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.

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)

TheK-trivial sets form an ideal in the Turing degrees, which is generated by its computably enumerable members and has an exact pair below the degree of the halting problem. The question of whether it has an exact pair in the c.e. degrees was first raised in [22, Question 4.2] and later in [25, Problem 5.5.8].We give a negative answer to this question. In fact, we show the following stronger statement in the c.e. degrees. There exists aK-trivial degreedsuch that for all (...) degreesa, bwhich are notK-trivial anda > d, b > dthere exists a degreevwhich is notK-trivial anda > v, b > v. This work sheds light to the question of the definability of theK-trivial degrees in the c.e. degrees. (shrink)

Is the existence of God a question of fact? To the majority of theists, both now and in the past, I think it has seemed clear that, if the phrase ‘God exists’ is to be meaningful, then it is a fact, either that God exists or that he does not. This assertion may even seem trivially true; and yet it has evidently been denied, in recent years, by many theologians. The reasons for such a denial are, in part, to be (...) found in the general reaction against metaphysical philosophy, which was characteristic of the early years of this century, and which is, in Britain, epitomised by A. J. Ayer's stipulation that no proposition can be factually significant unless it is verifiable; unless, in principle at least, some series of observations could conceivably show it to be true. By restricting ‘observation’ to the senses of the physical body, and by emphasising the fact that God, as transcendent by definition, was not a possible object of the senses, some philosophically sensitive theologians were startled into denying that ‘God’ was, even in principle, verifiable; and consequently into denying that propositions purporting to assert his existence were factual. (shrink)

In this letter we demonstrate that the fermionic zero modes on a superconducting domain wall can be associated to an one dimensional \ supersymmetry that contains non-trivial topological charges. In addition, the system also possesses three distinct \ supersymmetries with non-trivial topological charges and we also study some duality transformations of the supersymmetric algebras.

I argue that Quine''s famous claim, any statement can be held true come what may, demands an interpretation that implies that the meanings of the expressions in the held-true statement change. The intended interpretation of this claim is not clear from its context, and so it is often misunderstood by philosophers (and is misleadingly taught to their students). I explain Fodor and Lepore''s (1992) view that the above interpretation would render Quine''s assertion entirely trivial and reply, on both textual and (...) philosophical grounds, that only this trivial reading is consistent with Quine''s famous denial of analyticity. I also explain briefly how the trivial reading lends support to meaning holism, which, regardless of one''s views of its consequences, is an important position in the philosophy of language and mind. (shrink)

The author refers to his book “Die Selbstgestaltung des Lebendigen, synoptische Theorie des Lebens als ein Beitrag zu den philosophischen Grundlagen der Naturwissenschaft” and comments some basic theses of it, namely the body-soul-relation, the gestalt and the problem of the “soul” of the plant and the significance of these conceptions for the description and the understanding of the realizing itself of the living being. The Author prefers not to call a soul the transcendent order which is propre to a plant, (...) as to all what lives, and which is the consequence of their quality as a gestalt. He designs with J.Reinke the transcending order of the plant with the name “diaphysis”. He names his theory of life autotelism and has some hope that it can reconcile mecanism and vitalism in the idea of identity of the so called parallel processes in different ontological dimensions. The duality of body or organism and soul or “innen” is a result only of human thinking, really they are a whole together and do not work upon another, but what happens in the one of them, happens at the same moment in the other one like in a mirror. The unity of both has been stated so often, that it sounds rather trivial, but in the book cited above it is described and argued exactly in that sense that phenomene and gignomene in one and the same life process are correspondent structures in different ontological dimensions, coordinated to one another, as it were, like two mathematical co-ordinates.The author replies to a critic who has doubts about the possibility of our thinking to approximate to the problems in question. He designs that kind of thinking which is adequate to that purpose.L'auteur se réfère à son livre „Die Selbstgestaltung des Lebendigen, synoptiscne Theorie des Lebens als ein Beitrag zu den philosophischen Grundlagen der Naturwissenschaft” et commente quelques de ses thèses basales, à dire la relation du corps et de l'âme, la gestalt et le problème de l'âme de la plante et l'importance de ces notions pour la description et l'entendement de la réalisation de l'être vivant par lui-même. Il préfère de ne pas nommer un âme l'ordre transcendant qui est propre à la plante comme à tout être vivant et qui est la conséquence de leur qualité comme une gestalt. Il désigne avec J.Reinke l'ordre transcendant de la plante comme „diaphysis”. Il nomme sa théorie autotelisme et il a un peu d'espoir qu'elle peut reconcilier mécanisme et vitalisme par l'idée de l'identité des si-nommés „evénements parallèles” en dimensions ontologiques diverses. La dualité corps ou organisme et âme ou „innen” n'est qu'un résultat de la raison humaine, en réalité ils sont une unité ensemble et n'agissent pas l'un sur l'autre, mais qui arrive dans l'un d'eux, arrive dans l'autre au même instant comme dans un miroir. L'unité des deux est affirmée si souvent, que cela semble presque trivialement, mais dans le livre cité cela est décrit et argumenté exactement dans le sens, que phénomène et gignomène dans le même procès de vie sont structures correspondantes dans diverses dimensions ontologiques, qui sont coordonnées l'une à l'autre, pour ainsi dire, comme deux coordinates mathematiques.L'auteur replique à un critique qui doute qu'il y ait moyen pour notre méthode de penser de s'approcher aux problèmes en question. Il désigne la méthode de penser, qui est adéquate à ce propos. (shrink)

