There has recently been a renewed interest in the “force” interpretation of evolutionary biology. In this article, I present the general structure of the arguments for the force interpretation and identify a problem in its overly permissive conditions for being a Newtonian force. I then attempt a solution that (1) helps to illuminate the difference between forces and other types of causes and (2) makes room for random genetic drift as a force. In particular, I argue that forces are not (...) different in kind from other types of causes but rather that forces are situated on a continuum of causes distinguished by their unifying power. †To contact the author, please write to: Department of Philosophy, University of Wisconsin–Madison, 5185 Helen C. White Hall, Madison, WI 53706; e‐mail: [email protected]. (shrink)

In this paper, dealing with Platonic and Stoic dialectic in Philo, I wish to make a proposal that may offer some solution to the problem of the surprising absence of a proper use of the dialectic of the late Platonic dialogues in Philo’s works. Philonic scholars have not, to the best of my knowledge, raised this question; but Philo’s very rare allusions to Plato’s later dialogues were noted in David T. Runia’s comprehensive study on Philo and Plato’s Timaeus.

This work is concerned with the extent and magnitude of threat related to online radicalization in the context of terrorist acts and related offending. Online influences have been depicted as major drivers for the propagation and adoption of extremist ideologies, which often contain an element of collective grievance, and subsequent acts of violence. This is most pronounced in the discussion of so-called lone actor terrorism, but extends to all forms of extremist offending, and beyond. The present work situates online radicalization (...) leading to terrorist acts within the wider context of grievance-based beliefs and attitudes. Further, it addresses current positions and debates surrounding the relevance and mechanisms of online radicalization in terrorist offending. Recent evidence from quantitative studies is reviewed to estimate prevalence of online radicalization and the level of threat that results from it. This is followed by a discussion of plausible, but opposing, interpretations of the estimates presented. While online radicalization does occur, with and without reference to offline processes, the resulting threat is not overly high. This assessment, however, refers to the present only and is unlikely to hold for the future, given the general growth and acceleration of online activity among terrorist actors. (shrink)

Spanish philosopher José Ortega y Gasset and German philosopher Karl Jaspers were both born in 1883, and they both maintained the position that humans are principally historical beings. Therefore, as attested by this notion itself, there are points in which their philosophy coincides. Ortega argued that human beings have no nature, only history. His argument is that history as such is human nature; what is most natural about being human is the fact of being historical and thus always having historicity. (...) Jaspers maintained the same position that in contemplating historicity, one's focus should not be on human nature in the strictly hereditary sense, because it is one's traditions, not the genetic makeup, that most make one to be human. Jaspers emphasizes the conversion of an existential historic consciousness into a consciousness of historicity that is similar to what can be understood in Ortega as historical perspectivism imbued with pragmatism. (shrink)

In this book, Binder shows that at the heart of the most prominent arguments in favour of value-neutral approaches to overall freedom lies the value freedom has for human agency and development. Far from leading to the adoption of a value-neutral approach, however, ascribing importance to freedom’s agency value requires one to adopt a refined value-based approach. Binder employs an axiomatic framework in order to develop such an approach. She shows that a focus on freedom’s agency value has far reaching (...) consequences for existing results in the freedom ranking literature: it requires one to move beyond a person’s given all-things-considered preferences to the values underlying a person’s preference formation. Furthermore, it requires, as Binder argues, one to account for those differences between choice options which really matter to people. Binder illustrates the implications of her analysis for the evaluation of public policy and human development with the capability approach: only if sufficient importance is ascribed to freedom’s agency value can the capability approach keep its promises. ​. (shrink)

This book argues that Western philosophy's traditional understanding of Being as substance is incorrect, and demonstrates that Being is fundamentally Relationality. To make that argument, the book examines the history of Western philosophy's evolving conception of being, and shows how this tradition has been dominated by an Aristotelian understanding of substance and his corresponding understanding of relation. First, the book establishes that the original concept of Being in ancient Western philosophy was relational, and traces this relational understanding of Being through (...) the Neoplatonists. Then, it follows the substantial understanding of Being through Aristotle and the Scholastics to reach its crisis in Descartes. Finally, the book demonstrates that Heidegger represents a recovery of the original, relational understanding of Being. (shrink)

