Domain extension in mathematics occurs whenever a given mathematical domain is augmented so as to include new elements. Manders argues that the advantages of important cases of domain extension are captured by the model-theoretic notions of existential closure and model completion. In the specific case of domain extension via ideal elements, I argue, Manders’s proposed explanation does not suffice. I then develop and formalize a different approach to domain extension based on (...) Dedekind’s Habilitationsrede, to which Manders’s account is compared. I conclude with an examination of three possible stances towards extensions via ideal elements. (shrink)

Institute of Theoretical Astronomy, University of Cambridge, Cambridge, UK 3fS. received 28th August 1970, in final revised form 1st July 1971..

"La Nouvelle Alliance (1979), by physicist Ilya Prigogine and philosopher Isabelle Stengers, is both a book on the history and philosophy of thermodynamics and a philosophical essay on man's place in nature. It has given rise to lively interdisciplinary controversy.".

Bloom provides a masterful synthesis of recent advances in word-learning, placing them within the framework of abiding theoretical issues. I will augment and challenge his approach by underscoring the significance of word extension for questions concerning (a) the origin and evolution of infants' expectations, and (b) domain-specificity in word-learning.

Ludwig Wittgenstein’s practical incursions on the domain of art were many and well known. It is worth drawing attention to the design that he did together with Paul Engelmann for his sister Margarethe Stonborough-Wittgenstein’s house and the bust he made for, and was inspired by, the sculptor Michael Drobil. To attribute just an anecdotal character to Wittgenstein’s few artistic projects is a misunderstanding. The Austrian philosopher devoted himself to them with the fervor and rigor that characterize his philosophical writings. (...) He took care of every detail in his sister’s house, which sometimes ended up in difficult changes, although quite minimal (as when he had the ceilings lowered a few centimeters just before .. (shrink)

Assuming the indefinite extensibility of any domain of quantification leads to reasoning with extensible domain semantics. It is showed that some theorems (e.g. Thomson's) in conventional semantics logic are not theorems in a logic provided with this new semantics.

Ludwig Wittgenstein gave priority to aesthetics over other disciplines due to its invaluable capacities for revealing certain aspects of the nature of human understanding and for guiding our actions toward an ethical life. Although Wittgenstein did not focus on these issues in a systematic way, these worries were present in his philosophy during his lifetime. That is why I use a very wide range of his writings, from the Tractatus to letters and diaries. Aesthetic inquiries can throw light upon the (...) existence of an intransitive understanding–that is, a kind of knowledge that cannot be expressed by language shared by all competent users of a language. I also discuss Wittgenstein’s conception of ethics in order to explain the close relationship that he proposed between ethics and aesthetics. I hope this will clarify why he believed there was so much to learn from paintings, films, music, poems, or novels. I compare works of art and religious belief and how the former can help us to guide our lives as the latter does. I do this while extending the scope and claiming the importance of aesthetic education, given that it can help us to deal with our nature and with our history. (shrink)

Extensive measurement is the standard measurement-theoretic approach for constructing a ratio scale. It involves the comparison of objects that can be concatenated in an additively representable way. This paper studies the implications of extensively measurable welfare for social choice theory. We do this in two frameworks: an Arrovian framework with a fixed population and no interpersonal comparisons, and a generalized framework with variable populations and full interpersonal comparability. In each framework we use extensive measurement to introduce novel domain restrictions, (...) independence conditions, and constraints on social evaluation. We prove a welfarism theorem for these domains and characterize the social welfare functions that satisfy the axioms of extensive measurement at both the individual and social levels. The main results are simple axiomatizations of strong dictatorship in the Arrovian framework and classical utilitarianism in the generalized framework. (shrink)

