This is a survey of work on set-theoretical invariance criteria for logicality. It begins with a review of the Tarski-Sher thesis in terms, first, of permutation invariance over a given domain and then of isomorphism invariance across domains, both characterized by McGee in terms of definability in the language L∞,∞. It continues with a review of critiques of the Tarski-Sher thesis, and a proposal in response to one of those critiques via homomorphism invariance. That (...) has quite divergent characterization results depending on its formulation, one in terms of FOL, the other by Bonnay in terms of L∞,∞, both without equality. From that we move on to a survey of Bonnay’s work on similarity relations between structures and his results that single out invariance with respect to potential isomorphism among all such. Turning to the critique that calls for sameness of meaning of a logical operation across domains, the paper continues with a result showing that the isomorphism invariant operations that are absolutely definable with respect to KPU−Inf are exactly those definable in full FOL; this makes use of an old theorem of Manders. The concluding section is devoted to a critical discussion of the arguments for set-theoretical criteria for logicality. (shrink)

Invariance criteria are widely accepted as a means to demarcate the logical vocabulary of a language. In previous work, I proposed a framework of “semantic constraints” for model theoretic consequence which does not rely on a strict distinction between logical and nonlogical terms, but rather on a range of constraints on models restricting the interpretations of terms in the language in different ways. In this paper I show how invariance criteria can be generalized so as to (...) apply to semantic constraints on models. Some obviously unpalatable semantic constraints turn out to be invariant under isomorphisms. I shall connect our discussion to known counterexamples to invariance criteria for logical terms, and so the generalization will also shed light on the current existing debate on logicality. I analyse the failure of invariance to fulfil its role as a criterion for logicality, and argue that invariance conditions should best be thought of as merely methodological meta-constraints restricting the ways the model-theoretic apparatus should be used. (shrink)

My aim in this paper is to propose what seems to me a distinctive approach to set theoretic methodology. By ‘methodology’ I mean the study of the actual methods used by practitioners, the study of how these methods might be justified or reformed or extended. So, for example, when the intuitionist's philosophical analysis recommends a wholesale revision of the methods of proof used in classical mathematics, this is a piece of reformist methodology. In contrast with the intuitionist, I will focus (...) more narrowly on the methods of contemporary set theory, and, more importantly, I will certainly recommend no sweeping reforms. Rather, I begin from the assumption that the methodologist's job is to account for set theory as it is practiced, not as some philosophy would have it be. This credo lies at the very heart of the so-called ‘naturalism’ to be described here.A philosopher looking at set theoretic practice from the outside, so to speak, might notice any number of interesting methodological questions, beginning with the intuitionist's ‘why use classical logic?’, but this sort of question is not a live issue for most practicing set theorists. One central question on which the philosopher's and the practitioner's interests converge is this: what is the status of independent statements like the continuum hypothesis (CH)? A number of large questions arise in its wake: what criteria should guide the search for new axioms? For that matter, what reasons support our adoption of the old axioms? (shrink)

There is a natural map which assigns to every modelU of typeτ, (U ε Stτ) a groupG (U) in such a way that elementarily equivalent models are mapped into isomorphic groups.G(U) is a subset of a collection whose members are called Fraisse arrows (they are decreasing sequences of sets of partial isomorphisms) and which arise in connection with the Fraisse characterization of elementary equivalence. LetEC λ U be defined as {U εStr τ: ℬ ≡U and |ℬ|=λ; thenEG λ U can (...) be faithfully (i.e. 1-1) represented onto G(U) ×π *, whereπ *, is a collection of partitions over λ∪λ2∪.. (shrink)

We show that if κ is a weakly compact cardinal then the embeddability relation on trees of size κ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space View the MathML source there is an Lκ+κ-sentence φ such that the embeddability relation on its models of size κ, which are all trees, is Borel bi-reducible to R. In particular, this implies that the relation of embeddability on trees of size κ is complete for (...) analytic quasi-orders on View the MathML source. These facts generalize analogous results for κ=ω obtained in Louveau and Rosendal [17] and Friedman and Motto Ros [6], and it also partially extends a result from Baumgartner [3] concerning the structure of the embeddability relation on linear orders of size κ. (shrink)

The classification of clinical problems, such as AIDS, requires choices. Choices are made on epistemic (i.e., knowledge-based) and non-epistemic (i.e., action-based) grounds. That is, the ways in which we classify clinical problems, such as AIDS, involve a balancing of different understandings of clinical reality and of clinical values among participants of the clinical community. On this view, the interplay between epistemic and non-epistemic interests occurs within the embrace of particular clinical contexts.The ways in which we classify AIDS is the topic (...) of this paper. We consider the extent to which we construct clinical reality; we examine a suggested classification of AIDS; and we conclude suggesting that the choice regarding how to classify AIDS is the result of negotiation among participants in the clinical community. (shrink)

In the study of reference assignment, the question of what pragmatic criteria are used to evaluate the resulting interpretation seems not yet to have been properly dealt with. This paper addresses the issue by examining factors which affect the acceptability of various cases of bridging reference. It demonstrates that even highly successful accounts of reference assignment which place major emphasis on accessibility factors, e.g. the accessibility of candidate referents and the accessibility of contextual assumptions, must nonetheless involve some pragmatic (...) criterion with which hearers can evaluate the resulting interpretation. Moreover, it argues that the pragmatic criterion used in reference assignment is not the truth-based one, which has been widely accepted, but the criterion of consistency with the principle of relevance proposed by Sperber and Wilson. Various accounts are tested against a wide range of examples on which I have conducted experimental tests with several sets of subjects. (shrink)

