We report on some recent work centered on attempts to understand when one set is more random than another. We look at various methods of calibration by initial segment complexity, such as those introduced by Solovay [125], Downey, Hirschfeldt, and Nies [39], Downey, Hirschfeldt, and LaForte [36], and Downey [31]; as well as other methods such as lowness notions of Kučera and Terwijn [71], Terwijn and Zambella [133], Nies [101, 100], and Downey, Griffiths, and Reid [34]; (...) higher level randomness notions going back to the work of Kurtz [73], Kautz [61], and Solovay [125]; and other calibrations of randomness based on definitions along the lines of Schnorr [117].These notions have complex interrelationships, and connections to classical notions from computability theory such as relative computability and enumerability. Computability figures in obvious ways in definitions of effective randomness, but there are also applications of notions related to randomness in computability theory. For instance, an exciting by-product of the program we describe is a more-or-less naturalrequirement-freesolution to Post's Problem, much along the lines of the Dekker deficiency set. (shrink)

In this paper I argue that, by combining eliminativist and fictionalist approaches toward the sub-personal representational posits of predictive processing, we arrive at an empirically robust and yet metaphysically innocuous cognitive scientific framework. I begin the paper by providing a non-representational account of the five key posits of predictive processing. Then, I motivate a fictionalist approach toward the remaining indispensable representational posits of predictive processing, and explain how representation can play an epistemologically indispensable role within predictive processing explanations without thereby (...) requiring that representation metaphysically exists. Finally, I outline four consequences of accepting this approach and explain why they are beneficial: we arrive at a victory for metaphysical eliminativism in the ‘representation wars’; my account fits with extant empirical practice; my account provides guidance for future research; and, my account provides the beginnings of a response to Mark Sprevak’s IBE problem for fictionalist approaches toward sub-personal representation. (shrink)

We give a systematic technical exposition of the foundations of the theory of computably compact metric spaces. We discover several new characterizations of computable compactness and apply these characterizations to prove new results in computable analysis and effective topology. We also apply the technique of computable compactness to give new and less combinatorially involved proofs of known results from the literature. Some of these results do not have computable compactness or compact spaces in their statements, and thus these applications are (...) not necessarily direct or expected. (shrink)

It is common to think of medical and ethical modes of thought as different in kind. In such terms, some clinical situations are made more complicated by an additional ethical component. Against this picture, we propose that medical and ethical modes of thought are not different in kind, but merely different aspects of what it means to be human. We further propose that clinicians are uniquely positioned to synthesise these two aspects without prior knowledge of philosophical ethics.

We introduce a new hierarchy of computably enumerable degrees. This hierarchy is based on computable ordinal notations measuring complexity of approximation of${\rm{\Delta }}_2^0$functions. The hierarchy unifies and classifies the combinatorics of a number of diverse constructions in computability theory. It does so along the lines of the high degrees (Martin) and the array noncomputable degrees (Downey, Jockusch, and Stob). The hierarchy also gives a number of natural definability results in the c.e. degrees, including a definable antichain.

Robert I. Soare, Automorphisms of the Lattice of Recursively Enumerable Sets. Part I: Maximal Sets.Manuel Lerman, Robert I. Soare, $d$-Simple Sets, Small Sets, and Degree Classes.Peter Cholak, Automorphisms of the Lattice of Recursively Enumerable Sets.Leo Harrington, Robert I. Soare, The $\Delta^0_3$-Automorphism Method and Noninvariant Classes of Degrees.

Can we deploy creative practices to critically address the fatal interlocking of global surveillance technologies, neo-colonial expansionism, environmental degradation and the lethal threat of drone warfare? Throughout the following conversation, Shona Illingworth and Anthony Downey examine these and other questions in relation to the recent publication of Topologies of Air (Sternberg Press and The Power Plant, 2022). Edited by Downey, the book includes discussion and documentation of two major bodies of work by Illingworth, including Topologies of Air (2021) (...) and Lesions in the Landscape (2015), alongside an extended series of essays that analyse the psychological and environmental impact of military, industrial and corporate transformations of airspace and outer space. Employing interdisciplinary research and collaborative processes, Illingworth’s practice, as detailed in the discussion below, uses creative methodologies to visualize and interrogate this proliferating exploitation of airspace. The conversation between Illingworth and Downey also outlines the work of the Airspace Tribunal, an ongoing series of public hearings that brings together diverse disciplines, methodologies, knowledge and lived experiences to propose a new human right that will counter the colonization of the sky and, in time, protect individuals, communities and ecologies from ever-increasing threats from above. (shrink)

