We prove that degrees that are low for Kurtz randomness cannot be diagonally non-recursive. Together with the work of Stephan and Yu [16], this proves that they coincide with the hyperimmune-free non-DNR degrees, which are also exactly the degrees that are low for weak 1-genericity. We also consider Low(M, Kurtz), the class of degrees a such that every element of M is a-Kurtz random. These are characterised when M is the class of Martin-Löf random, computably random, or Schnorr random reals. (...) We show that Low(ML, Kurtz) coincides with the non-DNR degrees, while both Low(CR, Kurtz) and Low(Schnorr, Kurtz) are exactly the non-high, non-DNR degrees. (shrink)

We study the sets that are computable from both halves of some (Martin–Löf) random sequence, which we call 1/2-bases. We show that the collection of such sets forms an ideal in the Turing degrees that is generated by its c.e. elements. It is a proper subideal of the K-trivial sets. We characterize 1/2-bases as the sets computable from both halves of Chaitin’s Ω, and as the sets that obey the cost function c(x,s)=Ωs−Ωx−−−−−−−√. Generalizing these results yields a dense hierarchy of (...) subideals in the K-trivial degrees: For k<n, let Bk/n be the collection of sets that are below any k out of n columns of some random sequence. As before, this is an ideal generated by its c.e. elements and the random sequence in the definition can always be taken to be Ω. Furthermore, the corresponding cost function characterization reveals that Bk/n is independent of the particular representation of the rational k/n, and that Bp is properly contained in Bq for rational numbers p<q. These results are proved using a generalization of the Loomis–Whitney inequality, which bounds the measure of an open set in terms of the measures of its projections. The generality allows us to analyze arbitrary families of orthogonal projections. As it turns out, these do not give us new subideals of the K-trivial sets; we can calculate from the family which Bp it characterizes. We finish by studying the union of Bp for p<1; we prove that this ideal consists of the sets that are robustly computable from some random sequence. This class was previously studied by Hirschfeldt [D. R. Hirschfeldt, C. G. Jockusch, R. Kuyper and P. E. Schupp, Coarse reducibility and algorithmic randomness, J. Symbolic Logic81(3) (2016) 1028–1046], who showed that it is a proper subclass of the K-trivial sets. We prove that all such sets are robustly computable from Ω, and that they form a proper subideal of the sets computable from every (weakly) LR-hard random sequence. We also show that the ideal cannot be characterized by a cost function, giving the first such example of a Σ03 subideal of the K-trivial sets. (shrink)

We give two new characterizations of K-triviality. We show that if for all Y such that Ω is Y-random, Ω is -random, then A is K-trivial. The other direction was proved by Stephan and Yu, giving us the first titular characterization of K-triviality and answering a question of Yu. We also prove that if A is K-trivial, then for all Y such that Ω is Y-random, ≡LRY. This answers a question of Merkle and Yu. The other direction is immediate, so (...) we have the second characterization of K-triviality.The proof of the first characterization uses a new cupping result. We prove that if A≰LRB, then for every set X there is a B-random set Y such that X is computable from Y⊕A. (shrink)

There is a perfect thin class whose upward closure in the Turing degrees has full measure . Thus, in the Muchnik lattice of classes, the degree of 2-random reals is comparable with the degree of some perfect thin class. This solves a question of Simpson [S. Simpson, Mass problems and randomness, Bulletin of Symbolic Logic 11 1–27].

We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed that such functions are related to the proof-theoretic strength of the regularity of Lebesgue measure for Gδ sets. Our constructions essentially settle the reverse mathematical classification of this principle.

The hippocampal formation is a structure critical to navigation and many forms of memory. In mammals, the firing of place cells is widely regarded as the fundamental unit of HF information processing. Supporting homology between the avian and mammalian HF, context-specific patterns of Egr1 have been reported in birds that are comparable to those produced by place cell firing in mammals. Recent electrophysiological data, however, suggest that many avian species lack place cells, potentially undermining the correspondence between Egr1 and place (...) cell-related firing in the avian brain. To clarify this, the current study examines Egr1 expression in Japanese quail under conditions known to elicit only weakly spatially modulated firing patterns and report robust context-dependent Egr1 expression. These data confirm that context-dependent expression of Egr1 is not dependent on precise place fields and provide insight into how these birds are able to perform complex spatial tasks despite lacking mammalian-like place cells. (shrink)

EveryK-trivial set is computable from an incomplete Martin-Löf random set, i.e., a Martin-Löf random set that does not compute the halting problem.

