There are three questions associated with Simpson’s Paradox (SP): (i) Why is SP paradoxical? (ii) What conditions generate SP?, and (iii) What should be done about SP? By developing a logic-based account of SP, it is argued that (i) and (ii) must be divorced from (iii). This account shows that (i) and (ii) have nothing to do with causality, which plays a role only in addressing (iii). A counterexample is also presented against the causal account. Finally, the causal (...) and logic-based approaches are compared by means of an experiment to show that SP is not basically causal. (shrink)

There are three questions associated with Simpson’s paradox (SP): (i) Why is SP paradoxical? (ii) What conditions generate SP? and (iii) How to proceed when confronted with SP? An adequate analysis of the paradox starts by distinguishing these three questions. Then, by developing a formal account of SP, and substantiating it with a counterexample to causal accounts, we argue that there are no causal factors at play in answering questions (i) and (ii). Causality enters only in connection (...) with action. (shrink)

Simpson's Paradox is introduced and analysed via the mishaps of a researcher who at first falls afoul of the traps Simpson-reversals can set, and then he learns to exploit those traps to advantage. (Note: An error in the treatment of the Sure Thing Principle is corrected in "Simpson's Paradox: A Logically Benign, Empirically Treacherous Hydra").

It is shown that the Fisher smoking problem and Newcomb's problem are decisiontheoretically identical, each having at its core an identical case of Simpson's paradox for certain probabilities. From this perspective, incorrect solutions to these problems arise from treating them as cases of decisionmaking under risk, while adopting certain global empirical conditional probabilities as the relevant subjective probabihties. The most natural correct solutions employ the methodology of decisionmaking under uncertainty with lottery acts, with certain local empirical conditional probabilities adopted (...) as the relevant subjective probabilities. (shrink)

This article examines Simpson's paradox as applied to the theory of probabilites and percentages. The author discusses possible flaws in the paradox and compares it to the Sure Thing Principle, statistical inference, causal inference and probabilistic analyses of causation.

It is shown that the Fisher smoking problem and Newcomb's problem are decisiontheoretically identical, each having at its core an identical case of Simpson's paradox for certain probabilities. From this perspective, incorrect solutions to these problems arise from treating them as cases of decisionmaking under risk, while adopting certain global empirical conditional probabilities as the relevant subjective probabihties. The most natural correct solutions employ the methodology of decisionmaking under uncertainty with lottery acts, with certain local empirical conditional probabilities adopted (...) as the relevant subjective probabilities. (shrink)

This paper discusses Simpson's paradox and the problem of positive relevance in probabilistic causality. It is argued that Cartwright's solution to Simpson's paradox fails because it ignores one crucial form of the paradox. After clarifying different forms of the paradox, it is shown that any adequate solution to the paradox must allow a cause to be both a negative cause and a positive cause of..

There are three distinct questions associated with Simpson’s paradox. Why or in what sense is Simpson’s paradox a paradox? What is the proper analysis of the paradox? How one should proceed when confronted with a typical case of the paradox? We propose a “formal” answer to the first two questions which, among other things, includes deductive proofs for important theorems regarding Simpson’s paradox. Our account contrasts sharply with Pearl’s causal account of (...) the first two questions. We argue that the “how to proceed question?” does not have a unique response, and that it depends on the context of the problem. We evaluate an objection to our account by comparing ours with Blyth’s account of the paradox. Our research on the paradox suggests that the “how to proceed question” needs to be divorced from what makes Simpson’s paradox “paradoxical.”. (shrink)

There are three distinct questions associated with Simpson's paradox, (i) Why or in what sense is Simpson's paradox a paradox? (ii) What is the proper analysis of the paradox? (iii) How one should proceed when confronted with a typical case of the paradox? We propose a "formar" answer to the first two questions which, among other things, includes deductive proofs for important theorems regarding Simpson's paradox. Our account contrasts sharply with Pearl's causal (and questionable) (...) account of the first two questions. We argue that the "how to proceed question?" does not have a unique response, and that it depends on the context of the problem. We evaluate an objection to our account by comparing ours with Blyth's account of the paradox. Our research on the paradox suggests that the "how to proceed question" needs to be divorced from what makes Simpson's paradox "paradoxical.". (shrink)

