Topological relations such as inside, outside, or intersection are ubiquitous to our spatial thinking. Here, we examined how people reason deductively with topological relations between points, lines, and circles in geometric diagrams. We hypothesized in particular that a counterexample search generally underlies this type of reasoning. We first verified that educated adults without specific math training were able to produce correct diagrammatic representations contained in the premisses of an inference. Our first experiment then revealed that subjects who correctly (...) judged an inference as invalid almost always produced a counterexample to support their answer. Noticeably, even if the counterexample always bore a certain level of similarity to the initial diagram, we observed that an object was more likely to be varied between the two drawings if it was present in the conclusion of the inference. Experiments 2 and 3 then directly probed counterexample search. While participants were asked to evaluate a conclusion on the basis of a given diagram and some premisses, we modulated the difficulty of reaching a counterexample from the diagram. Our results indicate that both decreasing the counterexample density and increasing the counterexample distance impaired reasoning performance. Taken together, our results suggest that a search procedure for counterexamples, which proceeds object‐wise, could underlie diagram‐based geometric reasoning. Transposing points, lines, and circles to our spatial environment, the present study may ultimately provide insights on how humans reason about topological relations between positions, paths, and regions. (shrink)

There is currently much discussion about how decision making should proceed when an agent's degrees of belief are imprecise; represented by a set of probability functions. I show that decision rules recently discussed by Sarah Moss, Susanna Rinard and Rohan Sud all suffer from the same defect: they all struggle to rationalize diachronic ambiguity aversion. Since ambiguity aversion is among the motivations for imprecise credence, this suggests that the search for an adequate imprecise decision rule is not yet over.

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 41.4 (2003) 564-565 [Access article in PDF] Samuel J. Kerstein. Kant's Search for the Supreme Principle of Morality. New York: Cambridge University Press, 2002. Pp. xiv + 226. Cloth, $60.00. Summed up in a sentence, this book is both a critical examination of Kant's claim to have derived a supreme moral principle and a limited defense of Kant's project that appears to (...) depart significantly from Kant's own approach. Methodologically, a large portion of this book is devoted to analyzing various key concepts and claims about the nature of the categorical imperative in the Groundwork of the Metaphysics of Morals and the Critique of Practical Reason and testing them against "everyday" moral intuitions via numerous examples and counterexamples.Kerstein begins by distinguishing between the tasks of deduction and derivation of a supreme moral principle, whereby a deduction is understood as an argument for the existence of a supreme moral principle, and a derivation simply assumes this conditionally and goes on to argue that a particular principle and no other can claim the title. Kerstein waives the discussion of whether Kant succeeds in giving a deduction and hence satisfactorily refutes moral skepticism. His focus in the book is on the derivation proposed primarily in Groundwork I and II that if there is a supreme moral principle, then it must be that law expressed in his formulations of the Categorical Imperative (hereafter "CI"). Kerstein addresses Bruce Aune's "traditional" reading of Kant's derivation, wherein the argument utterly fails because Kant allegedly infers a relatively substantive command ("Act only on that maxim which you can at the same time will to be universal law") from a purely formal, higher order, and hence relatively empty one ("Conform your actions to universal law"). He then devotes a chapter each to two recent, influential defenses of Kant's derivation that seek his argument outside the Groundwork. Kerstein addresses Henry Allison's attempt to rescue the derivation via an account of agency rooted in Kant's notion of transcendental freedom, elaborated in the second Critique. Kerstein rejects Allison's reconstruction on the grounds that it remains too abstract to generate the degree of specificity arrived at in Kant's Groundwork universalization formulation. Chapter Three examines Christine Korsgaard's alternative reconstruction of Kant's argument that focuses on the formulation of humanity ("Act so as to treat humanity, whether in your own person or that of any other, always as an end and never as a means only"). He rejects her interpretation because, again, the move from the abstract command to conform to universal law to the substantive formulation about humanity as a universally recognizable unconditional good fails. From the former, more abstract claim, he argues, it does not follow that there is a universally recognizable unconditional good, let alone that this good is humanity in itself (and not, say, utility [happiness], or even some intrinsic value unrelated to humanity, both of which Kant rejects as candidates for the unconditionally good).In his own reconstruction, Kerstein dissolves the criticism of the traditional reading of Kant's Groundwork derivation of the formula of universal law by simply denying that Kant [End Page 564] argues directly from the more generic principle to the specific versions of the CI (91ff). Instead he suggests that Kant's argument is less direct, made by way of laying out a set of criteria that any supreme principle must meet, and showing how the CI meets them while no other candidates can. In the second half of the book he takes a painstaking and critical look at a total of eight criteria, each of which Kant holds to be necessary and together are jointly sufficient for establishing the CI as "the supreme norm for the evaluation of the moral permissability and requiredness of an action, but not of its moral value" (165). This statement includes Kerstein's most controversial and interesting reconstructive claim, namely his denial that moral worth attaches only to actions done in accordance with duty as determined by the test... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Rethinking Social Action through Music: The Search for Coexistence and Citizenship in Medellín’s Music Schools by Geoffrey BakerKim BoeskovGeoffrey Baker: Rethinking Social Action through Music: The Search for Coexistence and Citizenship in Medellín’s Music Schools (Cambridge, UK: Open Book Publishers, 2021)If indeed there exists, as Geir Johansen has proposed,1 a self-critical movement within the field of music education, Geoffrey Baker is undoubtedly one of its leading (...) figures. According to Johansen, the self-critical turn is characterized by an increasing number of music educators directing criticism toward music education itself by denouncing overly romantic visions of “the power of music” and scrutinizing how music educational practices are implicated in the propulsion of problematic social forces of inequality, oppression, and injustice. Baker’s widely read book from 2014, El Sistema: Orchestrating Venezuela’s Youth,2 thoroughly denounced the globally acclaimed Venezuelan ‘musical miracle’ as a myth. Based on extensive field work in Venezuela, Baker painstakingly dissected El Sistema as an archaic, morally, and pedagogically flawed model of music education veiled in the empty rhetoric of music as a tool of social redemption and documented how such models are ripe with opportunities for abuse, in every sad sense of the word.The continuation of Baker’s critical project is published in the form of a four-hundred-page book, entitled Rethinking Social Action Through Music. Baker presents the text as “a ‘post-El Sistema’ project”3 as it constitutes a departure [End Page 92] from his preoccupation with ‘The System’ toward potential alternatives to the Venezuelan model. He finds such an alternative in the Red (Red de Escuelas de Música de Medellín), a network of music schools in the Colombian city of Medellín, which figures as the center of attention in Baker’s institutional ethnography. However, even as the Colombian institution is invoked as a counterexample to its Venezuelan sibling, the Red is not held up for its ability to enact social change or display of innovative pedagogical principles. What Baker finds exemplary and worthy of investigation is the willingness he locates within the administrative layer of the Red to continuously engage in reflective work with regards to the tensions, conflicts, and pitfalls, as well as potentials of musical-social work. Particularly, Baker’s attention is directed to the decision-making processes and discussions among the leaders of the institution; how senior staff members adopt self-critical stances, how needs for institutional change are recognized, the recurring frictions between musical staff and members of the institution’s social team, and tensions between public statements expressing confidence in the program’s efficacy and internal conversations full of ambivalence and doubt.It is with reference to the reflexive processes of the Red that Baker conducts insightful and well-informed discussions of a range of themes relevant to music educators and scholars. Baker situates his analysis within the wider landscape of SATM (Social Action Through Music), which can roughly be understood as the field of musical practice emanating from the large socio-musical orchestral programs conceived in Latin America in the last decades of the 20th century and subsequently spread as ‘Sistema-inspired’ offspring throughout the globe in the beginning of the 21st century. Yet, the discussions found in this book are easily connected to pertinent themes of the philosophy and sociology of music education, such as the notion of artistic citizenship, decolonization, the political nature of music education, ‘the social’ in music education, to name a few. One of the striking qualities of this book is its ability to identify important issues in the local context of Medellín, extract the underlying mechanisms and through empirical examples unfold a much broader discussion of what social justice and social change might mean in the field of music education. Baker, originally educated as an ethnomusicologist, reflects on these topics with depth and nuance, underscoring the relevance of approaching these complex issues through ethnographic work, rather than basing such discussions solely on conceptual exploration.Appearing as a red line throughout the book is Baker’s insistence on foregrounding the ambiguity and ambivalence of musical-social work. In fact, his critical encounter with the field... (shrink)