A real is called properly n-generic if it is n-generic but not n+1-generic. We show that every 1-generic real computes a properly 1-generic real. On the other hand, if m > n ≥ 2 then an m-generic real cannot compute a properly n-generic real.

We investigate the role of continuous reductions and continuous relativization in the context of higher randomness. We define a higher analogue of Turing reducibility and show that it interacts well with higher randomness, for example with respect to van Lambalgen’s theorem and the Miller–Yu/Levin theorem. We study lowness for continuous relativization of randomness, and show the equivalence of the higher analogues of the different characterizations of lowness for Martin-Löf randomness. We also characterize computing higher [Formula: see text]-trivial sets by (...) higher random sequences. We give a separation between higher notions of randomness, in particular between higher weak 2-randomness and [Formula: see text]-randomness. To do so we investigate classes of functions computable from Kleene’s [Formula: see text] based on strong forms of the higher limit lemma. (shrink)

Despite the supreme political and economic significance of boundaries--and ongoing challenges to existing national boundaries--scant attention has been paid to their ethics. This volume explores how diverse ethical traditions understand the political and property rights reflected in territorial and jurisdictional boundaries. It is the first book to bring together thinkers from a range of traditions, both religious and secular, to discuss the ethics of boundaries. Each contributor represents a tradition's views on questions surrounding the use of boundaries to delimit property (...) and political rights. What does it mean to own something? What resources should not be privately owned? What justifies the erection of political boundaries between one people and another? How ''hard'' should such boundaries be? What rights extend to minorities within a state? Should territorial boundaries coincide with social ones? Does national autonomy have an ethical basis, or is it an aspect of modern power politics? Should we aim for a more inclusive community than that afforded by modern nation-states? Cross-chapter dialogue and a substantive conclusion draw out similarities and differences among the traditions represented, traditions that include Christianity, classical liberalism, Confucianism, international law, Islam, Judaism, liberal egalitarianism, and natural law. In addition to the editors, the contributors are Nigel Biggar, Joseph Boyle, Joseph Chan, Russell Hardin, Will Kymlicka, Loren Lomasky, Robert McCorquodale, Richard B. Miller, David Novak, Sulayman Nyang, Michael Nylan, Raul C. Pangalangan, Daniel Philpott, Jeremy Rabkin, Hillel Steiner, M. Raquibuz Zaman, and Noam J. Zohar. (shrink)

Review by: Johanna N. Y. Franklin The Bulletin of Symbolic Logic, Volume 19, Issue 1, Page 115-118, March 2013.

Journal of Philosophy of Education, EarlyView.

The example can be generalized. Suppose that ϕ and ψ are suffi- ciently different functions of an independent variable t. We may show that whenever X’s predictions for ϕ and ψ lie (weakly) between Z’s predictions and T’s predictions (the true values), then there are other quantities, interdefinable with ϕ and ψ, that reverse the ordering.

Crispin Wright is widely recognised as one of the most important and influential analytic philosophers of the twentieth and twenty-first centuries. This volume is a collective exploration of the major themes of his work in philosophy of language, philosophical logic, and philosophy of mathematics. It comprises specially written chapters by a group of internationally renowned thinkers, as well as four substantial responses from Wright. In these thematically organized replies, Wright summarizes his life's work and responds to the contributory essays collected (...) in this book. In bringing together such scholarship, the present volume testifies to both the enormous interest in Wright's thought and the continued relevance of Wright's seminal contributions in analytic philosophy for present-day debates. (shrink)

This article attacks the view that global justice should be understood in terms of a global principle of equality. The principle mainly discussed is global equality of opportunity – the idea that people of similar talent and motivation should have equivalent opportunity sets no matter to which society they belong. I argue first that in a culturally plural world we have no neutral way of measuring opportunity sets. I then suggest that the most commonly offered defences of global egalitarianism – (...) the cosmopolitan claim that human lives have equal value, the argument that a persons nationality is a morally arbitrary characteristic, and the more empirical claim that relationships among fellow-nationals are no longer special in a way that matters for justice – are all defective. If we fall back on the idea of equality as a default principle, then we have to recognize that pursuing global equality of opportunity systematically would leave no space for national self-determination. Finally, I ask whether global inequality might be objectionable for reasons independent of justice, and argue that the main reason for concern is the inequalities of power that are likely to emerge in a radically unequal world. (shrink)

A plausible desideratum for an account of the nature of objects, at, and across time, is that it accommodate the phenomenon of vagueness without locating vagueness in the world. A series of arguments have attempted to show that while universalist perdurantism – which combines a perdurantist account of persistence with an unrestricted mereological account of composition – meets this desideratum, endurantist accounts do not. If endurantists reject unrestricted composition then they must hold that vagueness is ontological. But if they embrace (...) unrestricted composition they are faced with the problem of the many, and cannot plausibly accommodate vagueness. This paper disambiguates two related sub-problems of the problem of the many, and argues that universalist perdurantism is not superior to universalist endurantism with respect to either of these. (shrink)

Examination of the target article for its relevance to the analysis of private behavior leads to three concerns: the absence of a new methodology for studying private behavior, the undisclosed possibility of interactions, and insufficient attention to the social context of idea generation. Regardless of these concerns, a larger issue remains: Can a science of n = 1 be credible?

