This paper proposes a uniform method for constructing tests, confidence regions and point estimates which is called exact since it reduces to Fisher's so-called exact test in the case of the hypothesis of independence in a 2 × 2 contingency table. All the wellknown standard tests based on exact sampling distributions are instances of the exact test in its general form. The likelihood ratio and x2 tests as well as the maximum likelihood estimate appears as asymptotic approximations to the corresponding (...) exact procedures. (shrink)

The term verificationism is used in two different ways: the first is in relation to the verification principle of meaning, which we usually and rightly associate with the logical empiricists, although, as we now know, it derives in reality from Wittgenstein, and the second is in relation to the theory of meaning for intuitionistic logic that has been developed, beginning of course with Brouwer, Heyting and Kolmogorov in the twenties and early thirties, but in much more detail lately, particularly in (...) connection with intuitionistic type theory. It is therefore very natural to ask how these two forms of verificationism are related to one another: was the verificationism that we had in the thirties a kind of forerunner of what we have now, or was it something entirely different? I would like to discuss this question by considering a very particular problem, which was at the heart of Schlick’s interests, namely, the problem whether there might exist undecidable propositions or, if you prefer, unsolvable problems or unanswerable questions: it is merely a matter of wording which of these terms you choose. As I said, it is a problem which was at the heart of Schlick’s interests: it is explicitly discussed already in his early, programmatic paper Die Wende in der Philosophie in the first volume of Erkenntnis from 1930, and there is a short late paper, which has precisely Unanswerable Questions? as its title, from 1935, and he discussed it on several occasions in between also. (shrink)

Editorial NoteThis lecture was given by Per Martin-Löf at Leiden University on August 25, 2001 at the invitation by Göran Sundholm to address the topic mentioned in the title and to reflect on Dummett’s earlier effort of almost a decade before. The lecture was part of a three-day conference on Gottlob Frege. Sundholm arranged for the lecture to be recorded and commissioned Bjørn Jespersen to make a transcript. The information in footnote 1, which Sundholm provided, has been independently confirmed (...) by Thomas Ricketts in an email to the author. The present version has been edited by Ansten Klev. Following the displayed text there is a lacuna in the original transcript corresponding to a pause in the recording when the tape was changed. The continuous text of the present version is the result of a few additions to the original transcript suggested by Klev and agreed to by the author. (shrink)

This volume contains several invited papers as well as a selection of the other contributions. The conference was the first meeting of the Soviet logicians interested in com- puter science with their Western counterparts. The papers report new results and techniques in applications of deductive systems, deductive program synthesis and analysis, computer experiments in logic related fields, theorem proving and logic programming. It provides access to intensive work on computer logic both in the USSR and in Western countries.

We formalise the notion of those infinite binary sequences z that admit a single program P which expresses the entire algorithmical structure of z. Such a program P minimizes the information which must be used in a relative computation for z. We propose two concepts with different strength for this notion, the learnable and the super-learnable sequences. We establish three different equivalent characterizations of learnable (super-learnable, resp.) sequences. In particular, we prove that a sequences z is learnable (super-learnable, resp.) if (...) and only if there is a computable probability measure p such that p is Schnorr (Martin-Lof, resp.) p-random. There is a recursively enumerable sequence which is not learnable. The learnable sequences are invariant with respect to all total and effective transformations of infinite binary sequences. (shrink)

In this survey we discuss work of Levin and V’yugin on collections of sequences that are non-negligible in the sense that they can be computed by a probabilistic algorithm with positive probability. More precisely, Levin and V’yugin introduced an ordering on collections of sequences that are closed under Turing equivalence. Roughly speaking, given two such collections $\mathcal {A}$ and $\mathcal {B}$, $\mathcal {A}$ is below $\mathcal {B}$ in this ordering if $\mathcal {A}\setminus \mathcal {B}$ is negligible. The degree structure associated (...) with this ordering, the Levin–V’yugin degrees, can be shown to be a Boolean algebra, and in fact a measure algebra. We demonstrate the interactions of this work with recent results in computability theory and algorithmic randomness: First, we recall the definition of the Levin–V’yugin algebra and identify connections between its properties and classical properties from computability theory. In particular, we apply results on the interactions between notions of randomness and Turing reducibility to establish new facts about specific LV-degrees, such as the LV-degree of the collection of 1-generic sequences, that of the collection of sequences of hyperimmune degree, and those collections corresponding to various notions of effective randomness. Next, we provide a detailed explanation of a complex technique developed by V’yugin that allows the construction of semi-measures into which computability-theoretic properties can be encoded. We provide two examples of the use of this technique by explicating a result of V’yugin’s about the LV-degree of the collection of Martin-Löf random sequences and extending the result to the LV-degree of the collection of sequences of DNC degree. (shrink)