Counterexamples are constructed for classical decision theory, turning on the fact that actions must often be chosen in groups rather than individually, i.e., the objects of rational choice are plans. It is argued that there is no way to define optimality for plans that makes the finding of optimal plans the desideratum of rational decision-making. An alternative called “locally global planning” is proposed as a replacement for classical decision theory. Decision-making becomes a non-terminating process without a precise target rather than (...) a terminating search for an optimal solution. (shrink)

In Causation, Prediction, and Search, we undertook a three part project. First, we characterized when causal models are indistinguishable by population conditional independence relations under several different assumptions relating causality to probability. Second, we proposed a number of algorithms that take sample data and optional background knowledge as input, and output a class of causal models compatible with the data and the background knowledge; the algorithms were accompanied by proofs of their correctness given assumptions that were clearly stated in (...) CPS, and that we will restate below. Finally, we offered a theory of how to predict the effects of interventions in causal structures, given only partial knowledge of causal structure. Freedman's objections are all directed against the causal inference algorithms we proposed. We do not have room here to discuss all of his criticisms, but we have answered his major points. With regard to the points we do not have room to discuss, the reader should be warned that Freedman is an unreliable interpreter of what we have written. For convenience, we have divided Freedman's objections into the following categories. 1.) Freedman questions some of the assumptions on which our correctness theorems are based. Some of his criticisms are based on covariance matrices that he constructed. None of the examples he constructed in sections 11.2, 11.3, or 12.3 are counterexamples to any theorem that we stated, nor are they even germane to the question of how probable are the assumptions we make. His examples only illustrate points discussed in detail in our book, in which we give similar examples. 2.) The most serious charge that Freedman makes is that the algorithms do not compute what we say they do. (shrink)

In Husserl's phenomenology, the concept of evidence remains inextricably linked with the concept of the possibility of doubt, because this possibility is the condition for distinguishing between the apodictic nature of transcendental experience and the non-apodictic nature of the experience of the world. The phenomenological method is aimed at checking the degree of validity of all knowledge by searching for its possible alternatives or counterexamples, and if all previous attempts at such a search end in failure, we are dealing (...) with knowledge of the rank of apodictic evidence, and there is always the possibility of eliminating all evidence in further course of the experience. As a consequence, lack of alternatives is not the final criterion of the evidence of knowledge. The concept of the possibility of doubt is directly related to the concept of the possibility of negative verifiability, i.e. the possibility of falsification, and consists in constructing in the imagination possible counterexamples of evidence given things and states of affairs, so that the verifiability of philosophical theses is not empirical and takes place in the area of ​​possibility, not reality. (shrink)

