Penelope Maddy’s Second Philosophy is one of the most well-known ap- proaches in recent philosophy of mathematics. She applies her second-philosophical method to analyze mathematical methodology by reconstructing historical cases in a setting of means-ends relations. However, outside of Maddy’s own work, this kind of methodological analysis has not yet been extensively used and analyzed. In the present work, we will make a first step in this direction. We develop a general framework that allows us to clarify the procedure (...) and aims of the Second Philosopher’s investigation into set-theoretic methodology; pro- vides a platform to analyze the Second Philosopher’s methods themselves; and can be applied to further questions in the philosophy of set theory. (shrink)

How do we come to like the things that we do? Each one of us starts from a relatively similar state at birth, yet we end up with vastly different sets of aesthetic preferences. These preferences go on to define us both as individuals and as members of our cultures. Therefore, it is important to understand how aesthetic preferences form over our lifetimes. This poses a challenging problem: to understand this process, one must account for the many factors at play (...) in the formation of aesthetic values and how these factors influence each other over time. A general framework based on basic neuroscientific principles that can also account for this process is needed. Here, we present such a framework and illustrate it through a model that accounts for the trajectories of aesthetic values over time. Our framework is inspired by meta-analytic data of neuroimaging studies of aesthetic appraisal. This framework incorporates effects of sensory inputs, rewards, and motivational states. Crucially, each one of these effects is probabilistic. We model their interactions under a reinforcement-learning circuitry. Simulations of this model and mathematical analysis of the framework lead to three main findings. First, different people may develop distinct weighing of aesthetic variables because of individual variability in motivation. Second, individuals from different cultures and environments may develop different aesthetic values because of unique sensory inputs and social rewards. Third, because learning is stochastic stemming from probabilistic sensory inputs, motivations, and rewards, aesthetic values vary in time. These three theoretical findings account for different lines of empirical research. Through our study, we hope to provide a general and unifying framework for understanding the various aspects involved in the formation of aesthetic values over time. (shrink)

This article explores recent developments in the sociology of the arts, namely the new theoretical framework set up by the French sociologist Pierre-Michel Menger in order to approach the artistic activity. It aims to show how he has shaped new tools of understanding and modelling for exploring the arts, as a particular world of action. Laying down the foundation of a conception of action related to symbolic interactionism and drawing on the economic analysis of risk and uncertainty, Menger (...) move towards a model where uncertainty, not determinism, is the default category. Uncertainty is thus the principle unifying the analysis and understanding of the particularities of the arts, and the main argument in integrating sociological and economic studies on artistic activities. Correlatively, artistic creation is qualified as an act of work, the creative work being defined as a system of action in uncertain horizon. Within this model, the artists are seen less like “rational fools” and more like Bayesian actors, adopting strategies for managing risk and uncertainty. (shrink)

This article explores recent developments in the sociology of the arts, namely the new theoretical framework set up by the French sociologist Pierre-Michel Menger in order to approach the artistic activity. It aims to show how he has shaped new tools of understanding and modelling for exploring the arts, as a particular world of action. Laying down the foundation of a conception of action related to symbolic interactionism and drawing on the economic analysis of risk and uncertainty, Menger (...) move towards a model where uncertainty, not determinism, is the default category. Uncertainty is thus the principle unifying the analysis and understanding of the particularities of the arts, and the main argument in integrating sociological and economic studies on artistic activities. Correlatively, artistic creation is qualified as an act of work, the creative work being defined as a system of action in uncertain horizon. Within this model, the artists are seen less like “rational fools” and more like Bayesian actors, adopting strategies for managing risk and uncertainty. (shrink)

This article explores recent developments in the sociology of the arts, namely the new theoretical framework set up by the French sociologist Pierre-Michel Menger in order to approach the artistic activity. It aims to show how he has shaped new tools of understanding and modelling for exploring the arts, as a particular world of action. Laying down the foundation of a conception of action related to symbolic interactionism and drawing on the economic analysis of risk and uncertainty, Menger (...) move towards a model where uncertainty, not determinism, is the default category. Uncertainty is thus the principle unifying the analysis and understanding of the particularities of the arts, and the main argument in integrating sociological and economic studies on artistic activities. Correlatively, artistic creation is qualified as an act of work, the creative work being defined as a system of action in uncertain horizon. Within this model, the artists are seen less like “rational fools” and more like Bayesian actors, adopting strategies for managing risk and uncertainty. (shrink)

In this paper, we explore the idea that sets depend on, or are grounded in, their members. It is said that a set depends on each of its members, and not vice versa. Members do not depend on the sets that they belong to. We show that the intuitive modal truth conditions for dependence, given in terms of possible worlds, do not accurately capture asymmetric dependence relations between sets and their members. We extend the modal truth conditions to include impossible (...) worlds and give a more satisfactory account of the dependence of a set on its members. Focusing on the case of singletons, we articulate a logical framework in which to evaluate set-theoretic dependence claims, using a normal first-order modal logic. We show that on this framework the dependence of a singleton on its single members follows from logic alone. However, the converse does not hold. (shrink)