We address the need for a model by considering two competing theories regarding the origin of life: (i) the Metabolism First theory, and (ii) the RNA World theory. We discuss two interrelated points, namely: (i) Models are valuable tools for understanding both the processes and intricacies of origin-of-life issues, and (ii) Insights from models also help us to evaluate the core objection to origin-of-life theories, called “the inefficiency objection”, which is commonly raised by proponents of both the Metabolism First theory (...) and the RNA World theory against each other. We use Simpson’s Paradox (SP) as a tool for challenging this objection. We will use models in various senses, ranging from taking them as representations of reality to treating them as theories/accounts that provide heuristics for probing reality. In this paper, we will frequently use models and theories interchangeably. Additionally, we investigate Conway’s Game of Life and contrast it with our SP-based approach to emergence-of-life issues. Finally, we discuss some of the consequences of our view. A scientific model is testable in three senses: (i) a logical sense, (ii) a nomological sense, and (iii) a current technological sense. The SP-based model is testable in the first two senses but it is not feasible to test it using current technology. (shrink)

Numerous papers have investigated the transitivity principle of ‘better-than.’ A recent argument appeals to the principle of mereological dominance for transitivity. However, writers have not treated mereological dominance in much detail. This paper sets out to evaluate the generality of mereological dominance and its effectiveness in supporting the transitivity principle. I found that the mereological dominance principle is vulnerable to a counterexample based on Simpson’s Paradox. The thesis concludes that the mereological dominance principle should be revised in certain (...) ways. (shrink)

Here is a simplified fiction which is based on a real case at a Californian University. The Faculty of Humanities decided to try to increase the number of women on their staff. There were 13 women and 13 men who applied for positions in the Faculty. All the positions were directed towards the study of either time or space, in the departments of History or Geography. There were 13 applicants for the positions in History and 13 applicants for the positions (...) in Geography. Of the women applicants 8 applied for positions in History, and 5 for Geography, whereas 5 of the men applied for positions in History and 8 in Geography. (shrink)

This paper argues for a claim made by Maria Carla Galavotti that the use of indeterministic causality involves one in Simpson's paradox. It is shown specifically that a consideration of Hesslow's well-known counter-example leads to Simpson's paradox.

A common way of analyzing the statistical relation between two tests of memory is to use contingency analyses. A potential problem with such analyses is known as Simpson’s paradox. The paradox is that collapsing two or more contingency tables may have the effect that the relationship expressed in the overall contingency table differs from the relationships expressed in the original tables. The paradox arises when covariates are correlated with each of the tests. It has been claimed (...) that the paradox has implications for the analysis of memory retrieval, and ways of solving the problem have been called for (Hintzman, 1980). In this paper, partial gamma (Davis, 1967) is suggested as one possible solution. This method can be used to compute a weighted average of the results in the original tables. The use of partial gamma is exemplified by applying it to hypothetical instances of Simpson’s paradox. (shrink)

ABSTRACTIn this paper, I review some recent treatments of Simpson's Paradox, and I propose a new rationalizing explanation of its paradoxicality.

Richard Otte (1985) has recently criticized the resolution of Simpson's paradox given by Nancy Cartwright (1979). He argues that there are difficulties with the version of the theory of probabilistic causality that Cartwright has developed, and that there is a way in which Simpson's paradox can arise that Cartwright's theory cannot handle. And Otte develops his own theory of probabilistic causality. I defend Cartwright's solution, and I argue that there are difficulties with the theory of probabilistic causality that (...) Otte proposes. (shrink)