Contemporary emotion research typically takes the phenomenon of emotion to be exhausted by a class of mental events that are intentional, conscious, and related to certain sorts of behaviour. Moreover, other affective phenomena, such as moods, are also considered to be relatively short-term, episodic, or occurrent states of the subject undergoing them. Emotions, and other putative emotional phenomena that common-sense takes as long-lasting, non-episodic, or dispositional are things that both philosophers and scientists sometimes recognise, but that are relatively neglected in (...) comparison to emotional episodes. This thesis aims at showing that this neglect is unjustified. I will argue that there is a class of entities, 'sentiments'—broadly characterised as dispositions to undergo emotional episodes—that are irreducible to emotional episodes or collections thereof and have properties that make them a suitable target of study by the emotion researcher.In the first chapter, I argue that an analysis of caring as a pattern of emotional episodes, while more plausible than alternative, non-emotional accounts, faces a number of counterexamples that motivate the search for an account of caring as related in a certain way to emotions but as irreducible to them. I argue that a dispositional account, according to which dispositions are conceived as distinct from their manifestations, is an account for which a strong case can be made.The second chapter is dedicated to defending a modest form of realism about dispositions in general and psychological dispositions in particular. According to realism, dispositions are genuine properties that, although perhaps reducible to non-dispositional properties, cannot be re-described in terms of events only. In the third chapter, I show in what ways emotional dispositions can positively contribute to the explanation of the occurrence of emotional episodes. In the fourth chapter, I argue that caring, understood as a species of sentiment, is not to be construed as a mere disposition to produce certain events; rather, we should allow that certain dispositions are genuinely mental or psychological. Assuming realism about the mental, I argue that some dispositions are mental in a way that others are not. I suggest that being intentional is the property that makes psychological dispositions genuinely mental. I end the chapter by drawing a connection between caring and the notion of character. On my view, caring is at least a necessary ingredient of certain character traits, in particular the virtues. In chapter five, I tackle a recent form of empirically informed scepticism about character and argue, on the basis of general considerations about psychological dispositions, that the sceptic’s case is not as strong as she makes out. Finally, in chapter six, I argue that at least certain forms of sentiment, for example romantic love, can be genuinely supported by reasons, thereby suggesting a way they can contribute to the value of our lives. Overall, the aim of this thesis is to establish the respectability of sentiments in a sophisticated account of the mind. (shrink)

This paper is about syllogistic reasoning, i.e., reasoning from such pairs of premises as, All the chefs are musicians; some of the musicians are painters. We present a computer model that implements the latest account of syllogisms, which is based on the theory of mental models. We also report four experiments that were designed to test this account. Experiments 1 and 2 examined the strategies revealed by the participants' use of paper and pencil as aids to reasoning. Experiment 3 used (...) a new technique to externalize thinking. The participants had to refute, if possible, putative conclusions by constructing external models that were examples of the premises but counterexamples of the conclusions. Experiment 4 used the same techniques to examine the participants' strategies as they drew their own conclusions from syllogistic premises. The results of the experiments showed that individuals not trained in logic can construct counterexamples, that they use similar operations to those implemented in the computer model, but that they rely on a much greater variety of interpretations of premises and of search strategies than the computer model does. We re‐evaluates current theories of syllogistic reasoning in the light of these results. (shrink)

This paper reviews current psychological theories of syllogistic inference and establishes that despite their various merits they all contain deficiencies as theories of performance. It presents the results of two experiments, one using syllogisms and the other using three-term series problems, designed to elucidate how the arrangement of terms within the premises affects performance. These data are used in the construction of a theory based on the hypothesis that reasoners construct mental models of the premises, formulate informative conclusions about the (...) relations in the model, and search for alternative models that are counterexamples to these conclusions. This theory, which has been implemented in several computer programs, predicts that two principal factors should affect performance: the figure of the premises, and the number of models that they call for. These predictions were confirmed by a third experiment. Dans cet article sont passées en revue les théories psychologiques du traitement des syllogismes. On établit qu'en dépit de leurs mérites variés toutes sont en défaut en tant que théories de la performance. On présente les résultats de deux expériences, une utilisant des syllogismes et l'autre des problémes avec des séries de trois termes conçues pour élucider comment l'arrangement des termes dans les premises affecte la performance. Ces données sont utilisées pour construire une théorie fondée sur l'hypothése que les gens construisent des modèles mentaux des premises, formulent des conclusions explicites sur les relations dans le modèle et cherchent des modèles qui seraient des contre-exemples pour leurs conclusions. Cette théorie, utilisée dans plusieurs programmes d'ordinateur, prédit que deux principaux facteurs affectent la performance: la figure des premises et le nombre de; modèles qu'ils mettent en jeu. Cette prédiction est confirmée dans les trois expériences. (shrink)

Faith is often regarded as having a fraught relationship with evidence. Lara Buchak even argues that it entails foregoing evidence, at least when this evidence would influence your decision to act on the proposition in which you have faith. I present a counterexample inspired by the book of Job, in which seeking evidence for the sake of deciding whether to worship God is not only compatible with faith, but is in fact an expression of great faith. One might still (...) think that foregoing evidence may make faith more praiseworthy than otherwise. But I argue against this claim too, once more drawing on Job. A faith that expresses itself by a search for evidence can be more praiseworthy than a faith that sits passively in the face of epistemic adversity. (shrink)