This paper argues that a principle manner in which Spanish philosopher José Ortega y Gasset’s historicist maxim ’man has no nature, what he has is history’ can be understood is through a pragmatist basis of anti-dualism, in part inherited from American philosopher John Dewey. The thesis here is that it is not that man has no nature, per se, rather that history is his nature because the two are anti-dualistic concepts; history is our nature because it is comprised of, as (...) famously posited by Ortega, “myself and my circumstance”. Hence, neither philosopher negates that “preparedness”, or “nature”, influences our being and behavior, rather they are arguing, more importantly, that “plasticity”, or nurture, plays a greater role given that this encompasses “nature” when we relate Ortega’s application of “history” to Dewey’s use of “experience”. Our “experiences” create our history, and interpretations as to the nature of them are not only potentially infinite, but they are primarily just that - interpretations, which can also change over time. Hence what we truly are is our history. Although these two philosophers are not frequently linked, there is a clear connection in this basic, Deweyen pragmatist anti-dualism as an elder of Ortega. And even though Dewey, unlike Ortega, wrote little on specifically the discipline of history, much of it was indeed fundamentally historical. (shrink)

A wide-ranging history of the intellectual developments that produced the modern idea of the algorithm. Bringing together the histories of mathematics, computer science, and linguistic thought, Language and the Rise of the Algorithm reveals how recent developments in artificial intelligence are reopening an issue that troubled mathematicians long before the computer age. How do you draw the line between computational rules and the complexities of making systems comprehensible to people? Here Jeffrey M. Binder offers a compelling tour of four visions (...) of universal computation that addressed this issue in very different ways: G. W. Leibniz's calculus ratiocinator; a universal algebra scheme Nicolas de Condorcet designed during the French Revolution; George Boole's nineteenth-century logic system; and the early programming language ALGOL, whose name is short for algorithmic language. These episodes show that symbolic computation has repeatedly become entangled in debates about the nature of communication. To what extent can meaning be controlled by individuals, like the values of a and b in algebra, and to what extent is meaning inevitably social? By attending to this long-neglected question, we come to see that the modern idea of the algorithm is implicated in a long history of attempts to maintain a disciplinary boundary separating technical knowledge from the languages people speak day to day. Machine learning, in its increasing dependence on words, now places this boundary in jeopardy, making its stakes all the more urgent to understand. The idea of the algorithm is a levee holding back the social complexity of language, and it is about to break. This book is about the flood that inspired its construction. (shrink)

Two decades ago, Rolf Landauer (1991) argued that “information is physical” and ought to have a role in the scientific analysis of reality comparable to that of matter, energy, space and time. This would also help to bridge the gap between biology and mathematics and physics. Although it can be argued that we are living in the ‘golden age’ of biology, both because of the great challenges posed by medicine and the environment and the significant advances that have been made—especially (...) in genetics and molecular and cell biology—we feel that information as an essential aspect of life has been neglected, or at least misunderstood. We therefore summon Maxwell’s Demon and its distant relative the ratchet, and apply these to biology. (shrink)

How has the international community responded to humanitarian crises after the end of the Cold War? While optimistic ideational perspectives on global governance stress the importance of humanitarian norms and argue that humanitarian crises have been increasingly addressed, more skeptical realist accounts point to material interests and maintain that these responses have remained highly selective. In empirical terms, however, we know very little about the actual extent of selectivity since, so far, the international community’s reaction to humanitarian crises has not (...) been systematically examined. This article addresses this gap by empirically examining the extent and the nature of the selectivity of humanitarian crises. To do so, the most severe humanitarian crises in the post-Cold War era are identified and examined for whether and how the international community responded. This study considers different modes of crisis response (ranging from inaction to military intervention) and different actors (including states, international institutions, and nonstate actors), yielding a more precise picture of the alleged “selectivity gap” and a number of theoretical implications for contemporary global security governance. (shrink)