The overall aim of this article is to discuss the organization of limit setting in healthcare in terms of legitimacy. We argue there is a strong ethical demand that such processes should be arranged to provide adversely affected people well-justified reasons to confer legitimacy to the processes despite favouring a different decision-making outcome. Two increasingly popular approaches, Accountability for Reasonableness (A4R) and Multi-Criteria Decision Analysis (MCDA), can both be applied to support legitimate decision-making processes. However, the role played by (...) ‘fair-minded people’ in the A4R framework can be shown to undermine an adequate conceptualization of legitimacy according to the ethical demand. We discuss and specify conditions enabling A4R to meet this ethical demand when being implemented in a real-world setting without having to renounce the aim of striving for ‘reasonableness’ and impartiality. The methodological approach MCDA describes how to arrange for transparent resource allocation. This approach does not encompass any normative guidance on how to shape limit settings in terms of ‘reasonableness’; it basically promotes ‘rational’ decisions. We discuss how this practical methodology can be integrated with the theoretical A4R framework to meet the ethical demand while at the same time promoting politically required impartiality of healthcare limit setting. (shrink)

It is shown to be consistent with set theory that every set of reals of size ℵ1 is null yet there are ℵ1 planes in Euclidean 3-space whose union is not null. Similar results will be obtained for other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain harmonic functions and a measure theoretic pigeonhole principle.

What is a logical constant? The question is addressed in the tradition of Tarski's definition of logical operations as operations which are invariant under permutation. The paper introduces a general setting in which invariance criteria for logical operations can be compared and argues for invariance under potential isomorphism as the most natural characterization of logical operations.

Noetherian type and Noetherian π-type are two cardinal functions which were introduced by Peregudov in 1997, capturing some properties studied earlier by the Russian School. Their behavior has been shown to be akin to that of the cellularity, that is the supremum of the sizes of pairwise disjoint non-empty open sets in a topological space. Building on that analogy, we study the Noetherian π-type of κ-Suslin Lines, and we are able to determine it for every κ up to the first (...) singular cardinal. We then prove a consequence of Changʼs Conjecture for ℵω regarding the Noetherian type of countably supported box products which generalizes a result of Lajos Soukup. We finish with a connection between PCF theory and the Noetherian type of certain Pixley–Roy hyperspaces. (shrink)

We propose a sheaf-theoretic framework for the representation of a quantum observable structure in terms of Boolean information sieves. The algebraic representation of a quantum observable structure in the relational local terms of sheaf theory effectuates a semantic transition from the axiomatic set-theoretic context of orthocomplemented partially ordered sets, la Birkhoff and Von Neumann, to the categorical topos-theoretic context of Boolean information sieves, la Grothendieck. The representation schema is based on the existence of a categorical adjunction, which is used as (...) a theoretical platform for the development of a functorial formulation of information transfer, between quantum observables and Boolean localisation devices in typical quantum measurement situations. We also establish precise criteria of integrability and invariance of quantum information transfer by cohomological means. (shrink)

An early, very preliminary edition of this book was circulated in 1962 under the title Set-theoretical Structures in Science. There are many reasons for maintaining that such structures play a role in the philosophy of science. Perhaps the best is that they provide the right setting for investigating problems of representation and invariance in any systematic part of science, past or present. Examples are easy to cite. Sophisticated analysis of the nature of representation in perception is to be (...) found already in Plato and Aristotle. One of the great intellectual triumphs of the nineteenth century was the mechanical explanation of such familiar concepts as temperature and pressure by their representation in terms of the motion of particles. A more disturbing change of viewpoint was the realization at the beginning of the twentieth century that the separate invariant properties of space and time must be replaced by the space-time invariants of Einstein's special relativity. Another example, the focus of the longest chapter in this book, is controversy extending over several centuries on the proper representation of probability. The six major positions on this question are critically examined. Topics covered in other chapters include an unusually detailed treatment of theoretical and experimental work on visual space, the two senses of invariance represented by weak and strong reversibility of causal processes, and the representation of hidden variables in quantum mechanics. The final chapter concentrates on different kinds of representations of language, concluding with some empirical results on brain-wave representations of words and sentences. (shrink)

The most promising contemporary form of epistemic scientific realism is based on the following intuition: Belief should be directed, not toward theories as wholes, but toward particular theoretical constituents that are responsible for, or deployed in, key successes. While the debate on deployment realism is quite fresh, a significant degree of confusion has already entered into it. Here I identify five criteria that have sidetracked that debate. Setting these distractions aside, I endeavor to redirect the attention of both (...) realists and non-realists to the fundamental intuition above. In more detail: I show that Stathis Psillos (1999) has offered an explicit criterion for picking out particular constituents, which, contrary to Kyle Stanford’s (2006a) criticisms, neither assumes the truth of theories nor requires hindsight. I contend, however, that, in Psillos’s case studies, Psillos has not successfully applied his explicit criterion. After clarifying the various alternative criteria at work (in those case studies and in a second line of criticism offered by Stanford), I argue that, irrespective of Stanford’s criticisms, the explicit criterion Psillos does offer is not an acceptable one. Nonetheless, the deployment realist’s fundamental intuition withstands all of these challenges. In closing, I point in a direction toward which I’ve elsewhere focused, suggesting that, despite the legitimacy and applicability of the deployment realist’s intuition, the historical threat that prompted it remains. (shrink)