Prevailing ethical thinking about informed consent to clinical research is characterized by theoretical confidence and practical disquiet. On the one hand, bioethicists are confident that informed consent is a fundamental norm. And, for the most part, they are confident that what makes consent to research valid is that it constitutes an autonomous authorization by the research participant. On the other hand, bioethicists are uneasy about the quality of consent in practice. One major source of this disquiet is substantial evidence of (...) the “therapeutic misconception”—the tendency of patient-subjects in clinical trials to confuse participating in research with receiving personalized medical care. This .. (shrink)

Emotion Review, Volume 14, Issue 1, Page 15-30, January 2022. We offer an account of mental health and well-being using the predictive processing framework. According to this framework, the difference between mental health and psychopathology can be located in the goodness of the predictive model as a regulator of action. What is crucial for avoiding the rigid patterns of thinking, feeling and acting associated with psychopathology is the regulation of action based on the valence of affective states. In PPF, valence (...) is modelled as error dynamics—the change in prediction errors over time. Our aim in this paper is to show how error dynamics can account for both momentary happiness and longer term well-being. What will emerge is a new neurocomputational framework for making sense of human flourishing. (shrink)

Plato's account of the famous trial of Socrates in 399 b.c., appeals to historians, philosophers, political scientists, and classicists. It is also essential reading for students of ancient Greek. Paul Allen Miller and Charles Platter provide running commentary, glosses of unfamiliar words, introductions that address historical and philosophical issues, and thought-provoking essays on each chapter.

Part is not a univocal term. Uses of parthood and composition that do not obey any supplementation principle have a long philosophical tradition and strong support from contemporary physics. We call such uses potential parts. This paper first shows why potential parts are important and incompatible with supplementation, then provides a formal mereology for such parts inspired by the path-integral approach to quantum electrodynamics.

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Triple Bottom Line (TBL) reports, outlining the economic, environmental and social impact of organisations, are increasingly viewed as a business requirement. Unfortunately, despite global frameworks, there is no one established standard against which to evaluate the social dimension. Thus, current social reporting is often disparagingly described as a public relations exercise with limited accountability, consistency or comparability. This article outlines the development of a generic TBL social impact framework and questionnaire designed to quantify an organisation's social impact. Based on valid (...) preexisting measures appropriate for organisations in the industrialised world, the proposed framework and questionnaire offers a comparable and objective social impact assessment tool for organisations. The aim is to prompt informed debate and discussion about current organisational social impact reporting, whilst providing organisations with a tool which enables the identification, quantification and comparability of social impact reporting. (shrink)

In this article, I will elucidate the Indian government’s two primary discourses concerning yoga since 2014 as right-wing Hindu Nationalist Prime Minister Narendra Modi and his Hindu Nationalist political party, the Bharatiya Janata Party, have interacted with both international and domestic audiences. These discourses can be broadly grouped into two categories, or what I refer to as Modi and the BJP’s “double discourse”: Yoga as a global soft power solution to counter the Global North’s climate change privilege on the international (...) stage and Yoga as biopower for the advancement of India’s depressed economy on the domestic front. Relying on yoga’s polyvalent character, Modi and the BJP are able to frame yoga in these two particular ways—which together signal their commitment to neoliberal economic ideology—by drawing from historical and contemporaneous precedents which I also outline in the article. I conclude with a brief visit to the preparations for the BJP’s 2019 International Day of Yoga, where this double discourse becomes most evident in the two divergent themes announced for the event. (shrink)

Let ϕ be a fixed numerical function. If the k-state Turing machine M with input string ϕ(k) (that is, started in its initial state scanning the leftmost 1 of a single string of ϕ(k) 1s on an otherwise blank tape) produces the output string m (that is, halts in its halting state scanning the leftmost 1 of a single string of m 1s on an otherwise blank tape), we shall say that the ϕ-fecundity of M is m. If M halts (...) in any other position or state, or fails to halt, its ϕ-fecundity is 0. (shrink)