Nicholas Shackel

Events between which we have no epistemic reason to discriminate have equal epistemic probabilities. Bertrand’s chord paradox, however, appears to show this to be false, and thereby poses a general threat to probabilities for continuum sized state spaces. Articulating the nature of such spaces involves some deep mathematics and that is perhaps why the recent literature on Bertrand’s Paradox has been almost entirely from mathematicians and physicists, who have often deployed elegant mathematics of considerable sophistication. At the same time, the philosophy of probability has been left out. In particular, left out entirely are the philosophical ground of the principle of indifference, the nature of the principle itself, the stringent constraint this places on the mathematical representation of the principle needed for its application to continuum sized event spaces, and what these entail for rigour in developing the paradox itself. This book puts the philosophy and its entailments back in and in so doing casts a new light on the paradox, giving original analyses of the paradox, its possible solutions, the source of the paradox, the philosophical errors we make in attempting to solve it and what the paradox proves for the philosophy of probability. The book finishes with the author’s proposed solution—a solution in the spirit of Bertrand’s, indeed—in which an epistemic principle more general than the principle of indifference offers a principled restriction of the domain of the principle of indifference.

Metaphysical graphical structuralism is the view that at some fundamental level the world is a mathematical graph of nodes and edges. Randall Dipert has advanced a graphical structuralist theory of fundamental particulars and Alexander Bird has advanced a graphical structuralist theory of fundamental properties. David Oderberg has posed a powerful challenge to graphical structuralism: that it entails the absurd inexistence of the world or the absurd cessation of all change. In this paper I defend graphical structuralism. A sharper formulation, some theorems about such structures, and careful attention to the interaction of metaphysical and mathematical features, shows that the absurdities depend on assumptions that are not essential to the view and brings to light a surprising fact about the necessary structure of fundamental properties.

The Nothing from Infinity paradox arises when the combination of two infinitudes of point particles meet in a supertask and disappear. Corral-Villate claims that my arguments for disappearance fail and concedes that this failure also produces an extreme kind of indeterminism, which I have called plenitudinous. So my supertask at least poses a dilemma of extreme indeterminism within Newtonian point particle mechanics. Plenitudinous indeterminism might be trivial, although easy attempts to prove it so seem to fail in the face of plausible continuity principles. However, the question of its triviality is here moot, since I show that, except in one case, Corral-Villate’s disproofs fail, and with a correction, the original arguments are unrefuted. Consequently, of the two contenders for the outcome of my supertask, the Nothing from Infinity paradox has won out.

In this chapter I show how challenges to our ability to tame the uncertainty of a pandemic leaves us vulnerable to uncertainty phobia. This is because not all the uncertainty that matters can be tamed by our knowledge of the relevant probabilities, contrary to what many believe. We are vulnerable because unrelievable wild uncertainty is a hard burden to bear, especially so when we must act in the face of it. The source of unrelievable wild uncertainty is that the nature of probability distributions matters for whether knowledge of them tames uncertainty. It matters because a warrant for the taming is provided by two theorems, but this warrant applies only to some kinds of probability distribution. Essentially, this is because the theorems are about what happens at a mathematical limit but real life never reaches the limit. Consequently, the warrant depends on how quickly the random processes producing the uncertainty converge towards their limit. If they are governed by one class of probability distributions, they converge quickly enough to possess the warrant. If they are governed by another class of probability distributions, they converge towards their limit too slowly and so do not possess that warrant. The random processes of pandemics involve the slow kind . Faced with such a burden, as we are in a pandemic, we are tempted to retreat into uncertainty phobia, leading to fixed definite opinions, precisely when the exercise of sound judgement to determine our responses requires our opinions to be hedged and mobile. Coping with a pandemic requires us to bear the burden of unrelievable wild uncertainty rather than give in to the temptation of uncertainty phobia. Pandemics require the virtue of epistemic forbearance.