Incongruity and anachronism characterize Ovid's treatment of the gods and mythological figures in the Metamorphoses; frequently the resulting discrepancy between the superhuman world of mythology and characteristic aspects of Roman society serves to pillory that society as well as to undermine the dignity of the traditional mythology. Linguistic parody is one of the tools Ovid uses to highlight these discrepancies. An example recently noted is that of the serenade delivered by Polyphemus the landlubber to his marine beloved, Galatea : by (...) casting this in the form of Gebetsparodie, Ovid mocks the literary topoi of the paraclausithyron as well as reducing the heroic status of the mythological protagonists. I suggest that in Tiresias’ brief appearance in Metamorphoses 3 Ovid imitates the pedantic locutions of jurists’ language in order to demonstrate how trivial and undignified are the preoccupations of the bickering Olympians. (shrink)

A new theory of spacetime is proposed in which translations are considered as a part of the de Sitter gauge group. The theory is built along the general principles of classical gauge field theories, which are outlined. Applications of gauge principles to linear and affine connections are also given in order to make the presentation self-sufficient. A de Sitter invariant Lagrangian is constructed, which yields approximately Einstein's vacuum equations when it is subjected to variation with respect to gauge potentials and (...) the result expressed in a specific gauge class. As a difference from the usual use of de Sitter groups, the radius of its translations must be small in the present approach, which probably has the meaning of an elementary subatomic length. The solution of the equations describing flat spacetime is not the trivial zero-curvature connection of the conventional approach. (shrink)

This paper discusses the relation between the minimal positive relevant logic B and intersection and union type theories. There is a marvelous coincidence between these very differently motivated research areas. First, we show a perfect fit between the Intersection Type Discipline ITD and the tweaking BT of B, which saves implication and conjunction but drops disjunction . The filter models of the -calculus (and its intimate partner Combinatory Logic CL) of the first author and her coauthors then become theory models (...) of these calculi. (The logician's "theory" is the algebraist's "filter".) The coincidence extends to a dual interpretation of key particles—the subtype translates to provable , type intersection to conjunction , function space to implication, and whole domain to the (trivially added but trivial) truth T. This satisfying ointment contains a fly. For it is right, proper, and to be expected that type union should correspond to the logical disjunction of B. But the simulation of functional application by a fusion (or modus ponens product) operation on theories leaves the key Bubbling lemma of work on ITD unprovable for the -prime theories now appropriate for the modeling. The focus of the present paper lies in an appeal to Harrop theories which are (a) prime and (b) closed under fusion. A version of the Bubbling lemma is then proved for Harrop theories, which accordingly furnish a model of and CL. (shrink)

Environmental problems often outstrip the abilities of any single scientist to understand, much less address them. As a result, collaborations within, across, and beyond the environmental sciences are an increasingly important part of the environmental science landscape. Here, we explore an insufficiently recognized and particularly challenging barrier to collaborative environmental science: value pluralism, the presence of non-trivial differences in the values that collaborators bring to bear on project decisions. We argue that resolving the obstacles posed by value pluralism to collaborative (...) environmental science requires detecting and coordinating the underlying problematic value differences. We identify five ways that a team might coordinate their problematic value differences and argue that, whichever mode is adopted, it ought to be governed by participatory virtues, pragmatic resolve, and moral concern. Relying on our experiences with the Toolbox Dialogue Initiative, as well as with other dialogical approaches that support team inquiry, we defend the claim that philosophical dialogue among collaborators can go a long way towards helping teams of environmental scientists and fellow travelers detect their problematic value differences. Where dialogical approaches fare less well is in helping teams coordinate these differences. We close by describing several principles for augmenting philosophical dialogue with other methods, and we list several of these methods in an appendix with brief descriptions and links for further learning. Overall, the article makes three main contributions to the research collaboration and values in science literatures: It deepens our understanding of problematic value pluralism in team science; It provides actionable guidance and methods for improving values-oriented philosophical dialogue interventions; and It demonstrates one way of doing engaged philosophy. (shrink)