We propose an alternative resolution of Simpson's paradox in multiple classification experiments, using a different maximum likelihood estimator. In the center of our analysis is a formal representation of free choice and randomization that is based on the notion of incompatible measurements.We first introduce a representation of incompatible measurements as a collection of sets of outcomes. This leads to a natural generalization of Kolmogoroff's axioms of probability. We then discuss the existence and uniqueness of the maximum likelihood estimator for (...) a probability weight on such a generalized sample space, given absolute frequency data.As a first example, we discuss an estimation problem with censured data that classically admits only biased ad hoc estimators.Next, we derive an explicit solution of the maximum likelihood estimation problem for a large class of experiments that arise from various kids of compositions of sample spaces. We identify the (categorical) direct sum of sample spaces as a representation of “free choice,” and the (categorical) direct product as a representation of “randomization.”Finally, we apply the foregoing discussion to the case of multiple classification experiments in order to show that there is no Simpson's paradox if the difference between free choice and randomization is recognized in the structure of the experiment.A comparison between our new estimator and the “usual” calculation can be summarized as follows: Pooling the data over one classification factor in the “usual” way in fact destroys or ignores the information contained in it, whereas our proposed maximum likelihood estimator is a proper marginal over this factor that “averages out” the information contained in it. The estimators agree with each other in the case of proportional sample sizes. (shrink)

One unique part of Rousseau's Social Contract is his argument that a just society must have a specific constitutional arrangement of powers centred around what he calls the Sovereign and the Prince. This makes his philosophy different from other contractualists, such as Hobbes and Locke, who think that the principles of good government are compatible with any number of institutional structures. Rousseau's constitutional theory is thus significant in a way that has no parallel in Hobbes or Locke. More to the (...) point, any problems that exist in his constitutional theory will have consequences for his political thought as a whole. This article argues that there is a contradiction at the center of Rousseau's theory of institutions that threatens the cogency of the Social Contract. Key Words: Rousseau • separation of powers • social contract • sovereignty. (shrink)

In Nietzsche’s later work the problem of the possibility of philosophy presents a significant interpretative and practical dilemma. Nietzsche attempts to undermine the idea of the absolute, as a source of value, meaning and truth, and to tease out the traces of this idea in our philosophising. He is thus one of those who has given us the means to complete the Kantian project of moving beyond metaphysical realism and a representational understanding of meaning. However, along with the gift comes (...) a paradox. For Nietzsche’s diagnosis seems to make it clear that desire for the absolute is intrinsic to the practice of philosophy – that in important respects, philosophy just is the (hopeless) attempt to frame or discover overarching, context-less objectivity. Furthermore, Nietzsche’s analysis of philosophy is accompanied by a recognition and critique of the nihilism that arises in reaction to the collapse of absolutism. I suggest that we can find (in Nietzsche) a resolution of this paradox that involves the continuation of philosophy, not through a consciousness for philosophy, but through understanding philosophy as a process without a subject. That is, understanding philosophy as a practice that does not involve a moment that is the resolution of the paradox at all, but which is a process, involving the continual crisis of its paradox (which we might see as a constitutive paradox). (shrink)

In this chapter I argue that emerging soldier enhancement technologies have the potential to transform the ethical character of the relationship between combatants, in conflicts between ‘Superpower’ militaries, with the ability to deploy such technologies, and technologically disadvantaged ‘Underdog’ militaries. The reasons for this relate to Paul Kahn’s claims about the paradox of riskless warfare. When an Underdog poses no threat to a Superpower, the standard just war theoretic justifications for the Superpower’s combatants using lethal violence against their opponents (...) breaks down. Therefore, Kahn argues, combatants in that position must approach their opponents in an ethical guise relevantly similar to ‘policing’. I argue that the kind of disparities in risk and threat between opposing combatants that Kahn’s analysis posits, don’t obtain in the context of face-to-face combat, in the way they would need to in order to support his ethical conclusions about policing. But then I argue that soldier enhancement technologies have the potential to change this, in a way that reactivates the force of those conclusions. (shrink)

