Jan Sprenger and Stephan Hartmann offer a fresh approach to central topics in philosophy of science, including causation, explanation, evidence, and scientific models. Their Bayesian approach uses the concept of degrees of belief to explain and to elucidate manifold aspects of scientific reasoning.

Responding to recent concerns about the reliability of the published literature in psychology and other disciplines, we formed the X-Phi Replicability Project to estimate the reproducibility of experimental philosophy. Drawing on a representative sample of 40 x-phi studies published between 2003 and 2015, we enlisted 20 research teams across 8 countries to conduct a high-quality replication of each study in order to compare the results to the original published findings. We found that x-phi studies – as represented in our sample (...) – successfully replicated about 70% of the time. We discuss possible reasons for this relatively high replication rate in the field of experimental philosophy and offer suggestions for best research practices going forward. (shrink)

This article introduces and defends a probabilistic measure of the explanatory power that a particular explanans has over its explanandum. To this end, we propose several intuitive, formal conditions of adequacy for an account of explanatory power. Then, we show that these conditions are uniquely satisfied by one particular probabilistic function. We proceed to strengthen the case for this measure of explanatory power by proving several theorems, all of which show that this measure neatly corresponds to our explanatory intuitions. Finally, (...) we briefly describe some promising future projects inspired by our account. (shrink)

Scientific theories are hard to find, and once scientists have found a theory, H, they often believe that there are not many distinct alternatives to H. But is this belief justified? What should scientists believe about the number of alternatives to H, and how should they change these beliefs in the light of new evidence? These are some of the questions that we will address in this article. We also ask under which conditions failure to find an alternative to H (...) confirms the theory in question. This kind of reasoning is frequently used in science and therefore deserves a careful philosophical analysis. 1 Introduction2 The Conceptual Framework3 The No Alternatives Argument4 Discussion I: A Quantitative Analysis of the No Alternatives Argument5 Discussion II: The Number of Alternatives and the Problem of Underdetermination6 ConclusionsAppendix AAppendix B. (shrink)

This paper explores trivalent truth conditions for indicative conditionals, examining the “defective” truth table proposed by de Finetti and Reichenbach. On their approach, a conditional takes the value of its consequent whenever its antecedent is true, and the value Indeterminate otherwise. Here we deal with the problem of selecting an adequate notion of validity for this conditional. We show that all standard validity schemes based on de Finetti’s table come with some problems, and highlight two ways out of the predicament: (...) one pairs de Finetti’s conditional with validity as the preservation of non-false values, but at the expense of Modus Ponens; the other modifies de Finetti’s table to restore Modus Ponens. In Part I of this paper, we present both alternatives, with specific attention to a variant of de Finetti’s table proposed by Cooper and Cantwell. In Part II, we give an in-depth treatment of the proof theory of the resulting logics, DF/TT and CC/TT: both are connexive logics, but with significantly different algebraic properties. (shrink)

The finding that intuitions about the reference of proper names vary cross-culturally was one of the early milestones in experimental philosophy. Many follow-up studies investigated the scope and magnitude of such cross-cultural effects, but our paper provides the first systematic meta-analysis of studies replicating. In the light of our results, we assess the existence and significance of cross-cultural effects for intuitions about the reference of proper names.

Bayesian epistemology addresses epistemological problems with the help of the mathematical theory of probability. It turns out that the probability calculus is especially suited to represent degrees of belief (credences) and to deal with questions of belief change, confirmation, evidence, justification, and coherence. Compared to the informal discussions in traditional epistemology, Bayesian epis- temology allows for a more precise and fine-grained analysis which takes the gradual aspects of these central epistemological notions into account. Bayesian epistemology therefore complements traditional epistemology; it (...) does not re- place it or aim at replacing it. (shrink)

In this paper we argue that there is a kind of moral disagreement that survives the Rawlsian veil of ignorance. While a veil of ignorance eliminates sources of disagreement stemming from self-interest, it does not do anything to eliminate deeper sources of disagreement. These disagreements not only persist, but transform their structure once behind the veil of ignorance. We consider formal frameworks for exploring these differences in structure between interested and disinterested disagreement, and argue that consensus models offer us a (...) solution concept for disagreements behind the veil of ignorance. (shrink)