Philosophical theories of various sorts rely on there being robust boundaries between kinds of content. One way of drawing such boundaries is to place them between subject matters, like physics and aesthetics, and the domains of sentences falling within them. Yet contemporary literature exploring the nature of discourse domains is relatively sparse. The goal of this paper is to articulate the core features of discourse domains for them to provide the sought-after explanatory utility of establishing robust boundaries between discursive contents. (...) Analyzing the role that discourse domains have under alethic theories yields valuable information about the ways in which domains subject themselves to being defined and how alethic theories can explain the variability of truth-aptness or truth across sentences from distinct domains. The concluding argument is that because of certain issues with defining domains as unambiguous classes of sentences when individuated on the grounds of topical subject matters, philosophers should consider a commitment to indeterminism about the extensions of fundamental domains. According to this view, although domains can be defined as relatively well-individuated classes of sentences based on topical distinctions, the temporal development of our conceptual frameworks and the phenomenon of mixed content compromise our ability to definitively account for the domain membership of all truth-apt sentences. Such an indeterminacy argument is relevant for all who rely on there being robust boundaries between topically individuated discursive contents. (shrink)

We argue that a comprehensive theory of reciprocals must rely on a general taxonomy of restrictions on the interpretation of relational expressions. Developing such a taxonomy, we propose a new principle for interpreting reciprocals that relies on the interpretation of the relation in their scope. This principle, the Maximal Interpretation Hypothesis (MIH), analyzes reciprocals as partial polyadic quantifiers. According to the MIH, the partial quantifier denoted by a reciprocal requires the relational expression REL in its scope to denote a maximal (...) relation in REL’s interpretation domain. In this way the MIH avoids a priori assumptions on the available readings of reciprocal expressions, which are necessary in previous accounts. Relying extensively on the work of Dalrymple et al. (Ling Philos 21:159–210, 1998), we show that the MIH also exhibits some observational improvements over Dalrymple et al.’s Strongest Meaning Hypothesis (SMH). In addition to deriving some attested reciprocal interpretations that are not expected by the SMH, the MIH offers a more restrictive account of the way context affects the interpretation of reciprocals through its influence on relational domains. Further, the MIH generates a reciprocal interpretation at the predicate level, which is argued to be advantageous to Dalrymple et al.’s propositional selection of reciprocal meanings. More generally, we argue that by focusing on restrictions on relational domains, the MIH opens the way for a more systematic study of the ways in which lexical meaning, world knowledge and contextual information interact with the interpretation of quantificational expressions. (shrink)

As the National Information Infra- structure develops new avenues for information products and services will open. Creating, identifying and protecting the information market space is a critical component to the success of information product and service developments. As a result, the producers of those products and service seek to protect their proprietary interest in the underlying information. However, these actions have broader consequences: Attempts to extend legal protection to basic facts and other public domain information demonstrate that the public (...) information space is slowly being reduced. Reviewing several information controversies as case studies is fruitful. Not only for assessing current trends in information disputes (warfare) but can form the basis for establishing several predictors useful in determining when future information ownership controversies may develop and result in the loss of public information space. One set of predictors helps define the information controversy and is descriptive. The second set of predictors characterizes the marketplace environment. It provides further nuance to the economic factors that force information owners to seek legal remedy. Identifying instances where both models suggest a change in information rights or ownership is most likely to result in the critical loss of access to public information space. (shrink)

Each science has its own domain of investigation, but one and the same science can be formalized in different languages with different universes of discourse. The concept of the domain of a science and the concept of the universe of discourse of a formalization of a science are distinct, although they often coincide in extension. In order to analyse the presuppositions and implications of choices of domain and universe, this article discusses the treatment of omega arguments (...) in three very different formalizations of arithmetic. In Peano's formalization the domain is a restricted class of individuals, while the universe of discourse is the unrestricted class of all individuals. In Gödel's formalization the domain is a restricted class of individuals as in Peano's formalization, but the universe of discourse coincides with the domain. In Whitehead-Russell's formalization the domain is a class of logical notions in Tarski's sense, that are necessarily not individuals, whereas the universe of discourse is the unrestricted class of individuals as in Peano's formalization. The present approach emphasizes the viewpoint that the universe of discourse of a given discourse is important in determining which propositions are expressed by which sentences. (shrink)