As a natural example of a 1-random real, Chaitin proposed the halting probability Ω of a universal prefix-free machine. We can relativize this example by considering a universal prefix-free oracle machine U. Let [Formula: see text] be the halting probability of UA; this gives a natural uniform way of producing an A-random real for every A ∈ 2ω. It is this operator which is our primary object of study. We can draw an analogy between the jump operator from computability theory (...) and this Omega operator. But unlike the jump, which is invariant under the choice of an effective enumeration of the partial computable functions, [Formula: see text] can be vastly different for different choices of U. Even for a fixed U, there are oracles A =* B such that [Formula: see text] and [Formula: see text] are 1-random relative to each other. We prove this and many other interesting properties of Omega operators. We investigate these operators from the perspective of analysis, computability theory, and of course, algorithmic randomness. (shrink)

Plato seems to have been pessimistic about how most people stand with regard to virtue. However, unlike the Stoics, he did not conclude that most people are vicious. Rather, as we know from discussions across several dialogues, he countenanced decent ethical conditions that fall short of genuine virtue, which he limited to the philosopher. Despite Plato's obvious interest in this issue, commentators rarely follow his lead by investigating in detail such conditions in the dialogues. When scholars do investigate what kind (...) of virtue, if any, Plato thinks is open to non-philosophers, they typically look to the Republic. But in the Republic Plato sets out an ideal city; therefore, the virtue available to non-philosophers there is likely different from what he thinks is available to them in the real world. If we want to determine Plato's thoughts about the virtue of actual non-philosophers, we must look elsewhere. In this paper, I set my sights on the Phaedo. (shrink)

A splitting of an r.e. set A is a pair A1, A2 of disjoint r.e. sets such that A1 A2 = A. Theorems about splittings have played an important role in recursion theory. One of the main reasons for this is that a splitting of A is a decomposition of A in both the lattice, , of recursively enumerable sets and in the uppersemilattice, R, of recursively enumerable degrees . Thus splitting theor ems have been used to obtain results about (...) the structure of , the structure of R, and the relationship between the two structures. Furthermore it is fair to say that questions about splittings have often generated important new technical developments in recursion theory. In this article we survey most of the results and techniques associated with splitting properties of r.e. sets in ordinary recursion theory. (shrink)

On one hand, it is commonly accepted that clinicians should not deceive their patients, yet on the other there are many instances in which deception could be in a patient’s best interest. In this paper, I propose that this conflict is in part driven by a narrow conception of deception as contingent on belief. I argue that we cannot equate non-deceptive care solely with introducing or sustaining a patient’s true belief about their condition or treatment, because there are many instances (...) of clinical care which are non-doxastic and non-deceptive. Inasmuch as this is true, better understanding of non-doxastic attitudes, such as hope and pretence, could improve our understanding of deception in clinical practice. (shrink)

Current mainstream approaches to conscientious objection either uphold the standards of public health care by preventing objections or protect the consciences of health‐care professionals by accommodating objections. Public justification approaches are a compromise position that accommodate conscientious objections only when objectors can publicly justify the grounds of their objections. Public justification approaches require objectors and assessors to speak a common normative language and to this end it has been suggested that objectors should be required to cast their objection in terms (...) of public reason. We provide critical support for such a public reason condition and argue that it would be neither too demanding nor too permissive. We also respond to objections that it unfairly favours secular over religious objectors and that public reasons cannot be held with the kind of sincerity thought to characterize conscientious objections. (shrink)

In this paper, we explicate the method of Investigative Ordinary Language Philosophy (IOLP). The term was coined by John Cook to describe the unique philosophical approach of Frank Ebersole. We argue that (i) IOLP is an overlooked yet valuable philosophical method grounded in our everyday experiences and concerns; and (ii) as such, Frank Ebersole is an important but neglected figure in the history of ordinary language philosophy.