Drawing on interviews with 140 young British males, this article explores the ways in which men talk about their own bodies and bodily practices, and those of other men. The specific focus of interest is a variety of body modification practices. We argue, however, that the significance of this analysis extends beyond the topic of body modification. In discussing the appearance of their bodies, the men we interviewed talked less about muscle and skin than about their own selves located within (...) particular social, cultural and moral universes. This article shows that, in talking about seemingly trivial questions such as whether to have one’s nose pierced or join a gym, men are actively engaged in regulating normative masculinity. Our analysis lends support to the claim that the body has become a new (identity) project in high/late/post-modernity, but shows how fraught with difficulties this project is for young men who must simultaneously work on and discipline their bodies while disavowing any (inappropriate) interest in their own appearance. The analysis highlights the pervasive individualism of young men’s discourses, and the absence of alternative ways of making sense of embodied experiences. (shrink)

Over 60% of all epidemics have a zoonotic origin, that is, they result from the transmission of infectious diseases from animals to humans. The spill-over of diseases often happens because humans exploit and use animals. In this article, I outline the four most common interfaces that favour the emergence and spread of zoonotic infectious diseases: wildlife hunting, small-scale farming, industrialised farming practices and live animal markets. I analyse which practices serve human food security – and thus have a non-trivial purpose (...) – and which predominantly have an economic purpose or serve as a symbol of wealth. I conclude that many practices that increase the risk for zoonotic infectious disease outbreaks actually do not contribute to the survival and food security of humans. I make two arguments in turn. First, I argue that in cases where the consumption and use of animal products does not contribute to the food security of a population, then this population has a duty to abstain from them, since they impose a grave and avoidable risk to themselves as well as to innocent third-parties. However, some communities must sometimes rely on practices that increase the risk of emergence and transmission of zoonotic infectious diseases, because they have no healthy alternatives. That is, the food security of the local population depends on the consumption and use of risky animal products. The second argument I advance is that, in such scenarios, the international community has a duty to provide the communities concerned with alternative food options, as well as economic and educational opportunities and technologies, in order to reduce the spill-over risk of infectious diseases. Given that abstention from such practices contributes signifcantly to the public good and benefits the international community, the latter has a corresponding duty to provide local communities which abandon such practices with alternatives. (shrink)

Context: Few professional philosophers have addressed in any detail radical constructivism, but have focused instead on the related assumptions and limitations of postmodern epistemology, various anti-realisms, and subjective relativism. Problem: In an attempt to supply a philosophical answer to the guest editors’ question, “Why isn’t everyone a radical constructivist?” I address the realist (hence non-radical) implications of the theory’s invocation of “others” as an invariable, observer-independent, “external” constraint. Results: I argue that constructivists cannot consistently defend a radically subjectivist theory of (...) knowing while remaining entirely agnostic about the nature and existence of the larger world (including independent others). That is, any non-solipsistic account of human experience must explicitly acknowledge its extra-subjective, ontological dimension. Implications: It follows that no pedagogical, social, philosophical, or commonsensical insight associated with so-called “trivial” or “social” constructivism survives or receives any support from the move to radical constructivism. (shrink)

At least since the time of Plato’s writings, epideictic rhetoric has been criticized as deceptive, as epistemologically bankrupt, and as politically irrelevant. Aristotle himself emphasizes that the key ‘topic’of epideictic is amplification and stresses that the epideictic orator chiefly adds ‘size’ and ‘beauty’ to widely shared memories. This paper reinterprets Aristotle’s statements and argues that Aristotle’s account brings to light significant civic resources embodied in epideictic. A genuine statesman uses ceremonial speech to articulate and explain a regime’s underlying ethos and (...) purposes; thus he defines the regime’s telos and orients the citizenry toward it. In that way, it is argued, epideictic oratory is not the trivial cousin of deliberative and judicial rhetoric, but rather the rhetorical genre whose essential function is to explain and defend the fundamental building blocks of the regime. (shrink)