Many of our concepts are introduced to us via, and seem only to be constrained by, roughand-ready explanations and some sample paradigm positive and negative applications. This happens even in informal logic and mathematics. Yet in some cases, the concepts in question – although only informally and vaguely characterized – in fact have, or appear to have, entirely determinate extensions. Here’s one familiar example. When we start learning computability theory, we are introduced to the idea of an algorithmically computable function (...) (from numbers to numbers) – i.e. one whose value for any given input can be determined by a step-by-step calculating procedure, where each step is fully determined by some antecedently given finite set of calculating rules. We are told that we are to abstract from practical considerations of how many steps will be needed and how much ink will be spilt in the process, so long as everything remains finite. We are also told that each step is to be ‘small’ and the rules governing it must be ‘simple’, available to a cognitively limited calculating agent: for we want an algorithmic procedure, step-by-minimal-step, to be idiot-proof. For a classic elucidation of this kind, see e.g. Rogers (1967, pp. 1–5). Church’s Thesis, in one form, then claims this informally explicated concept in fact has a perfectly precise extension, the set of recursive functions. Church’s Thesis can be supported in a quasi-empirical way, by the failure of our searches for counterexamples. It can be supported too in a more principled way, by the observation that different appealing ways of sharpening up the informal chararacterization of algorithmic computability end up specifying the same set of recursive functions. But such considerations fall short of a demonstration of the Thesis. So is there a different argumentative strategy we could use, one that could lead to a proof? Sometimes it is claimed that there just can’t be, because you can never really prove results involving an informal concept like algorithmic computability.. (shrink)

Policymakers who seek to make scientifically informed decisions are constantly confronted by scientific uncertainty and expert disagreement. This thesis asks: how can policymakers rationally respond to expert disagreement and scientific uncertainty? This is a work of non-ideal theory, which applies formal philosophical tools developed by ideal theorists to more realistic cases of policymaking under scientific uncertainty. I start with Bayesian approaches to expert testimony and the problem of expert disagreement, arguing that two popular approaches— supra-Bayesianism and the standard model of (...) expert deference—are insufficient. I develop a novel model of expert deference and show how it can deal with many of these problems raised for them. I then turn to opinion pooling, a popular method for dealing with disagreement. I show that various theoretical motivations for pooling functions are irrelevant to realistic policymaking cases. This leads to a cautious recommendation of linear pooling. However, I then show that any pooling method relies on value judgements, that are hidden in the selection of the scoring rule. My focus then narrows to a more specific case of scientific uncertainty: multiple models of the same system. I introduce a particular case study involving hurricane models developed to support insurance decision-making. I recapitulate my analysis of opinion pooling in the context of model ensembles, confirming that my hesitations apply. This motivates a shift of perspective, to viewing the problem as a decision theoretic one. I rework a recently developed ambiguity theory, called the confidence approach, to take input from model ensembles. I show how it facilitates the resolution of the policymaker’s problem in a way that avoids the issues encountered in previous chapters. This concludes my main study of the problem of expert disagreement. In the final chapter, I turn to methodological reflection. I argue that philosophers who employ the mathematical methods of the prior chapters are modelling. Employing results from the philosophy of scientific models, I develop the theory of normative modelling. I argue that it has important methodological conclusions for the practice of formal epistemology, ruling out popular moves such as searching for counterexamples. (shrink)

This article examines cognitive process models of human sentence comprehension based on the idea of informed search. These models are rational in the sense that they strive to find a good syntactic analysis quickly. Informed search derives a new account of garden pathing that handles traditional counterexamples. It supports a symbolic explanation for local coherence as well as an algorithmic account of entropy reduction. The models are expressed in a broad framework for theories of human sentence comprehension.

In this paper, I try to develop an account of functions which might be of use to a biologist engaged in classifying and explaining natural phenomena. The most pressing difficulty facing such an account is the need to reconcile the normativity of function statements with their explanatory force. I consider two familiar accounts of function statements, offered by Andrew Woodfield and Larry Wright . I examine both accounts in search of the strongest possible formulation of each type of theory. (...) I fail to find a formulation of the ACT which is completely immune from counterexamples, but I do find a satisfactory formulation of the HT. In particular, I argue that the HT should incorporate a point central to the ACT that functions involve a means/end relationship between two devices. I then consider functional explanations: I argue that an ACT which holds that function statements are intrinsically explanatory cannot offer a satisfactory account. The account offered by Woodfield, who rejects this assumption, is adequate, though less attractive than the account suggested by the HT. I conclude by considering how the HT can account for functional norms. (shrink)

This article argues, against contemporary experimentalist criticism, that conceptual analysis has epistemic value, with a structure that encourages the development of interesting hypotheses which are of the right form to be valuable in diverse areas of philosophy. The article shows, by analysis of the Gettier programme, that conceptual analysis shares the proofs and refutations form Lakatos identified in mathematics. Upon discovery of a counterexample, this structure aids the search for a replacement hypothesis. The search is guided by (...) heuristics. The heuristics of conceptual analysis are similar to those in other interesting areas of scholarship, and so hypotheses generated by it are of the right form to be applicable to diverse areas. The article shows that the explanationist criterion in epistemology was developed and applied in this way. The epistemic value of conceptual analysis is oblique because it contributes not towards the main purpose of conceptual analysis but towards the reliable development of epistemically valuable hypotheses in philosophy and scholarship. (shrink)