This study set out to investigate intellectual domains as well as the use of measurement and validation methods in language assessment research and second language acquisition published in English in peer-reviewed journals. Using Scopus, we created two datasets: a dataset of core journals consisting of 1,561 articles published in four language assessment journals, and a dataset of general journals consisting of 3,175 articles on language assessment published in the top journals of SLA and applied linguistics. We applied document co-citation analysis (...) to detect thematically distinct research clusters. Next, we coded citing papers in each cluster based on an analytical framework for measurement and validation. We found that the focus of the core journals was more exclusively on reading and listening comprehension assessment, facets of speaking and writing performance such as raters and validation, as well as feedback, corpus linguistics, and washback. By contrast, the primary focus of assessment research in the general journals was on vocabulary, oral proficiency, essay writing, grammar, and reading. The secondary focus was on affective schemata, awareness, memory, language proficiency, explicit vs. implicit language knowledge, language or semantic awareness, and semantic complexity. With the exception of language proficiency, this second area of focus was absent in the core journals. It was further found that the majority of citing publications in the two datasets did not carry out inference-based validation on their instruments before using them. More research is needed to determine what motivates authors to select and investigate a topic, how thoroughly they cite past research, and what internal and external factors lead to the sustainability of a Research Topic in language assessment. (shrink)

This essay clarifies quantifier variance and uses it to provide a theory of indefinite extensibility that I call the variance theory of indefinite extensibility. The indefinite extensibility response to the set-theoretic paradoxes sees each argument for paradox as a demonstration that we have come to a different and more expansive understanding of ‘all sets’. But indefinite extensibility is philosophically puzzling: extant accounts are either metasemantically suspect in requiring mysterious mechanisms of domain expansion, or metaphysically suspect in requiring nonstandard assumptions (...) about mathematical objects. Happily, the view of quantifier meanings that underwrites quantifier variance can be used to provide an account of indefinite extensibility that is both metasemantically and metaphysically satisfying. Section 1 introduces the puzzle of indefinite extensibility; section 2 develops and clarifies the metasemantics of quantifier variance; section 3 solves section 1's puzzle of indefinite extensibility by applying section 2's account of quantifier meanings; and section 4 compares the theory developed in section 3 to several other theories in the literature. (shrink)

As an important component to promote the development of affective brain–computer interfaces, the study of emotion recognition based on electroencephalography (EEG) has encountered a difficult challenge; the distribution of EEG data changes among different subjects and at different time periods. Domain adaptation methods can effectively alleviate the generalization problem of EEG emotion recognition models. However, most of them treat multiple source domains, with significantly different distributions, as one single source domain, and only adapt the cross-domain marginal distribution (...) while ignoring the joint distribution difference between the domains. To gain the advantages of multiple source distributions, and better match the distributions of the source and target domains, this paper proposes a novel multi-source joint domain adaptation (MSJDA) network. We first map all domains to a shared feature space and then align the joint distributions of the further extracted private representations and the corresponding classification predictions for each pair of source and target domains. Extensive cross-subject and cross-session experiments on the benchmark dataset, SEED, demonstrate the effectiveness of the proposed model, where more significant classification results are obtained on the more difficult cross-subject emotion recognition task. (shrink)

This paper reassesses the domain of the business ethics curriculum and, drawing on recent shifts in the business environment, maps out some suggestions for extending the core ground of the discipline. It starts by assessing the key elements of the dominant English- language business ethics textbooks and identifying the domain as reflected by those publications as where the law ends and beyond the legal minimum. Based on this, the paper identifies potential gaps and new areas for the discipline (...) by drawing on four main aspects. First, it argues that the domain of business ethics requires extensions dependent on the particular geographic region where the subject is taught. A second factor for broadening the scope is the impact of recent scandals, which arguably direct the focus of ethical inquiry towards the nature of the business systems in which individuals and corporations operate. Third, the impact of globalization and its result on growing corporate involvement in regulatory processes is discussed. Fourth and finally, business ethics reaches beyond the traditional constituencies of economic stakeholders as new actors from civil society enter the stage of ethical decision making in business. We conclude by suggesting that a reconsideration of the domain of business ethics, especially in Europe, is timely, but that to do so represents a major challenge to business ethics educators. (shrink)