Why are conditional degrees of belief in an observation E, given a statistical hypothesis H, aligned with the objective probabilities expressed by H? After showing that standard replies are not satisfactory, I develop a suppositional analysis of conditional degree of belief, transferring Ramsey’s classical proposal to statistical inference. The analysis saves the alignment, explains the role of chance-credence coordination, and rebuts the charge of arbitrary assessment of evidence in Bayesian inference. Finally, I explore the implications of this analysis for Bayesian (...) reasoning with idealized models in science. (shrink)

Subjective Bayesianism is a major school of uncertain reasoning and statistical inference. It is often criticized for a lack of objectivity: it opens the door to the influence of values and biases, evidence judgments can vary substantially between scientists, it is not suited for informing policy decisions. My paper rebuts these concerns by connecting the debates on scientific objectivity and statistical method. First, I show that the above concerns arise equally for standard frequentist inference with null hypothesis significance tests. Second, (...) the criticisms are based on specific senses of objectivity with unclear epistemic value. Third, I show that Subjective Bayesianism promotes other, epistemically relevant senses of scientific objectivity—most notably by increasing the transparency of scientific reasoning. (shrink)

The enduring replication crisis in many scientific disciplines casts doubt on the ability of science to estimate effect sizes accurately, and in a wider sense, to self-correct its findings and to produce reliable knowledge. We investigate the merits of a particular countermeasure—replacing null hypothesis significance testing with Bayesian inference—in the context of the meta-analytic aggregation of effect sizes. In particular, we elaborate on the advantages of this Bayesian reform proposal under conditions of publication bias and other methodological imperfections that are (...) typical of experimental research in the behavioral sciences. Moving to Bayesian statistics would not solve the replication crisis single-handedly. However, the move would eliminate important sources of effect size overestimation for the conditions we study. (shrink)

This paper develops axiomatic foundations for a probabilistic-interventionist theory of causal strength. Transferring methods from Bayesian confirmation theory, I proceed in three steps: I develop a framework for defining and comparing measures of causal strength; I argue that no single measure can satisfy all natural constraints; I prove two representation theorems for popular measures of causal strength: Pearl's causal effect measure and Eells' difference measure. In other words, I demonstrate these two measures can be derived from a set of plausible (...) adequacy conditions. The paper concludes by sketching future research avenues. (shrink)

This paper explores the scope and limits of rational consensus through mutual respect, with the primary focus on the best known formal model of consensus: the Lehrer–Wagner model. We consider various arguments against the rationality of the Lehrer–Wagner model as a model of consensus about factual matters. We conclude that models such as this face problems in achieving rational consensus on disagreements about unknown factual matters, but that they hold considerable promise as models of how to rationally resolve non-factual disagreements.

One of the most troubling and persistent challenges for Bayesian Confirmation Theory is the Problem of Old Evidence. The problem arises for anyone who models scientific reasoning by means of Bayesian Conditionalization. This article addresses the problem as follows: First, I clarify the nature and varieties of the POE and analyze various solution proposals in the literature. Second, I present a novel solution that combines previous attempts while making weaker and more plausible assumptions. Third and last, I summarize my findings (...) and put them into the context of the general debate about POE and Bayesian reasoning. (shrink)

The aggregation of consistent individual judgments on logically interconnected propositions into a collective judgment on those 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 that problem as the discursive dilemma. In this paper, we motivate that many groups do not only want to reach a factually right conclusion, but also want to correctly evaluate the reasons for that conclusion. In (...) other words, we address the problem of tracking the true situation instead of merely selecting the right outcome. We set up a probabilistic model analogous to Bovens and Rabinowicz (2006) and compare several aggregation procedures by means of theoretical results, numerical simulations and practical considerations. Among them are the premise-based, the situation-based and the distance-based procedure. Our findings confirm the conjecture in Hartmann, Pigozzi and Sprenger (2008) that the premise-based procedure is a crude, but reliable and sometimes even optimal form of judgment aggregation. (shrink)

Data from medical research are typically summarized with various types of outcome measures. We present three arguments in favor of absolute over relative outcome measures. The first argument is from cognitive bias: relative measures promote the reference class fallacy and the overestimation of treatment effectiveness. The second argument is decision-theoretic: absolute measures are superior to relative measures for making a decision between interventions. The third argument is causal: interpreted as measures of causal strength, absolute measures satisfy a set of desirable (...) properties, but relative measures do not. Absolute outcome measures outperform relative measures on all counts. (shrink)

