now be seen as more than mere negligence. Recent demonstrations frequently contain signs that make the point succinctly: “Silence is violence.” A central claim of my January lecture was that philosophy’s status quo is no longer tenable. Even before the pandemic slashed university budgets and staff, our employers were cutting philosophy programs, enrollments were shrinking, and jobs were increasingly hard to find. Despite energetic attempts on the part of many of our colleagues to promote a more inclusive approach to our research and teaching, the depressing truth remains: -
The Bayesian Approach to the Philosophy of Science
The Bayesian Approach to the Philosophy of Science Michael Strevens For the Macmillan Encyclopedia of Philosophy, second edition The posthumous publication, in 1763, of Thomas Bayes’ “Essay Towards Solving a Problem in the Doctrine of Chances” inaugurated a revolution in the understanding of the confirmation of scientific hypotheses—two hun- dred years later. Such a long period of neglect, followed by such a sweeping revival, ensured that it was the inhabitants of the latter half of the twentieth century above all who determined what it was to take a “Bayesian approach” to scientific reasoning. Like most confirmation theorists, Bayesians alternate between a descrip- tive and a prescriptive tone in their teachings: they aim both to describe how scientific evidence is assessed, and to prescribe how it ought to be assessed. This double message will be made explicit at some points, but passed over quietly elsewhere. Subjective Probability The first of the three fundamental tenets of Bayesianism is that the scientist’s epistemic attitude to any scientifically significant proposition is, or ought to be, exhausted by the subjective probability the scientist assigns to the propo- sition. A subjective probability is a number between zero and one that re- 1 flects in some sense the scientist’s confidence that the proposition is true. (Subjective probabilities are sometimes called degrees of belief or credences.) A scientist’s subjective probability for a proposition is, then, more a psy- chological fact about the scientist than an observer-independent fact about the proposition. Very roughly, it is not a matter of how likely the truth of the proposition actually is, but about how likely the scientist thinks it to be. -
“Strange” Limits
c Gabriel Nagy “Strange” Limits Notes from the Functional Analysis Course (Fall 07 - Spring 08) Prerequisites: The reader is assumed to be familiar with the Hahn-Banach Theorem and/or the theory of (ultra)filter convergence. References to some of my notes will be added later. Notations. Given a non-empty set S, we denote by `∞(S) the vector space of all bounded R functions x : S → . On occasion an element x ∈ `∞(S) will also be written as an “S-tuple” R R x = (xs)s∈S. If S is infinite, we denote by Pfin(S) the collection of all finite subsets of S. Assuming S is infinite, for an element x = (x ) ∈ `∞(S), we can define the quantities s s∈S R lim sup xs = inf sup xs , F ∈P (S) S fin s∈SrF lim inf xs = sup inf xs . S s∈S F F ∈Pfin(S) r Definition. Suppose S is infinite. A limit operation on S is a linear map Λ : `∞(S) → , R R satisfying the inequalities: lim inf x ≤ Λ(x) ≤ lim sup x , ∀ x = (x ) ∈ `∞(S). (1) s s s s∈S R S S Comment. Of course, the quantities lim supS xs and lim infS xs can be defined for ar- bitrary “S-tuples,” in which case these limits might equal ±∞. In fact, this allows one to use S-tuples which might take infinite values. In other words, one might consider maps x : S → [−∞, ∞]. We will elaborate on his point of view later in this note. With the above terminology, our problem is to construct limit operations. -
University of Oklahoma Graduate College The
UNIVERSITY OF OKLAHOMA GRADUATE COLLEGE THE VIRTUES OF BAYESIAN EPISTEMOLOGY A DISSERTATION SUBMITTED TO THE GRADUATE FACULTY in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY By MARY FRANCES GWIN Norman, Oklahoma 2011 THE VIRTUES OF BAYESIAN EPISTEMOLOGY A DISSERTATION APPROVED FOR THE DEPARTMENT OF PHILOSOPHY BY ______________________________ Dr. James Hawthorne, Chair ______________________________ Dr. Chris Swoyer ______________________________ Dr. Wayne Riggs ______________________________ Dr. Steve Ellis ______________________________ Dr. Peter Barker © Copyright by MARY GWIN 2011 All Rights Reserved. For Ted How I wish you were here. Acknowledgements I would like to thank Dr. James Hawthorne for his help in seeing me through the process of writing my dissertation. Without his help and encouragement, I am not sure that I would have been able to complete my dissertation. Jim is an excellent logician, scholar, and dissertation director. He is also a kind person. I am very lucky to have had the opportunity to work with him. I cannot thank him enough for all that he has given me. I would also like to thank my friends Patty, Liz, Doren, Murry, Becky, Jim, Rebecca, Mike, Barb, Susanne, Blythe, Eric, Ty, Rosie, Bob, Judy, Lorraine, Paulo, Lawrence, Scott, Kyle, Pat, Carole, Joseph, Ken, Karen, Jerry, Ray, and Daniel. Without their encouragement, none of this would have been possible. iv Table of Contents Acknowledgments iv Abstract vi Chapter 1: Introduction 1 Chapter 2: Bayesian Confirmation Theory 9 Chapter 3: Rational Analysis: Bayesian Logic and the Human Agent 40 Chapter 4: Why Virtue Epistemology? 