Over the past two or three decades there has arisen a fairly widespread dissatisfaction with the usefulness of the formal logical techniques of Frege and Russell for teaching the evaluation of reasoning in natural language. The problem with these formal methods, I argue, lies in certain questionable presuppositions that underlie these methods. ;My own case will rest mainly on the following three claims: first, that the millennia old belief that there are two fundamentally different types of argument, viz., induction and (...) deduction, is crucially mistaken and hampers rather than helps our understanding and evaluating of reasoning; second, that the belief that only formally valid arguments are deductively valid is a false dogma, one that is pernicious not only in its own right, but also because it has lent support to the other dogmas we must set aside if we are to have any radical progress in our understanding of and instruction in reasoning in natural language; and third, that logic is better seen not as a body of knowledge, but as a practice or art; and, as with any art, we come to perform it better not through the learning of a system of propositional knowledge but through appropriate training. In addition, I shall outline what I take the appropriate training to be, one that makes use of the imaginative search for counterexamples, and which makes logical training a more practical enterprise. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Literature 27.1 (2003) 1-20 [Access article in PDF] Crime, Compassion, and The Reader John E. MacKinnon IN "WRITING AFTER AUSCHWITZ," Günter Grass describes how at the age of seventeen he stubbornly refused to believe the evidence arrayed before him and his classmates of Nazi atrocities, the photographs showing piles of eyeglasses, shoes, hair, and bones. "Germans never could have done, never did do a thing like that," (...) he recalls thinking. And "when the Never collapsed," he and other young Germans were assured that they were free of responsibility. "It took several more years," Grass reports, "before I began to realize: This will not go away; our shame cannot be repressed or come to terms with." Later, he confesses that those of his age "belonged to the Auschwitz generation—not as criminals, to be sure, but in the camp of the criminals." 1Although one would be hard-pressed to find a novel more dramatically different in style and narrative development than Grass's own dark, audacious fables, Bernhard Schlink's The Reader 2 nonetheless shares this fascination with the notion of collective guilt, in particular the burden of inheritance endured by Germany's "second generation." Set in a university town in the late 1950s, the novel tells the story of a high school student, Michael Berg, who falls in love with Hanna Schmitz, a woman twenty years his senior. Though capable of genuine concern and deep affection, Hanna is oddly secretive, often brusque and officious, occasionally given to cruelty and sarcasm, and prone to unpredictable fits of pique. She also has an endearing quirk: she likes Michael to read to her. Eventually, and without explanation, she vanishes from Michael's life. It is seven years before he sees her again, when, as a law student, he attends a trial of concentration-camp guards, one of whom is Hanna. She stands accused, along with four other [End Page 1] women, of serving at both Auschwitz and a satellite camp near Cracow, of being actively involved in the selection of prisoners for almost certain death, and of locking several hundred women in a church, where they burned to death during a bombing raid.In light of testimony given at trial, Hanna's puzzling conduct, and his own recollections of their time together, it finally dawns on Michael that Hanna cannot read, that she was and remains illiterate. But while in retrospect this may explain certain episodes of frustration, incomprehension, and overreaction on her part, does it have any bearing on her guilt? What would Michael himself have done if similarly situated, and similarly limited? Thus, the story of Michael and Hanna serves as an allegory of the relation between the two generations of Germans, those who were Nazis, or who at least assisted or accommodated them, and the "lucky late-born." It is against this background that Schlink is able most searchingly to investigate questions of complicity, recrimination, and atonement.As much courtroom drama as rite-of-passage tale, The Reader is also something of a philosophical investigation. According to Eva Hoffman, in fact, Schlink has managed to compose "a fictionalized essay... a kind of philosophical parable." For all the briskly evocative prose, she adds, The Reader "gradually acquires some of the severity of a legal or logical argument, in which propositions are set forth and then tested from various points of view." 3 In a recent article, Jeremiah P. Conway offers a sympathetic interpretation of the novel, attending in particular to the philosophical issues that it raises. Conway challenges Martha Nussbaum's claim that it is logically incoherent for us to feel compassion for one whom we hold responsible for a morally reprehensible act. 4 For in Hanna, he suggests, we have a compelling counterexample, a fit object of our compassion no less than our condemnation. And yet, Conway's challenge is grounded in an even more fundamental denial that The Reader is at all concerned to establish the circumstances of Hanna's life as mitigating her crime.In what follows, I will argue that this interpretation is untenable... (shrink)

Ravit Dotan argues that a No Free Lunch theorem from machine learning shows epistemic values are insufficient for deciding the truth of scientific hypotheses. She argues that NFL shows that the best case accuracy of scientific hypotheses is no more than chance. Since accuracy underpins every epistemic value, non-epistemic values are needed to assess the truth of scientific hypotheses. However, NFL cannot be coherently applied to the problem of theory choice. The NFL theorem Dotan’s argument relies upon is a member (...) of a family of theorems in search, optimization, and machine learning. They all claim to show that if no assumptions are made about a search or optimization problem or learning situation, then the best case performance of an algorithm is that of random search or random guessing. A closer inspection shows that these theorems all rely upon assigning uniform probabilities over problems or learning situations, which is just the Principle of Indifference. A counterexample can be crafted that shows that NFL cannot be coherently applied across different descriptions of the same learning situation. To avoid this counterexample, Dotan needs to privilege some description of the learning situation faced by scientists. However, this means that NFL cannot be applied since an important assumption about the problem is being made. So Dotan faces a dilemma: either NFL leads to incoherent best-case partial beliefs or it is inapplicable to the problem of theory choice. This negative result has implications for the larger debate over theory choice. (shrink)