Healthcare systems around the world are struggling to maintain a sufficient workforce to provide adequate care during the COVID-19 pandemic. Staffing problems have been exacerbated by healthcare workers (HCWs) refusing to work out of concern for their families. I sketch a deontological framework for assessing when it is morally permissible for HCWs to abstain from work to protect their families from infection and when it is a dereliction of duty to patients. I argue that it is morally permissible for HCWs (...) to abstain from work when their duty to treat is outweighed by the combined risks and burdens of that work. For HCWs who live with their families, the obligation to protect one’s family from infection contributes significantly to those burdens. There are, however, a range of complicating factors including the strength of duty to treat which varies according to the HCW’s role, the vulnerability of family members to the disease, the willingness of family members to risk infection and the resources available to the HCW to protect their family. In many cases, HCWs in ‘frontline’ roles with a weak duty to treat and families at home will be morally permitted to abstain from work given the risks posed by COVID-19; therefore, society should provide additional incentives to maintain sufficient staff in these roles. (shrink)

We consider the computably enumerable sets under the relation of Q-reducibility. We first give several results comparing the upper semilattice of c.e. Q-degrees, RQ, Q, under this reducibility with the more familiar structure of the c.e. Turing degrees. In our final section, we use coding methods to show that the elementary theory of RQ, Q is undecidable.

We examine the randomness and triviality of reals using notions arising from martingales and prefix-free machines.

Journal of Political Philosophy, EarlyView.

ABSTRACT‘Self-ambiguity’, we suggest, is best understood as an uncertainty about how strongly a given feature reflects who one truly is. When this understanding of self-ambiguity is applied to a view of the self as having both essential and shapable components, self-ambiguity can be seen to have two aspects: (1) uncertainty about one's essential or relatively unchangeable characteristics, e.g. one's sexuality, and (2) uncertainty about how to shape oneself, e.g. which values to commit to, actions to pursue, or essential features to (...) identify with. We explain how a narrative account of agency can accommodate these forms of self-ambiguity and argue that such an account also reveals another kind of self-ambiguity, namely, (3) uncertainty about whether one's established self-narrative represents who one really is. We illustrate this third form of self-ambiguity in the context of addiction where people's established addiction self-narratives make it difficult to identify with recovery. We argue that recovery will require embracing, especially, our third form of self-ambiguity as a chance for positive self-transformation. Treatment for addiction should, therefore, support people in going through and ultimately narratively resolving the inevitable self-ambiguities of the recovery process. (shrink)

Robert Card’s “Reasonability View” is a significant contribution to the debate over the place of conscientious objection in health care. In his view, conscientious objections can only be accommodated if the grounds for the objection meet a reasonability standard. I identify inconsistencies in Card’s description of the reasonability standard and argue that each version he specifies is unsatisfactory. The criteria for reasonability that Card sets out most frequently have no clear underpinning principle and are too permissive of immoral objections. Card (...) has also claimed that petitioners must justify their positions with Rawlsian public reason. I argue that, although the resulting reasonability standard is principled, it is overly restrictive. I also show that a reasonability standard built on Rawls’ more lenient conception of reasonableness would be overly permissive of objections at odds with professional healthcare standards. Finally, I argue for my favored solution, which bases the reasonability standard on minimal professional standards. (shrink)

In this paper I argue that split-brain syndrome is best understood within an extended mind framework and, therefore, that its very existence provides support for an externalist account of conscious perception. I begin by outlining the experimental aberration model of split-brain syndrome and explain both: why this model provides the best account of split-brain syndrome; and, why it is commonly rejected. Then, I summarise Susan Hurley’s argument that split-brain subjects could unify their conscious perceptual field by using external factors to (...) stand-in for the missing corpus callosum. I next provide an argument that split-brain subjects do unify their perceptual fields via external factors. Finally, I explain why my account provides one with an experimental aberration model which avoids the problems typically levelled at such views, and highlight some empirical predictions made by the account. The nature of split-brain syndrome has long been considered mysterious by proponents of internalist accounts of consciousness. However, in this paper I argue that externalist theories can provide a straightforward explanation of the condition. I therefore conclude that the ability of externalist accounts to explain split-brain syndrome gives us strong reason to prefer them over internalist rivals. (shrink)