The enduring replication crisis in many scientific disciplines casts doubt on the ability of science to estimate effect sizes accurately, and in a wider sense, to self-correct its findings and to produce reliable knowledge. We investigate the merits of a particular countermeasure—replacing null hypothesis significance testing with Bayesian inference—in the context of the meta-analytic aggregation of effect sizes. In particular, we elaborate on the advantages of this Bayesian reform proposal under conditions of publication bias and other methodological imperfections that are (...) typical of experimental research in the behavioral sciences. Moving to Bayesian statistics would not solve the replication crisis single-handedly. However, the move would eliminate important sources of effect size overestimation for the conditions we study. (shrink)

This paper develops a probabilistic reconstruction of the No Miracles Argument in the debate between scientific realists and anti-realists. The goal of the paper is to clarify and to sharpen the NMA by means of a probabilistic formalization. In particular, we demonstrate that the persuasive force of the NMA depends on the particular disciplinary context where it is applied, and the stability of theories in that discipline. Assessments and critiques of "the" NMA, without reference to a particular context, are misleading (...) and should be relinquished. This result has repercussions for recent anti-realist arguments, such as the claim that the NMA commits the base rate fallacy. It also helps to explain the persistent disagreement between realists and anti-realists. (shrink)

This paper focuses on the question of how to resolve disagreement and uses the Lehrer-Wagner model as a formal tool for investigating consensual decision-making. The main result consists in a general definition of when agents treat each other as epistemic peers (Kelly 2005; Elga 2007), and a theorem vindicating the “equal weight view” to resolve disagreement among epistemic peers. We apply our findings to an analysis of the impact of social network structures on group deliberation processes, and we demonstrate their (...) stability with the help of numerical simulations. (shrink)

Hypothetico-deductive (H-D) confirmation builds on the idea that confirming evidence consists of successful predictions that deductively follow from the hypothesis under test. This article reviews scope, history and recent development of the venerable H-D account: First, we motivate the approach and clarify its relationship to Bayesian confirmation theory. Second, we explain and discuss the tacking paradoxes which exploit the fact that H-D confirmation gives no account of evidential relevance. Third, we review several recent proposals that aim at a sounder and (...) more comprehensive formulation of H-D confirmation. Finally, we conclude that the reputation of hypothetico-deductive confirmation as outdated and hopeless is undeserved: not only can the technical problems be addressed satisfactorily, the hypothetico-deductive method is also highly relevant for scientific practice. (shrink)

The interpretation of tests of a point null hypothesis against an unspecified alternative is a classical and yet unresolved issue in statistical methodology. This paper approaches the problem from the perspective of Lindley's Paradox: the divergence of Bayesian and frequentist inference in hypothesis tests with large sample size. I contend that the standard approaches in both frameworks fail to resolve the paradox. As an alternative, I suggest the Bayesian Reference Criterion: it targets the predictive performance of the null hypothesis in (...) future experiments; it provides a proper decision-theoretic model for testing a point null hypothesis and it convincingly accounts for Lindley's Paradox. (shrink)

A descriptive norm is a behavioral rule that individuals follow when their empirical expectations of others following the same rule are met. We aim to provide an account of the emergence of descriptive norms by first looking at a simple case, that of the standing ovation. We examine the structure of a standing ovation, and show it can be generalized to describe the emergence of a wide range of descriptive norms.

According to influential accounts of scientific method, such as critical rationalism, scientific knowledge grows by repeatedly testing our best hypotheses. But despite the popularity of hypothesis tests in statistical inference and science in general, their philosophical foundations remain shaky. In particular, the interpretation of non-significant results—those that do not reject the tested hypothesis—poses a major philosophical challenge. To what extent do they corroborate the tested hypothesis, or provide a reason to accept it? Popper sought for measures of corroboration that could (...) adequately answer this question. According to Popper, corroboration is different from probability-raising, and grounded in the predictive success and testability of a hypothesis. As such, corroboration becomes an indicator of the scientific value of a hypothesis and guides our practical preferences over hypotheses that have been subjected to severe tests. This article proves two impossibility results for corroboration measures based on statistical relevance. The generality of these results shows that Popper’s qualitative characterization of corroboration must be misguided. I explore what a more promising and scientifically useful concept of corroboration could look like. (shrink)

Appendix 1 was incomplete in the initial online publication. The original article has been corrected.