This paper reassesses the domain of the business ethics curriculum and, drawing on recent shifts in the business environment, maps out some suggestions for extending the core ground of the discipline. It starts by assessing the key elements of the dominant English-language business ethics textbooks and identifying the domain as reflected by those publications as 'where the law ends' and 'beyond the legal minimum'. Based on this, the paper identifies potential gaps and new areas for the discipline by (...) drawing on four main aspects. First, it argues that the domain of business ethics requires extensions dependent on the particular geographic region where the subject is taught. A second factor for broadening the scope is the impact of recent scandals, which arguably direct the focus of ethical inquiry towards the nature of the business systems in which individuals and corporations operate. Third, the impact of globalization and its result on growing corporate involvement in regulatory processes is discussed. Fourth and finally, business ethics reaches beyond the traditional constituencies of 'economic stakeholders' as new actors from civil society enter the stage of ethical decision making in business. We conclude by suggesting that a reconsideration of the domain of business ethics, especially in Europe, is timely, but that to do so represents a major challenge to business ethics educators. (shrink)

We give an extension of the Jónsson‐Tarski representation theorem for both normal and non‐normal modal algebras so that it preserves countably many infinite meets and joins. In order to extend the Jónsson‐Tarski representation to non‐normal modal algebras we consider neighborhood frames instead of Kripke frames just as Došen's duality theorem for modal algebras, and to deal with infinite meets and joins, we make use of Q‐filters, which were introduced by Rasiowa and Sikorski, instead of prime filters. By means of (...) the extended representation theorem, we show that every predicate modal logic, whether it is normal or non‐normal, has a model defined on a neighborhood frame with constant domains, and we give a completeness theorem for some predicate modal logics with respect to classes of neighborhood frames with constant domains. Similarly, we show a model existence theorem and a completeness theorem for infinitary modal logics which allow conjunctions of countably many formulas. (shrink)

The structuralist theory of truth approximation essen-tially deals with truth approximation by theory revision for a fixed domain. However, variable domains can also be taken into account, where the main changes concern domain extensions and restrictions. In this paper I will present a coherent set of definitions of “more truth-likeness”, “empirical progress” and “truth approximation” due to a revision of the domain of intended applications. This set of definitions seems to be the natural counterpart of the basic (...) definitions of similar notions as far as theory revision is concerned. The formal aspects of theory revision strongly suggest an analogy between truth approximation and design research, for example, drug research. Whereas a new drug may be better for a certain disease than an old one, a certain drug may be better for another disease than for the original target disease, a phenomenon which was nicely captured by the title of a study by Rein Vos [1991]: Drugs Looking for Diseases. Similarly, truth approximation may not only take the shape of theory revision but also of domain revision, naturally suggesting the phenomenon of “Theories looking for domains”. However, whereas Vos documented his title with a number of examples, so far, apart from plausible cases of “truth accumulation by domain extension”, I did not find clear-cut empirical instantiations of the analogy, only, as such, very interesting, non-empirical examples. (shrink)

The qualitative nature of co-deployed or “associated” metaphorical source domains in discourse has been extensively researched, often in terms of whether they share common conceptual roots. There are however few empirical studies on the strength of these associations and their implications. In mental healthcare activities like psychological interviews and counseling, for example, strongly associated sources may suggest unique “conceptual assemblages” that highlight underexplored (dis)similarities between clients or client groups, beyond the typical focus on isolated sources and frequencies. Our case study (...) compares source domain associations produced by two interviewee groups – one meeting the diagnostic threshold for acute stress disorder (ASD), the other not – relating their first-hand experiences of traumatic events during the 2019 Hong Kong social unrest. The machine learning method of association rule mining is used to extract and quantify the top association rules in both groups. Results show i) a shared rule suggesting a highly schematic construal of trauma that nevertheless varies in instantiating details, and ii) several distinct rules that corroborates and lends further insight into the “agentive vs. non-agentive” characteristics exhibited by ASD vs. non-ASD individuals. Implications and future directions, including potential extensions to other discourse contexts, are discussed. (shrink)