The isomorphism invariance criterion of logical nature has much to commend it. It can be philosophically motivated by the thought that logic is distinctively general or topic neutral. It is capable of precise set-theoretic formulation. And it delivers an extension of ‘logical constant’ which respects the intuitively clear cases. Despite its attractions, the criterion has recently come under attack. Critics such as Feferman, MacFarlane and Bonnay argue that the criterion overgenerates by incorrectly judging mathematical notions as logical. We consider (...) five possible precisifications of the overgeneration argument and find them all unconvincing. (shrink)

Although the invariance criterion of logicality first emerged as a criterion of a purely mathematical interest, it has developed into a criterion of considerable linguistic and philosophical interest. In this paper I compare two different perspectives on this criterion. The first is the perspective of natural language. Here, the invariance criterion is measured by its success in capturing our linguistic intuitions about logicality and explaining our logical behavior in natural-linguistic settings. The second perspective is more theoretical. Here, (...) the invariance criterion is used as a tool for developing a theoretical foundation of logic, focused on a critical examination, explanation, and justification of its veridicality and modal force. (shrink)

Summary By introducing the concept of “invariants”, Koffka endowed perceptual psychology with a flexible theoretical tool, which is suitable for representing vision situations in which a definite part of the stimulus pattern is relevant but not sufficient to determine a corresponding part of the perceived scene. He characterised his “invariance principle” as a principle conclusively breaking free from the “old constancy hypothesis”, which rigidly surmised point-to-point relations between stimulus and perceptual properties. In this paper, we explain the basic (...) terms and assumptions implicit in Koffka’s concept, by representing them in a set-theoretic framework. Then, we highlight various aspects and implications of the concept in terms of answers to six separate questions: forms of invariants, heuristic paths to them, what is invariant in an invariant, roots of conditional indeterminacy, variability vs. indeterminacy, and overcoming of the indeterminacy. Lastly, we illustrate the lasting value and theoretical power of the concept, by showing that Koffka’s insights relating to it do occur in modern perceptual psychology and by highlighting its role in a model of perceptual transparency. (shrink)

We introduce the class of unshreddable theories, which contains the simple and NIP theories, and prove that such theories have exactly saturated models in singular cardinals, satisfying certain set-theoretic hypotheses. We also give criteria for a theory to have singular compactness.

Health-promoting nudges have been put into practice by different agents, in different contexts and with different aims. This article formulates a set of criteria that enables a thorough ethical evaluation of such nudges. As such, it bridges the gap between the abstract, theoretical debates among academics and the actual behavioral interventions being implemented in practice. The criteria are derived from arguments against nudges, which allegedly disrespect nudgees, as these would impose values on nudgees and/or violate their rationality (...) and autonomy. Instead of interpreting these objections as knock-down arguments, I take them as expressing legitimate worries that can often be addressed. I analyze six prototypical nudge cases, such as Google’s rearrangement of fridges and the use of defaults in organ donation registration. I show how the ethical criteria listed are satisfied by most—but not all—nudges in most—but not all—circumstances. (shrink)

My aim has been to adapt Quine's criterion of the ontological commitment of theories couched in standard quantificational idiom to a much broader class of theories by focusing on the set-theoretic structure of the models of those theories. For standard first-order theories, the two criteria coincide on simple entities. Divergences appear as they are applied to higher-order theories and as composite entities are taken into account. In support of the extended criterion, I appeal to its fruits in treating the (...) various examples considered above and to the healthy intuitions of the non-noneists among us. Don't O(m) and E(m) comprise just the things we should have though existed according to a particular interpretation m of a language or a theory? Whatever the answer (and it will hardly be unanimous), I hope to have pointed the way towards a recognition of ontology as a worthwhile branch of modern theory. (shrink)

Existing theoretical interpretations contend that modernization is a global but diverse and multidimensional process. Yet, a systematic analysis of multiple forms of modernity and modernization ‘is the major challenge to current social and political theory’ (Wagner). The paper aims at revealing limitations of current theoretical interpretations of modernization and demonstrating systematically essential features of modernity. I describe the crucial criteria of modernization and suggest an integrated approach within which the most influential theories are simultaneously applied as coherent (...) explanations. Such a synergic application allows identifying concomitant dimensions of modernity and modernization. The proposed approach significantly differs from a prevailing theoretical discourse on modernity and modernization. It demonstrates universal features of modernization along with civilization variability and uniqueness of cultural programs, antinomic emancipation trends, permanence of change and innovations, increase of efficiency, competitiveness and the quality of life. All these dimensions relate to a set of consistent interpretations of a complex multi-effects phenomenon, which minimizes the existing conceptual contradictions. The elaborated model supports the notion of modernization as a continuous transition to novel forms of social order which respond to emerging challenges and global competition while an invariant criterion of modernization is the boost in the capabilities of people to make their choices. (shrink)