The aim of this paper is to defend the Principle of Sufficient Reason (PSR). I analyse various versions of this principle (and their structure) and various ways of justifying it. Then I present and attempt to challenge some counterexamples allegedly refuting a universal application of the PSR. One can distinguish three versions of the PSR: for each state of affairs there is a sufficient reason for its obtaining (PSR-O); for each true proposition there is a direct or indirect justification (PSR-E); (...) for each state of affairs there exists an intellectual duty to search for a sufficient reason for obtaining of that state of affairs, and for each known true proposition - to search for its justification (PSR-M). There are standard and non-standard versions of the PSR-O. The PSR-O stand can be only valid if there are no chains of contingent reasons and outcomes with first modules, i.e. all chains are actually infinite. However, there are serious (yet debatable) arguments against the possibility of such infinity. The necessary condition of the PSR-O non-stand universal working is the existence of a necessary substance (thing): that substance would be a (free) reason of certain contingent states of affairs obtaining in its domain, and those states of affairs would then be reasons for all other contingent state of affairs and things. (The principle does not require a necessary substance to have its own reason). Limiting the chain of reasons to contingent states of affairs ended with a necessary state (if this state is a reason by necessity) would lead to the paradox of universal necessarism. An ontological \"cost\" of holding the PSR-O non-stand is a postulate of the existence of the necessary, supra-natural substance, i.e. God. However, there are serious advantages of the view which accepts that the PSR-O non-stand is generally valid: a nomological unity of the world and explanatory simplicity. (shrink)

We exhibit several counterexamples showing that the famous Serrin's symmetry result for semilinear elliptic overdetermined problems may not hold for partially overdetermined problems, that is when both Dirichlet and Neumann boundary conditions are prescribed only on part of the boundary. Our counterexamples enlighten subsequent positive symmetry results obtained by the first two authors for such partially overdetermined systems and justify their assumptions as well.

Corcoran, J. 2005. Counterexamples and proexamples. Bulletin of Symbolic Logic 11(2005) 460. -/- John Corcoran, Counterexamples and Proexamples. Philosophy, University at Buffalo, Buffalo, NY 14260-4150 E-mail: corcoran@buffalo.edu Every perfect number that is not even is a counterexample for the universal proposition that every perfect number is even. Conversely, every counterexample for the proposition “every perfect number is even” is a perfect number that is not even. Every perfect number that is odd is a proexample for the existential proposition (...) that some perfect number is odd. Conversely, every proexample for the proposition “some perfect number is odd” is a perfect number that is odd. As trivial these remarks may seem, they can not be taken for granted, even in mathematical and logical texts designed to introduce their respective subjects. One well-reviewed book on counterexamples in analysis says that in order to demonstrate that a universal proposition is false it is necessary and sufficient to construct a counterexample. It is easy to see that it is not necessary to construct a counterexample to demonstrate that the proposition “every true proposition is known to be true” is false–necessity fails. Moreover the mere construction of an object that happens to be a counterexample does not by itself demonstrate that it is a counterexample–sufficiency fails. In order to demonstrate that a universal proposition is false it is neither necessary nor sufficient to construct a counterexample. Likewise, of course, in order to demonstrate that an existential proposition is true it is neither necessary nor sufficient to construct a proexample. This article defines the above relational concepts of counterexample and of proexample, it discusses their surprising history and philosophy, it gives many examples of uses of these and related concepts in the literature and it discusses some of the many errors that have been made as a result of overlooking the challenging subtlety of the proper use of these two basic and indispensable concepts. (shrink)

In this essay, I present a straightforward counterexample to the Uniqueness Thesis, which holds, roughly speaking, that there is a unique rational response to any particular body of evidence.

Ethicists and economists commonly assume that if A is all things considered better than B, and B is all things considered better than C, then A is all things considered better than C. Call this principle Transitivity. Although it has great conceptual, intuitive, and empirical appeal, I argue against it. Larry S. Temkin explains how three types of ethical principle, which cannot be dismissed a priori, threaten Transitivity: (a) principles implying that in some cases different factors are relevant to comparing (...) A to C than to comparing A to B or B to C; (b) principles of limited scope; (c) principles implying that morally relevant differences in degree can amount to differences in kind. My counterexamples employ a principle of type (c): pleasures and pains enormously different in intensity differ in kind. Temkin has also endorsed this type of counterexample, using arguments based on earlier drafts of this paper. (shrink)

This paper defends a counterexample to Modus Tollens, and uses it to draw some conclusions about the logic and semantics of indicative conditionals and probability operators in natural language. Along the way we investigate some of the interactions of these expressions with 'knows', and we call into question the thesis that all knowledge ascriptions have truth-conditions.

A symposium on my *Objects: Nothing Out of the Ordinary* (2015). In response to Wallace, I attempt to clarify the dialectical and epistemic role that my arguments from counterexamples were meant to play, I provide a limited defense of the comparison to the Gettier examples, and I embrace the comparison to Moorean anti-skeptical arguments. In response to deRosset, I provide a clearer formulation of conservatism, explain how a conservative should think about the interaction between intuition and science, and discuss what (...) conservatives should say about scattered territories, clonal colonies, and arbitrary systems. In response to Tillman and Spencer, I fortify my original presentation of the debunking arguments by clarifying why, even while trees (if they exist) are paradigmatically causal, conservatives are meant to be rationally obstructed from believing that it is trees that are causing our tree beliefs. (shrink)

According to a bedrock assumption in the current methodology of armchair philosophy, we may refute a theory aiming at analyzing a concept by providing a counterexample in which it intuitively seems that a hypothetical or real situation does not fit with what the theory implies. In this paper, we shall argue that this assumption is at most either untenable or otherwise useless in bringing about what is commonly expected from it.