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.". (shrink)

This paper synthesizes confirmation by instances and confirmation by successful predictions, and thereby the Hempelian and the hypothetico-deductive traditions in confirmation theory. The merger of these two approaches is subsequently extended to the piecemeal confirmation of entire theories. It is then argued that this synthetic account makes a useful contribution from both a historical and a systematic perspective.

This paper applies Causal Modeling Semantics (CMS, e.g., Galles and Pearl 1998; Pearl 2000; Halpern 2000) to the evaluation of the probability of counterfactuals with disjunctive antecedents. Standard CMS is limited to evaluating (the probability of) counterfactuals whose antecedent is a conjunction of atomic formulas. We extend this framework to disjunctive antecedents, and more generally, to any Boolean combinations of atomic formulas. Our main idea is to assign a probability to a counterfactual ( A ∨ B ) > C at (...) a causal model M by looking at the probability of C in those submodels that truthmake A ∨ B (Briggs 2012; Fine 2016, 2017). The probability of p (( A ∨ B ) > C ) is then calculated as the average of the probability of C in the truthmaking submodels, weighted by the inverse distance to the original model M. The latter is calculated on the basis of a proposal by Eva et al. (2019). Apart from solving a major problem in the research on counterfactuals, our paper shows how work in semantics, causal inference and formal epistemology can be fruitfully combined. (shrink)

Inductive logic generalizes the idea of logical entailment and provides standards for the evaluation of non-conclusive arguments. A main application of inductive logic is the generalization of observational data to theoretical models. In the empirical sciences, the mathematical theory of statistics addresses the same problem. This paper argues that there is no separable purely logical aspect of statistical inference in a variety of complex problems. Instead, statistical practice is often motivated by decision-theoretic considerations and resembles empirical science.

To what extent does the design of statistical experiments, in particular sequential trials, affect their interpretation? Should postexperimental decisions depend on the observed data alone, or should they account for the used stopping rule? Bayesians and frequentists are apparently deadlocked in their controversy over these questions. To resolve the deadlock, I suggest a three‐part strategy that combines conceptual, methodological, and decision‐theoretic arguments. This approach maintains the pre‐experimental relevance of experimental design and stopping rules but vindicates their evidential, postexperimental irrelevance. †To (...) contact the author, please write to: Tilburg Center for Logic and Philosophy of Science, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands; e‐mail: [email protected] (shrink)

Precaution is a relevant and much-invoked value in environmental risk analysis, as witnessed by the ongoing vivid discussion about the precautionary principle (PP). This article argues (i) against purely decision-theoretic explications of PP; (ii) that the construction, evaluation, and use of scientific models falls under the scope of PP; and (iii) that epistemic and decision-theoretic robustness are essential for precautionary policy making. These claims are elaborated and defended by means of case studies from climate science and conservation biology.

Group judgements are often – implicitly or explicitly – influenced by their members’ individual expertise. However, given that expertise is seldom recognized fully and that some distortions may occur (bias, correlation, etc.), it is not clear that differential weighting is an epistemically advantageous strategy with respect to straight averaging. Our paper characterizes a wide set of conditions under which differential weighting outperforms straight averaging and embeds the results into the multidisciplinary group decision-making literature.

In many scientific, economic and policy-related problems, pieces of information from different sources have to be aggregated. Typically, the sources are not equally competent. This raises the question of how the relative weights and competences should be related to arrive at an optimal final verdict. Our paper addresses this question under a more realistic perspective of measuring the practical loss implied by an inaccurate verdict.

The application of probabilistic arguments to rational decisions in a single case is a contentious philosophical issue which arises in various contexts. Some authors (e.g. Horgan, Philos Pap 24:209–222, 1995; Levy, Synthese 158:139–151, 2007) affirm the normative force of probabilistic arguments in single cases while others (Baumann, Am Philos Q 42:71–79, 2005; Synthese 162:265–273, 2008) deny it. I demonstrate that both sides do not give convincing arguments for their case and propose a new account of the relationship between probabilistic reasoning (...) and rational decisions. In particular, I elaborate a flaw in Baumann’s reductio of rational single-case decisions in a modified Monty Hall Problem. (shrink)