Context: Ernst von Glasersfeld’s radical constructivism has been very influential in education, particularly in mathematics and science education. Problem: There is limited guidance available for educational researchers who wish to design research that is consistent with constructivist thinking. Von Glasersfeld’s radical constructivism, together with the theoretical perspectives outlined by constructivist educational researchers such as Guba and Lincoln, can be considered as a source of guidance. Method: The paper outlines a constructivist knowledge framework that could be adopted for educational research. (...) The discussion considers how judgement of what counts as knowledge could be made, and how the set of procedures chosen could enable the researcher to represent the findings of the inquiry as knowledge. Results: An argument is made for researchers to explicate the criteria for judging an inquiry. Each criterion can then be linked to the standards to be reached and the techniques for generating data. The joint satisfaction of criteria and techniques for a constructivist inquiry creates conditions that indicate the “trustworthiness” or “authenticity” of an educational research study. Implications: The illustration of how a constructivist inquiry could be judged recognises how the contribution of von Glasersfeld’s radical constructivism can be used to inform the practice of educational research. Constructivist content: The argument presented in the paper links to radical constructivism and suggests ways in which it can be applied in the context of educational research. (shrink)

Numerous learning tasks can be described as the process of extrapolating patterns from observed data. One of the driving intuitions behind the theory of algorithmic randomness is that randomness amounts to the absence of any effectively detectable patterns: it is thus natural to regard randomness as antithetical to inductive learning. Osherson and Weinstein [11] draw upon the identification of randomness with unlearnability to introduce a learning-theoretic framework (in the spirit of formal learning theory) for modelling algorithmic randomness. They define two (...) success criteria—specifying under what conditions a pattern may be said to have been detected by a computable learning function—and prove that the collections of data sequences on which these criteria cannot be satisfied respectively correspond to the set of weak 1-randoms and the set of weak 2-randoms. This learning-theoretic approach affords an intuitive perspective on algorithmic randomness, and it invites the question of whether restricting attention to learning-theoretic success criteria comes at an expressivity cost. In other words, is the framework expressive enough to capture most core algorithmic randomness notions and, in particular, Martin-Löf randomness—arguably, the most prominent algorithmic randomness notion in the literature? In this paper, we answer the latter question in the affirmative by providing a learning-theoretic characterisation of Martin-Löf randomness. We then show that Schnorr randomness, another central algorithmic randomness notion, also admits a learning-theoretic characterisation in this setting. (shrink)

We show that recurrence conditions do not yield invariant Borel probability measures in the descriptive set-theoretic milieu, in the strong sense that if a Borel action of a locally compact Polish group on a standard Borel space satisfies such a condition but does not have an orbit supporting an invariant Borel probability measure, then there is an invariant Borel set on which the action satisfies the condition but does not have an invariant Borel probability measure.

This article discusses explanatory theories of normative concepts and argues for a set of criteria of adequacy by which such theories may be evaluated. The criteria offered fall into four categories: ontological, theoretical, pragmatic, and moral. After defending the criteria and discussing their relative weighting, this article uses them to prune the set of available explanatory theories of oppression. Functionalist theories, including Hegelian recognition theory and Foucauldian social theory, are rejected, as are psychoanalytic theory and social (...) dominance theory. Finally, the article defends structural rational choice theory as the most promising methodology for explaining oppression. Key Words: oppression • explanation • rational choice theory. (shrink)

The overall aim of this thesis is to present a model for rational goal-setting and to illustrate how it can be applied in evaluations of public policies, in particular policies concerning sustainable development and environmental quality. The contents of the thesis are divided into two sections: a theoretical section and an empirical section. Paper I identifies a set of rationality criteria for single goals and discusses them in relation to the typical function of goals. It is argued that (...) goals are typically set to enhance goal achievement. A goal that successfully furthers its achievement is “achievement-inducing”. It holds for each of the identified criteria that, ceteris paribus, improved satisfaction of a criterion makes a goal better in the achievement-inducing sense.Paper II contains an analysis of the notion of goal system coherence. It is argued that the coherence of a goal system is determined by the relations that hold among the goals in the system, in particular the relations of operationalization, means and ends, support, and conflict. Paper III investigates the rationality of utopian goals. The paper analyzes four arguments that support the normative criterion of attainability: that utopian goals are too imprecise and too far-reaching to guide action effectively, counterproductive, and morally objectionable. A tentative defence of utopian goal-setting is built on counter-arguments that can be put forward to weaken each of the four objections. Paper IV investigates the nature of self-defeating goals. The paper identifies three types of situations in which self-defeating mechanisms obstruct goal achievement: situations in which the goal itself carries the seeds of its own non-fulfilment, situations in which the activity of goal-setting contributes to goal failure, and situations in which disclosure of the goal interferes with progress. Paper V provides a brief description of the Swedish system of environmental objectives and a preliminary inventory of the management difficulties that attach to this goal system.Paper VI contains an investigation into the rationality of five Swedish environmental objectives through an application of the rationality criteria identified in Papers I-II. The paper identifies and discusses some difficulties that are associated with management by objectives and the use of goals in environmental policy. Paper VII analyses the rationality of the Swedish environmental quality objective A good built environment. Among the conclusions drawn in the paper are that some of the sub-goals to the objective are formulated in terms that are unnecessarily vague from an action-guiding standpoint and that others are problematic from the viewpoint of evaluability. (shrink)

