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 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)

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 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.

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].

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 define the Scott complexity of a countable structure to be the least complexity of a Scott sentence for that structure. This is a finer notion of complexity than Scott rank: it distinguishes between whether the simplest Scott sentence is $\Sigma _{\alpha }$, $\Pi _{\alpha }$, or $\mathrm {d-}\Sigma _{\alpha }$. We give a complete classification of the possible Scott complexities, including an example of a structure whose simplest Scott sentence is $\Sigma _{\lambda + 1}$ for $\lambda $ a limit (...) ordinal. This answers a question left open by A. Miller.We also construct examples of computable structures of high Scott rank with Scott complexities $\Sigma _{\omega _1^{CK}+1}$ and $\mathrm {d-}\Sigma _{\omega _1^{CK}+1}$. There are three other possible Scott complexities for a computable structure of high Scott rank: $\Pi _{\omega _1^{CK}}$, $\Pi _{\omega _1^{CK}+1}$, $\Sigma _{\omega _1^{CK}+1}$. Examples of these were already known. Our examples are computable structures of Scott rank $\omega _1^{CK}+1$ which, after naming finitely many constants, have Scott rank $\omega _1^{CK}$. The existence of such structures was an open question. (shrink)

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.

From a theoretical perspective, most discussions of statistical learning have focused on the possible “statistical” properties that are the object of learning. Much less attention has been given to defining what “learning” is in the context of “statistical learning.” One major difficulty is that SL research has been monitoring participants’ performance in laboratory settings with a strikingly narrow set of tasks, where learning is typically assessed offline, through a set of two-alternative-forced-choice questions, which follow a brief visual or auditory familiarization (...) stream. Is that all there is to characterizing SL abilities? Here we adopt a novel perspective for investigating the processing of regularities in the visual modality. By tracking online performance in a self-paced SL paradigm, we focus on the trajectory of learning. In a set of three experiments we show that this paradigm provides a reliable and valid signature of SL performance, and it offers important insights for understanding how statistical regularities are perceived and assimilated in the visual modality. This demonstrates the promise of integrating different operational measures to our theory of SL. (shrink)

Andrew Chignell and Omri Boehm have recently argued that Kant’s pre-Critical proof for the existence of God entails a Spinozistic conception of God and hence substance monism. The basis for this reading is the assumption common in the literature that God grounds possibilities by exemplifying them. In this article I take issue with this assumption and argue for an alternative Leibnizian reading, according to which possibilities are grounded in essences united in God’s mind (later also described as Platonic ideas intuited (...) by God). I show that this view about the distinction between God’s cognition of essences as the ground of possibility and the actual world is not only explicitly stated by Kant, but is also consistent with his metaphysical picture of teleology in nature and causality during the pre-Critical period. Finally, I suggest that the distinction between the conceptual order of essences embodied in the idea of God and the order of the objects of experience plays a role in the transition into the Critical system, where it is transformed into the distinction between the intelligible and the sensible worlds. (shrink)

In the ‘Appendix to the Transcendental Dialectic’ of the Critique of Pure Reason, Kant contends that the idea of God has a positive regulative role in the systematization of empirical knowledge. But why is this regulative role assigned to this specific idea? Kant’s account is rather opaque and this question has also not received much attention in the literature. In this paper I argue that an adequate understanding of the regulative role of the idea of God depends on the specific (...) metaphysical content Kant attributes to it in the Critique and other writings. I show that neither a heuristic principle of conceptual systematicity, nor conceiving God as a hypothesis of an intelligent designer, can satisfy the demands of reason to make the unity and necessity of the laws of nature intelligible. Regarding the positive account about the metaphysical content of the idea of God, I support my argument by referring to Kant’s pre-critical discussion of the usefulness of the conception of God for the project of science, and by expounding Kant’s critical account of the necessity of the laws of nature. Thus my account sheds light on the continuity of Kant’s conception of God and his appropriation of his own rationalistic metaphysics. (shrink)

All new, unpublished interviews with world-renowned dissident Noam Chomsky.