How Moore’s open question argument works, insofar as it does, remains a matter of controversy. My purpose here is to construct an open question argument based on a novel interpretation of how Moore’s argument might work. In order to sidestep exegetical questions, I do not claim here to be offering Moore’s own argument. Rather, I offer a reconstruction making use of important elements of Moore’s methodology and assumptions that could be reasonable within a Moorean viewpoint. The crucial role within the argument is played by what I call the real thought move. I show that the reconstructed argument is more defensible from some standard objections than the common construction. I finish by drawing attention to a neglected objective that would make sense within Moore’s viewpoint, showing that it fits with a major commitment of Bonjour’s moderate rationalism, and show how the package might be important for the non-naturalist today.

Previous claims to have resolved the two-envelope paradox have been premature. The paradoxical argument has been exposed as manifestly fallacious if there is an upper limit to the amount of money that may be put in an envelope; but the paradoxical cases which can be described if this limitation is removed do not involve mathematical error, nor can they be explained away in terms of the strangeness of infinity. Only by taking account of the partial sums of the infinite series of expected gains can the paradox be resolved.

Benardete presents a version of Zeno's dichotomy in which an infinite sequence of gods each intends to raise a barrier iff a traveller reaches the position where they intend to raise their barrier. In this paper, I demonstrate the abstract form of the Benardete Dichotomy. I show that the diagnosis based on that form can do philosophical work not done by earlier papers rejecting Priest's version of the Benardete Dichotomy, and that the diagnosis extends to a paradox not normally classified as a dichotomy. I show how the form is exploited to generate paradox. Introduction The form of the Benardete dichotomy 2.1 The unsatisfiable pair diagnosis Applying the unsatisfiable pair diagnosis 3.1 Perez Laraudogoitia 3.2 Hawthorne 3.3 Angel 3.4 Yablo and Sorensen Exploiting the form

The principle of indifference is supposed to suffice for the rational assignation of probabilities to possibilities. Bertrand advances a probability problem, now known as his paradox, to which the principle is supposed to apply; yet, just because the problem is ill‐posed in a technical sense, applying it leads to a contradiction. Examining an ambiguity in the notion of an ill‐posed problem shows that there are precisely two strategies for resolving the paradox: the distinction strategy and the well‐posing strategy. The main contenders for resolving the paradox, Marinoff and Jaynes, offer solutions which exemplify these two strategies. I show that Marinoff’s attempt at the distinction strategy fails, and I offer a general refutation of this strategy. The situation for the well‐posing strategy is more complex. Careful formulation of the paradox within measure theory shows that one of Bertrand’s original three options can be ruled out but also shows that piecemeal attempts at the well‐posing strategy will not succeed. What is required is an appeal to general principle. I show that Jaynes’s use of such a principle, the symmetry requirement, fails to resolve the paradox; that a notion of metaindifference also fails; and that, while the well‐posing strategy may not be conclusively refutable, there is no reason to think that it can succeed. So the current situation is this. The failure of Marinoff’s and Jaynes’s solutions means that the paradox remains unresolved, and of the only two strategies for resolution, one is refuted and we have no reason to think the other will succeed. Consequently, Bertrand’s paradox continues to stand in refutation of the principle of indifference.

Neuroscience and psychology have recently turned their attention to the study of the subpersonal underpinnings of moral judgment. In this article we critically examine an influential strand of research originating in Greene's neuroimaging studies of ‘utilitarian’ and ‘non-utilitarian’ moral judgement. We argue that given that the explananda of this research are specific personal-level states—moral judgments with certain propositional contents—its methodology has to be sensitive to criteria for ascribing states with such contents to subjects. We argue that current research has often failed to meet this constraint by failing to correctly ‘fix’ key aspects of moral judgment, criticism we support by detailed examples from the scientific literature

Neuroimaging studies on moral decision-making have thus far largely focused on differences between moral judgments with opposing utilitarian (well-being maximizing) and deontological (duty-based) content. However, these studies have investigated moral dilemmas involving extreme situations, and did not control for two distinct dimensions of moral judgment: whether or not it is intuitive (immediately compelling to most people) and whether it is utilitarian or deontological in content. By contrasting dilemmas where utilitarian judgments are counterintuitive with dilemmas in which they are intuitive, we were able to use functional magnetic resonance imaging to identify the neural correlates of intuitive and counterintuitive judgments across a range of moral situations. Irrespective of content (utilitarian/deontological), counterintuitive moral judgments were associated with greater difficulty and with activation in the rostral anterior cingulate cortex, suggesting that such judgments may involve emotional conflict; intuitive judgments were linked to activation in the visual and premotor cortex. In addition, we obtained evidence that neural differences in moral judgment in such dilemmas are largely due to whether they are intuitive and not, as previously assumed, to differences between utilitarian and deontological judgments. Our findings therefore do not support theories that have generally associated utilitarian and deontological judgments with distinct neural systems.