82 Chapter 5: Reliability, Credit, Virtue 122 Chapter 6: Epistemic Virtues, Prior Probabilities, and Norms for Human Performance 145 Bibliography 171 v Abstract The aim of this dissertation is to address the intersection of two normative epistemologies, Bayesian confirmation theory (BCT) and virtue epistemology (VE). -
Most Honourable Remembrance. the Life and Work of Thomas Bayes
MOST HONOURABLE REMEMBRANCE. THE LIFE AND WORK OF THOMAS BAYES Andrew I. Dale Springer-Verlag, New York, 2003 668 pages with 29 illustrations The author of this book is professor of the Department of Mathematical Statistics at the University of Natal in South Africa. Andrew I. Dale is a world known expert in History of Mathematics, and he is also the author of the book “A History of Inverse Probability: From Thomas Bayes to Karl Pearson” (Springer-Verlag, 2nd. Ed., 1999). The book is very erudite and reflects the wide experience and knowledge of the author not only concerning the history of science but the social history of Europe during XVIII century. The book is appropriate for statisticians and mathematicians, and also for students with interest in the history of science. Chapters 4 to 8 contain the texts of the main works of Thomas Bayes. Each of these chapters has an introduction, the corresponding tract and commentaries. The main works of Thomas Bayes are the following: Chapter 4: Divine Benevolence, or an attempt to prove that the principal end of the Divine Providence and Government is the happiness of his creatures. Being an answer to a pamphlet, entitled, “Divine Rectitude; or, An Inquiry concerning the Moral Perfections of the Deity”. With a refutation of the notions therein advanced concerning Beauty and Order, the reason of punishment, and the necessity of a state of trial antecedent to perfect Happiness. This is a work of a theological kind published in 1731. The approaches developed in it are not far away from the rationalist thinking. -
Hahn Banach Theorem
HAHN-BANACH THEOREM 1. The Hahn-Banach theorem and extensions We assume that the real vector space X has a function p : X 7→ R such that (1.1) p(ax) = ap(x), a ≥ 0, p(x + y) ≤ p(x) + p(y). Theorem 1.1 (Hahn-Banach). Let Y be a subspace of X on which there is ` such that (1.2) `y ≤ p(y) ∀y ∈ Y, where p is given by (1.1). Then ` can be extended to all X in such a way that the above relation holds, i.e. there is a `˜: X 7→ R such that ˜ (1.3) `|Y = `, `x ≤ p(x) ∀x ∈ X. Proof. The proof is done in two steps: we first show how to extend ` on a direction x not contained in Y , and then use Zorn’s lemma to prove that there is a maximal extension. If z∈ / Y , then to extend ` to Y ⊕ {Rx} we have to choose `z such that `(αz + y) = α`x + `y ≤ p(αz + y) ∀α ∈ R, y ∈ Y. Due to the fact that Y is a subspace, by dividing for α if α > 0 and for −α for negative α, we have that `x must satisfy `y − p(y − x) ≤ `x ≤ p(y + x) − `y, y ∈ Y. Thus we should choose `x by © ª © ª sup `y − p(y − x) ≤ `x ≤ inf p(y + x) − `y . y∈Y y∈Y This is possible since by subadditivity (1.1) for y, y0 ∈ R `(y + y0) ≤ p(y + y0) ≤ p(y + x) + p(y0 − x). Thus we can extend ` to Y ⊕ {Rx} such that `(y + αx) ≤ p(y + αx). -
Maty's Biography of Abraham De Moivre, Translated
Statistical Science 2007, Vol. 22, No. 1, 109–136 DOI: 10.1214/088342306000000268 c Institute of Mathematical Statistics, 2007 Maty’s Biography of Abraham De Moivre, Translated, Annotated and Augmented David R. Bellhouse and Christian Genest Abstract. November 27, 2004, marked the 250th anniversary of the death of Abraham De Moivre, best known in statistical circles for his famous large-sample approximation to the binomial distribution, whose generalization is now referred to as the Central Limit Theorem. De Moivre was one of the great pioneers of classical probability the- ory. He also made seminal contributions in analytic geometry, complex analysis and the theory of annuities. The first biography of De Moivre, on which almost all subsequent ones have since relied, was written in French by Matthew Maty. It was published in 1755 in the Journal britannique. The authors provide here, for the first time, a complete translation into English of Maty’s biography of De Moivre. New