Anger, indignation, guilt, rumination, victim compensation, and perpetrator punishment are considered primary responses associated with justice sensitivity. However, injustice and high JS may predispose to further responses. We had N = 293 adults rate their JS, 17 potential responses toward 12 unjust scenarios from the victim’s, observer’s, beneficiary’s, and perpetrator’s perspectives, and several control variables. Unjust situations generally elicited many affective, cognitive, and behavioral responses. JS generally predisposed to strong affective responses toward injustice, including sadness, pity, disappointment, and helplessness. It (...) impaired trivialization, victim-blaming, or justification, which may otherwise help cope with injustice. It predisposed to conflict solutions and victim compensation. Particularly victim and beneficiary JS had stronger effects in unjust situations from the corresponding perspective. These findings add to a better understanding of the main and interaction effects of unjust situations from different perspectives and the JS facets, differences between the JS facets, as well as the links between JS and behavior and well-being. (shrink)

The philosophy of physics literature contains conflicting claims on the heuristic significance of general covariance. Some authors maintain that Einstein's general relativity distinguishes itself from other theories in that it must be generally covariant, for example, while others argue that general covariance is a physically vacuous and trivial requirement applicable to virtually any theory. Moreover, when general covariance is invested with heuristic significance, that significance as a rule is assigned to so-called “active” general covariance, underwritten by the principle of background (...) independence, rather than to general covriance as Einstein understood it. While agreeing with the latter group of commentators that general covariance indeed carries heuristic significance, I argue that a background independent theory need not be generally covariant and that instead the Principle of Equivalence as Einstein understood it provides the key to understanding the heuristic power of general covariance as a “mathematical sieve” for determining the gravitational field law. However, heuristic significance accrues to general covariance only indirectly, by its encompassing the relativity of frames underwritten by Einstein's principle of equivalence. (shrink)

The system B+ is the minimal positive relevant logic. B+ is trivially extended to B+T on adding a greatest truth (Church constant) T. If we leave ∨ out of the formation apparatus, we get the fragment B∧T. It is known that the set of ALL B∧T theories provides a good model for the combinators CL at Level-I, which is the theory level. Restoring ∨ to get back B+T was not previously fruitful at Level-I, because the set of all B+T theories (...) is NOT a model of CL. It was to be expected from semantic completeness arguments for relevant logics that basic combinator laws would hold when restricted to PRIME B+T theories. Overcoming some previous difficulties, we show that this is the case, at Level I. But this does not form a model for CL. This paper also looks for corresponding results at Level-II, where we deal with sets of theories that we call propositions. We adapt work by Ghilezan to note that at Level-II also there is a model of CL in B∧T propositions. However, the corresponding result for B+T propositions extends smoothly to Level-II only in part. Specifically, only some of the basic combinator laws are proved here. We accordingly leave some work for the reader. (shrink)

Donald Lee has claimed that of three ethical values, freedom, justice, and security-survival, involved in the effects of population growth on the future and the survival of all human beings, security-survival is the most fundamental. As such, it should have priority over freedom and justice. Based on this hierarchy, Lee draws the conclusion that one does not have the right to unlimited procreation, and that ultimately it is the duty of government to impose limits on population growth. I accept Lee’s (...) argument that personal rights must be balanced by personal responsibility, but I argue that justice is the fundamental ethical principle in this discussion. This is not a trivial distinction, for it leads to two significant conclusions. First, by focusing proper attention on justice, the threat to survival of the race from overpopulation is reduced to reasonable and realistic proportions. Second, and particulady important with regard to Lee’s position, the recognition of the need for justice brings to light the fact that the primary responsibility of government is to address itself to redressing injustice in society, injustice which does pose a very real threat to the survival of mankind. In this context, I argue that under no circumstances should government have the right or the responsibility to enforce limits on procreation. (shrink)

This paper outlines a heterodox and largely unexplored conception of objecthood according to which the notion of an individual object is a determinable. §1 outlines the view. §2 argues that the view is incompatible with a natural analysis of kind membership and, as a consequence, undermines the Quinean distinction between ontology and ideology. The view is then used to alleviate one source of Quinean hostility towards non-trivial restrictions on de re possibility in §3, and to elucidate Fine’s neo-Aristoteltian, non-modal conception (...) of essence in §4. §5 concludes. (shrink)

This book presents the most complete translation to date of Erwin Schrödinger's work on colorimetry. In his work Schrödinger proposed a projective geometry of color space, rather than a Euclidean line-element. He also proposed new (at the time) colorimetric methods - in detail and at length - which represented a dramatic conceptual shift in colorimetry. Schrödinger shows how the trichromatic (or Young-Helmholtz) theory of color and the opponent-process (or Hering) theory of color are formally the same theory, or at least (...) only trivially different. These translations of Schrödinger's bold concepts for color space have a fresh resonance and importance for contemporary color theory. (shrink)

An ethics of imitation risks trivializing, aggrandizing, and homogenizing the company of disciples. Should followers of Jesus even try to walk this way?