I discuss three recent counterexamples to the transitivity of grounding due to Jonathan Schaffer. I argue that the counterexamples don’t work and draw some conclusions about the relationship between grounding and explanation.

Kopec and Titelbaum collect five alleged counterexamples to Uniqueness, the thesis that it is impossible for agents who have the same total evidence to be ideally rational in having different doxastic attitudes toward the same proposition. I argue that four of the alleged counterexamples fail, and that Uniqueness should be slightly modified to accommodate the fifth example.

Most ordinary folks think that there are ordinary objects such as trees and frogs. They do not think there are extraordinary objects such as the mereological sum of trees and frogs, as the permissivist does. Nor do they deny the existence of ordinary composite objects such as tables, as the eliminativist does. In his recent book, Objects: Nothing Out of the Ordinary, Korman positions himself alongside ordinary folk. He deftly defends the common sense view of ordinary objects, and argues against (...) the permissivist and eliminativist. One of the ways he does this is by launching what he calls ‘arguments from counterexample’. In this paper, I focus on the dialectical effectiveness of such arguments generally. I discuss whether and how we might sometimes be justified in using arguments from counterexample and commonsense in philosophical debate, even if they appear to be (or are) question-begging. I conclude that, given some of Korman’s other commitments, there is little reason for his opponents to take his arguments from counterexample seriously. (shrink)

Manley and Wasserman (2008) join the chorus of opposition to the possibility of conditional analysis of dispositions. But that score cannot be settled without more careful attention to the implicit philosophical methodology. Some of the opposition to such an analysis badly overestimates the effect of counterexamples, as if the Gettier example were sufficient to refute the possibility of conjunctive analysis of knowledge. A general objection to a form of analysis must satisfy a number of constraints, and Manley and Wasserman join (...) the chorus too in failing to satisfy them. Most significant is the optional presupposition that the conditional used in analysis will exhibit some sort of centring. We show that even a careful effort to repair and reform Manley and Wasserman's objection to provide a satisfactory argument requires, ultimately, appeal to centring. Worse, the particular positive proposal they offer is vulnerable to a minor variant of their own counterexample. (shrink)

The theory of causal Bayes nets [15, 19] is, from an empirical point of view, currently one of the most promising approaches to causation on the market. There are, however, counterexamples to its core axiom, the causal Markov condition. Probably the most serious of these counterexamples are EPR/B experiments in quantum mechanics (cf. [13, 23]). However, these are also the only counterexamples yet known from the quantum realm. One might therefore wonder whether they are the only phenomena in the quantum (...) realm that create problems for causal Bayes nets. The aim of this paper is to demonstrate that not only the phenomenon of quantum correlations in EPR/B experiments create problems for causal Bayes nets, but also the temporal evolution of quantum systems, which is described as dualistic by quantum mechanics. For this purpose, it is shown that single photon experiments in a Mach-Zehnder interferometer (MZI) violate the causal Markov condition as well. It is then argued, however, that the Markov violation does not occur under the de Broglie-Bohm interpretation of Bohmian mechanics. (shrink)

One strategy in recent discussions of theological fatalism is to draw on Harry Frankfurt’s famous counterexamples to the principle of alternate possibilities (PAP) to defend human freedom from divine foreknowledge. For those who endorse this line, “Frankfurt counterexamples” are supposed to show that PAP is false, and this conclusion is then extended to the foreknowledge case. This makes it critical to determine whether Frankfurt counterexamples perform as advertised, an issue recently debated in this journal via a pair of articles by (...) David Widerker and John Martin Fischer. I suggest that this debate can be avoided: divine foreknowledge is itself aparadigmatic counterexample to PAP, requiring no support from suspect Frankfurt counterexamples. (shrink)

Frankfurt sets out to refute the principle according to which ‘a person is morally responsible for what he has done only if he could have done otherwise’ . Frankfurt devises a counterexample in which an agent is intuitively responsible even though she could not have done otherwise: Suppose someone – Black, let us say – wants Jones to perform a certain action. Black is prepared to go to considerable lengths to get his way, but he prefers to avoid showing (...) his hand unnecessarily. So he waits until Jones is about to make up his mind what to do, and he does nothing unless it is clear to him that Jones is going to decide to do something other than what he wants him to do. If it does become clear that Jones is going to decide to do something else, Black takes effective steps to ensure that Jones decides to do, and that he does do, what he wants him to do. Whatever Jones's initial preferences and inclinations, then, Black will have his way … Now suppose that Black never has to show his hand because Jones, for reasons of his own, decides to perform and does perform the very action Black wants him to perform. In that case, it seems clear, Jones will bear precisely …. (shrink)

We provide counterexamples to some purported characterizations of dilation due to Pedersen and Wheeler :1305–1342, 2014, ISIPTA ’15: Proceedings of the 9th international symposium on imprecise probability: theories and applications, 2015).