In this work we study dimensional theoretical properties of some a±ne dynamical systems. By dimensional theoretical properties we mean Hausdor® dimension and box- counting dimension of invariant sets and ergodic measures on theses sets. Especially we are interested in two problems. First we ask whether the Hausdor® and box- counting dimension of invariant sets coincide. Second we ask whether there exists an ergodic measure of full Hausdor® dimension on these invariant sets. If this is not the case we (...) ask the question, whether at least the variational principle for Haus- dor® dimension holds, which means that there is a sequence of ergodic measures such that their Hausdor® dimension approximates the Hausdor® dimension of the invariant set. It seems to be well accepted by experts that these questions are of great importance in developing a dimension theory of dynamical systems (see the book of Pesin about dimension theory of dynamical systems [PE2]). Dimensional theoretical properties of conformal dynamical systems are fairly well understood today. For example there are general theorems about conformal repellers and hyperbolic sets for conformal di®eomorphisms (see chapter 7 of [PE2]). On the other hand the existence of two di®erent rates of expansion or contraction forces problems that are not captured by a general theory this days. At this stage of de- velopment of the dimension theory of dynamical systems it seems natural to study non conformal examples. This is the ¯rst step to understand the mechanisms that determine dimensional theoretical properties of non conformal dynamical systems. A±ne dynamical systems represent simple examples of non conformal systems. They are easy to de¯ne, but studying their dimensional theoretical properties does never- theless provide challenging mathematical problems and exemplify interesting phe- nomena. We consider here a special class of self-a±ne repellers in dimension two, depending on four parameters (see 2.1.). Furthermore we study a class of attractors of piecewise a±ne maps in dimension three depending on four parameters as well. The last object of our work are projections of these maps that are known as gener- alized Baker's transformations (see 2.2.). The contents of our work is the following: In chapter two we give an overview about some main results in the area of di- mension theory of a±ne dynamical systems and de¯ne the systems we study in this work. We will explain, what is known about the dimensional theoretical properties of these systems and describe what our new results are. In chapter three we then apply symbolic dynamics to our systems. We will introduce explicit shift codings 4 and ¯nd representations of all ergodic measures for our systems using these codings. From chapter four to chapter eight we study dimensional theoretical properties, which our systems generally or generically have. In chapter four we will prove a formula for the box-counting dimension of the repellers and the attractors (see the- orem 4.1.). Then in chapter ¯ve we apply general dimensional theoretical results for ergodic measures found by Ledrappier and Young [LY] and Barreira, Schmeling and Pesin [BPS] to our systems. These results relate the dimension of ergodic measures to metric entropy and Lyapunov exponents. Using this approach we will be able to reduce questions about the dimension of ergodic measures in our context to ques- tions about certain overlapping and especially overlapping self-similar measures on the line. These overlapping self-similar measures are studied in chapter six. Our main theorem extends a result of Peres and Solomyak [PS2] concerning the absolute continuity resp. singularity of symmetric self-similar measures to asymmetric ones (see theorem 6.1.3.). In chapter seven we bring our results together. We prove that we generically (in the sense of Lebesgue measure on a part of the parameter space) have the iden- tity of box-counting and Hausdor® dimension for the repellers and the attractors. (see theorem 7.1.1. and corollary 7.1.2.). This result suggest that one can expect that the identity of box-counting dimension and Hausdor® dimension holds at least generically in some natural classes of non conformal dynamical systems. Furthermore we will see in chapter seven that there generically exists an ergodic measure of full Hausdor® dimension for the repellers. On the other hand the vari- ational principle for Hausdor® dimension is not generic for the attractors. It holds only if we assume a certain symmetry (see theorem 7.1.1.). For generalized Baker's transformations we will ¯nd a part of the parameter space where there generically is an ergodic measure of full dimension and a part where the variational principle for Hausdor® dimension does not hold (see theorem 7.1.3.). Roughly speaking the reason why the variational principle does not hold here is, that if there exists both a stable and an unstable direction one can not generically maximize the dimension in the stable and in the unstable direction at the same time. In an other setting this phenomenon was observed before by Manning and McCluskey [MM]. In chapter eight we extend some results of the last section to invariant sets that correspond to special Markov chains instead of full shifts (see theorem 8.1.1.). In the last two chapters of our work we are interested in number theoretical excep- tions to our generic results. The starting point of our considerations in section nine are results of ErdÄos [ER1] and Alexander and Yorke [AY] that establish singularity and a decrease of dimension for in¯nite convolved Bernoulli measures under special conditions. Using a generalized notion of the Garsia entropy ([GA1/2]) we are able 5 to understand the consequences of number theoretical peculiarities in broader class of overlapping measures (see theorem 9.1.1.). In chapter ten we then analyze number theoretical peculiarities in the context of our dynamical systems. We restrict our attention to a symmetric situation where we generically have the existence of a Bernoulli measure of full dimension and the identity of Hausdor® and box-counting dimension for all of our systems. In the ¯rst section of chapter ten we ¯nd parameter values such that the variational principle for Hausdor® dimension does not hold for the attractors and for the Fat Baker's transformations (see theorem 10.1.1.). These are the ¯rst known examples of dynamical systems for which the variational principle for Hausdor® dimension does not hold because of number theoretical peculiarities of parameter values. For the repellers we have been able to show that under certain number theoretical conditions there is at least no Bernoulli measure of full Hausdor® dimension; the question if the variational principle for Hausdor® dimension holds remains open in this situation. In the second section of chapter ten we will show that the identity for Hausdor® and box-counting dimension can drops because there are number theoretical pecu- liarities. In the context of Weierstrass-like functions this phenomenon was observed by Przytycki and Urbanski [PU]. Our theorem extends this result to a larger class of sets, invariant under dynamical systems (see theorem 10.2.1). At the end of this work the reader will ¯nd two appendices, a list of notations and the list of references. In appendix A we introduce the notions of dimension we use in this work and collect some general facts in dimension theory. In appendix B we state the facts about Pisot-Vijayarghavan number, we need in our analysis of number theoretical peculiarities. The list of notations contains general notations and a table with a summary of notations we use to describe the dynamical systems that we study. Acknowledgments I wish to thank my supervisor JÄorg Schmeling for a lot of valuable discussion and all his help. Also thanks to Luis Barreira for his great hospitality in Lisboa and many interesting comments. This work was done while I was supported by "Promotionstipendium gem. NaFÄoG der Freien UniversitÄat Berlin". (shrink)