A study of the elementary theory of quotients of symmetric groups is carried out in a similar spirit to Shelah . Apart from the trivial and alternating subgroups, the normal subgroups of the full symmetric group S on an infinite cardinal μ are all of the form Sκ = the subgroup consisting of elements whose support has cardinality 20, cƒ 20 < κ, 0 < κ < 20, and κ = 0, we make a further analysis of the first order (...) theory of Sλ/Sκ. introducing many-sorted second order structures , all of whose sorts have cardinality at most 20, and in terms of which we can completely characterize the elementary theory of the groups Sλ/Sκ. (shrink)

Throughout the biological and biomedical sciences there is a growing need for, prescriptive ‘minimum information’ (MI) checklists specifying the key information to include when reporting experimental results are beginning to find favor with experimentalists, analysts, publishers and funders alike. Such checklists aim to ensure that methods, data, analyses and results are described to a level sufficient to support the unambiguous interpretation, sophisticated search, reanalysis and experimental corroboration and reuse of data sets, facilitating the extraction of maximum value from data sets (...) them. However, such ‘minimum information’ MI checklists are usually developed independently by groups working within representatives of particular biologically- or technologically-delineated domains. Consequently, an overview of the full range of checklists can be difficult to establish without intensive searching, and even tracking thetheir individual evolution of single checklists may be a non-trivial exercise. Checklists are also inevitably partially redundant when measured one against another, and where they overlap is far from straightforward. Furthermore, conflicts in scope and arbitrary decisions on wording and sub-structuring make integration difficult. This presents inhibit their use in combination. Overall, these issues present significant difficulties for the users of checklists, especially those in areas such as systems biology, who routinely combine information from multiple biological domains and technology platforms. To address all of the above, we present MIBBI (Minimum Information for Biological and Biomedical Investigations); a web-based communal resource for such checklists, designed to act as a ‘one-stop shop’ for those exploring the range of extant checklist projects, and to foster collaborative, integrative development and ultimately promote gradual integration of checklists. (shrink)

A prodigious amount of material is contained within the pages of this short book. The various chapters comprise a quick but rigorous survey of the main results presented in advanced level courses in mathematical logic. The accent here is on the development of proofs for theorems, and not upon topics in the philosophy of mathematics or in "foundational studies." This is not a weakness. No worthwhile investigation of the philosophy or foundations of mathematics can today take place except on the (...) prior basis of such material as is so succinctly presented in this book. Lyndon has thus rendered a service to those who find that they must digest a large dose of modern mathematical logic and then move on to rather more esoteric topics. Among the subjects dealt with are: interpretations and structures, semantic implication, sentential logic, boolean algebra, variants of sentential logic, decidable theories, the compactness theorem, the Löwenheim-Skolem theorem, categoricity [[sic]], the Herbrand-Gentzen theorem, Craig's theorem, diagonal arguments and the paradoxes, Church's theorem, and the Gödel theorems on incompleteness and consistency. The particular methods used in this book reflect the recent interest in the algebraic properties of formal logical structures. The text includes numerous non-trivial exercises which are left to the student, and a short but well-chosen bibliography. Finally, one would think that the price being asked for this admittedly high-quality book is a bit high.—H. P. K. (shrink)

This is undoubtedly one of the most well-conceived and nicely executed introductory books on special relativity ever written. The authors take the view that relativity theory is no longer an advanced and esoteric branch of physics, but ought to be part of the basic intellectual equipment of any bright college student. To this end, the theory of special relativity is presented as a complete and unified set of concepts and not merely as a kind of gloss on classical Newtonian mechanics. (...) This is accomplished in a manner which captures the flavor and excitement of theoretical physics and makes it available to persons of modest mathematical means, and at the same time avoids the overpopularizing and lack of rigor which so often afflict works of this kind. The text is accompanied by numerous non-trivial problems, many with solutions, and a large number of helpful drawings. There is also a reasonable amount of discussion of matters of philosophic interest, and of the most recent experimental inquiries, including a section on the significance of the Mössbauer effect. Altogether, a first-rate job.—H. P. K. (shrink)