Recent research on moral decision-making has suggested that many common moral judgments are based on immediate intuitions. However, some individuals arrive at highly counterintuitive utilitarian conclusions about when it is permissible to harm other individuals. Such utilitarian judgments have been attributed to effortful reasoning that has overcome our natural emotional aversion to harming others. Recent studies, however, suggest that such utilitarian judgments might also result from a decreased aversion to harming others, due to a deficit in empathic concern and social emotion. The present study investigated the neural basis of such indifference to harming using functional neuroimaging during engagement in moral dilemmas. A tendency to counterintuitive utilitarian judgment was associated both with ‘psychoticism’, a trait associated with a lack of empathic concern and antisocial tendencies, and with ‘need for cognition’, a trait reflecting preference for effortful cognition. Importantly, only psychoticism was also negatively correlated with activation in the subgenual cingulate cortex (SCC), a brain area implicated in empathic concern and social emotions such as guilt, during counterintuitive utilitarian judgments. Our findings suggest that when individuals reach highly counterintuitive utilitarian conclusions, this need not reflect greater engagement in explicit moral deliberation. It may rather reflect a lack of empathic concern, and diminished aversion to harming others.

Nicholas Rescher claims that rational decision theory “may leave us in the lurch”, because there are two apparently acceptable ways of applying “the standard machinery of expected-value analysis” to his Dr. Psycho paradox which recommend contradictory actions. He detects a similar contradiction in Newcomb’s problem. We consider his claims from the point of view of both Bayesian decision theory and causal decision theory. In Dr. Psycho and in Newcomb’s Problem, Rescher has used premisses about probabilities which he assumes to be independent. From the former point of view, we show that the probability premisses are not independent but inconsistent, and their inconsistency is provable within probability theory alone. From the latter point of view, we show that their consistency can be saved, but then the contradictory recommendations evaporate. Consequently, whether one subscribes to evidential or causal decision theory, rational decision theory is not in any way vitiated by Rescher’s arguments.

Dretske proposes a theory of knowledge in terms of a theory of information, but wishes to deny that empirical knowledge settles the large question of scepticism. This leads him to deny the closure of knowledge under known entailment. In a recent paper Jäger argues that Dretske's theory of information entails closure for knowledge, 'at least for the kind of propositions here at issue'. If Jäger is right, Dretske is seriously embarrassed and must give something up. In this paper I show that there are two flaws in Jäger's argument. The principle of informational closure considered by Jäger is incompatible with Dretske's theory of information, and Jäger's argument that Dretske is committed to a certain kind of substitution instance of that principle of informational closure is invalid. I propose adequacy conditions on signalled information and use them to motivate a formulation of a general closure principle for signalled information. I show that Dretske's account of information satisfies the adequacy conditions, but in a way which commits him to an instance of the general closure principle. I argue that Dretske is consequently committed to closure for some cases of knowledge for which he wishes to deny closure. Finally, I sketch how, on the basis of the closure principle to which Dretske is committed, Jäger's broader argument may yet go through.

In his important recent book Schroeder proposes a Humean theory of reasons that he calls hypotheticalism. His rigourous account of the weight of reasons is crucial to his theory, both as an element of the theory and constituting his defence to powerful standard objections to Humean theories of reasons. In this paper I examine that rigourous account and show it to face problems of vacuity and consonance. There are technical resources that may be brought to bear on the problem of vacuity but implementation is not simple and philosophical motivation a further difficulty. Even supposing vacuity is fixed, the problems of consonance bring to light a different obstruction lying in Schroeder’s path. There is a difference between the general weighing of reasons and the context specificity of the correct placing of weight on them in deliberation and this difference cannot be fixed by the resources in the account. For these reasons we are still waiting for a plausible Humean theory of reasons

Many of the philosophical doctrines purveyed by postmodernists have been roundly refuted, yet people continue to be taken in by the dishonest devices used in proselytizing for postmodernism. I exhibit, name, and analyse five favourite rhetorical manoeuvres: Troll's Truisms, Motte and Bailey Doctrines, Equivocating Fulcra, the Postmodernist Fox Trot, and Rankly Relativising Fields. Anyone familiar with postmodernist writing will recognise their pervasive hold on the dialectic of postmodernism and come to judge that dialectic as it ought to be judged.