Randomized Controlled Trials are currently the gold standard within evidence-based medicine. Usually, they are conducted as sequential trials allowing for monitoring for early signs of effectiveness or harm. However, evidence from early stopped trials is often charged with being biased towards implausibly large effects. To our mind, this skeptical attitude is unfounded and caused by the failure to perform appropriate conditioning in the statistical analysis of the evidence. We contend that a shift from unconditional hypothesis tests in the style of (...) Neyman and Pearson to conditional hypothesis tests gives a superior appreciation of the obtained evidence and significantly improves the practice of sequential medical trials, while staying firmly rooted in frequentist methodology. (shrink)

Scientific and statistical inferences build heavily on explicit, parametric models, and often with good reasons. However, the limited scope of parametric models and the increasing complexity of the studied systems in modern science raise the risk of model misspecification. Therefore, I examine alternative, data-based inference techniques, such as bootstrap resampling. I argue that their neglect in the philosophical literature is unjustified: they suit some contexts of inquiry much better and use a more direct approach to scientific inference. Moreover, they make (...) more parsimonious assumptions and often replace theoretical understanding and knowledge about mechanisms by careful experimental design. Thus, it is worthwhile to study in detail how nonparametric models serve as inferential engines in science. (shrink)

The question of how judgments of explanatory value inform probabilistic inference is well studied within psychology and philosophy. Less studied are the questions: How does probabilistic information affect judgments of explanatory value? Does probabilistic information take precedence over causal information in determining explanatory judgments? To answer these questions, we conducted two experimental studies. In Study 1, we found that probabilistic information had a negligible impact on explanatory judgments of event-types with a potentially unlimited number of available, alternative explanations; causal credibility (...) was the main determinant of explanatory value. In Study 2, we found that, for event-token explanations with a definite set of candidate alternatives, probabilistic information strongly affected judgments of explanatory value. In the light of these findings, we reassess under which circumstances explanatory inference is probabilistically sound. (shrink)

Bayesian model selection has frequently been the focus of philosophical inquiry (e.g., Forster, Br J Philos Sci 46:399–424, 1995; Bandyopadhyay and Boik, Philos Sci 66:S390–S402, 1999; Dowe et al., Br J Philos Sci 58:709–754, 2007). This paper argues that Bayesian model selection procedures are very diverse in their inferential target and their justification, and substantiates this claim by means of case studies on three selected procedures: MML, BIC and DIC. Hence, there is no tight link between Bayesian model selection and (...) Bayesian philosophy. Consequently, arguments for or against Bayesian reasoning based on properties of Bayesian model selection procedures should be treated with great caution. (shrink)

Explanation is a central concept in human psychology. Drawing upon philosophical theories of explanation, psychologists have recently begun to examine the relationship between explanation, probability and causality. Our study advances this growing literature in the intersection of psychology and philosophy of science by systematically investigating how judgments of explanatory power are affected by the prior credibility of a potential explanation, the causal framing used to describe the explanation, the generalizability of the explanation, and its statistical relevance for the evidence. Collectively, (...) the results of our five experiments support the hypothesis that the prior credibility of a causal explanation plays a central role in explanatory reasoning: first, because of the presence of strong main effects on judgments of explanatory power, and second, because of the gate-keeping role it has for other factors. Highly credible explanations were not susceptible to causal framing effects. Instead, highly credible hypotheses were sensitive to the effects of factors which are usually considered relevant from a normative point of view: the generalizability of an explanation, and its statistical relevance for the evidence. These results advance current literature in the philosophy and psychology of explanation in three ways. First, they yield a more nuanced understanding of the determinants of judgments of explanatory power, and the interaction between these factors. Second, they illuminate the close relationship between prior beliefs and explanatory power. Third, they clarify the relationship between abductive and probabilistic reasoning. (shrink)

The origins of testing scientific models with statistical techniques go back to 18th century mathematics. However, the modern theory of statistical testing was primarily developed through the work of Sir R.A. Fisher, Jerzy Neyman, and Egon Pearson in the inter-war period. Some of Fisher's papers on testing were published in economics journals (Fisher, 1923, 1935) and exerted a notable influence on the discipline. The development of econometrics and the rise of quantitative economic models in the mid-20th century made statistical significance (...) testing a commonplace, albeit controversial tool within economics. -/- In the debate about significance testing, methodological controversies intertwine with epistemological issues and sociological developments. Our aim in this chapter is to expound these connections and to show how the use of, and the debate about, significance testing in economics differs from other social sciences, such as psychology. (shrink)