A pragmatist philosopher of history asks what practical difference it makes for this or that historical “fact” to be taken as “useful and meaningful,” and then consider that the principal motivation behind what is recorded, what continues to circulate, and to what extent, in the annals of historical texts. Part of the methodology of pragmatism is derived from history, since usefulness is attested over time. History and historiography are shaped, in part, by pragmatic interests. This discussion is indispensable for the (...) era of digital humanities that is upon us, with all the new materials and interpretations that are to come as a result. (shrink)

Abstract:The history of metaphysical thought has been dominated by the notion of substance as the ground of being, with substance, primarily following Aristotle, being understood in terms of independent/separate existence. This understanding raises fundamental problems, a primary one being the one–many problem. As Plato recognizes in both Parmenides and the Sophist, to assert being to be fundamentally either one or many leads to contradictions. However, there is an alternative understanding of the ground of being which can be traced to some (...) of the earliest Presocratic thinkers. The author examines two of those philosophers, Anaximander and Heraclitus, and explores how they understand the ground of being primarily in terms of relation. (shrink)

My main argument will be that Plotinus’ notion of The One is best understood as “pure relation”. I will argue that Aristotle’s understanding of relation as being determined by relata has been the dominant understanding and that this is wrong-headed, that relata are actually determined by relationality. Thus, relation is actually ontologically prior to the relata. This entails that the First Principle of Being is pure relation.

It has always been an important task of economics to assess individual and social welfare. The traditional approach has assumed that the measuring rod for welfare is the satisfaction of the individual’s given and unchanging preferences, but recent work in behavioural economics has called this into question by pointing out the inconsistencies and context-dependencies of human behaviour. When preferences are no longer consistent, we have to ask whether a different measure for individual welfare can, and should, be found. This book (...) goes beyond the level of preference and instead considers whether a hedonistic view of welfare represents a viable alternative, and what its normative implications are. Offering a welfare theory with stronger behavioural and evolutionary foundations, Binder follows a naturalistic methodology to examine the foundations of welfare, connecting the concept with a dynamic theory of preference learning, and providing a more realistic account of human behaviour. This book will be of interest to researchers and those working in the fields of welfare economics, behavioural and evolutionary economics. (shrink)

This essay considers Shaul Magid's book on Meir Kahane and American Jewish politics through a discussion of the idea of modern Jewish “civility.” I argue that Magid's book enacts an argument that contemporary liberal Zionist politics are not as foreign to Kahanism as they might appear.

What is an Author? What’s in a name? Marnie Binder asks if it matters who’s writing, and other questions of authorship.

Scholars in both digital humanities and media studies have noted an apparent disconnect between computation and the interpretive methods of the humanities. Alan Liu has argued that literary scholars employing digital methods encounter a “meaning problem” due to the difficulty of reconciling algorithmic methods with interpretive ones. Conversely, the media scholar Friedrich Kittler has questioned the adequacy of hermeneutics as a means of studying computers. This paper argues that that this disconnect results from a set of contingent decisions made in (...) both humanistic and mathematical disciplines in the first half of the nineteenth century that delineated, with implications that continue to resonate in the present day, which aspects of human activity would come to be formalized in algorithms and which would not. I begin with a discussion of Nicolas de Condorcet, who attempted, at the height of the 1789 revolution, to turn algebra into a universal language; his work, I argue, exemplifies the form of algorithmic thinking that existed before the Romantic turn. Next, I discuss William Wordsworth’s arguments about the relationship of poetry and science. While Wordsworth is sometimes viewed as a critic of science, I argue that his polemic is specifically targeted at highly politicized projects like Condorcet’s that sought to supplant existing modes of thought with scientific rationality. Finally, I demonstrate the importance of Romantic thought for George Boole, creator of the logic system that would eventually form the basis of digital electronics. The reason Boole was able to succeed where Condorcet had failed, I argue, was that Romantic notions of culture enabled him to reconcile a mechanical view of mathematical reasoning with an organic view of the development of meaning—a dichotomy that remains a key assumption of computer interfaces in the twenty-first century. (shrink)