In 2011, Ingmar Riedel‐Kruse's bioengineering laboratory at Stanford University publicized an application that uses paramecia for what the researchers termed “biotic games.” These games make use of living organisms, computer programs, and lab equipment to implement games like Pong, Pac‐man, and soccer. Gamesand related activities are often considered nonserious or trivial, whereas life, biological systems, and science are treated very seriously in moral analysis and public perception. The manipulation of living matter frequently engenders at least some controversy in the marketplace (...) of ideas, and using living things in games is no exception. Some of the objections lodged against biotic games have appeared in the ethics literature on similar topics; however, the addition of an entertainment element introduces some objections distinct from those about similar cases, as the online comments vividly illustrate. We aim to explore and address the objections in this paper, using the comments to organize and launch the discussion. In scientific work, there is typically a presumption of some prospect of translation and application of generated knowledge for public benefit. In the case of biotic games, these applications are not self‐evident. Because of this, a serious analysis of the justifications, limitations, and features of biotic games is warranted. To this end, we outline key ethical limits that ought to be placed on these activities as well as the obligations that these activities generate for researchers, other professionals, and lay people who design, implement, use, and play them. (shrink)

A real is Martin-Löf (Schnorr) random if it does not belong to any effectively presented null ${\Sigma^0_1}$ (recursive) class of reals. Although these randomness notions are very closely related, the set of Turing degrees containing reals that are K-trivial has very different properties from the set of Turing degrees that are Schnorr trivial. Nies proved in (Adv Math 197(1):274–305, 2005) that all K-trivial reals are low. In this paper, we prove that if ${{\bf h'} \geq_T {\bf 0''}}$ , then h (...) contains a Schnorr trivial real. Since this concept appears to separate computational complexity from computational strength, it suggests that Schnorr trivial reals should be considered in a structure other than the Turing degrees. (shrink)

We study the connection between Schnorr triviality and genericity. We show that while no 2-generic is Turing equivalent to a Schnorr trivial and no 1-generic is tt-equivalent to a Schnorr trivial, there is a 1-generic that is Turing equivalent to a Schnorr trivial. However, every such 1-generic must be high. As a corollary, we prove that not all K-trivials are Schnorr trivial. We also use these techniques to extend a previous result and show that the bases of cones of Schnorr (...) trivial Turing degrees are precisely those whose jumps are at least 0". (shrink)

Pediatricians provided expert testimony that, in the case of Baby K, provision of ventilator support goes beyond accepted standards of care for anencephalic infants and so is medically futile. This argument, however reasonable, does not persuade those who believe in the absolute value of even a fraction of human life. In Baby K, court records indicate that Ms. H, Baby K's mother, persistently adheres to the sanctity-of-life principle on religious grounds.While I think that quality-of-life considerations have a role in medical (...) decision making, those who reject such considerations must be respected. This article makes the following claims, on behalf of religious dissenters: the Baby K case should be interpreted in light of the freedom of religion guaranteed in the First Amendment ; religious beliefs should not be trivialized in clinical ethics, even if reason, not belief, is the language of the public forum; the time-honored free exercise clause is essential to the American experiment in liberty and should not be overridden in the name of a concept as vague as futility; and free exercise of religion deserves serious discussion in the futility debate, and significant religious accommodation must be included in any hospital or societal futility policies. In the concluding section, issues are raised regarding the balance between religious consideration and resource allocation. (shrink)

Pediatricians provided expert testimony that, in the case of Baby K, provision of ventilator support goes beyond accepted standards of care for anencephalic infants and so is medically futile. This argument, however reasonable, does not persuade those who believe in the absolute value of even a fraction of human life. In Baby K, court records indicate that Ms. H, Baby K's mother, persistently adheres to the sanctity-of-life principle on religious grounds.While I think that quality-of-life considerations have a role in medical (...) decision making, those who reject such considerations must be respected. This article makes the following claims, on behalf of religious dissenters: the Baby K case should be interpreted in light of the freedom of religion guaranteed in the First Amendment ; religious beliefs should not be trivialized in clinical ethics, even if reason, not belief, is the language of the public forum; the time-honored free exercise clause is essential to the American experiment in liberty and should not be overridden in the name of a concept as vague as futility; and free exercise of religion deserves serious discussion in the futility debate, and significant religious accommodation must be included in any hospital or societal futility policies. In the concluding section, issues are raised regarding the balance between religious consideration and resource allocation. (shrink)

We show that there is a low T-upper bound for the class of K-trivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in $\Delta _2^0 $ T-degrees for which there is a low T-upper bound.

The purpose of this note is to formulate some weaker versions of the so called Ramsey test that do not entail the following unacceptable consequenceIf A and C are already accepted in K, then if A, then C is also accepted in K. and to show that these versions still lead to the same triviality result when combined with a preservation criterion.