We examine conceptual and methodological problems that arise in the course of the scientific study of possible influences of religious belief on the experience of physical pain. We start by attempting to identify a notion of religious belief that might enter into interesting psychological generalizations involving both religious belief and pain. We argue that it may be useful to think of religious belief as a complex dispositional property that relates believers to a sufficiently thick belief system that encompasses both cognitive and non-cognitive elements. Such a conception of religious belief is more likely to correlate with psychological properties of believers that are both sufficiently shared and sufficiently unique to distinguish their psychology from believers in another religion or from non-believers. If the dispositional psychological property that constitutes religious belief does influence pain, then our analysis suggests that it doesn’t do so directly but rather through one of its occurrent manifestations. We offer a taxonomy of the different ways in which occurrent states of belief or experience may interact with physical pain, and we try to identify those that are more interesting or promising. We then proceed to employ the conceptual framework we developed to some of the existing evidence about the neural and psychological correlates of religious belief and experience, and about the cognitive modulation of physical pain. Finally we turn to analyse two experiments that directly investigated the relation between religious belief and pain. We draw attention to the limitations of existing evidence and end by suggesting directions for future conceptual and empirical inquiry.

An important suggestion of objective Bayesians is that the maximum entropy principle can replace a principle which is known to get into paradoxical difficulties: the principle of indifference. No one has previously determined whether the maximum entropy principle is better able to solve Bertrand’s chord paradox than the principle of indifference. In this paper I show that it is not. Additionally, the course of the analysis brings to light a new paradox, a revenge paradox of the chords, that is unique to the maximum entropy principle.

A prolonged debate about the nature of norms has been conducted in terms of the scope of a modal operator. Here I argue that the features of what I call Normative Focus are more fundamental than scope. We shall see limitations of scope contrasted with better analysis in terms of Normative Focus. Some authors address such limitations by extending what they mean by scope. I show that scope is still not doing the work: what does it is their elicitation of our tacit knowledge of Normative Focus. Finally, I show that scope cannot capture Normative Focus because scope allows us to make only one distinction where we need to make three. So we should leave scope to the philosophers of language and turn instead to the ontology of Normative Focus.

In this paper I present a novel supertask in a Newtonian universe that destroys and creates infinite masses and energies, showing thereby that we can have infinite indeterminism. Previous supertasks have managed only to destroy or create finite masses and energies, thereby giving cases of only finite indeterminism. In the Nothing from Infinity paradox we will see an infinitude of finite masses and an infinitude of energy disappear entirely, and do so despite the conservation of energy in all collisions. I then show how this leads to the Infinity from Nothing paradox, in which we have the spontaneous eruption of infinite mass and energy out of nothing. I conclude by showing how our supertask models at least something of an old conundrum, the question of what happens when the immovable object meets the irresistible force.

We call something a paradox if it strikes us as peculiar in a certain way, if it strikes us as something that is not simply nonsense, and yet it poses some difficulty in seeing how it could make sense. When we examine paradoxes more closely, we find that for some the peculiarity is relieved and for others it intensifies. Some are peculiar because they jar with how we expect things to go, but the jarring is to do with imprecision and misunderstandings in our thought, failures to appreciate the breadth of possibility consistent with our beliefs. Other paradoxes, however, pose deep problems. Closer examination does not explain them away. Instead, they challenge the coherence of certain conceptual resources and hence challenge the significance of beliefs which deploy those resources. I shall call the former kind weak paradoxes and the latter, strong paradoxes. Whether a particular paradox is weak or strong is sometimes a matter of controversy—sometimes it has been realised that what was thought strong is in fact weak, and vice versa,— but the distinction between the two kinds is generally thought to be worth drawing. In this Cchapter, I shall cover both weak and strong probabilistic paradoxes.