The theory of history is a central topic in Ortega’s writings. The American pragmatists wrote little on history. Ortega had fervent critiques of American Pragmatism. The argument presented here is that there are similarities in his theory on history with a pragmatist view, nonetheless, which can be summarized as a pragmatist perspectivism on the philosophy of history and historiography. Historical data selected for recording is largely determined by pragmatic reasoning; the historical details are useful, meaningful, relevant, and interesting –and continue (...) to be– as they help us solve problems, since a study of the past is effective in helping us better prepare for the future. There is important epistemological and metaphysical value to be found in a pragmatist perspective on history and historiography. (shrink)

Spanish Philosopher José Ortega y Gasset advanced a number of strong criticisms of American pragmatism, yet some pragmatist notions can also be detected in his own philosophy. Within Ortega’s pragmatist perspectivism one can locate the possibility of overcoming one of the principal perceived problems of pragmatism: namely, its tendency toward relativism. This paper focuses on the ways in which Ortega’s discussion of pragmatism pertains to history and historiography. Ortega’s position that history is written from a select number of perspectives is (...) congenial to pragmatist pluralism. What is recorded and continues to thrive in the annals of history is, within a pragmatist framework, whatever continues to be interesting, relevant, useful, and meaningful because it makes a difference – from and for these perspectives. The more of these perspectives we study, the closer we approach what, on Ortega’s view, constitutes the “eternal truth which every period has lived,” because materials initially gathered and framed for pragmatic reasons can later on provide important opportunities for the reflexive analysis of historical knowledge. (shrink)

The most widely repeated retributivist argument against the utilitarian theory of punishment is that utilitarianism permits punishment of the innocent. While defenders of utilitarianism have shown that a publicly announced policy of punishing the innocent is unlikely to serve utility, critics have insisted that utilitarianism morally obliges officials to deceive the public by framing the innocent. Yet philosophers and legal scholars have heretofore failed to test this claim against the writings of the theory's originators. We directly examine the writings of (...) Jeremy Bentham and other eighteenth and nineteenth century utilitarians and demonstrate that the originators of utilitarian penology clearly opposed both punishment of the innocent and deception of the public. We argue that utilitarianism originated as a legal theory that emphasized several institutional conditions for the public pursuit of utility, including security of person and property, legality, legislative supremacy, democratic accountability, publicity and transparency. These institutional conditions would preclude both systematic and ad hoc framing of the innocent. We show that the original utilitarians considered individuals incompetent to determine and pursue the public welfare, and that the contemporary conception of utilitarianism as an ethical standard governing individuals is a modern innovation. Bentham's theory of punishment did not derive from any general ethical theory. Bentham, like his chief forebears Hume, Helvetius and Beccaria, thought of public utility as a standard of value for public action, such as legislation. He assumed that private action was ruled by self-interest (i.e. private utility) so that there was little point in directing arguments about the general welfare to individual ethical actors. In assessing public action, Bentham was far less concerned about consequences than has generally been supposed and far more concerned about process. He identified utility with security of expectations and the rule of law. As a consequence, he endorsed public actions that could be seen to have emerged from a rational and well-informed debate about their consequences for the public welfare. Utility was not a definition of the good or a guide to conscience, but a standard designed for use in public, deliberative debate. Utilitarianism was not so much a philosophical theory as a rhetorical practice, understood as a transparent language of analysis and argument for use in political deliberation.We elaborate the procedural conditions presupposed by utilitarian discourse, beginning with Bentham's commitment to a conception of legality involving legislative promulgation of formal rules faithfully applied by rigidly constrained bureaucrats and judges. Next, we demonstrate Bentham's commitment to democratic representation of the public in devising legislation. Finally, we emphasize Bentham's commitment to publicity in government decisionmaking. Utilitarian policy required public scrutiny of all decisions and the information and reasons considered in making them. In sum, Bentham's utilitarianism was primarily concerned with the problem of how to design government so that it could accurately identify and faithfully pursue the public good while being openly seen to do so. It follows that Bentham's utility principle does not require him to endorse deceiving the public and framing an innocent person. (shrink)