Adam Smith is perceived, through his best-known book, An Inquiry into the Nature and Causes of the Wealth of Nations, as the founder of economics as a science. His thought has shaped modern ideas about the market economy and the role of the state in relation to it. Yet Smith needs to be recognized as more than this, as a man of letters, moralist, historian, and critic, as well as an economist, if we are to get full value for his (...) ideas and perspectives in contemporary applications. Ian Simpson Ross is the biographer of Lord Kames, Smith's patron, and of the Scottish poet William Dunbar, and has edited, with E C Mossner, Smith's correspondence for the Glasgow edition of his works. In this, the first full-scale biography of Adam Smith for a hundred years, Ross brings his subject in to historical light as a thinker and author by examining his family circumstance, education, career, and social and intellectual circle, including David Hume and Francois Quesnay, revealed through his correspondence, archival documents, the reports of contemporaries, and the record of his publications. Readers will meet Smith as a student at a lively Glasgow and sleepy Oxford; freelance lecturer in rhetoric; innovative university teacher; tutor travelling abroad with a Duke; acclaimed political economist; policy advisor to governments during and after the American crisis; and finally, if paradoxically in view of his tenets, a Commissioner of Customs coping with the free traders in the smuggling business. This is the life of a Scottish moral philosopher whose legacy of thought concerns and affects us all. Its lively and informed account will appeal to those interested in the social and intellectual milieu of the eighteenth century, and in scottish history. Economists and philosophers will find much to read about the history of their disciplines, supported by full documentation. (shrink)

Smith moved from Kirkcaldy to Edinburgh late in 1778, after his appointment as a Commissioner for managing His Majesty's Customs in Scotland. We may think it a paradox that this prominent advocate of free trade should end up enforcing the mercantile system, but there was a family tradition of Customs service, and while WN does attack restraints on some branches of trade and encouragement for others, especially in the form of monopolies, Smith was not an across the board economic (...) libertarian. He urged that there were reasons of state, as well as economic considerations, for implementing an efficient collection of moderate duties on imports and exports. In Edinburgh, Smith regularly entertained the intelligentsia and their visitors at a Friday Club and Sunday suppers. (shrink)

There is no more important question in thinking about life-and actually living-in political community than whether it is to be permeated by, and purposefully oriented around, the main truths about human flourishing. It is at least paradoxical that, precisely when the state and its law and political life are shaping people's lives more and more, the professed roots of all this influence are growing thinner, more shallow. Lawmakers who profess and in many cases even think they should be "neutral" about (...) values are more involved with how persons' lives go than, perhaps, ever before. Of course, any community which has lost faith in the truth about human wellbeing is going to be quite befuddled by the question. But the perennial centrality of the question about truth and politics can be obscured for understandable reasons. One is the immediate urgency of so many practical challenges, such as settling the defense or state budget, protecting against terrorist attack, or deciding what to do about a nuclear-armed North Korea. Attention to the fundamental question is attenuated, too, by focus upon diffuse problems which affect nearly everyone daily, such as getting good healthcare coverage or creating jobs. Significant public debates about moral issues affecting policy are still common. But they tend to be episodic, and wary of getting down to foundations. When Americans argue today about immigration policy and capital punishment, for instance, they rely upon profound ethical beliefs. All too often, however, they link these commitments not to the objective moral truth at the heart of the matter, but instead to past practices, or to communal identity as a matter of fact who we are!"). (shrink)

I describe epistemic versions of the contrast between causal and conventionally probabilistic decision theory, including an epistemic version of Newcomb's paradox.

The theory of lower previsions is designed around the principles of coherence and sure-loss avoidance, thus steers clear of all the updating anomalies highlighted in Gong and Meng's "Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss, and Simpson's Paradox" except dilation. In fact, the traditional problem with the theory of imprecise probability is that coherent inference is too complicated rather than unsettling. Progress has been made simplifying coherent inference by demoting sets of probabilities from fundamental building blocks to secondary (...) representations that are derived or discarded as needed. (shrink)