Propositional canonical Gentzen-type systems, introduced in [2], are systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a connective is introduced and no other connective is mentioned. [2] provides a constructive coherence criterion for the non-triviality of such systems and shows that a system of this kind admits cut-elimination iff it is coherent. The semantics of such systems is provided using two-valued non-deterministic matrices (2Nmatrices). [23] extends these results (...) to systems with unary quantifiers of a very restricted form. In this paper we substantially extend the characterization of canonical systems to (n, k)-ary quantifiers, which bind k distinct variables and connect n formulas, and show that the coherence criterion remains constructive for such systems. Then we focus on the case of k ∈ {0, 1} and show that the following statements concerning a canonical calculus G are equivalent: (i) G is coherent, (ii) G has a strongly characteristic 2Nmatrix, and (iii) G admits strong cut-elimination. We also show that coherence is not a necessary condition for standard cut-elimination, and then characterize a subclass of canonical systems for which this property does hold. (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)

What is philosophy of mathematics and what is it about? The most popular answer, I suppose, to this question would be that philosophers should provide a justification for our presently most cherished mathematical theories and for the most important tool to develop such theories, namely logico-mathematical proof. In fact, it does cover a large part of the activity of philosophers that think about mathematics. Discussions about the merits and faults of classical logic versus one or other ‘deviant’ logics as the (...) logical basis for mathematical theories, ranging from intuitionist over modal logic to paraconsistent logic, typically belong to this area, as do debates about the natural-number concept, its ‘nature’, its properties, especially its uniqueness, and so on. No doubt sociologists of knowledge could explain why philosophers of mathematics came to select these particular problems and deal with them the way they do. What it does imply, however, is that the question is meaningful whether different kinds of philosophies of mathematics are possible, and—why not?—perhaps even desirable. To a certain extent, the answer is trivial: look at what, e.g., phenomenologists have to say about mathematics and you must notice it does not fit into the scheme sketched above.1 Rather, the question should be whether there are forms that, on the one hand, reinterpret the whole enterprise, and, on the other hand, somehow remain related to the work of ‘mainstream’ philosophers of mathematics today.The book under review here does precisely that. Let me be a bit more precise. The general proposal, present throughout all the contributions, is that even at a first, superficial, glance at what mathematicians do when they do mathematics, it is far more complex than ‘just thinking about and …. (shrink)

We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is n-random if it is Martin-Löf random relative to ∅. We show that a set is 2-random if and only if there is a constant c such that infinitely many initial segments x of the set are c-incompressible: C ≥ |x|-c. The ‘only if' direction was obtained independently by Joseph Miller. This characterization can be extended to the case of time-bounded C-complexity. Next we prove some results (...) on lowness. Among other things, we characterize the 2-random sets as those 1-random sets that are low for Chaitin's Ω. Also, 2-random sets form minimal pairs with 2-generic sets. The r.e. low for Ω sets coincide with the r.e. K-trivial ones. Finally we show that the notions of Martin-Löf randomness, recursive randomness, and Schnorr randomness can be separated in every high degree while the same notions coincide in every non-high degree. We make some remarks about hyperimmune-free and PA-complete degrees. (shrink)

Given two infinite binary sequences A,B we say that B can compress at least as well as A if the prefix-free Kolmogorov complexity relative to B of any binary string is at most as much as the prefix-free Kolmogorov complexity relative to A, modulo a constant. This relation, introduced in Nies [14] and denoted by A≤LKB, is a measure of relative compressing power of oracles, in the same way that Turing reducibility is a measure of relative information. The equivalence classes (...) induced by ≤LK are called LK degrees and there is a least degree containing the oracles which can only compress as much as a computable oracle, also called the ‘low for K’ sets. A well-known result from Nies [14] states that these coincide with the K-trivial sets, which are the ones whose initial segments have minimal prefix-free Kolmogorov complexity. We show that with respect to ≤LK, given any non-trivial sets X,Y there is a computably enumerable set A which is not K-trivial and it is below X,Y. This shows that the local structures of and Turing degrees are not elementarily equivalent to the corresponding local structures in the LK degrees. It also shows that there is no pair of sets computable from the halting problem which forms a minimal pair in the LK degrees; this is sharp in terms of the jump, as it is known that there are sets computable from which form a minimal pair in the LK degrees. We also show that the structure of LK degrees below the LK degree of the halting problem is not elementarily equivalent to the or structures of LK degrees. The proofs introduce a new technique of permitting below a set that is not K-trivial, which is likely to have wider applications. (shrink)

Let f be a computable function from finite sequences of 0ʼs and 1ʼs to real numbers. We prove that strong f-randomness implies strong f-randomness relative to a PA-degree. We also prove: if X is strongly f-random and Turing reducible to Y where Y is Martin-Löf random relative to Z, then X is strongly f-random relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including non-K-triviality and autocomplexity. We prove that f-randomness relative to a (...) PA-degree implies strong f-randomness, hence f-randomness does not imply f-randomness relative to a PA-degree. (shrink)

We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2-random. We also study the class $superhigh^\diamond$ and show that it contains some, but not all, of the noncomputable K-trivial sets.