In a recent paper, Breckenridge and Magidor argue for an interesting and counterintuitive account of instantial reasoning. According to this account, in arguments such as one beginning with 'There is some x such that x is mortal. Let O be such an x. ...', the 'O' refers to a particular object, although we cannot know which. I give and defend a simple counterexample involving the notion of an unreferred-to object.

Frankfurt sets out to refute the principle according to which ‘a person is morally responsible for what he has done only if he could have done otherwise’. Frankfurt devises a counterexample in which an agent is intuitively responsible even though she could not have done otherwise: Suppose someone – Black, let us say – wants Jones to perform a certain action. Black is prepared to go to considerable lengths to get his way, but he prefers to avoid showing his (...) hand unnecessarily. So he waits until Jones is about to make up his mind what to do, and he does nothing unless it is clear to him that Jones is going to decide to do something other than what he wants him to do. If it does become clear that Jones is going to decide to do something else, Black takes effective steps to ensure that Jones decides to do, and that he does do, what he wants him to do. Whatever Jones's initial preferences and inclinations, then, Black will have his way … Now suppose that Black never has to show his hand because Jones, for reasons of his own, decides to perform and does perform the very action Black wants him to perform. In that case, it seems clear, Jones will bear precisely …. (shrink)

I present two counterexamples to the recently back-in-favour truth-tracking account of knowledge: one involving a true belief resting on a counterfactually robust delusion, one involving a true belief acquired alongside a bunch of false beliefs. These counterexamples carry over to a recent modification of the theory due to Briggs and Nolan (2012), and seem invulnerable to a recent defence of the theory against known counterexamples, by Adams and Clarke (2005).

The paper deals with the epistemological problems posed by the application of the modus ponens rule in cognitive processing of distributed information. Distinguishing the deductive correctness of a rule of inference from its normativity for rational belief, we consider ‘counterexamples’ of modus ponens proposed in modern epistemology of logic that endanger not its model-theoretic validity, but epistemic rationality for boundedly rational cognitive agents. We explain the epistemic lacunas in the closure of knowledge according to modus ponens revealed by Dretske’s ‘counterexamples’ (...) by a variety of standards of validity that go beyond ‘common sense’. However, the epistemic ‘counterexamples’ of modus ponens proposed by McGee, Kolodny and MacFarlane require, in our opinion, an appeal to less obvious epistemic phenomena, which can be discovered by reasoning for interpretation. We demonstrate that the supposed ‘counterexamples’ to modus ponens do not conflict with either the epistemic neutrality of logic or its epistemic relevance to boundedly rational agents. On the contrary, the appeal to reasoning for interpretation, which is needed to resolve the difficulties caused by these ‘counterexamples’, helps to achieve a balance between the epistemic constraints of reasoning agents and the logical requirements for the correctness of their interpretive and deductive cognitive operations. (shrink)

A great deal of attention has been paid recently to the claim that traditional Frankfurt-type counterexamples to the Principle of Alternative Possibilities (PAP), which depend for their success on the presence of a perfectly reliable indicator (or prior sign ) of what an agent will freely do if left to act on his own, are guilty of begging the question against incompatibilists, since such indicators seem to presuppose a deterministic relation between an agent’s free action and its causal antecedents. Objections (...) of this sort have given rise to considerable efforts to construct alternative Frankfurt-type counterexamples that do not rely on prior signs of this kind and so do not presuppose determinism in a way that incompatibilists should find objectionable. One consequence of this shift in the way Frankfurt-type counterexamples are formulated is that it provides an opportunity for the forceful resurgence of certain versions of the so-called flicker defense of PAP. In this paper I develop two versions of the flicker defense, indicate their advantages over other versions of this strategy, and defend them against objections. Insofar as either of these is successful, it will show not only that PAP has yet to be falsified by any of the modified Frankfurt-type counterexamples currently on offer but that cases of this sort are in principle incapable of falsifying PAP. (shrink)

Derek Parfit argues that everyone ought to follow the principles whose universal acceptance would make things go best. I present a counterexample: a world in which no one's moral beliefs have any motivating force. I explain how Parfit's metaethical commitments imply that such a world is possible, and why this possibility is a problem for Parfit's project of reconciling Kantianism, contractualism, and consequentialism. I consider two of Parfit's responses to my counterexample.

In recent years there has been a growing consensus that ordinary reasoning does not conform to the laws of classical logic, but is rather nonmonotonic in the sense that conclusions previously drawn may well be removed upon acquiring further information. Even so, rational belief formation has up to now been modelled as conforming to some fundamental principles that are classically valid. The counterexample described in this paper suggests that a number of the most cherished of these principles should not (...) be regarded as valid for commonsense reasoning. An explanation of this puzzling failure is given, arguing that a problem in the theory of rational choice transfers to the realm of belief formation. (shrink)

This paper considers five counterexamples to Mary Douglas's definition of dirt, one of which is extracted from a scene from George Eliot's novel Middlemarch and another from Marilyn Strathern's essay on anthropology at home.

We give a counterexample to the claim that every provably total function of Basic Arithmetic is a polynomially bounded primitive recursive function.

The Direct Argument is an important argument for demonstrating that moral responsibility is incompatible with determinism because it makes no presuppositions about the nature of free will. One of the inference rules employed in the Direct Argument is rule A: If a proposition is broadly logically necessary, then it is true and no one is, nor ever has been, even partially morally responsible for the fact that the proposition is true. While inference rule A is assumed by all parties to (...) the debate, I demonstrate that the rule is invalid. (shrink)