We present an axiomatic framework for semantics that can be applied to natural and formal languages. Our main goal is to suggest a very simple mathematical model that describes fundamental cognitive aspects of the human brain and that can still be applied to artificial intelligence. One of our main results is a theorem that allows us to infer syntactical properties of a language out of its corresponding semantics. The role of pragmatics in semantics in our mathematical framework is (...) also discussed. (shrink)

Quasi-set theory provides us a mathematical background for dealing with collections of indistinguishable elementary particles. In this paper, we show how to obtain the usual statistics (Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac) into the scope of quasi-set theory. We also show that, in order to derive Maxwell–Boltzmann statistics, it is not necessary to assume that the particles are distinguishable or individuals. In other words, Maxwell–Boltzmann statistics is possible even in an ensamble of indistinguishable particles, at least from the theoretical point of (...) view. The main goal of this paper is to provide the mathematical grounds of a quasi-set theoretical framework for statistical mechanics. (shrink)

An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the `width' of the set theoretic universe, such as Cantor's continuum hypothesis. Within a higher-order framework I show that contingency about (...) the width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the "Leibniz biconditionals" stating that what is possible, in the broadest sense of possible, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent and has attractions. (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)

Osteopathy is a person-centred healthcare discipline that emphasizes the body’s structure-function interrelationship—and its self-regulatory mechanisms—to inform a whole-person approach to health and wellbeing. This paper aims to provide a theoretical framework for developing an integrative hypothesis in osteopathy, which is based on the enactivist and active inference accounts. We propose that osteopathic care can be reconceptualised under active inference as a unifying framework. Active inference suggests that action-perception cycles operate to minimize uncertainty and optimize an individual’s internal (...) model of the lived world and, crucially, the consequences of their behaviour. We argue that active inference offers an integrative framework for osteopathy, which can evince the mechanisms underlying dyadic and triadic exchanges and osteopathic care outcomes. We propose that this theoretical framework can underpin osteopathic care across the lifespan, from preterm infants to the elderly and those with persistent pain and other physical symptoms. In situations of chronicity, as an ecological niche, the patient-practitioner dyad provides the osteopath and the patient with a set of affordances, i.e., possibilities for action provided by the environment, that through shared intentionally, can promote adaptations and restoration of productive agency. Through a dyadic therapeutic relationship, as they engage with their ecological niche’s affordances—a structured set of affordances shared by agents—osteopath and patient actively construct a shared sense-making narrative and realise a shared generative model of their relation to the niche. In general, touch plays a critical role in developing a robust therapeutic alliance, mental state alignment, and biobehavioural synchrony between patient and practitioner. However, its role is particularly crucial in the fields of neonatology and paediatrics, where it becomes central in regulating allostasis and restoring homeostasis. We argue that from an active inference standpoint, the dyadic shared ecological niche underwrites a robust therapeutic alliance, which is crucial to the effectiveness of osteopathic care. Considerations and implications of this model—to clinical practice and research, both within- and outside osteopathy—are critically discussed. (shrink)

This article introduces a novel theoretical framework for addressing epistemic injustice—a phenomenon where certain groups or individuals are systematically excluded from knowledge creation and dissemination processes—by employing ontological relativity and conceptual analysis. “Ontological relativity” refers to a philosophical perspective that posits our understanding of reality as being shaped by our toolbox of concepts, categories, language, and social practices; “conceptual analysis” is a method of inquiry that involves the rigorous examination and deconstruction of a particular concept or set of (...) concepts in order to uncover their constituent elements, relationships, and underlying assumptions. To exemplify the effectiveness of the ontology‐based approach, two paradigmatic applications are explored: (a) educational practices and (b) clinical practice and access to health care. Through the presentation of these applications and the step‐by‐step illustration of the applied methodology, the aim of the article is to showcase the efficacy of ontology in tackling epistemic injustices, suggesting innovative paths for future research in this domain. (shrink)

Vardi, M.Y., Verification of concurrent programs: the automata-theoretic framework, Annals of Pure and Applied Logic 51 79–98. We present an automata-theoretic framework to the verification of concurrent and nondeterministic programs. The basic idea is that to verify that a program P is correct one writes a program A that receives the computation of P as input and diverges only on incorrect computations of P. Now P is correct if and only if a program PA, obtained by combining P (...) and A, terminates. We formalize this idea in a framework of ω-automata with a recursive set of states. This unifies previous works on verification of fair termination and verification of temporal properties. (shrink)