In a widely cited article Boyle suggests that a movement against the growing propertization of knowledge should develop a mobilization frame centered around the idea of the public domain. Based on an analysis of the framing strategies in the two most important protest mobilizations against extensive intellectual property rights I discuss whether and to which degree these movements did actually put the concept of the public domain at the center of their argumentation. The article uses political claims analysis (...) and discourse analysis to show that the actual framing strategies relied on other frames. It closes with a discussion, explaining why the idea of the public domain is essentially a defensive concept with a limited potential for movements that primarily address the production of knowledge. (shrink)

This engaging, interactive and pedagogical introduction to ethics combines the best features of a textbook and an anthology. The Moral Domain: Guided Readings in Philosophical and Literary Texts contains numerous readings from key philosophical writings in ethics along with captivating literary selections that bring the ethical issues to life. Offering extensive excerpts from major figures in the history of Western ethics--Aquinas, Aristotle, Hobbes, Hume, Kant, Mill and Plato--the book also integrates work from non-Western perspectives, including selections from the Bhagavad (...) Gita, Confucian views and Hsun-Tzu. It also represents women's voices with readings by Julia Annas, Sarah Broadie, Carol Gilligan, Martha Nussbaum and others. Literary selections--including work from the Bible, Camus, Dostoevsky, Golding, Sophocles, Tolstoy, Twain and Wharton--enable students to grasp deep ethical concepts at an intuitive level. The Moral Domain features a unique built-in study guide that helps students to better comprehend and interact with the material. It introduces each selection with orienting questions and then intersperses explanations, commentary and study questions (designed to test comprehension and provoke reflection) throughout the readings. Each chapter includes a "Further Discussion and Applications" section that demonstrates how ethical theory affects such contemporary moral debates and problems as abortion, euthanasia, feminism, hunger, warfare and more. An exemplary text for introduction to ethics and moral philosophy courses, The Moral Domain provides a comprehensive and accessible introduction to all facets of ethics; its foundations, history, debates and current real-life controversies. (shrink)

This paper presents a new logical analysis of quantity comparatives (i.e. More linguists than philosophers came to the party.) within the Delineation Semantics approach to gradability and comparison (McConnell-Ginet in Comparison constructions in English. PhD thesis, University of Rochester, Rochester, 1973; Kamp in Formal semantics of natural language. Cambridge University Press, Cambridge, 1975; Klein in Linguist Philos 4:1–45, 1980) among many others. Along with the Degree Semantics framework (Cresswell in Montague grammar. Academic Press, New York, 1976; von Stechow in J (...) Semant 3:1–77, 1984; Kennedy in Projecting the adjective. PhD thesis. University of California, Santa Cruz, 1997, among many others) Delineation Semantics is one of the dominant logical frameworks for analyzing the meaning of gradable constituents of the adjectival syntactic category; however, there has been very little work done investigating the application of this framework to the analysis of gradability outside the adjectival domain. This state of affairs distinguishes the Delineation Semantics framework from its Degree Semantics counterpart, where such questions have been investigated in great deal since the beginning of the twenty-first century. Nevertheless, it has been observed [for example, by Doetjes (Seventeenth amsterdam colloquium. University of Amsterdam, Amsterdam, 2011); van Rooij (The vagueness handbook. Springer, Dordrecht, 2011c)] that there is nothing inherently adjectival about the way that the interpretations of scalar predicates are calculated in Delineation Semantics, and therefore that there is enormous potential for this approach to shed light on the nature of gradability and comparison in the nominal and verbal domains. This paper is a first contribution to realizing this potential within a Mereological extension of a simple version of the DelS system. (shrink)