Levin and Schnorr (independently) introduced the monotone complexity, Km(α), of a binary string α. We use monotone complexity to define the relative complexity (or relative randomness) of reals. We define a partial ordering ≤Km on 2ω by α ≤Km β iff there is a constant c such that Km(α ↾ n) ≤ Km(β ↾ n) + c for all n. The monotone degree of α is the set of all β such that α ≤Km β and β ≤Km α. We (...) show the monotone degrees contain an antichain of size 2N0, a countable dense linear ordering (of degrees of cardinality 2N0), and a minimal pair. Downey, Hirschfeldt, LaForte, Nies and others have studied a similar structure, the K -degrees, where K is the prefix-free Kolmogorov complexity. A minimal pair of K -degrees was constructed by Csima and Montalbán. Of particular interest are the noncomputable trivial reals, first constructed by Solovay. We define a real to be (Km, K)-trivial if for some constant c, Km(α ↾ n) ≤ K(n)+c for all n. It is not known whether there is a Km-minimal real, but we show that any such real must be (Km, K)-trivial. Finally, we consider the monotone degrees of the computably enumerable (c.e.) and strongly computably enumerable (s.c.e.) reals. We show there is no minimal c.e. monotone degree and that Solovay reducibility does not imply monotone reducibility on the c.e. reals. We also show the s.c.e. monotone degrees contain an infinite antichain and a countable dense linear ordering. (shrink)

Let ω be the set of natural numbers. For functions f, g: ω → ω, we say f is dominated by g if f < g for all but finitely many n ∈ ω. We consider the standard “fair coin” probability measure on the space 2ω of in-finite sequences of 0's and 1's. A Turing oracle B is said to be almost everywhere dominating if, for measure 1 many X ∈ 2ω, each function which is Turing computable from X is (...) dominated by some function which is Turing computable from B. Dobrinen and Simpson have shown that the almost everywhere domination property and some of its variant properties are closely related to the reverse mathematics of measure theory. In this paper we exposit some recent results of Kjos-Hanssen, Kjos-Hanssen/Miller/Solomon, and others concerning LR-reducibility and almost everywhere domination. We also prove the following new result: If B is almost everywhere dominating, then B is superhigh, i. e., 0″ is truth-table computable from B ′, the Turing jump of B. (shrink)

We prove a number of results in effective randomness, using methods in which Π⁰₁ classes play an essential role. The results proved include the fact that every PA Turing degree is the join of two random Turing degrees, and the existence of a minimal pair of LR degrees below the LR degree of the halting problem.

K-axiom-based epistemic closure for explicit knowledge is rejected for even the most trivial cases of deductive inferential reasoning on account of the fact that the closure axiom does not extend beyond a raw consequence relation. The recognition that deductive inference concerns interaction as much as it concerns consequence allows for perspectives from logics of multi-agent information flow to be refocused onto mono-agent deductive reasoning. Instead of modeling the information flow between different agents in a communicative or announcement setting, we model (...) the information flow between different states of a single agent as that agent reasons deductively. The resource management of the database of agent states for the deductive reasoning fragment in question is covered by the residuated structure that encodes the nonassociative Lambek Calculus with permutation, bottom, and identity: NLP01. (shrink)

We review the current knowledge concerning strong jump-traceability. We cover the known results relating strong jump-traceability to randomness, and those relating it to degree theory. We also discuss the techniques used in working with strongly jump-traceable sets. We end with a section of open questions.

This paper articulates the structure of a two species of weakly aggregative necessity in a common idiom, neighbourhood semantics, using the notion of a k-filter of propositions. A k-filter on a non-empty set I is a collection of subsets of I which contains I, is closed under supersets on I, and contains ∪{Xi ≤ Xj : 0 ≤ i < j ≤ k} whenever it contains the subsets X0,…, Xk. The mathematical content of the proof that weakly aggregative modal logic (...) is complete relative to k-ary frame theory, the standard semantic idiom for weakly aggregative modal logic is presented in language-independent terms as a representation theorem for k-filters: every non-trivial k-filter is included in the union of ≤ k non-trivial filters. The elementary theory of k-filters is developed and then applied in the form of an ultrafilter extension result for k-ary frame theory. (shrink)