What are the basic building blocks of the world? This book presents a naturalistic theory saying that the universe and everything in it can be reduced to three fundamental entities: a field, a set of values that can be actualized at different places in the field, and an actualizer of the values. The theory is defended by using it to answer the main questions in metaphysics, such as: What is causality, existence, laws of nature, consciousness, thinking, free will, time, mathematical (...) entities, ethical values, etc.? The theory is compared with the main alternatives and argued to solve problems better than the existing theories. Several new theories are suggested, such as how to understand mental causation, free will and the truth of ethics and mathematics. (shrink)

We present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF . It allows the use of set terms, but provides a static check of their validity. Like the inconsistent “ideal calculus” for set theory, it is essentially based on just two set-theoretical principles: extensionality and comprehension (to which we add ∈-induction and optionally the axiom of choice). Comprehension is formulated as: x ∈ {x | ϕ} ↔ (...) ϕ, where {x | ϕ} is a legal set term of the theory. In order for {x | ϕ} to be legal, ϕ should be safe with respect to {x}, where safety is a relation between.. (shrink)

Many science systems are witnessing the rise of intermediary organizations with a coordinating mission, but to date a systematic understanding of their function and effects is lacking. The aim of this paper is to contribute to the understanding of the coordinating efforts of intermediary organizations. Starting from the definition of coordination as the establishment or strengthening of a relationship among the activities in a system, with the aim to enhance their common effectiveness, I develop a heuristic framework that facilitates (...) the systematic analysis of coordination in science. I illustrate and substantiate my framework with the empirical case study of a Dutch coordination task force in the area of chemical technologies. Thanks to the framework I could disentangle a number of functions that this task force fulfils concerning research programming, funding allocation and supporting interactions and collaborations. This approach enabled me to systematically analyse a very heterogeneous set of processes that each deserve to be called coordination. The analysis yields a clear overview of eight coordination processes that are each described in terms of activities, intervention, relationships, mechanisms and performance. I conclude my paper with suggestions for further research on coordination in the science system. (shrink)

This theoretical exploration is an attempt to conceptualize the link between gender and urban environmental health. The proposed ecofeminist framework enables an understanding of the link between the urban physical and social environments and health inequities mediated by gender and socioeconomic status. This framework is proposed as a theoretical magnifying glass to reveal the underlying logic that connects environmental exploitation on the one hand, and gendered health inequities on the other. Ecofeminism has the potential to reveal (...) an inherent, normative conceptual analysis and argumentative justification of western society that permits the oppression of women and the exploitation of the environment. This insight will contribute to a better understanding of the mechanisms underlying gendered environmental health inequities and inform healthy public policy that is supportive of urban environmental health, particularly for low‐income mothers. (shrink)

It is usually stated that quantum mechanics presents problems with the identity of particles, the most radical position—supported by E. Schrödinger—asserting that elementary particles are not individuals. But the subject goes deeper, and it is even possible to obtain states with an undefined particle number. In this work we present a set theoretical framework for the description of undefined particle number states in quantum mechanics which provides a precise logical meaning for this notion. This construction goes in the (...) line of solving a problem posed by Y. Manin, namely, to incorporate quantum mechanical notions at the foundations of mathematics. We also show that our system is capable of representing quantum superpositions. (shrink)

We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending ZFC by using his multiverse axioms MV and the ‘core hypothesis’. In the first part, we examine the evidential framework for MV, in particular the use of large cardinals and of ‘worlds’ obtained through forcing to ‘represent’ alternative extensions of ZFC. In the second part, we address the existence and the possible features of the core of MV_T (where T is ZFC+Large Cardinals). (...) In the last part, we discuss the hypothesis that the core is Ultimate-L, and examine whether and how, based on this fact, the Core Universist can justify V=Ultimate-L as the best (and ultimate) extension of ZFC. To this end, we take into account several strategies, and assess their prospects in the light of MV’s evidential framework. (shrink)

$${{{\mathcal {F}}}}$$ -systems are useful digraphs to model sentences that predicate the falsity of other sentences. Paradoxes like the Liar and the one of Yablo can be analyzed with that tool to find graph-theoretic patterns. In this paper we studied this general model consisting of a set of sentences and the binary relation ‘ $$\ldots $$ affirms the falsity of $$\ldots $$ ’ among them. The possible existence of non-referential sentences was also considered. To model the sets of all the (...) sentences that can jointly be valued as true we introduced the notion of conglomerate, the existence of which guarantees the absence of paradox. Conglomerates also enabled us to characterize referential contradictions, i.e., sentences that can only be false under a classical valuation due to the interactions with other sentences in the model. A Kripke-style fixed-point characterization of groundedness was offered, and complete (meaning that every sentence is deemed either true or false) and consistent (meaning that no sentence is deemed true and false) fixed points were put in correspondence with conglomerates. Furthermore, argumentation frameworks are special cases of $$\mathcal{F}$$ -systems. We showed the relation between local conglomerates and admissible sets of arguments and argued about the usefulness of the concept for the argumentation theory. (shrink)

