Stephan Hartmann

In a famous experiment by Tversky and Kahneman (Psychol Rev 90:293–315, 1983), featuring Linda the bank teller, the participants assign a higher probability to a conjunction of propositions than to one of the conjuncts, thereby seemingly committing a probabilistic fallacy. In this paper, we discuss a slightly different example featuring someone named Walter, who also happens to work at a bank, and argue that, in this example, it is rational to assign a higher probability to the conjunction of suitably chosen propositions than to one of the conjuncts. By pointing out the similarities between Tversky and Kahneman’s experiment and our example, we argue that the participants in the experiment may assign probabilities to the propositions in question in such a way that it is also rational for them to give the conjunction a higher probability than one of the conjuncts

A federal assembly consists of a number of representatives for each of the nations (states, Länder, cantons,...) that make up the federation. How many representatives should each nation receive? What makes this issue worth quibbling about is that the model of representation that is instituted will have an impact on the welfare distribution over the nations in the federation that will ensue over due course. We will investigate what models of representation yield welfare distributions that score higher on a utilitarian measure. First, we construct a continuum of models of representation ranging from equal to proportional representation. In between these extremes are models of degressive proportionality. We run a Monte-Carlo simulation in which a large number of motions are voted up or down within different contexts of evaluation and investigate how well the resulting welfare distributions score on the utilitarian measure. Subsequently, we will provide matching analytical results for a slightly idealized case. We conclude with a discussion of the significance of our results and of the role of simulations and analytical results and point to further work.

The variety of evidence thesis in confirmation theory states that more varied supporting evidence confirms a hypothesis to a greater degree than less varied evidence. Under a very plausible interpretation of this thesis, positive test results from multiple independent instruments confirm a hypothesis to a greater degree than positive test results from a single instrument. We invoke Bayesian Networks to model confirmation on grounds of evidence that is obtained from less than fully reliable instruments and show that the variety of evidence thesis is not sacrosanct when testing is conducted with less than fully reliable instruments: under certain conditions, a hypothesis receives more confirmation from evidence that is obtained from one rather than from more independent instruments. In the appendix, we prove certain convergence results for large numbers of positive test results from single versus multiple less than fully reliable instruments.

There is a long philosophical tradition of addressing questions in philosophy of science and epistemology by means of the tools of Bayesian probability theory (see Earman (1992) and Howson and Urbach (1993)). In the late '70s, an axiomatic approach to conditional independence was developed within a Bayesian framework. This approach in conjunction with developments in graph theory are the two pillars of the theory of Bayesian Networks, which is a theory of probabilistic reasoning in artificial intelligence. The theory has been very successful over the last two decades and has found a wide array of applications ranging from medical diagnosis to safety systems for hazardous industries.

The aggregation of consistent individual judgments on logically interconnected propositions into a collective judgment on the same propositions has recently drawn much attention. Seemingly reasonable aggregation procedures, such as propositionwise majority voting, cannot ensure an equally consistent collective conclusion. The literature on judgment aggregation refers to such a problem as the \textit{discursive dilemma}. In this paper we assume that the decision which the group is trying to reach is factually right or wrong. Hence, we address the question of how good the various approaches are at selecting the right conclusion. We focus on two approaches: distance-based procedures and a Bayesian analysis. They correspond to group-internal and group-external decision-making, respectively. We compare those methods in a probabilistic model, demonstrate the robustness of our results over various generalizations and discuss their applicability in different situations. The findings vindicate (i) that in judgment aggregation problems, reasons should carry higher weight than conclusions and (ii) that considering members of an advisory board to be highly competent is a better strategy than to underestimate their advice.".