The continuum form of the Gauss-Hertz principle is extended to include the time domain as well as space. The Schrödinger equation and general relativity are derived by this method. The equivalence of the principle is shown to that of the Hamiltonian method where the energy is the expression −[φ∇ 2 φ+A·∇2 A], with φ being the difference between the acceleration potential and potential energy density, andA being the difference between the vector potentials of the acceleration field and the force (...) field. The goal of Hertz to “demonstrate a third arrangement of the principles of mechanics...which starts with... time, space and mass” has apparently been achieved for relativity and for quantum mechanics, in addition to those classical equations previously found. (shrink)

Certain extensions of Nelson's constructive logic N with strong negation have recently become important in arti.cial intelligence and nonmonotonic reasoning, since they yield a logical foundation for answer set programming (ASP). In this paper we look at some extensions of Nelson's .rst-order logic as a basis for de.ning nonmonotonic inference relations that underlie the answer set programming semantics. The extensions we consider are those based on 2-element, here-and-there Kripke frames. In particular, we prove completeness for .rst-order here-and-there logics, and their (...) minimal strong negation extensions, for both constant and varying domains. We choose the constant domain version, which we denote by QNc5, as a basis for de.ning a .rst-order nonmonotonic extension called equilibrium logic. We establish several metatheoretic properties of QNc5, including Skolem forms and Herbrand theorems and Interpolation, and show that the .rst-oder version of equilibrium logic can be used as a foundation for answer set inference. (shrink)

We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space without isolated points, X, in such a way that every sequentially continuous function from Cantor space to ℤ extends to a sequentially continuous function from X to ℝ. The second asserts an analogous property for Baire space relative to any inhabited locally non-compact CSM. Both results rely on having (...) careful constructive formulations of the concepts involved.As a first application, we derive new relationships between “continuity principles” asserting that all functions between specified metric spaces are pointwise continuous. In particular, we give conditions that imply the failure of the continuity principle “all functions from X to ℝ are continuous”, when X is an inhabited CSM without isolated points, and when X is an inhabited locally non-compact CSM. One situation in which the latter case applies is in models based on “domain realizability”, in which the failure of the continuity principle for any inhabited locally non-compact CSM, X, generalizes a result previously obtained by Escardó and Streicher in the special case X = C[0, 1].As a second application, we show that, when the notion of inhabited complete separable metric space without isolated points is interpreted in a recursion-theoretic setting, then, for any such space X, there exists a Banach-Mazur computable function from X to the computable real numbers that is not Markov computable. This generalizes a result obtained by Hertling in the special case that X is the space of computable real numbers. (shrink)

We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X, in such a way that every sequentially continuous function from Cantor space to ℤ extends to a sequentially continuous function from X to ℝ. The second asserts an analogous property for Baire space relative to any inhabited locally non‐compact CSM. Both results rely on (...) having careful constructive formulations of the concepts involved.As a first application, we derive new relationships between “continuity principles” asserting that all functions between specified metric spaces are pointwise continuous. In particular, we give conditions that imply the failure of the continuity principle “all functions from X to ℝ are continuous”, when X is an inhabited CSM without isolated points, and when X is an inhabited locally non‐compact CSM. One situation in which the latter case applies is in models based on “domain realizability”, in which the failure of the continuity principle for any inhabited locally non‐compact CSM, X, generalizes a result previously obtained by Escardó and Streicher in the special case X = C[0, 1].As a second application, we show that, when the notion of inhabited complete separable metric space without isolated points is interpreted in a recursion‐theoretic setting, then, for any such space X, there exists a Banach‐Mazur computable function from X to the computable real numbers that is not Markov computable. This generalizes a result obtained by Hertling in the special case that X is the space of computable real numbers. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