It is usually stated that quantum mechanics presents problems with the identity of particles, the most radical position—supported by E. Schrödinger—asserting that elementary particles are not individuals. But the subject goes deeper, and it is even possible to obtain states with an undefined particle number. In this work we present a set theoretical framework for the description of undefined particle number states in quantum mechanics which provides a precise logical meaning for this notion. This construction goes in the (...) line of solving a problem posed by Y. Manin, namely, to incorporate quantum mechanical notions at the foundations of mathematics. We also show that our system is capable of representing quantum superpositions. (shrink)

The focus of this paper is to develop a comprehensive conceptual framework for the relationship between internal employer brand image and talent retention. An extensive and semi-systematic literature review identified a number of antecedents and consequences that have been empirically tested in various cutting-edge research studies that were conducted around the world. The existing literature is reviewed using a topical approach, and 66 research studies, most recent from various repositories, were carefully chosen and reviewed based on the criteria. Such (...) studies have been discerned and synthesized in order to establish a complete and accurate understanding of the phenomenon. Finally, a comprehensive and relatively rich conceptual framework has been proposed for future empirical explanations in various social settings via prospective research. (shrink)

“Small” large cardinal notions in the language of ZFC are those large cardinal notions that are consistent with V = L. Besides their original formulation in classical set theory, we have a variety of analogue notions in systems of admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics and recursive ordinal notations (as used in proof theory). On the face of it, it is surprising that such distinctively set-theoretical notions have analogues in such disaparate (...) and relatively constructive contexts. There must be an underlying reason why that is possible (and, incidentally, why “large” large cardinal notions have not led to comparable analogues). My long term aim is to develop a common language in which such notions can be expressed and can be interpreted both in their original classical form and in their analogue form in each of these special constructive and semi-constructive cases. This is a program in progress. What is done here, to begin with, is to show how that can be done to a considerable extent in the settings of classical and admissible set theory (and thence, admissible recursion theory). The approach taken here is to expand the language of set theory to allow us to talk about (possibly partial) operations applicable both to sets and to operations and to formulate the large cardinal notions in question in terms of operational closure conditions; at the same time only minimal existence axioms are posited for sets. The resulting system, called Operational Set Theory, is a partial adaptation to the set-theoretical framework of the explicit mathematics framework Feferman (1975). The.. (shrink)

Set theory is a branch of mathematics with a special subject matter, the infinite, but also a general framework for all modern mathematics, whose notions figure in every branch, pure and applied. This Element will offer a concise introduction, treating the origins of the subject, the basic notion of set, the axioms of set theory and immediate consequences, the set-theoretic reconstruction of mathematics, and the theory of the infinite, touching also on selected topics from higher set theory, controversial axioms (...) and undecided questions, and philosophical issues raised by technical developments. (shrink)

We show that one can solve Post's Problem by constructing generic sets in the usual set theoretic framework applied to tiny universes. This method leads to a new class of recursively enumerable sets: r.e. generic sets. All r.e. generic sets are low and simple and therefore of Turing degree strictly between 0 and 0'. Further they supply the first example of a class of low recursively enumerable sets which are automorphic in the lattice E of recursively enumerable sets with (...) inclusion. We introduce the notion of a promptly simple set. This describes the essential feature of r.e. generic sets with respect to automorphism constructions. (shrink)

Research on collective emotion spans social sciences, psychology and philosophy. There are detailed case studies and diverse theories of collective emotion. However, experimental evidence regarding the universal characteristics, antecedents and consequences of collective emotion remains sparse. Moreover, current research mainly relies on emotion self-reports, accounting for the subjective experience of collective emotion and ignoring their cognitive and physiological bases. In response to these challenges, we argue for experimental research on collective emotion. We start with an overview of theoretical frameworks (...) to identify a set of three characteristics of collective emotion. Based on research in cognitive and affective sciences, we then examine the corresponding candidate mechanisms. Finally, we highlight outstanding questions, review experimental evidence, and suggest ideas for future experimental research. (shrink)

Quillen [17] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success and—as, e.g., the work of Voevodsky on the homotopy theory of schemes [15] or the work of Joyal [11, 12] and Lurie [13] on quasicategories seem to indicate—it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematical logic, (...) inspired by the groupoid model of (intensional) Martin–Löf type theory [14] due to Hofmann and Streicher [9]. In particular, we show that a form of Martin–Löf type theory can be soundly modelled in any model category. This result indicates moreover that any model category has an associated “internal language” which is itself a form of Martin-Löf type theory. This suggests applications both to type theory and to homotopy theory. Because Martin–Löf type theory is, in one form or another, the theoretical basis for many of the computer proof assistants currently in use, such as Coq and Agda (cf. [3] and [5]), this promise of applications is of a practical, as well as theoretical, nature. This paper provides a precise indication of this connection between homotopy theory and logic; a more detailed discussion of these and further results will be given in [20]. (shrink)