Karl Popper developed a theory of deductive logic in the late 1940s. In his approach, logic is a metalinguistic theory of deducibility relations that are based on certain purely structural rules. Logical constants are then characterized in terms of deducibility relations. Characterizations of this kind are also called inferential definitions by Popper. In this paper, we expound his theory and elaborate some of his ideas and results that in some cases were only sketched by him. Our focus is on Popper's (...) notion of duality, his theory of modalities, and his treatment of different kinds of negation. This allows us to show how his works on logic anticipate some later developments and discussions in philosophical logic, pertaining to trivializing connectives, the duality of logical constants, dual-intuitionistic logic, the conservativeness of language extensions, the existence of a bi-intuitionistic logic, the non-logicality of minimal negation, and to the problem of logicality in general. (shrink)

The lumen naturale plays an important role in the philosophy of Descartes, particularly in the Meditations. Yet, while its analysis has not been completely neglected, the lumen naturale has hardly received the philosophical examination it deserves. While it is not possible in so short a pa­per to entirely remedy this deficiency, I do hope in this article to provide some insights into Descartes’ understanding of this concept. In this light, I will seek to examine Descartes’ understanding of the lumen naturale (...) in terms of how it functions and its object(s) in order to understand what, if anything, the lumen naturale tells us about the world around us. (shrink)

The pleasure of burning books consumed Montag, consumed him until the day the books burned back, their possibilities enflaming his curiosity to the point of existential immolation. Yet from these ashes, he rises. Fahrenheit 451 is a novel of ascent, an ascent to freedom that can be found only in knowledge. Superficially, the relationship between freedom and knowledge seems antagonistic; however, examining Bradbury’s novel in Plato’s light—particularly focusing on the images of the Cave and Line—can provide piercing insights into the (...) inherent harmony uniting knowledge and freedom, rendering this antagonism illusory. (shrink)

The understanding of Being in terms of substance has given rise to many philosophical problems, the most obvious and persistent of which is subject/object dualism. Heidegger recognises the problems substance ontology has created and rejects the ontological primacy of the subject. In doing so, he discovers an alternate ontological understanding, one that ultimately constitutes a return to a Neoplatonic ontology in which Being is understood in terms of relation. Heidegger’s ontology is, therefore, a recovery of this Neoplatonic relational ontology.

Recently, there has been a move in contemporary epistemological philosophy toward a virtue epistemology, which sees certain character traits of the rational agent as critical in the acquisition of knowledge. This attempt to introduce virtue into epistemological investigations has, however, relied almost exclusively on anAristotelian account of virtue. In this paper, I attempt to take a new tack and examine a virtue epistemological account grounded in Platonic thought. Taking seriously the distinction between knowledge and opinion found in the Republic, I (...) then draw upon two virtues, humility and what I call sincerity, to flesh out this account. (shrink)

While seeking novel food sources to feed the increasing population of the globe, several alternatives have been discussed, including algae, fungi or in vitro meat. The increasingly propagated usage of farmed insects for human nutrition raises issues regarding food safety, consumer information and animal protection. In line with law, insects like any other animals must not be reared or manipulated in a way that inflicts unnecessary pain, distress or harm on them. Currently, there is a great need for research in (...) the area of insect welfare, especially regarding species-specific needs, health, farming systems and humane methods of killing. Recent results from neurophysiological, neuroanatomical and behavioral sciences prompt caution when denying consciousness and therefore the likelihood of presence of pain and suffering or something closely related to it to insects. From an animal protection point of view, these issues should be satisfyingly solved before propagating and establishing intensive husbandry systems for insects as a new type of mini-livestock factory farming. (shrink)