Abductive reasoning assigns special status to the explanatory power of a hypothesis. But how do people make explanatory judgments? Our study clarifies this issue by asking: How does the explanatory power of a hypothesis cohere with other cognitive factors? How does probabilistic information affect explanatory judgments? In order to answer these questions, we conducted an experiment with 671 participants. Their task was to make judgments about a potentially explanatory hypothesis and its cognitive virtues. In the responses, we isolated three constructs: (...) Explanatory Value, Rational Acceptability, and Entailment. Explanatory judgments strongly cohered with judgments of causal relevance and with a sense of understanding. Furthermore, we found that Explanatory Value was sensitive to manipulations of statistical relevance relations between hypothesis and evidence, but not to explicit information about the prior probability of the hypothesis. These results indicate that probabilistic information about statistical relevance is a strong determinant of Explanatory Value. More generally, our study suggests that abductive and probabilistic reasoning are two distinct modes of inference. (shrink)

Statistical inference is often justified by long-run properties of the sampling distributions, such as the repeated sampling rationale. These are frequentist justifications of statistical inference. I argue, in line with existing philosophical literature, but against a widespread image in empirical science, that these justifications are flawed. Then I propose a novel interpretation of probability in statistics, the artefactual interpretation. I believe that this interpretation is able to bridge the gap between statistical probability calculations and rational decisions on the basis of (...) observed data. The artefactual interpretation is able to justify statistical inference without making any assumptions about probability in the material world. (shrink)

Over the years, mathematics and statistics have become increasingly important in the social sciences1 . A look at history quickly confirms this claim. At the beginning of the 20th century most theories in the social sciences were formulated in qualitative terms while quantitative methods did not play a substantial role in their formulation and establishment. Moreover, many practitioners considered mathematical methods to be inappropriate and simply unsuited to foster our understanding of the social domain. Notably, the famous Methodenstreit also concerned (...) the role of mathematics in the social sciences. Here, mathematics was considered to be the method of the natural sciences from which the social sciences had to be separated during the period of maturation of these disciplines. All this changed by the end of the century. By then, mathematical, and especially statistical, methods were standardly used, and their value in the social sciences became relatively uncontested. The use of mathematical and statistical methods is now ubiquitous: Almost all social sciences rely on statistical methods to analyze data and form hypotheses, and almost all of them use (to a greater or lesser extent) a range of mathematical methods to help us understand the social world. Additional indication for the increasing importance of mathematical and statistical methods in the social sciences is the formation of new subdisciplines, and the establishment of specialized journals and societies. Indeed, subdisciplines such as Mathematical Psychology and Mathematical Sociology emerged, and corresponding journals such as The Journal of Mathematical Psychology (since 1964), The Journal of Mathematical Sociology (since 1976), Mathematical Social Sciences (since 1980) as well as the online journals Journal of Artificial Societies and Social Simulation (since 1998) and Mathematical Anthropology and Cultural Theory (since 2000) were established. What is more, societies such as the Society for Mathematical Psychology (since 1976) and the Mathematical Sociology Section of the American Sociological Association (since 1996) were founded. Similar developments can be observed in other countries. The mathematization of economics set in somewhat earlier (Vazquez 1995; Weintraub 2002). However, the use of mathematical methods in economics started booming only in the second half of the last century (Debreu 1991). Contemporary economics is dominated by the mathematical approach, although a certain style of doing economics became more and more under attack in the last decade or so. Recent developments in behavioral economics and experimental economics can also be understood as a reaction against the dominance (and limitations) of an overly mathematical approach to economics. There are similar debates in other social sciences. It is, however, important to stress that problems of one method (such as axiomatization or the use of set theory) can hardly be taken as a sign of bankruptcy of mathematical methods in the social sciences tout court. This chapter surveys mathematical and statistical methods used in the social sciences and discusses some of the philosophical questions they raise. It is divided into two parts. Sections 1 and 2 are devoted to mathematical methods, and Sections 3 to 7 to statistical methods. As several other chapters in this handbook provide detailed accounts of various mathematical methods, our remarks about the latter will be rather short and general. Statistical methods, on the other hand, will be discussed in-depth. (shrink)