Models are a principle instrument of modern science. They are built, applied, tested, compared, revised and interpreted in an expansive scientific literature. Throughout this paper, I will argue that models are also a valuable tool for the philosopher of science. In particular, I will discuss how the methodology of Bayesian Networks can elucidate two central problems in the philosophy of science. The first thesis I will explore is the variety-of-evidence thesis, which argues that the more varied the supporting evidence, the greater the degree of confirmation for a given hypothesis. However, when investigated using Bayesian methodology, this thesis turns out not to be sacrosanct. In fact, under certain conditions, a hypothesis receives more confirmation from evidence that is obtained from one rather than more instruments, and from evidence that confirms one rather than more testable consequences of the hypothesis. The second challenge that I will investigate is scientific theory change. This application highlights a different virtue of modeling methodology. In particular, I will argue that Bayesian modeling illustrates how two seemingly unrelated aspects of theory change, namely the (Kuhnian) stability of (normal) science and the ability of anomalies to over turn that stability and lead to theory change, are in fact united by a single underlying principle, in this case, coherence. In the end, I will argue that these two examples bring out some metatheoretical reflections regarding the following questions: What are the differences between modeling in science and modeling in philosophy? What is the scope of the modeling method in philosophy? And what does this imply for our understanding of Bayesianism?

In his entry on "Quantum Logic and Probability Theory" in the Stanford Encyclopedia of Philosophy, Alexander Wilce (2012) writes that "it is uncontroversial (though remarkable) the formal apparatus quantum mechanics reduces neatly to a generalization of classical probability in which the role played by a Boolean algebra of events in the latter is taken over the 'quantum logic' of projection operators on a Hilbert space." For a long time, Patrick Suppes has opposed this view (see, for example, the paper collected in Suppes and Zanotti (1996). Instead of changing the logic and moving from a Boolean algebra to a non-Boolean algebra, one can also 'save the phenomena' by weakening the axioms of probability theory and work instead with upper and lower probabilities. However, it is fair to say that despite Suppes' efforts upper and lower probabilities are not particularly popular in physics as well as in the foundations of physics, at least so far. Instead, quantum logic is booming again, especially since quantum information and computation became hot topics. Interestingly, however, imprecise probabilities are becoming more and more popular in formal epistemology as recent work by authors such as James Joye (2010) and Roger White (2010) demonstrates.

The formalism of quantum mechanics provides us with the probabilities for certain events to occur—this much is uncontroversial. But how are we to understand these probabilities? The essays in this special issue approach this question from different angles. The first three contributions take as their point of departure the philosophy of probability and discuss what the two main outlooks—objective and subjective interpretations of probability—have to offer in the context of quantum mechanics. The following five papers explore the question of how probabilities in particular interpretations of quantum mechanics—modal interpretations, relative state interpretations, Bohmian mechanics, and GRW theory—can be interpreted. The last three essays address specific issues: non-standard probability and algebraic quantum field theory, quantum propensities, and the relation between decoherence and the interpretation of probability.

This book successfully achieves to serve two different purposes. On the one hand, it is a readable physics-based introduction into the philosophy of science, written in an informal and accessible style. The author, himself a professor of physics at the University of Notre Dame and active in the philosophy of science for almost twenty years, carefully develops his metatheoretical arguments on a solid basis provided by an extensive survey along the lines of the historical development of physics. On the other hand, this book supplies one long argument for Cushing´s own attitude in the philosophy of science. While former studies of the author, from which this book draws in part, focused each on one special episode in the history of science, this book gathers case material from many different parts of physics and epochs. The main goal of this book is ”to impress upon the reader the essential and ineliminable role that philosophical considerations have played in the actual practice of science” (p. xv). The book is beautifully edited and produced; it contains a wealth of illustrative figures, well-chosen short quotations from original sources and contemporary commentators (some longer quotations are relegated in an appendix at the end of a chapter) and does not dispense with insightful mathematical arguments in the main text (some advanced deductions are, however, relegated in the appendices). It contains nine parts, whereas only the first and the last one are exclusively devoted to philosophical issues. The seven remaining parts, each subdivided into three chapters, centre around one major episode (a theory, a world view, etc.) in the history of physics. The author presents this material in a clear and philosophically unbiased way so that also readers who do not share Cushing’s subsequent philosophical conclusions will find this inspiring book extremely useful. Part 1 (”The scientific enterprise”) discusses some traditional (”objectivist”) views concerning the status of scientific knowledge, ”the” scientific method, and the relation....