This article presents a quantitative way of modeling the interim decisions of clinical trials. While statistical approaches tend to focus on the epistemic aspects of statistical monitoring rules, often overlooking ethical considerations, ethical approaches tend to neglect key epistemic dimension. The proposal is a second-order decision theoretic framework. The framework provides means for retrospective assessment of interim decisions based on a clear and consistent set of criteria that combines both ethical and epistemic considerations. The framework is broadly (...) Bayesian and addresses a fundamental question behind many concerns about clinical trials: What does it take for an interim decision (e.g., whether to stop the trial, or continue) to be a good decision? I provide simulation illustrating the modeling of interim decisions counterfactually using the framework. (shrink)

Completed in 1983, this work culminates nearly half a century of the late Alfred Tarski's foundational studies in logic, mathematics, and the philosophy of science. Written in collaboration with Steven Givant, the book appeals to a very broad audience, and requires only a familiarity with first-order logic. It is of great interest to logicians and mathematicians interested in the foundations of mathematics, but also to philosophers interested in logic, semantics, algebraic logic, or the methodology of the deductive sciences, and to (...) computer scientists interested in developing very simple computer languages rich enough for mathematical and scientific applications. The authors show that set theory and number theory can be developed within the framework of a new, different, and simple equational formalism, closely related to the formalism of the theory of relation algebras. There are no variables, quantifiers, or sentential connectives. Predicates are constructed from two atomic binary predicates (which denote the relations of identity and set-theoretic membership) by repeated applications of four operators that are analogues of the well-known operations of relative product, conversion, Boolean addition, and complementation. All mathematical statements are expressed as equations between predicates. There are ten logical axiom schemata and just one rule of inference: the one of replacing equals by equals, familiar from high school algebra. Though such a simple formalism may appear limited in its powers of expression and proof, this book proves quite the opposite. The authors show that it provides a framework for the formalization of practically all known systems of set theory, and hence for the development of all classical mathematics. The book contains numerous applications of the main results to diverse areas of foundational research: propositional logic; semantics; first-order logics with finitely many variables; definability and axiomatizability questions in set theory, Peano arithmetic, and real number theory; representation and decision problems in the theory of relation algebras; and decision problems in equational logic. (shrink)

The extant marketing literature provides little guidance for theory development or practice with regard to questions of ethical conformity and the resulting market response. To begin to bridge this research gap, we advance a theoretical framework of ethical conformity in marketing, appealing to marketing ethics, management strategy, and sociological foundations. We set the stage for our theoretical arguments by considering the role of normative expectations related to marketing practices and behaviors held by societal constituents. Against this backdrop, (...) we propose drivers of conformity in marketing, including practices consistent with both overconformity and underconformity. The framework allows us to advance testable research propositions by which questions of ethical conformity may be explored. We conclude by suggesting additional future research needed to develop the domain, specifically in the form of empirical inquiries uncovering firm strategic decisions with ethical implications. (shrink)

This article aims to highlight the importance of organizational climate with both destructive and constructive deviance behaviour in different cultural setting with workplace as a common ground. First, we discuss the need for research in workplace deviance especially destructive and constructive deviance behaviour with the review of previous studies from deviance literature. Next, we present the importance of climate and culture with both destructive and constructive deviance by proposing relationship among them with the help of a framework. The presented (...) theoretical framework can be useful for conducting future empirical research. Finally, we present the conclusion and future research in conducting cross-national research with respect to deviance. (shrink)

There has been much discussion about how to obtain legitimacy at macro-level priority setting in health care by use of fair procedures, but how should we consider priority setting by individual clinicians or health workers at the micro-level? Despite the fact that just health care totally hinges upon their decisions, surprisingly little attention seems being paid to the legitimacy of these decisions. This paper addresses the following question: what are the conditions that have to be met in order to ensure (...) that individual claims on health care are well aligned with an overall concept of just health care? Drawing upon a distinction between individual and aggregated needs, I argue that even though we assume the legitimacy of macro-level guidelines, this legitimacy is not directly transferable to decisions at micro-level simply by adherence to the guidelines’ recommendation. Further, I argue that individual claims are subject to the formal principle of equality and the demands of vertical and horizontal equity in a way that gives context- and patient-related equity concerns precedence over equity concerns captured at the macro-level. I conclude that if we aim to achieve just health care, we need to develop a complementary framework for legitimising individual judgment of patients’ claims on health care resources. Moreover, I suggest the basic structure of such a framework. (shrink)