Dummctt argues that classical quantification is illegitimate when the domain is given as the objects which fall under an indefinitely extensible concept, since in such cases the objects are not the required definite totality. The chief problem in understanding this complex argument is the crucial but unexplained phrase 'definite totality' and the associated claim that it follows from the intuitive notion of set that the objects over which a classical quantifier ranges form a set. 'Definite totality' is best understood (...) as disguised plural talk like Cantor's 'consistent multiplicity', although this does not help in understanding how a totality could be anything other than definite. Moreover, contrary to his claims, Dummett's own notion of set is not intuitive and he does not demystify the set-theoretic paradoxes. In conclusion, it is argued that Dummett's context principle is responsible for the incoherent projection of the haziness of a conception of some objects onto reality. (shrink)

Hermeneutic rationality arises from the idea that experience is a cumulative process, in which differences are not eliminated but preserved. The universality which derives from this process is an “intensional universality”, which follows a law of direct proportionality between extension and intension: the more an educated individual enriches her experiences, the more she is able to universalize her understanding of others. Experience is then inevitably open and never closed, that is, free for other experiences. If we use the word (...) “domain” to describe the condition of a closed and predetermined system or ontology, the entities of which are pre-assumed, we can therefore call this open universality a “universality without domain”.This description of hermeneutical experience leads to a question about its logical and ontological presuppositions. In the second part of the paper I refer to a particular practice which is central for hermeneutical theory in order to show how I understand these presuppositions: the practice of translation. In translation we experience the impossibility of rendering completely the sense of the source language in a target language, that is, we experience a difference which cannot be suppressed. I compare this experience to the discovery of the incommensurable magnitudes in ancient Greek mathematics, a discovery that shook Parmenidean and Pythagorean ontology: in my opinion, philosophical hermeneutics more or less implicitly develops the consequences of this incredible discovery, which marks the passage to a differential, comparative and integrative rationality and ontology. (shrink)

We consider the domain of applicability of general relativity (GR), as a classical theory of gravity, by considering its applications to a variety of settings of physical interest as well as its relationship with real observations. We argue that, as it stands, GR is deficient whether it is treated as a microscopic or a macroscopic theory of gravity. We briefly discuss some recent attempts at removing this shortcoming through the construction of a macroscopic theory of gravity. We point out (...) that such macroscopic extensions of GR are likely to be nonunique and involve non-Riemannian geometrical frameworks. (shrink)

In this paper, I will argue for a new account of presuppositions which is based on double indexing as well as minimal representational contexts providing antecedent material for anaphoric presuppositions, rather than notions of context defined in terms of the interlocutors’ pragmatic presuppositions or the information accumulated from the preceding discourse. This account applies in particular to new phenomena concerning the presupposition of quantifier domains. But it is also intended to be an account of presuppositions in general. The account differs (...) from the Satisfaction Theory and the Binding Theory of presuppositions in that it can be viewed as a conservative extension of traditional static semantics and in that it does not involve the notion of pragmatic presupposition. (shrink)

This paper develops an idea of saving ordinary uses of vague predicates from the Sorites by means of domain restriction. A tolerance level for a predicate, along a dimension, is a difference with respect to which the predicate is semantically insensitive. A central gap for the predicate+dimension in a domain is a segment of an associated scale, larger than this difference, where no object in the domain has a measure, and such that the extension of the (...) predicate has measures on one side of the gap and the anti-extension on the other. The domain restriction imposes a central gap. (shrink)

Dummctt argues that classical quantification is illegitimate when the domain is given as the objects which fall under an indefinitely extensible concept, since in such cases the objects are not the required definite totality. The chief problem in understanding this complex argument is the crucial but unexplained phrase 'definite totality' and the associated claim that it follows from the intuitive notion of set that the objects over which a classical quantifier ranges form a set. 'Definite totality' is best understood (...) as disguised plural talk like Cantor's 'consistent multiplicity', although this does not help in understanding how a totality could be anything other than definite. Moreover, contrary to his claims, Dummett's own notion of set is not intuitive and he does not demystify the set-theoretic paradoxes. In conclusion, it is argued that Dummett's context principle is responsible for the incoherent projection of the haziness of a conception of some objects onto reality. (shrink)