BackgroundAssessing consciousness in other subjects, particularly in non-verbal and behaviourally disabled subjects (e.g., patients with disorders of consciousness), is notoriously challenging but increasingly urgent. The high rate of misdiagnosis among disorders of consciousness raises the need for new perspectives in order to inspire new technical and clinical approaches. Main bodyWe take as a starting point a recently introduced list of operational indicators of consciousness that facilitates its recognition in challenging cases like non-human animals and Artificial Intelligence to explore their relevance (...) to disorders of consciousness and their potential ethical impact on the diagnosis and healthcare of relevant patients. Indicators of consciousness mean particular capacities that can be deduced from observing the behaviour or cognitive performance of the subject in question (or from neural correlates of such performance) and that do not define a hard threshold in deciding about the presence of consciousness, but can be used to infer a graded measure based on the consistency amongst the different indicators. The indicators of consciousness under consideration offer a potential useful strategy for identifying and assessing residual consciousness in patients with disorders of consciousness, setting the theoretical stage for an operationalization and quantification of relevant brain activity.ConclusionsOur heuristic analysis supports the conclusion that the application of the identified indicators of consciousness to its disorders will likely inspire new strategies for assessing three very urgent issues: the misdiagnosis of disorders of consciousness; the need for a gold standard in detecting consciousness and diagnosing its disorders; and the need for a refined taxonomy of disorders of consciousness. (shrink)

Global governance, central to international rule-making, is rapidly evolving; thus, there is a need for a way to evaluate whether institutions have the capacity to address the problems of the contemporary era. Current methods of evaluating the democratic quality of contemporary governance are closely linked to legitimacy, about which there are competing definitional theories. This article uses a theoretical approach based around “new“ governance and the environmental policy arena to argue that contemporary governance is best understood as social-political interaction (...) built on “participation as structure“ and “deliberation as process“, with the level of interaction ultimately determining legitimacy. It presents a new arrangement of the accepted attributes of “good“ governance using a set of principles, criteria and indicators, and relates these to the structures and processes of governance. The implications and application of the analytical framework are also discussed. (shrink)

Purpose of the article is the reconstruction of ancient Greek and ancient Roman models of religiosity as anthropological invariants that determine the patterns of thinking and being of subsequent eras. Theoretical basis. The author applied the statement of Protagoras that "Man is the measure of all things" to the reconstruction of the religious sphere of culture. I proceed from the fact that each historical community has a set of inherent ideas about the principles of reality, which found unique "universes (...) of meanings". The historical space acquires anthropological properties that determine the specific mythology of the respective societies, as well as their spiritual successors. In particular, the religious models of ancient Greece and ancient Rome had a huge influence on formation of the worldview of the Christian civilization of the West. Originality. Multiplicity of the Olympic mythology contributed to the diversity of the expression forms of the Greek genius, which manifested itself in different fields of cultural activity, not reducible to political, philosophical or religious unity. The poverty of Roman mythology was compensated by a clear awareness of the unity of the community, which for all historical vicissitudes had always remained an unchanging ideal, and which was conceived as a reflection of the unity of the heavens. These two approaches to the divine predetermined the formation of two interacting, but conceptually different anthropological paradigms of Antiquity. Conclusions. Western concepts of divinity are invariants of two basic theological concepts – "Greek" and "Roman". These are ideal types, so these two tendencies can co-exist in one society. The Roman trend continued to be realized by the anti-Roman religion, which took Roman forms and Roman name. Iconoclasm was a Byzantine version of the Reformation, promoted by the Isaurian emperors and failed due to the strong Hellenistic naturalistic lobby. Modern "Romans" are trying to get rid of the last elements of religious naturalism, and modern "Greeks" are trying to preserve the Hellenic elements in Christianity. Patterns can be transformed, but the observational view will still be able to identify their lineage. The developed model allows a deeper understanding of the culture of both ancient societies, as well as the outlook of Western man. (shrink)