It is well known that, in the terminology of Moschovakis, Descriptive set theory (1980), every adequate normed pointclass closed under ∀ω has an effective version of the generalized reduction property (GRP) called the easy uniformization property (EUP). We prove a dual result: every adequate normed pointclass closed under ∃ω has the EUP. Moschovakis was concerned with the descriptive set theory of subsets of Polish topological spaces. We set up a general framework for parts of descriptive set theory and prove (...) results that have as special cases not only the just-mentioned topological results, but also corresponding results concerning the descriptive set theory of classes of structures. Vaught (1973) asked whether the class of cPCδ classes of countable structures has the GRP. It does. A cPC(A) class is the class of all models of a sentence of the form ¬∃K̄φ, where φ is a sentence of L∞ω that is in A and K̄ is a set of relation symbols that is in A. Vaught also asked whether there is any primitive recursively closed set A such that some effective version of the GRP holds for the class of cPC(A) classes of countable structures. There is: The class of cPC(A) classes of countable structures has the EUP if ω ∈ A and A is countable and primitive recursively closed. Those results and some extensions are obtained by first showing that the relevant classes of classes of structures, which Vaught showed normed, are in a suitable sense adequate and closed under ∃ω, and then applying the dual easy uniformization theorem. (shrink)

There is a small but growing community of researchers spanning a spectrum of disciplines which are united in rejecting the still dominant computationalist paradigm in favor of the enactive approach. The framework of this approach is centered on a core set of ideas, such as autonomy, sense-making, emergence, embodiment, and experience. These concepts are finding novel applications in a diverse range of areas. One hot topic has been the establishment of an enactive approach to social interaction. The main purpose (...) of this paper is to serve as an advanced entry point into these recent developments. It accomplishes this task in a twofold manner: it provides a succinct synthesis of the most important core ideas and arguments in the theoretical framework of the enactive approach, and it uses this synthesis to refine the current enactive approach to social interaction. A new operational definition of social interaction is proposed which not only emphasizes the cognitive agency of the individuals and the irreducibility of the interaction process itself, but also the need for jointly co-regulated action. It is suggested that this revised conception of ‘socio-cognitive interaction’ may provide the necessary middle ground from which to understand the confluence of biological and cultural values in personal action. (shrink)

There is a small but growing community of researchers spanning a spectrum of disciplines which are united in rejecting the still dominant computationalist paradigm in favor of theenactive approach. The framework of this approach is centered on a core set of ideas, such as autonomy, sense-making, emergence, embodiment, and experience. These concepts are finding novel applications in a diverse range of areas. One hot topic has been the establishment of an enactive approach to social interaction. The main purpose of (...) this paper is to serve as an advanced entry point into these recent developments. It accomplishes this task in a twofold manner: it provides a succinct synthesis of the most important core ideas and arguments in the theoretical framework of the enactive approach, and it uses this synthesis to refine the current enactive approach to social interaction. A new operational definition of social interaction is proposed which not only emphasizes the cognitive agency of the individuals and the irreducibility of the interaction process itself, but also the need for jointly co-regulated action. It is suggested that this revised conception of ‘socio-cognitive interaction’ may provide the necessary middle ground from which to understand the confluence of biological and cultural values in personal action. (shrink)

One of the principal challenges to human survival will be for human beings, embedded in a plurality of cultural contexts, to engage with and learn from one another respectfully in the continuing task of creating a more liveable world. I argue here that theoretical psychology can contribute to setting some of the terms for this effort through the kind of conception it advances of the person as agent. I discuss broadly two philosophical perspectives toward human agency which have become (...) prominent in psychology in the last few decades, the humanistic and the hermeneutic. I argue that these two different frameworks make complementary contributions to the project of understanding ourselves and furthering the development of our humanity. 2012 APA, all rights reserved). (shrink)

'Mass terms', words like water, rice and traffic, have proved very difficult to accommodate in any theory of meaning since, unlike count nouns such as house or dog, they cannot be viewed as part of a logical set and differ in their grammatical properties. In this study, motivated by the need to design a computer program for understanding natural language utterances incorporating mass terms, Harry Bunt provides a thorough analysis of the problem and offers an original and detailed solution. An (...) extension of classical set theory, Ensemble Theory, is defined, and this provides the conceptual basis of a framework for the analysis of natural language meaning which Dr Bunt calls Two-level model-theoretic semantics. The validity of the framework is convincingly demonstrated by the formal analysis of a fragment of English including sentences with quantified and modified mass terms. Separate chapters of the book are devoted to an axiomatic definition of Ensemble Theory and a detailed discussion of its status as a mathematical formalism. (shrink)