This paper explores the role of existential vulnerability in the experience of personal identity and how identity is found and created. Existential vulnerabilities mark a boundary between what humans can bring about willfully or manipulate to their advantage and what is resistant to such actions. These vulnerabilities have their origin, on an ontological level, in fundamental conditions of human existence. At the same time, they have implications on a psychological level when it comes to self-experience and identity formation. Narrative and (...) value-based identity depend on how a person relates to finitude and the ambiguous side of lived experience. Relational identity depends on how a person relates to existential aloneness and the fact that the meaning and value of our actions are partly out of our control; they are always also dependent on other people’s responses to us. Bodily identity makes us feel continuous and real, but at the same time vulnerable to death and the gaze and actions of others. Being ‘thrown’ into an arbitrary life context is also a form of existential vulnerability. Authentic psychological identities can develop by giving meaning to these circumstances and balancing acceptance of existential vulnerability with the courage to make choices and act. (shrink)

Spanish Philosopher José Ortega y Gasset advanced a number of strong criticisms of American pragmatism, yet some pragmatist notions can also be detected in his own philosophy. Within Ortega’s pragmatist perspectivism one can locate the possibility of overcoming one of the principal perceived problems of pragmatism: namely, its tendency toward relativism. This paper focuses on the ways in which Ortega’s discussion of pragmatism pertains to history and historiography. Ortega’s position that history is written from a select number of perspectives is (...) congenial to pragmatist pluralism. What is recorded and continues to thrive in the annals of history is, within a pragmatist framework, whatever continues to be interesting, relevant, useful, and meaningful because it makes a difference – from and for these perspectives. The more of these perspectives we study, the closer we approach what, on Ortega’s view, constitutes the “eternal truth which every period has lived,” because materials initially gathered and framed for pragmatic reasons can later on provide important opportunities for the reflexive analysis of historical knowledge. (shrink)

In the present paper we empirically investigate the psychometric properties of some of the most famous statistical and logical cognitive illusions from the “heuristics and biases” research program by Daniel Kahneman and Amos Tversky, who nearly 50 years ago introduced fascinating brain teasers such as the famous Linda problem, the Wason card selection task, and so-called Bayesian reasoning problems (e.g., the mammography task). In the meantime, a great number of articles has been published that empirically examine single cognitive illusions, theoretically (...) explaining people’s faulty thinking, or proposing and experimentally implementing measures to foster insight and to make these problems accessible to the human mind. Yet these problems have thus far usually been empirically analyzed on an individual-item level only (e.g., by experimentally comparing participants’ performance on various versions of one of these problems). In this paper, by contrast, we examine these illusions as a group and look at the ability to solve them as a psychological construct. Based on an sample ofN= 2,643 Luxembourgian school students of age 16–18 we investigate the internal psychometric structure of these illusions (i.e., Are they substantially correlated? Do they form a reflexive or a formative construct?), their connection to related constructs (e.g., Are they distinguishable from intelligence or mathematical competence in a confirmatory factor analysis?), and the question of which of a person’s abilities can predict the correct solution of these brain teasers (by means of a regression analysis). (shrink)

This paper examines the existential context of physical and mental health. Hans Georg Gadamer and The World Health Organization’s conceptualizations are discussed, and current medicalized and idealized views on health are critically examined. The existential dimension of health is explored in the light of theories of selfhood consisting of different parts, Irvin Yalom’s approach to “ultimate concerns” and Martin Heidegger’s conceptualization of “existentials.” We often become aware of health as an existential concern during times of illness, and health and illness (...) can co-exist. The paper discusses how existential suffering in Western culture is described, to an increasing degree, as disorders or psychological deficits, and perfectionistic health goals easily can become a problem. We seek to avoid suffering rather than relate to it, with all the tension that may create. The paper argues that suffering is an unavoidable aspect of people’s experience of their lives, and actively relating to suffering must be regarded as a fundamental aspect of health. The need and usefulness of a concept of “existential health” is discussed. (shrink)

This article posits a pragmatist philosophy of history as exemplified in the work of British Philosopher F.C.S. Schiller (1864–1937). Part of this argument for a pragma-tist philosophy of history resides on pragmatism’s key notion of “experience” be-ing presented here as both related to human forces that are operant in history, and the particularly important “temporal” nature within the term, making it also in part “historical.” The goal is to more generally broaden scholarship in pragmatism as both containing important elements of (...) a unique and coherent philosophy of history, and to bring Schiller closer into the academic circle of the history of pragmatist thought. Available for download. (shrink)