In this paper, we solve a long-standing open question, about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering [Formula: see text] has infinitely many successivities, then the spectrum of the successivity relation is closed upwards in the computably enumerable Turing degrees. To do this, we use a new method of constructing [Formula: see text]-isomorphisms, which has already found other applications such as Downey, Kastermans and Lempp [9] and is (...) of independent interest. It would seem to promise many further applications. (shrink)

We study the Turing degrees which contain a real of effective packing dimension one. Downey and Greenberg showed that a c.e. degree has effective packing dimension one if and only if it is not c.e. traceable. In this paper, we show that this characterization fails in general. We construct a real $A\leq_T\emptyset''$ which is hyperimmune-free and not c.e. traceable such that every real $\alpha\leq_T A$ has effective packing dimension 0. We construct a real $B\leq_T\emptyset'$ which is not c.e. traceable (...) such that every real $\alpha\leq_T B$ has effective packing dimension 0. (shrink)

We describe punctual categoricity in several natural classes, including binary relational structures and mono-unary functional structures. We prove that every punctually categorical structure in a finite unary language is${\text {PA}}(0')$-categorical, and we show that this upper bound is tight. We also construct an example of a punctually categorical structure whose degree of categoricity is$0''$. We also prove that, with a bit of work, the latter result can be pushed beyond$\Delta ^1_1$, thus showing that punctually categorical structures can possess arbitrarily complex (...) automorphism orbits.As a consequence, it follows that binary relational structures and unary structures are not universal with respect to primitive recursive interpretations; equivalently, in these classes every rich enough interpretation technique must necessarily involve unbounded existential quantification or infinite disjunction. In contrast, it is well-known that both classes are universal for Turing computability. (shrink)

We study connections between classical asymptotic density, computability and computable enumerability. In an earlier paper, the second two authors proved that there is a computably enumerable set A of density 1 with no computable subset of density 1. In the current paper, we extend this result in three different ways: The degrees of such sets A are precisely the nonlow c.e. degrees. There is a c.e. set A of density 1 with no computable subset of nonzero density. There is a (...) c.e. set A of density 1 such that every subset of A of density 1 is of high degree. We also study the extent to which c.e. sets A can be approximated by their computable subsets B in the sense that A\B has small density. There is a very close connection between the computational complexity of a set and the arithmetical complexity of its density and we characterize the lower densities, upper densities and densities of both computable and computably enumerable sets. We also study the notion of "computable at density r" where r is a real in the unit interval. Finally, we study connections between density and classical smallness notions such as immunity, hyperimmunity, and cohesiveness. (shrink)

We construct the set of the title, answering a question of Cholak, Jockusch, and Slaman [1], and discuss its connections with the study of the proof-theoretic strength and effective content of versions of Ramsey's Theorem. In particular, our result implies that every ω-model of RCA 0 + SRT 2 2 must contain a nonlow set.

Why do some addicted people chronically fail in their goal to recover, while others succeed? On one established view, recovery depends, in part, on efforts of intentional planning agency. This seems right, however, firsthand accounts of addiction suggest that the agent’s self-narrative also has an influence. This paper presents arguments for the view that self-narratives have independent, self-fulfilling momentum that can support or undermine self-governance. The self-narrative structures of addicted persons can entrench addiction and alienate the agent from practically feasible (...) recovery plans. Strategic re-narration can redirect narrative momentum and therefore support recovery in ways that intentional planning alone cannot. (shrink)

The 2023 doctors’ strikes in the UK have elicited a familiar moral outcry that such strikes are morally wrong. We consider five arguments that might be thought to show doctors’ strikes are morally impermissible but show that they all fail. The most we can conclude from such arguments is that doctors’ strikes are morally permissible in a narrower range of circumstances than strikes in other sectors.We then outline two independent but compatible justifications for doctors’ strikes, one that appeals to doctors’ (...) interests in fair pay and working conditions and one that appeals to doctors’ duty to protect public health. We also suggest that doctors’ strikes can be supererogatory when they aim to correct a government failing in its own duty to protect public health. Finally, we assess the 2023 UK doctors’ strikes. We conclude that they are justified and there is a case for considering them supererogatory. (shrink)