The spacing effect is among the most widely replicated empirical phenomena in the learning sciences, and its relevance to education and training is readily apparent. Yet successful applications of spacing effect research to education and training is rare. Computational modeling can provide the crucial link between a century of accumulated experimental data on the spacing effect and the emerging interest in using that research to enable adaptive instruction. In this paper, we review relevant literature and identify 10 criteria for (...) rigorously evaluating computational models of the spacing effect. Five relate to evaluating the theoretic adequacy of a model, and five relate to evaluating its application potential. We use these criteria to evaluate a novel computational model of the spacing effect called the Predictive Performance Equation. Predictive Performance Equation combines elements of earlier models of learning and memory including the General Performance Equation, Adaptive Control of Thought—Rational, and the New Theory of Disuse, giving rise to a novel computational account of the spacing effect that performs favorably across the complete sets of theoretic and applied criteria. We implemented two other previously published computational models of the spacing effect and compare them to PPE using the theoretic and applied criteria as guides. (shrink)

On the basis of an analysis of everyday experience and practice, criteria of legitimate assertions of existence and truth are offered. A specific object, like a newspaper, can be asserted to exist if it has some invariant characteristics and is present in actual perception. A statement, like “This newspaper is black and white,” can be accepted as true if it is well-established in some empirical domain. Each of these criteria provides a sufficient condition for acceptance of existence and (...) truth, respectively, at the empirical level. Following Herman Weyl, it is argued that these criteria can be extended to the scientific theoretical level to support a selective and moderate version of scientific realism according to which entities like the electromagnetic and gravitational fields, but not crystalline spheres or some topological manifolds, can legitimately be asserted to exist. (shrink)

Linkage (or genetic) maps are graphs, which are intended to represent the linear ordering of genes on the chromosomes. They are constructed on the basis of statistical data concerning the transmission of genes. The invention of this technique in 1913 was driven by Morgan's group's adoption of a set of hypotheses concerning the physical mechanism of heredity. These hypotheses were themselves grounded in Morgan's defense of the chromosome theory of heredity, according to which chromosomes are the physical basis of genes. (...) In this paper, I analyze the 1919 debate between William Castle and Morgan's group, about the construction of genetic maps. The official issue of the debate concerns the arrangement of genes on chromosomes. However, the disputants tend to carry out the discussions about how one should model the data in order to draw predictions concerning the transmission of genes; the debate does not bear on the data themselves, nor does it focus on the hypotheses explaining these data. The main criteria that are appealed to by the protagonists are simplicity and predictive efficacy. However, I show that both parties' assessments of the simplicity and predictive efficacy of different ways of modeling the data themselves depend on background theoretical positions. I aim at clarifying how preference for a given model and theoretical commitments articulate. (shrink)

We explore Shelah's model‐theoretic dividing line methodology. In particular, we discuss how problems in model theory motivated new techniques in model theory, for example classifying theories by their potential (consistently with Zermelo–Fraenkel set theory with the axiom of choice (ZFC)) spectrum of cardinals in which there is a universal model. Two other examples are the study (with Malliaris) of the Keisler order leading to a new ZFC result on cardinal invariants and attempts to clarify the “main gap” by reducing the (...) dependence of certain versions on (highly independent) cardinal arithmetic. (shrink)

This work applies the competitive exclusion principle and the concept of potential competitors as simple axiomatic tools to generalized situations in ecology. These tools enable apparent competition and its dual counterpart to be explicitly evaluated in poorly understood ecological systems. Within this set-theory framework we explore theoretical symmetries and invariances, De Morgan’s laws, frozen evolutionary diversity and virtual processes. In particular, we find that the exclusion principle compromises the geometrical growth of the number of species. By theoretical extending (...) this principle, we can describe interspecific depredation in the dual case. This study also briefly considers the debated situation of intraspecific competition. The ecological consequences of our findings are discussed; particularly, the use of our framework to reinterpret coupled mathematical differential equations describing certain ecological processes. (shrink)

Set-theoretic pluralism is an increasingly influential position in the philosophy of set theory (Balaguer [1998], Linksy and Zalta [1995], Hamkins [2012]). There is considerable room for debate about how best to formulate set-theoretic pluralism, and even about whether the view is coherent. But there is widespread agreement as to what there is to recommend the view (given that it can be formulated coherently). Unlike set-theoretic universalism, set-theoretic pluralism affords an answer to Benacerraf’s epistemological challenge. The purpose of this paper is (...) to determine what Benacerraf’s challenge could be such that this view is warranted. I argue that it could not be any of the challenges with which it has been traditionally identified by its advocates, like of Benacerraf and Field. Not only are none of the challenges easier for the pluralist to meet. None satisfies a key constraint that has been placed on Benacerraf’s challenge. However, I argue that Benacerraf’s challenge could be the challenge to show that our set-theoretic beliefs are safe – i.e., to show that we could not have easily had false ones. Whether the pluralist is, in fact, better positioned to show that our set-theoretic beliefs are safe turns on a broadly empirical conjecture which is outstanding. If this conjecture proves to be false, then it is unclear what the epistemological argument for set-theoretic pluralism is supposed to be. (shrink)

This book introduces readers to the concept of the neutrosophic set which can deal with dynamic and complex decision-making problems. With the complexity of the socio-economic environment, today's decision-making is one of the most notable ventures, whose mission is to decide the best alternative under numerous known or unknown criteria. This book provides a large amount of theoretical and practical information about the latest research in the field, allowing readers to gain an extensive understanding of both the fundamentals (...) and applications of neutrosophic sets to solve different kinds of decision-making problems and mathematical programming such as medical diagnosis, pattern recognition, construction problems, technology selection etc. (shrink)

In this article we study several reduction principles in the context of Simpson’s set theory ATR0S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ATR_{0}^{S}$$\end{document} and Kripke-Platek set theory KP (with infinity). Since ATR0S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ATR_{0}^{S}$$\end{document} is the set-theoretic version of ATR0 there is a direct link to second order arithmetic and the results for reductions over ATR0S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ATR_{0}^{S}$$\end{document} are as expected and more or less (...) straightforward. However, over KP we obtain several interesting new results and are lead to some open questions. (shrink)

The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set-theoretic possibilities, a phenomenon that challenges the universe (...) view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for. (shrink)

We calculate the universal Euler characteristic and universal dimension function on semilinear and semibounded sets and obtain some criteria for definable equivalence of semilinear and semibounded sets in terms of these invariants.