In a recent Analysis piece, John Shand (2014) argues that the Predictive Theory of Mind provides a unique explanation for why one cannot play chess against oneself. On the basis of this purported explanatory power, Shand concludes that we have an extra reason to believe that PTM is correct. In this reply, we first rectify the claim that one cannot play chess against oneself; then we move on to argue that even if this were the case, Shand’s argument does not (...) give extra weight to the Predictive Theory of Mind. (shrink)

Existing accounts of hypothetico-deductive confirmation are able to circumvent the classical objections, but the confirmation of conjunctions of hypotheses brings them into trouble. Therefore this paper develops a new, falsificationist account of qualitative confirmation by means of Ken Gemes ' theory of content parts. The new approach combines the hypothetico-deductive view with falsificationist and instance confirmation principles. It is considerably simpler than the previous suggestions and gives a better treatment of conjunctive hypotheses while solving the tacking problems equally well.

There is considerable confusion about the role of p-values in statistical model checking. To clarify that point, I introduce the distinction between measures of surprise and measures of evidence which come with different epistemological functions. I argue that p-values, often understood as measures of evidence against a null model, do not count as proper measures of evidence and are closer to measures of surprise. Finally, I sketch how the problem of old evidence may be tackled by acknowledging the epistemic role (...) of surprise indices. (shrink)

According to influential accounts of scientific method, e.g., critical rationalism, scientific knowledge grows by repeatedly testing our best hypotheses. In comparison to rivaling accounts of scientific reasoning such as Bayesianism, these accounts are closer to crucial aspects of scientific practice. But despite the preeminence of hypothesis tests in statistical inference, their philosophical foundations are shaky. In particular, the interpretation of "insignificant results"---outcomes where the tested hypothesis has survived the test---poses a major epistemic challenge that is not sufficiently addressed by the (...) standard methodology for conducting such tests. In this paper, I argue that a quantitative explication of degree of corroboration can fill this important methodological and epistemological gap. First, I argue that this concept is distinct from the Bayesian notion of evidential support and that it plays an independent role in scientific reasoning. Second, I demonstrate that degree of corroboration cannot be suitably explicated in a probabilistic relevance framework, as proposed by Popper. Third, I derive two measures of corroboration that possess a large number of attractive properties, establish an insightful relation between corroboration and evidential support and are not committed to a Bayesian or a frequentist framework. In sum, the paper rethinks the foundations of inductive inference by providing a novel logic of hypothesis testing. (shrink)

The relation between probabilistic and explanatory reasoning is a classical topic in philosophy of science. Most philosophical analyses are concerned with the compatibility of Inference to the Best Explanation with probabilistic, Bayesian inference, and the impact of explanatory considerations on the assignment of subjective probabilities. This paper reverses the question and asks how causal and explanatory considerations are affected by probabilistic information. We investigate how probabilistic information determines the explanatory value of a hypothesis, and in which sense folk explanatory practice (...) can be said to be rational. Our study identifies three main factors in reasoning about a explanatory hypothesis: cognitive salience, rational acceptability and logical entailment. This corresponds well to the variety of philosophical accounts of explanation. Moreover, we show that these factors are highly sensitive to manipulations of probabilistic information. This finding suggests that probabilistic reasoning is a crucial part of explanatory inferences, and it motivates new avenues of research in the debate about Inference to the Best Explanation and probabilistic measures of explanatory power. (shrink)

In Part I of this paper, we identified and compared various schemes for trivalent truth conditions for indicative conditionals, most notably the proposals by de Finetti and Reichenbach on the one hand, and by Cooper and Cantwell on the other. Here we provide the proof theory for the resulting logics DF/TT and CC/TT, using tableau calculi and sequent calculi, and proving soundness and completeness results. Then we turn to the algebraic semantics, where both logics have substantive limitations: DF/TT allows for (...) algebraic completeness, but not for the construction of a canonical model, while CC/TT fails the construction of a Lindenbaum-Tarski algebra. With these results in mind, we draw up the balance and sketch future research projects. (shrink)

Philosophy of science has seen a passionate debate over the influence of non-cognitive values on theory choice. In this paper, we argue against a dichotomous divide between cognitive and non-cognitive values and for the possibility of a dual role for feminist values. By analyzing the influence of feminist values on evolutionary psychology and evolutionary biology, we show how they have cognitive and non-cognitive functions at the same time.