The use of computers to support learning at work has for long been propagated. Although a large bulk of experience exists in this field, it is still an open question what role computer applications play and can play in the process of learning. It can even be questioned if the learning processes themselves are sufficiently well understood to enable designers and others to provide relevant support. In this article these questions are addressed with reference to experience gained with two projects (...) on computer support for shop floor learning. The projects involved the future users actively in the design process and it is argued that this involvement reached far beyond the mere shaping of the learning artefacts. (shrink)

Changing the information format from probabilities into frequencies as well as employing appropriate visualizations such as tree diagrams or 2 × 2 tables are important tools that can facilitate people’s statistical reasoning. Previous studies have shown that despite their widespread use in statistical textbooks, both of those visualization types are only of restricted help when they are provided with probabilities, but that they can foster insight when presented with frequencies instead. In the present study, we attempt to replicate this effect (...) and also examine, by the method of eye tracking, why probabilistic 2 × 2 tables and tree diagrams do not facilitate reasoning with regard to Bayesian inferences (i.e., determining what errors occur and whether they can be explained by scan paths), and why the same visualizations are of great help to an individual when they are combined with frequencies. All ten inferences of N = 24 participants were based solely on tree diagrams or 2 × 2 tables that presented either the famous “mammography context” or an “economics context” (without additional textual wording). We first asked participants for marginal, conjoint, and (non-inverted) conditional probabilities (or frequencies), followed by related Bayesian tasks. While solution rates were higher for natural frequency questions as compared to probability versions, eye-tracking analyses indeed yielded noticeable differences regarding eye movements between correct and incorrect solutions. For instance, heat maps (aggregated scan paths) of distinct results differed remarkably, thereby making correct and faulty strategies visible in the line of theoretical classifications. Moreover, the inherent structure of 2 × 2 tables seems to help participants avoid certain Bayesian mistakes (e.g., “Fisherian” error) while tree diagrams seem to help steer them away from others (e.g., “joint occurrence”). We will discuss resulting educational consequences at the end of the paper. (shrink)

In teaching statistics in secondary schools and at university, two visualizations are primarily used when situations with two dichotomous characteristics are represented: 2×2 tables and tree diagrams. Both visualizations can be depicted either with probabilities or with frequencies. Visualizations with frequencies have been shown to help students significantly more in Bayesian reasoning problems than probability visualizations do. Because tree diagrams or double-trees (which are largely unknown in school) are node-branch-structures, these two visualizations (compared to the 2×2 table) can even simultaneously (...) display probabilities on branches and frequencies inside the nodes. This is a teaching advantage as it allows the frequency concept to be used to better understand probabilities. However, 2×2 tables and (double-) trees have a decisive disadvantage: While joint probabilities (e.g., P(AB)) are represented in 2×2 tables but no conditional probabilities (e.g., P(A|B)), it is exactly the other way around with (double-) trees. Therefore, a visualization that is equally suitable for the representation of joint probabilities and conditional probabilities is desirable. In this article, we present a new visualization—the frequency net—in which all absolute frequencies and all types of probabilities can be depicted. In addition to a detailed theoretical analysis of the frequency net, we report the results of a study with 249 university students that shows that “net diagrams” can improve reasoning without previous instruction to a similar extent as 2×2 tables and double-trees. Regarding questions about conditional probabilities, frequency visualizations (2×2 table, double-tree, or net diagram with natural frequencies) are consistently superior to probability visualizations. The frequency net performs as well as the frequency double-tree. Only the 2×2 table with frequencies—the only visualization that participants were already familiar with —led to higher performance rates. If, on the other hand, a question about a joint probability had to be answered, all implemented visualizations clearly supported participants’ performance, and no uniform format effect becomes visible. Participants reached the highest performance in the versions with probability 2×2 tables and probability net diagrams. Furthermore, after conducting a detailed error analysis, we report interesting error shifts between the two information formats and the different visualizations and give recommendations for teaching probability. (shrink)