We show that for each computable ordinal $\alpha > 0$ it is possible to find in each Martin-Löf random ${\rm{\Delta }}_2^0 $ degree a sequence R of Cantor-Bendixson rank α, while ensuring that the sequences that inductively witness R’s rank are all Martin-Löf random with respect to a single countably supported and computable measure. This is a strengthening for random degrees of a recent result of Downey, Wu, and Yang, and can be understood as a randomized version of (...) it. (shrink)

In this paper we argue that knowledge in health care is a multidimensional dynamic construct, in contrast to the prevailing idea of knowledge being an objective state. Polanyi demonstrated that knowledge is personal, that knowledge is discovered, and that knowledge has explicit and tacit dimensions. Complex adaptive systems science views knowledge simultaneously as a thing and a flow, constructed as well as in constant flux. The Cynefin framework is one model to help our understanding of knowledge as a personal construct (...) achieved through sense making. Specific knowledge aspects temporarily reside in either one of four domains – the known, knowable, complex or chaotic, but new knowledge can only be created by challenging the known by moving it in and looping it through the other domains. Medical knowledge is simultaneously explicit and implicit with certain aspects already well known and easily transferable, and others that are not yet fully known and must still be learned. At the same time certain knowledge aspects are predominantly concerned with content, whereas others deal with context. Though in clinical care we may operate predominately in one knowledge domain, we also will operate some of the time in the others. Medical knowledge is inherently uncertain, and we require a context-driven flexible approach to knowledge discovery and application, in clinical practice as well as in health service planning. (shrink)

We are never illness or disease, but, rather, always their sum in the world of day-to-day experience. Disease and illness are not closed systems, but mutually constitutive and continuously interacting worlds. In the patient’s case it is always experience as well. Pain, sickness and death help make that particular experienced identity unavoidable, and at some level ultimately inaccessible to medicine’s changing understanding of disease and tools for managing it. Health—rather than cost containment, specific conditions, or technologies—should be the central focus (...) for health care and health-care reform. A compelling reason to focus on health comes from the observation that the prevalence of disease over the .. (shrink)

Background Metaphors are central to the human understanding of complex issues; through the immediate associations they evoke and frame problems and suggest solutions. Our suggestion of Music in the Park as a metaphor for health systems reform brings to the forefront the environmentally diverse but bounded spaces of health services that offer a variety of attractors within their confines, while pushing into the background organizational and economic concerns.Reflections Parks, like health services, are embedded in their local landscape, serving their communities, (...) but most importantly parks are public spaces, publically funded, ideally offering universal access and equity and to be shared by all who want to go there.Music, like health, is tangible, technical and scientific, yet ultimately experiential and based on meaning. While it encompasses a wide range of styles and approaches, music making requires as its most important skill active listening which brings with it to be ‘in the moment’, to take personal risks and to draw energy and inspiration from the participants. Hence ‘audiences’ are equally active participants because music only has meaning if it internally resonates with the listener and only can exist in what is a co-constructed experience.Conclusions Music in the Park is a metaphor for primary health care systems based on shared values of experts and unique local communities. Health professionals are players in this arena, who develop and practise the full range of their skills in response to individual and community needs and preferences. Their leadership works through inspiration and empowerment, making patients ‘co-producers’ of their own health and ‘co-shapers’ of their health services. (shrink)

It is known that the box dimension of any Martin-Löf random closed set of ${\{0,1\}^\mathbb{N}}$ is ${\log_2(\frac{4}{3})}$ . Barmpalias et al. [J Logic Comput 17(6):1041–1062, 2007] gave one method of producing such random closed sets and then computed the box dimension, and posed several questions regarding other methods of construction. We outline a method using random recursive constructions for computing the Hausdorff dimension of almost every random closed set of ${\{0,1\}^\mathbb{N}}$ , and propose a general method for random closed (...) sets in other spaces. We further find both the appropriate dimensional Hausdorff measure and the exact Hausdorff dimension for such random closed sets. (shrink)

This paper examines the relationship between university mission statements and sustainability practices by institutions of higher education. We examine mission statement constructs and the degree to which higher educational institutions meet specific sustainability criteria in line with the College Sustainability Report Card. Our sample consists of 347 universities from the Sustainable Endowment Institute's (2011) Green Report Card. Previous research suggests that mission statements are essential for superior organizational performance outcomes. We examine the relationship between university mission statement content and sustainability (...) practices. Findings indicate that the greater the number of specific terms used in the university mission statements, the higher the statistical likelihood that those universities had higher sustainability ratings. Findings also indicate that private institutions and nonreligious‐affiliated institutions are more likely to include sustainability constructs in their mission statements than colleges and universities with religious affiliation and public institutions. Several propositions to guide future research are discussed. (shrink)

A semimeasure is a generalization of a probability measure obtained by relaxing the additivity requirement to superadditivity. We introduce and study several randomness notions for left-c.e. semimeasures, a natural class of effectively approximable semimeasures induced by Turing functionals. Among the randomness notions we consider, the generalization of weak 2-randomness to left-c.e. semimeasures is the most compelling, as it best reflects Martin-Löf randomness with respect to a computable measure. Additionally, we analyze a question of Shen, a positive answer to which (...) would also have yielded a reasonable randomness notion for left-c.e. semimeasures. Unfortunately, though, we find a negative answer, except for some special cases. (shrink)