mate- rial, much of it taken from modern sources, is given in footnotes, along with numerous annotations designed to provide additional clarity to Maty’s biography for contemporary readers. INTRODUCTION ´emigr´es that both of them are known to have fre- Matthew Maty (1718–1776) was born of Huguenot quented. In the weeks prior to De Moivre’s death, parentage in the city of Utrecht, in Holland. He stud- Maty began to interview him in order to write his ied medicine and philosophy at the University of biography. De Moivre died shortly after giving his Leiden before immigrating to England in 1740. Af- reminiscences up to the late 1680s and Maty com- ter a decade in London, he edited for six years the pleted the task using only his own knowledge of the Journal britannique, a French-language publication man and De Moivre’s published work. -
Probabilities, Random Variables and Distributions A
Probabilities, Random Variables and Distributions A Contents A.1 EventsandProbabilities................................ 318 A.1.1 Conditional Probabilities and Independence . ............. 318 A.1.2 Bayes’Theorem............................... 319 A.2 Random Variables . ................................. 319 A.2.1 Discrete Random Variables ......................... 319 A.2.2 Continuous Random Variables ....................... 320 A.2.3 TheChangeofVariablesFormula...................... 321 A.2.4 MultivariateNormalDistributions..................... 323 A.3 Expectation,VarianceandCovariance........................ 324 A.3.1 Expectation................................. 324 A.3.2 Variance................................... 325 A.3.3 Moments................................... 325 A.3.4 Conditional Expectation and Variance ................... 325 A.3.5 Covariance.................................. 326 A.3.6 Correlation.................................. 327 A.3.7 Jensen’sInequality............................. 328 A.3.8 Kullback–LeiblerDiscrepancyandInformationInequality......... 329 A.4 Convergence of Random Variables . 329 A.4.1 Modes of Convergence . 329 A.4.2 Continuous Mapping and Slutsky’s Theorem . 330 A.4.3 LawofLargeNumbers........................... 330 A.4.4 CentralLimitTheorem........................... 331 A.4.5 DeltaMethod................................ 331 A.5 ProbabilityDistributions............................... 332 A.5.1 UnivariateDiscreteDistributions...................... 333 A.5.2 Univariate Continuous Distributions . 335 -
Epistemology Formalized
Philosophical Review Epistemology Formalized Sarah Moss University of Michigan As we often tell our undergraduates, epistemology is the study of knowl- edge. Given just this simple definition, ‘formal epistemology’ seems like a misnomer for the philosophical program inspired by Thomas Bayes and developed in the twentieth century by Ramsey (1926), de Finetti ([1937] 1993), Jeffrey (1983), and others. Bayesians articulate constraints on rational credences: synchronic constraints on what credences you may have, and diachronic constraints on how your credences must evolve. Like traditional epistemologists, Bayesians are concerned with norms governing your doxastic state. But in modeling your doxastic state, Bayes- ians do not represent what full beliefs you have.1 And so they do not have the resources to talk about which of those beliefs constitute knowledge. This paper develops a formal extension of traditional episte- mology for which ‘epistemology’ is not a misnomer.I accept the traditional claim that beliefs can constitute knowledge. But I argue for an apparently radical thesis about the doxastic states that Bayesians care about: some of these states can also constitute knowledge. For example, suppose you are playing an ordinary poker game, and you have just been dealt some mid- dling cards face down. Your justifiedly low credence that you have been Thanks to Dave Chalmers, Branden Fitelson, Allan Gibbard, John Hawthorne, Jim Joyce, Ted Sider, Jason Stanley, Seth Yalcin, and the Michigan Crop Circle for helpful comments, as well as audiences at the University of Pittsburgh, Princeton University, Rutgers Univer- sity, University of St Andrews, Syracuse University, Yale University, the 2011 Orange Beach Epistemology Workshop, and the 2011 Formal Epistemology Workshop. -
(Introduction to Probability at an Advanced Level) - All Lecture Notes