In his paper "Are There Set Theoretic Possible Worlds?", Selmer Bringsjord argued that the set theoretic definition of possible worlds proffered by, among others, Robert Adams and Alvin Plantinga is incoherent. It is the purpose of this note to evaluate that argument. The upshot: these set theoretic accounts can be preserved, but only by abandoning the power set axiom.

The semantic tradition in logic descends from Tarski's seminal work on truth and logical consequence. In the introduction to this volume, Sagi and Woods remind us that this tradition prominently uses model theory to study languages and their interpretations. Tarski's model-theoretic definition of logical consequence is the prime example of this approach, seeking as it does to reduce logical properties to a class of operations on classical, iterative (ZF) sets. Sagi and Woods explain with admirable clarity the origins, implications, and (...) philosophical questions surrounding this project. The contributions to the volume go on to address whether this approach successfully elucidates logicality and how it intersects with the foundations of mathematics and natural language semantics. (shrink)

The purpose of this article is to shape a theoretical framework of attentiveness in care, which may function as a background to study attentiveness in a health care setting empirically. More insight into the functions, forms, and aspects of attentiveness in a particular health care setting is important, as there is a lack of indicators and criteria that enable a sharp picture of the caring side of health provision. The concept of attentiveness and its relation to care have (...) seldom been examined thoroughly and broadly. This article argues that attentiveness is constitutive for good care, as it can create a space in which a relationship may arise. (shrink)

In this paper I argue in favour of a moderate and selective version of scientific realism with respect to the existence of some physical theoretical objects and the truth of some statements about them. The analysis of common sense or ordinary experience reveals that existence and truth assertions concerning familiar objects are warranted if they satisfy what we call the criteria of presence and invariance. Ordinary objects exemplify a form or a structure determined by constant and changing (...) features with respect to the variation of the conditions of observation and a content given in sensory presence. It is claimed that some theoretical physical objects that possess a structure or a form determined by invariant and variable characteristics relatively to a specified group of transformations and a content provided by observations on which the measurement of those characteristics is grounded can also be justifiably asserted to exist. We further show that this defence of scientific realism in physics doesn't fall prey to the objections that have been raised against the usual vindication of scientific realism based on the no-miracle argument. Some standard objections to scientific realism, such as the underdetermination of theories by observations and the pessimistic meta-induction, are examined and shown to be unconvincing. (shrink)

Psychological essentialism is a cognitive bias through which human beings conceive the entities around them as having inner essences and basic natures. Social scientists routinely generate flawed inferences because their methods require the truth of psychological essentialism. This article develops set-theoretic analysis as a scientific-constructivist approach that overcomes the bias of psychological essentialism. With this approach, the “sets” of set-theoretic analysis are mental phenomena that establish boundaries and identify similarities and differences among entities whose natural kind composition is not known. (...) The approach is illustrated through a consideration of research on intelligence, race, and poverty in the United States. (shrink)

Personality theories often identify sets of primary parts. These are sets of a few personality parts expansive enough to collectively describe the total personality. Examples of such sets include the trilogy of mind , Freud’s structural set , and the recently-introduced systems set . These groups may be of unrecognized importance in understanding human personality. The defining characteristics of such sets are identified, their history is reviewed, their theoretical contributions considered, and then, criteria for distinguishing good from bad (...) sets of primary parts are proposed. Finally, the potential contribution of primary parts sets to personality psychology is revisited. (shrink)

A theistic science would have to represent the integration of all kinds of knowledge intent on explaining the whole of reality. These would include, at least, history, metaphysics, theology, formal logic, mathematics, and experimental sciences. However, what is the whole of reality that one wants to explain? :.

The interrelations of science and technology as an object of study seem to have drawn the attention of a number of disciplines: the history of both science and technology, sociology, economics and economic history, and even the philosophy of science. The question that comes to mind is whether the phenomenon itself is new or if advances in the disciplines involved account for this novel interest, or, in fact, if both are intercon nected. When the editors set out to plan this (...) volume, their more or less explicit conviction was that the relationship of science and technology did reveal a new configuration and that the disciplines concerned with 1tS analysis failed at least in part to deal with the change because of conceptual and methodological preconceptions. To say this does not imply a verdict on the insufficiency of one and the superiority of any other one disciplinary approach. Rather, the situation is much more complex. In economics, for example, the interest in the relationship between science and technology is deeply influenced by the theoretical problem of accounting for the factors of economic growth. The primary concern is with technology and the problem is whether the market induces technological advances or whether they induce new demands that explain the subsequent diffusion of new technologies. Science is generally considered to be an exogenous factor not directly subject to market forces and, therefore, appears to be of no interest. (shrink)