This chapter outlines the main ideas of my book National responsibility and global justice. It begins with two widely held but conflicting intuitions about what global justice might mean on the one hand, and what it means to be a member of a national community on the other. The first intuition tells us that global inequalities of the magnitude that currently exist are radically unjust, while the second intuition tells us that inequalities are both unavoidable and fair once national responsibility (...) is allowed to operate. This conflict might be resolved either by adopting a cosmopolitan theory of justice (which leaves no room for national responsibility) or by adopting a ?political? theory of justice (which denies that questions of distributive justice can arise beyond the walls of the sovereign state). Since neither resolution is satisfactory, the chapter defends the idea of national responsibility and proposes a new theory of global justice, whose main elements are the protection of basic human rights worldwide, and fair terms of interaction between independent political communities. (shrink)

In his lectures on Logic and Metaphysics, Kant distinguishes between logical and real essences. While the former is related to concepts and is knowable, the latter is related to things and is unknowable. In this paper, I argue that the unknowability is explained by the modal characteristic of real essences as a necessitating ground on which a priori knowledge is impossible. I also show how this claim is related to the unknowable necessity of particular laws of nature. Since laws of (...) nature are conceived as grounded in real essences, the unknowability of the latter is equivalent to Kant's other claim that there can be no knowledge of the necessity of particular laws of nature. Necessity can only be known a priori, and therefore, the necessity of particular laws is only assumed and conceived as grounded in something unknowable, a real essence. This conclusion will allow me to attribute to Kant a position I label as “regulative essentialism”, meaning that real essences have an indispensable role in accordance with the rational interest to explain nature as a system of laws and natural kinds, combined with an epistemic humility about the correspondence of our empirical concepts to real essences. (shrink)

This report examines the issue of restraining prisoners undergoing medical treatment from several angles: Cases illustrating the situation regarding shackling since Physicians for Human Rights-Israel's establishment; the issues and ramification raised by this phenomenon, both in terms of human rights and medical ethics, as well as from the governmental point of view, and the PHR-Israel point of view.

We show that the class of strongly jump-traceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of well-behaved cost functions, called benign. This characterisation implies the containment of the class of strongly jump-traceable c.e. Turing degrees in a number of lowness classes, in particular the classes of the degrees which lie below incomplete random degrees, indeed all LR-hard random degrees, and all ω-c.e. random degrees. The last result implies recent results of (...) Diamondstone's and Ng's regarding cupping with superlow c.e. degrees and thus gives a use of algorithmic randomness in the study of the c.e. Turing degrees. (shrink)

This book investigates law's interaction with practical reasons. What difference can legal requirements—e.g. traffic rules, tax laws, or work safety regulations—make to normative reasons relevant to our action? Do they give reasons for action that should be weighed among all other reasons? Or can they, instead, exclude and take the place of some other reasons? The book critically examines some of the existing answers and puts forward an alternative understanding of law's interaction with practical reasons. -/- At the outset, two (...) competing positions are pitted against each other: Joseph Raz's view that (legitimate) legal authorities have pre-emptive force, namely that they give reasons for action that exclude some other reasons; and an antithesis, according to which law-making institutions (even those that meet prerequisites of legitimacy) can at most provide us with reasons that compete in weight with opposing reasons for action. These two positions are examined from several perspectives, such as justified disobedience cases, law's conduct-guiding function in contexts of bounded rationality, and the phenomenology associated with authority. -/- It is found that, although each of the above positions offers insight into the conundrum at hand, both suffer from significant flaws. These observations form the basis on which an alternative position is put forward and defended. According to this position, the existence of a reasonably just and well-functioning legal system constitutes a reason that fits neither into a model of ordinary reasons for action nor into a pre-emptive paradigm—it constitutes a reason to adopt an (overridable) disposition that inclines its possessor towards compliance with the system's requirements. (shrink)

It has been a common assumption that words are substances that instantiate or have properties. In this paper, I question the assumption that our ontology of words requires posting substances by outlining a bundle theory of words, wherein words are bundles of various sorts of properties (such as semantic, phonetic, orthographic, and grammatical properties). I argue that this view can better account for certain phenomena than substance theories, is ontologically more parsimonious, and coheres with claims in linguistics.

This chapter charts a metamorphosis in Djarin's ethical decision‐making across the first two seasons of The Mandalorian. Philosophy and psychology have centered on moral reasoning and developmental processes based on logic, reasoning, and a primary concern with oneself, sometimes tempered by respect for others. Female development typically orients women toward an “ethics of care”: People have responsibilities toward others as well as themselves. The Mandalorian is the story of one man's moral overhaul toward care. Early Mando's moral decisions are based (...) on hierarchy, status, rules, fairness, maximizing benefit, and values. The shift in Mando's decision‐making approach occurred gradually, beginning with the impact of a rampaging blurrg. The hypothetical scenario about the pharmacy illuminates something crucial for understanding Djarin's gradual adoption of ethics of care: often, our choice to frame an ethical dilemma through one perspective or another winds up determining our moral decision. (shrink)