Fall 2018 Statistics 201A (Introduction to Probability at an advanced level) - All Lecture Notes Aditya Guntuboyina August 15, 2020 Contents 0.1 Sample spaces, Events, Probability.................................5 0.2 Conditional Probability and Independence.............................6 0.3 Random Variables..........................................7 1 Random Variables, Expectation and Variance8 1.1 Expectations of Random Variables.................................9 1.2 Variance................................................ 10 2 Independence of Random Variables 11 3 Common Distributions 11 3.1 Ber(p) Distribution......................................... 11 3.2 Bin(n; p) Distribution........................................ 11 3.3 Poisson Distribution......................................... 12 4 Covariance, Correlation and Regression 14 5 Correlation and Regression 16 6 Back to Common Distributions 16 6.1 Geometric Distribution........................................ 16 6.2 Negative Binomial Distribution................................... 17 7 Continuous Distributions 17 7.1 Normal or Gaussian Distribution.................................. 17 1 7.2 Uniform Distribution......................................... 18 7.3 The Exponential Density...................................... 18 7.4 The Gamma Density......................................... 18 8 Variable Transformations 19 9 Distribution Functions and the Quantile Transform 20 10 Joint Densities 22 11 Joint Densities under Transformations 23 11.1 Detour to Convolutions...................................... -
Truth and Justification in Knowledge Representation ⋆
Truth and Justification in Knowledge Representation ? Andrei Rodin1;2[0000−0002−3541−8867] and Serge Kovalyov3[0000−0001−5707−5730] 1 Institute of Philosophy RAS, 12/1 Goncharnaya Str., Moscow, 109240, Russian Federation 2 HSE, 20 Myasnitskaya Ulitsa, Moscow, 101000, Russian Federation [email protected] https://www.hse.ru/org/persons/218712080 3 ICS RAS, 65 Profsoyuznaya street, Moscow 117997, Russian Federation [email protected] https://www.ipu.ru/node/16660 Abstract. While the traditional philosophical epistemology stresses the importance of distinguishing knowledge from true beliefs, the formalisa- tion of this distinction with standard logical means turns out to be prob- lematic. In Knowledge Representation (KR) as a Computer Science disci- pline this crucial distinction is largely neglected. A practical consequence of this neglect is that the existing KR systems store and communicate knowledge that cannot be verified and justified by users of these systems without external means. Information obtained from such systems does not qualify as knowledge in the sense of philosophical epistemology. Recent advances in the research area at the crossroad of the compu- tational mathematical logic, formal epistemology and computer science open new perspectives for an effective computational realisation of jus- tificatory procedures in KR. After exposing the problem of justification in logic, epistemology and KR, we sketch a novel framework for rep- resenting knowledge along with relevant justificatory procedures, which is based on the Homotopy Type theory (HoTT). This formal framework supports representation of both propositional knowledge, aka knowledge- that, and non-propositional knowledge, aka knowledge-how or procedu- ral knowledge. The default proof-theoretic semantics of HoTT allows for combining the two sorts of represented knowledge at the formal level by interpreting all permissible constructions as justification terms (wit- nesses) of associated propositions. -
Inference to the Best Explanation, Bayesianism, and Knowledge
Inference to the Best Explanation, Bayesianism, and Knowledge Alexander Bird Abstract How do we reconcile the claim of Bayesianism to be a correct normative the- ory of scientific reasoning with the explanationists’ claim that Inference to the Best Explanation provides a correct description of our inferential practices? I first develop Peter Lipton’s approach to compatibilism and then argue that three challenges remain, focusing in particular on the idea that IBE can lead to knowl- edge. Answering those challenges requires renouncing standard Bayesianism’s commitment to personalism, while also going beyond objective Bayesianism regarding the constraints on good priors. The result is a non-standard, super- objective Bayesianism that identifies probabilities with evaluations of plausibil- ity in the light of the evidence, conforming to Williamson’s account of evidential probability. Keywords explanationism, IBE, Bayesianism, evidential probability, knowledge, plausibility, credences. 1 Introduction How do scientists reason? How ought they reason? These two questions, the first descriptive and the second normative, are very different. However, if one thinks, as a scientific realist might, that scientists on the whole reason as they ought to then one will expect the two questions to have very similar answers. That fact bears on debates between Bayesians and explanationists. When we evaluate a scientific hypothesis, we may find that the evidence supports the hypoth- esis to some degree, but not to such a degree that we would be willing to assert out-