We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending $\mathsf {ZFC}$ by using his multiverse axioms $\mathsf {MV}$ and the ‘core hypothesis’. In the first part, we examine the evidential framework for $\mathsf {MV}$, in particular the use of large cardinals and of ‘worlds’ obtained through forcing to ‘represent’ alternative extensions of $\mathsf {ZFC}$. In the second part, we address the existence and the possible features of the core of $\mathsf {MV}_T$ (...) (where T is $\mathsf {ZFC}$ +Large Cardinals). In the last part, we discuss the hypothesis that the core is Ultimate-L, and examine whether and how, based on this fact, the Core Universist can justify V=Ultimate-L as the best (and ultimate) extension of $\mathsf {ZFC}$. To this end, we take into account several strategies, and assess their prospects in the light of $\mathsf {MV}$ ’s evidential framework. (shrink)

A theoretical term gets its meaning from a set of meaning-constitutive or 'analytic' sentences of the relevant theory. The meanings of theoretical terms may change when the theories change. After a discussion of Kant and Frege, I propose a broadly Quinean view of analyticity, without adopting Quine's meaning skepticism. A sentence of a given theory in a certain language is called analytic if revising the theory so that this sentence is lost entails the abandonment of the given linguistic (...) (alternatively, of the given theoretical) framework. This solution to the problem of defining analyticity is rooted in intuitive judgements of the members of a linguistic community about the identity of languages or theories. (shrink)

The aim of this paper is to formulate and study two weak axiom systems for the conceptual framework of constructive set theory . Arithmetical CST is just strong enough to represent the class of von Neumann natural numbers and its arithmetic so as to interpret Heyting Arithmetic. Rudimentary CST is a very weak subsystem that is just strong enough to represent a constructive version of Jensenʼs rudimentary set theoretic functions and their theory. The paper is a contribution to the (...) study of formal systems for CST that capture significant stages in the development of constructive mathematics in CST. (shrink)

Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) in chaos theory, (ii) for paradoxical decompositions of solid three-dimensional objects, (iii) in the theory of effective computability (Church-Turing thesis) related to the possible “solution of supertasks,” and (iv) for weak solutions. Several approaches to set theory and their advantages and disadvatages for physical (...) applications are discussed: Canlorian “naive” (i.e., nonaxiomatic) set theory, contructivism, and operationalism. In the author's opinion, an attitude of “suspended attention” (a term borrowed from psychoanalysis) seems most promising for progress. Physical and set theoretical entities must be operationalized wherever possible. At the same time, physicists should be open to “bizarre” or “mindboggling” new formalisms, which need not be operationalizable or testable at the lime of their creation, but which may successfully lead to novel fields of phenomenology and technology. (shrink)

There is a small but growing community of researchers spanning a spectrum of disciplines which are united in rejecting the still dominant computationalist paradigm in favor of the enactive approach. The framework of this approach is centered on a core set of ideas, such as autonomy, sense-making, emergence, embodiment, and experience. These concepts are finding novel applications in a diverse range of areas. One hot topic has been the establishment of an enactive approach to social interaction. The main purpose (...) of this paper is to serve as an advanced entry point into these recent developments. It accomplishes this task in a twofold manner: it provides a succinct synthesis of the most important core ideas and arguments in the theoretical framework of the enactive approach, and it uses this synthesis to refine the current enactive approach to social interaction. A new operational definition of social interaction is proposed which not only emphasizes the cognitive agency of the individuals and the irreducibility of the interaction process itself, but also the need for jointly co-regulated action. It is suggested that this revised conception of `socio-cognitive interaction' may provide the necessary middle ground from which to understand the confluence of biological and cultural values in personal action. (shrink)

For the modern set theorist the empty set Ø, the singleton {a}, and the ordered pair 〈x, y〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building locks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and (...) Choice are broached in the formal elaboration of the ‘set of’f {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary set-theoretic concepts serves as amotif that reflects and illuminates larger and more significant developments in mathematical logic: the shift from the intensional to the extensional viewpoint, the development of type distinctions, the logical vs. the iterative conception of set, and the emergence of various concepts and principles as distinctively set-theoretic rather than purely logical. Here there is a loose analogy with Tarski's recursive definition of truth for formal languages: The mathematical interest lies mainly in the procedure of recursion and the attendant formal semantics in model theory, whereas the philosophical interest lies mainly in the basis of the recursion, truth and meaning at the level of basic predication. Circling back to the beginning, we shall see how central the empty set, the singleton, and the ordered pair were, after all. (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)

The dominant school of logic, semantics, and the foundation of mathematics construct its theories within the framework of set theory. There are three strategies by means of which a member of this school might attempt to justify his ontology of sets. One strategy is to show that sets are already included in the naturalistic part of our everyday ontology. If they are, then one may assume that whatever justifies the everyday ontology justifies the ontology of sets. Another strategy is (...) to show that set theory is already part of logic. In this case, the ontology of sets would be justified in the sam way logic is justified. The third strategy is to show that set theory plays some unique role in theoretical work. If it does, then its ontology would be justified pragmatically. In this paper it is shown that none of these strategies is successful. One properly constructs foundations, not within set theory. bit within an intensional logic that takes properties, relations, propositions as basic. (shrink)