Background The American Association of Medical Colleges has called for incorporation of the health humanities into medical education, and many medical schools now offer formal programs or content in this field. However, there is growing recognition among educators that we must expand beyond empathy and wellness and apply the health humanities to questions of social justice – that is, critical health humanities. In this paper we demonstrate how this burgeoning field offers us tools for integrating social justice into medical education, (...) utilizing the frameworks of critical consciousness and structural competency. Practice of health humanities Critical health humanities can be applied at multiple levels of learners, and in a variety of contexts. We are two physician-writers who have developed several educational programs that demonstrate this. We taught a seminar that introduced first-year and second-year undergraduates to concepts such as social determinants of health, intergenerational trauma, intersectionality, resilience, and cross-cultural care through works of fiction, poetry, film, podcasts, stand-up comedy, and more. Through creative projects and empathic reflection, students engaged with the complexities of structural forces that create and maintain health disparities. Medical students in their clinical years can engage in critical health humanities learning experiences as well. We teach several multidisciplinary electives that address social (in)justice in medicine, as well as mentor fourth-year students engaged in independent electives that foster critical awareness around health equity and ethics. Beyond the classroom, we have actively engaged in critical health humanities practices through story slams, literary journal clubs, conference presentations, and Grand Rounds. Through these activities we have included learners at GME and CME levels. These examples also demonstrate how community engagement and multidisciplinary partnerships can contribute to the practice of critical health humanities. Conclusion In this paper, we explore the growing field of critical health humanities and its potential for teaching health equity through narrative practices. We provide concrete examples of educational activities that incorporate critical consciousness and structural competency – frameworks we have found useful for conceptualizing critical health humanities as a pedagogical practice. We also discuss the strengths and challenges of this work and suggest future directions. (shrink)

I defend a new reading of Spinoza's account of causation that reconciles the strengths of the mechanist and formal cause interpretations by locating instances of nature's fixed and unchanging laws inside individual natures; natures are efficacious because that's where the laws are. God's necessity, for instance, follows from certain logical principles contained within God's nature. Causes between finite particulars likewise stem entirely from finite natures. They do so, I argue, because finite instances of nature's fixed and unchanging laws are inscribed (...) within those natures. In each of these instances, effects follow from natures on account of laws contained within them. (shrink)

This work contributes to the program of studying effective versions of “almost-everywhere” theorems in analysis and ergodic theory via algorithmic randomness. Consider the setting of Cantor space {0,1}N with the uniform measure and the usual shift. We determine the level of randomness needed for a point so that multiple recurrence in the sense of Furstenberg into effectively closed sets P of positive measure holds for iterations starting at the point. This means that for each k∈N there is an n such (...) that n,2n,…,kn shifts of the point all end up in P. We consider multiple recurrence into closed sets that possess various degrees of effectiveness: clopen, Π10 with computable measure, and Π10. The notions of Kurtz, Schnorr, and Martin-Löf randomness, respectively, turn out to be sufficient. We obtain similar results for multiple recurrence with respect to the k commuting shift operators on {0,1}Nk. (shrink)

ABSTRACTThe proportion of the American electorate that is “constrained” by ideology has risen dramatically since Philip E. Converse suggested, in the early 1960s, that ideology is the province of only a small fraction of the mass public. In part, the rise of ideological voters has been obscured by the tendency of scholars after Converse to equate them with those who use terms referring to ideological concepts, such as liberal and conservative, in open-ended interviews. These “concept ideologues,” however, are not the (...) only members of the public whose political opinions show evidence of ideological constraint. There is also a growing segment of the public, the “policy ideologues,” who take positions on three or more policy issues but do not happen to mention ideological concepts. Policy ideologues prove to be as politically knowledgeable as concept ideologues; their attitudes are nearly as constrained across issues and over time; and their policy stances usually “make sense” as liberal or conservative—not surprisingly, as they are almost as willing as concept ideologues to label themselves ideologically, when asked. By 2012, the portion of the public consisting of concept and policy ideologues had reached 42 percent. It declined to 38 percent in 2016, but was still a far cry from the 12 percent of the electorate whom Converse identified as ideologues or near ideologues as of 1956. (shrink)

In this paper we analyze an extension of Martin-Löf s intensional set theory by means of a set contructor P such that the elements of P are the subsets of the set S. Since it seems natural to require some kind of extensionality on the equality among subsets, it turns out that such an extension cannot be constructive. In fact we will prove that this extension is classic, that is “ true holds for any proposition A.

NINE Morality and Prospection -- TEN Prospection Gone Awry: Depression -- ELEVEN Creativity and Aging: What We Can Make With What We Have Left -- Afterword -- Author Index -- Subject Index.