The introduction of statistical models represented by directed acyclic graphs (DAGs) has proved fruitful in the construction of expert systems, in allowing efficient updating algorithms that take advantage of conditional independence relations (Pearl, 1988, Lauritzen et al. 1993), and in inferring causal structure from conditional independence relations (Spirtes and Glymour, 1991, Spirtes, Glymour and Scheines, 1993, Pearl and Verma, 1991, Cooper, 1992). As a framework for representing the combination of causal and statistical hypotheses, DAG models have shed light on a (...) number of issues in statistics ranging from Simpson’s Paradox to experimental design (Spirtes, Glymour and Scheines, 1993). The relations of DAGs with statistical constraints, and the equivalence and distinguishability properties of DAG models, are now well understood, and their characterization and computation involves three properties connecting graphical structure and probability distributions: (i) a local directed Markov property, (ii) a global directed Markov property, (iii) and factorizations of joint densities according to the structure of a graph (Lauritizen, et al., 1990). (shrink)

Probab ility (probability; subjective and objective probability; the Principal Principle; independence and correlation; conditional probability; material, indicative and subjunctive conditionals; correlation and causation; screening off; Simpson’s paradox; Bayes’ theorem; Bayesian updating).

It has recently been suggested that philosophy – in particular epistemology – has a contribution to make to the analysis of criminal and military intelligence. The present article pursues this suggestion, taking three phenomena that have recently been studied by philosophers, and showing that they have important implications for the gathering and sharing of intelligence, and for the use of intelligence in the determining of military strategy. The phenomena discussed are: (1) Simpson's Paradox, (2) the distinction between resiliency and (...) reliability of data, and (3) the Causal Markov Condition. (shrink)

Twenty years ago, Nancy Cartwright wrote a perceptive essay in which she clearly distinguished causal relations from associations, introduced philosophers to Simpson’s paradox, articulated the difficulties for reductive probabilistic analyses of causation that flow from these observations, and connected causal relations with strategies of action (Cartwright 1979). Five years later, without appreciating her essay, I and my (then) students began to develop formal representations of causal and probabilistic relations, which, subsequently informed by the work of computer scientists and (...) statisticians, led eventually to a practical theory of causal inference and prediction, a theory incorporating some of the sensibilities Cartwright had voiced (Glymour et al. 1987; Spirtes et al. 1993). That theory, and ideas related to it, have become a subfield of computer science with contributions far deeper than mine from many sources, and its inferential and predictive techniques have been successfully applied in biology, economics, educational research, geology and space physics. (shrink)

Economics prefers complete explanations: general over partial equilibrium, microfoundational over aggregate. Similarly, probabilistic accounts of causation frequently prefer greater detail to less as in typical resolutions of Simpson’s paradox. Strategies of causal refinement equally aim to distinguish direct from indirect causes. Yet, there are countervailing practices in economics. Representative-agent models aim to capture economic motivation but not to reduce the level of aggregation. Small structural vector-autoregression and dynamic stochastic general-equilibrium models are practically preferred to larger ones. The distinction (...) between exogenous and endogenous variables suggests partitioning the world into distinct subsystems. The tension in these practices is addressed within a structural account of causation inspired by the work of Herbert Simon’s, which defines cause with reference to complete systems adapted to deal with incomplete systems and piecemeal evidence. The focus is on understanding the constraints that a structural account of causation places on the freedom to model complex or lower-order systems as simpler or higher-order systems and on to what degree piecemeal evidence can be incorporated into a structural account. (shrink)

Causal modeling methods such as path analysis, used in the social and natural sciences, are also highly relevant to philosophical problems of probabilistic causation and statistical explanation. We show how these methods can be effectively used (1) to improve and extend Salmon's S-R basis for statistical explanation, and (2) to repair Cartwright's resolution of Simpson's paradox, clarifying the relationship between statistical and causal claims.