In this paper, I analyze how frameworks shape historiographical explanations. I argue that, in order to identify a sequence of events as relevant to a historical outcome, assumptions about the workings of the relevant domain have to be made. By extending Lakatosian considerations, I argue that these assumptions are provided by a framework that contains a set of factors and intertwined principles that (supposedly) govern how a historical phenomenon works. I connect frameworks with a counterfactual account of historical explanation. (...) Frameworks enable us to explain the past by providing a backbone of explanatory patterns of counterfactual dependency. I conclude by noting that both counterfactual scenarios and scenarios of the future require frameworks and, therefore, historiographical explanation generates a set of possible futures. Analyzing these possible futures enables us to reveal the theoretical commitments of historiography. (shrink)

BackgroundThe idea that individuals are responsible for their health has been the focus of debate in the theoretical literature and in its concrete application to healthcare policy in many countries. Controversies persist regarding the form, substance and fairness of allocating health responsibility to the individual, particularly in universal, need-based healthcare systems.ObjectiveTo examine how personal health responsibility has been framed and rationalised in Norwegian key policy documents on priority setting.MethodsDocuments issued or published by the Ministry of Health and Care Services (...) between 1987 and 2018 were thematically analysed. We developed a predefined conceptual framework that guided the analysis. The framework included: the subject and object of responsibility, the level of conceptual abstraction, temporality, normative justificatory arguments and objections to the application of personal health responsibility.ResultsAs an additional criterion, personal health responsibility has been interpreted as relevant if: the patient’s harmful behaviour is repeated after receiving treatment, and if the success of the treatment is conditional on the patient’s behavioural change. When discussed as a retrospective criterion, considerations of reciprocal fairness have been dominant. When discussed as a prospective criterion, the expected benefit of treatment justified its relevance.ConclusionPersonal health responsibility appears to challenge core values of equality, inclusion and solidarity in the Norwegian context and has been repeatedly rejected as a necessary criterion for priority setting. However, the responsibility criterion seems to have some relevance in particular priority setting decisions. (shrink)

Utilizing Carnap's notions of "questions [or assertions] within a framework" and "questions [or assertions] about a framework" and his account of A-truth (analyticity in the broad sense), a theory of the ontological status of entities--in particular, that of theoretical entities--is adumbrated. In addition to the usual L-rules, each conceptual framework considered embodies a set of sentences whose truth value is quickly decidable on other than purely linguistic grounds, a set of A-true formulae, and a set of (...) rules for the confirmation and disconfirmation of non-L-true, non-A-true formulae which are not quickly decidable. Criteria which acceptable sets of A-true formulae must fulfill are given. Using this apparatus, a schema for a realistic interpretation of scientific theories is outlined. (shrink)

Ernst Friedrich Ferdinand Zermelo transformed the set theory of Cantor and Dedekind in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization setting out generative set-existence principles. Zermelo thereby tempered the ontological thrust of early set theory, initiated the delineation of what is to be regarded as set-theoretic, drawing out the combinatorial aspects from the logical, and established the basic conceptual framework for the development of modern set theory. (...) Two decades later Zermelo promoted a distinctive cumulative hierarchy view of models of set theory and championed the use of infinitary logic, anticipating broad modern developments. In this paper Zermelo's published mathematical work in set theory is described and analyzed in its historical context, with the hindsight afforded by the awareness of what has endured in the subsequent development of set theory. Elaborating formulations and results are provided, and special emphasis is placed on the to and fro surrounding the Schröder-Bernstein Theorem and the correspondence and comparative approaches of Zermelo and Gödel. Much can be and has been written about philosophical and biographical issues and about the reception of the Axiom of Choice, and we will refer and defer to others, staying the course through the decidedly mathematical themes and details. (shrink)

Kurt Gödel with his work on the constructible universeLestablished the relative consistency of the Axiom of Choice and the Continuum Hypothesis. More broadly, he ensured the ascendancy of first-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of set theoretic (...) constructions and speculated about how problems might be settled with new axioms. We here chronicle this development from the point of view of the evolution of set theory as a field of mathematics. Much has been written, of course, about Gödel's work in set theory, from textbook expositions to the introductory notes to his collected papers. The present account presents an integrated view of the historical and mathematical development as supported by his recently published lectures and correspondence. Beyond the surface of things we delve deeper into the mathematics. What emerges are the roots and anticipations in work of Russell and Hilbert, and most prominently the sustained motif of truth as formalizable in the “next higher system”. We especially work at bringing out how transforming Gödel's work was for set theory. It is difficult now to see what conceptual and technical distance Gödel had to cover and how dramatic his re-orientation of set theory was. (shrink)