We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X, in such a way that every sequentially continuous function from Cantor space to ℤ extends to a sequentially continuous function from X to ℝ. The second asserts an analogous property for Baire space relative to any inhabited locally non‐compact CSM. Both results rely on (...) having careful constructive formulations of the concepts involved.As a first application, we derive new relationships between “continuity principles” asserting that all functions between specified metric spaces are pointwise continuous. In particular, we give conditions that imply the failure of the continuity principle “all functions from X to ℝ are continuous”, when X is an inhabited CSM without isolated points, and when X is an inhabited locally non‐compact CSM. One situation in which the latter case applies is in models based on “domain realizability”, in which the failure of the continuity principle for any inhabited locally non‐compact CSM, X, generalizes a result previously obtained by Escardó and Streicher in the special case X = C[0, 1].As a second application, we show that, when the notion of inhabited complete separable metric space without isolated points is interpreted in a recursion‐theoretic setting, then, for any such space X, there exists a Banach‐Mazur computable function from X to the computable real numbers that is not Markov computable. This generalizes a result obtained by Hertling in the special case that X is the space of computable real numbers. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

The existence of singularities alerts that one of the highest priorities of a centennial perspective on general relativity should be a careful re-thinking of the validity domain of Einstein’s field equations. We address the problem of constructing distinguishable extensions of the smooth spacetime manifold model, which can incorporate singularities, while retaining the form of the field equations. The sheaf-theoretic formulation of this problem is tantamount to extending the algebra sheaf of smooth functions to a distribution-like algebra sheaf in which (...) the former may be embedded, satisfying the pertinent cohomological conditions required for the coordinatization of all of the tensorial physical quantities, such that the form of the field equations is preserved. We present in detail the construction of these distribution-like algebra sheaves in terms of residue classes of sequences of smooth functions modulo the information of singular loci encoded in suitable ideals. Finally, we consider the application of these distribution-like solution sheaves in geometrodynamics by modeling topologically-circular boundaries of singular loci in three-dimensional space in terms of topological links. It turns out that the Borromean link represents higher order wormhole solutions. (shrink)

Reichenbach’s early solution to the scientific problem of how abstract mathematical representations can successfully express real phenomena is rooted in his view of coordination. In this paper, I claim that a Reichenbach-inspired, ‘layered’ view of coordination provides us with an effective tool to systematically analyse some epistemic and conceptual intricacies resulting from a widespread theorising strategy in evolutionary biology, recently discussed by Okasha (2018) as ‘endogenization’. First, I argue that endogenization is a form of extension of natural selection theory (...) that comprises three stages: quasi-axiomatisation, functional extension, and semantic extension. Then, I argue that the functional extension of one core principle of natural selection theory, namely, the principle of heritability, requires the semantic extension of the concept of inheritance. This is because the semantic extension of ‘inheritance’ is necessary to establish a novel form of coordination between the principle of heritability and the extended domain of phenomena that it is supposed to represent. Finally, I suggest that – despite the current lack of consensus on the right semantic extension of ‘inheritance’ – we can fruitfully understand the reconceptualization of ‘inheritance’ provided by niche construction theorists as the result of a novel form of coordination. (shrink)

The brain-computer interface of steady-state visual evoked potential is one of the fundamental ways of human-computer communication. The main challenge is that there may be a nonlinear relationship between different SSVEP in other states. For improving the performance of SSVEP BCI, a novel CNN algorithm model is proposed in this study. Based on the discrete Fourier transform to calculate the signal's power spectral density, we perform zero-padding in the signal's time domain to improve its performance on the PSD and (...) make it more refined. In this way, the frequency point interval in the PSD of the SSVEP is consistent with the minimum gap between the stimulation frequency. Combining the nonlinear transformation capabilities of CNN in deep learning, a zero-padding frequency domain convolutional neural network model is proposed. Extensive experiments based on the SSVEP dataset validate the effectiveness of our method. The study verifies that the proposed ZPFDCNN method can improve the effectiveness of the SSVEP-based high-speed BCI ITR. It has massive potential in the application of BCI. (shrink)