In what follows, I critique the interpretation that Sober and Wilson offer of their group selection model in Unto Others. Sober and Wilson mistakenly claim that their model operates as an example of Simpson's paradox and defend an interpretation of their model according to which groups are operated upon by natural selection. In the place of their interpretation, I offer one that parallels the mathematical calculation of the model's outcome and does not depend on the postulation of a force (...) of group selection or a value for group fitness. (shrink)

BackgroundWinning funding for health and medical research usually involves a lengthy application process. With success rates under 20%, much of the time spent by 80% of applicants could have been better used on actual research. An alternative funding system that could save time is using democracy to award the most deserving researchers based on votes from the research community. We aimed to pilot how such a system could work and examine some potential biases.MethodsWe used an online survey with a convenience (...) sample of Australian researchers. Researchers were asked to name the 10 scientists currently working in Australia that they thought most deserved funding for future research. For comparison, we used recent winners from large national fellowship schemes that used traditional peer review.ResultsVoting took a median of 5 min (inter-quartile range 3 to 10 min). Extrapolating to a national voting scheme, we estimate 599 working days of voting time (95% CI 490 to 728), compared with 827 working days for the current peer review system for fellowships. The gender ratio in the votes was a more equal 45:55 (female to male) compared with 34:66 in recent fellowship winners, although this could be explained by Simpson’s paradox. Voters were biased towards their own institution, with an additional 1.6 votes per ballot (inter-quartile range 0.8 to 2.2) above the expected number. Respondents raised many concerns about the idea of using democracy to fund research, including vote rigging, lobbying and it becoming a popularity contest.ConclusionsThis is a preliminary study of using voting that does not investigate many of the concerns about how a voting system would work. We were able to show that voting would take less time than traditional peer review and would spread the workload over many more reviewers. Further studies of alternative funding systems are needed as well as a wide discussion with the research community about potential changes. (shrink)

Although the underrepresentation of women in academic philosophy is well-known and hard to overlook, philosophers started to pay more attention to the actual data on and potential explanations of the underrepresentation of women in philosophy only in the last decade and predominately in anglophone countries. The data I present in this paper deal with introductory logic courses at German universities (gender of textbook authors, gender of logic course instructors, student grades by gender at one university). Two results are particularly noteworthy: (...) For one, logic teaching in Germany is currently predominantly male and even more male than philosophy teaching as a whole. For another, there is no evidence for any differences between grades of female and male students. The student grade data, however, illustrate how statistical phenomena like Simpson's paradox can explain how male students can appear to achieve better grades than female students even if there is no underlying difference: When correcting for students’ major (philosophy vs. STEM), logic grades of male and female students are remarkably similar. Although the scope of the data presented here is limited, they allow to add further details to an emerging map of the profession. I close with some suggestions for avoiding stereotyping logic as “male”. (shrink)

Several authors have argued that non-extreme probabilities used in special sciences such as chemistry and biology can be objective chances, even if the true microphysical description of the world is deterministic. This article examines an influential version of this argument and shows that it depends on a particular methodology for defining the relationship between coarse-grained and fine-grained events. An alternative methodology for coarse-graining is proposed. This alternative methodology blocks this argument for the existence of emergent chances, and makes better sense (...) of two well-known subjects of philosophical discussion: the Miners Puzzle and Simpson’s Paradox. (shrink)

I compare the failure of counterfactual dependence as a criterion of event causation to the failure of stochastic dependence as a criterion of causal law. Counterexamples to the stochastic analysis arise from cases of Simpson's Paradox, and Nancy Cartwright has suggested a way of transforming the stochastic analysis into something that avoids these counterexample. There is an analogical relationship between cases of Simpson's Paradox and cases of causal overdetermination. I exploit this analogical relationship to motivate my own view (...) about the connection between counterfactuals and event causation. According to my view, counterfactuals and event causation are intimately connected by a principle which falls short of being an analysis but which, as I show, correctly handles not only overdetermination but common causes as well. (shrink)

This comment discusses the suggestions made in Mahoney’s “Constructivist Set-Theoretic Analysis: An Alternative to Essentialist Social Science” (2023). Mahoney presents an approach to cases of intersectionality or confounding which produce causal results unlike those that result from traditional net effects causal modeling. He presents it as an alternative to “essentialism,” which he describes as a cognitive error. These alternatives have the same problems as those he attributes to net effects analysis, with one exception: the method does allow for the identification (...) of subpopulations with different causal mechanisms producing different outcomes for this subpopulation. (shrink)