Schnorr randomness is a notion of algorithmic randomness for real numbers closely related to Martin-Löf randomness. After its initial development in the 1970s the notion received considerably less attention than Martin-Löf randomness, but recently interest has increased in a range of randomness concepts. In this article, we explore the properties of Schnorr random reals, and in particular the c.e. Schnorr random reals. We show that there are c.e. reals that are Schnorr random but not Martin-Löf random, and provide a new (...) characterization of Schnorr random real numbers in terms of prefix-free machines. We prove that unlike Martin-Löf random c.e. reals, not all Schnorr random c.e. reals are Turing complete, though all are in high Turing degrees. We use the machine characterization to define a notion of "Schnorr reducibility" which allows us to calibrate the Schnorr complexity of reals. We define the class of "Schnorr trivial" reals, which are ones whose initial segment complexity is identical with the computable reals, and demonstrate that this class has non-computable members. (shrink)

Loïc Wacquant argues for a radicalization of the habitus concept provided by Pierre Bourdieu, suggesting that habitus is a site and mode for conducting research, not simply an explanatory or theoretical mechanism. Taking seriously this call to examine skills and communities of practice through apprenticeship, however, requires that the theoretical account of habitus be subject to empirical testing. Moreover, enquiry into communities of practice, especially the subtle psychological, behavioural and even neurological consequences of skill acquisition, means that claims about the (...) habitus can be scrutinized by fields such as psychology, neurology and even human biology. Habitus as a truly open path of inquiry will demand, not wholly novel concepts, but a recognition of when claims about practice are simultaneously testable, and falsifiable, by other forms of empirical enquiry, including the human sciences. (shrink)

The placebo effect is now generally defined widely as an individual’s response to the psychosocial context of a clinical treatment, as distinct from the treatment’s characteristic physiological effects. Some researchers, however, argue that such a wide definition leads to confusion and misleading implications. In response, they propose a narrow definition restricted to the therapeutic effects of deliberate placebo treatments. Within the framework of modern medicine, such a scope currently leaves one viable placebo treatment paradigm: the non-deceptive and non-concealed administration of (...) “placebo pills” or open-label placebo (OLP) treatment. In this paper, I consider how the placebo effect occurs in OLP. I argue that a traditional, belief-based account of OLP is paradoxical. Instead, I propose an account based on the non-doxastic attitude of pretence, understood within a fictionalist framework. (shrink)

We prove that there exists a noncomputable c.e. real which is low for weak 2-randomness, a definition of randomness due to Kurtz, and that all reals which are low for weak 2-randomness are low for Martin-Löf randomness.

In this article we show that it is possible to completely classify the degrees of r.e. bases of r.e. vector spaces in terms of weak truth table degrees. The ideas extend to classify the degrees of complements and splittings. Several ramifications of the classification are discussed, together with an analysis of the structure of the degrees of pairs of r.e. summands of r.e. spaces.

We characterize the class of c.e. degrees that bound a critical triple as those degrees that compute a function that has no ω-c.e. approximation.

Patrick Downey finds comedy at the heart of the Western philosophical and theological tradition. In Serious Comedy Downey tracks tragedy and comedy from the beginning of Western thought to the twentieth century, beginning with an in-depth examination of Aristotle and three Platonic dialogues: the Republic, the Phaedrus, and the Symposium. In the book's second section Downey argues that the Bible is at heart a comedic narrative and analyzes the philosophical and theological implications of this comedy. In the (...) third section Downey traces comedy and tragedy in classic and modern philosophers such as Machiavelli, Dante, Hegel, Nietzsche, and Kierkegaard. Readers with an interest in classical philosophy and theology, political philosophy, and narrative theory will be particularly interested in this insightful and thoughtful perspective on the Western tradition. (shrink)

I argue that Aristotle believes that virtue comes in degrees. After dispatching with initial concerns for the view, I argue that we should accept it because Aristotle conceives of heroic virtue as the highest degree of virtue. I support this interpretation of heroic virtue by considering and rejecting alternative readings, then showing that heroic virtue characterized as the highest degree of virtue is consistent with the doctrine of the mean.