A complete theory T has the Schröder–Bernstein property or simply the SB-property if any pair of elementarily bi-embeddable models are isomorphic. This property has been studied in the discrete first-order setting and can be seen as a first step towards classification theory. This paper deals with the SB-property on continuous theories. Examples of complete continuous theories that have this property include Hilbert spaces and any completion of the theory of probability algebras. We also study a weaker notion, the SB-property up (...) to perturbations. This property holds if any two elementarily bi-embeddable models are isomorphic up to perturbations. We prove that the theory of Hilbert spaces expanded with a bounded self-adjoint operator has the SB-property up to perturbations of the operator and that the theory of atomless probability algebras with a generic automorphism have the SB-property up to perturbations of the automorphism. We also study how the SB-property behaves with respect to randomizations. Finally we prove, in the continuous setting, that if T is a strictly stable theory then T does not have the SB-property. (shrink)

This book is open access under a CC BY 4.0 license. This timely book addresses the conflict between globalism and nationalism. It provides a liberal communitarian response to the rise of populism occurring in many democracies. The book highlights the role of communities next to that of the state and the market. It spells out the policy implications of liberal communitarianism for privacy, freedom of the press, and much else. In a persuasive argument that speaks to politics today from Europe (...) to the United States to Australia, the author offers a compelling vision of hope. Above all, the book offers a framework for dealing with moral challenges people face as they seek happiness but also to live up to their responsibilities to others and the common good. At a time when even our most basic values are up for question in policy debates riddled with populist manipulation, Amitai Etzioni’s bold book creates a new frame which introduces morals and values back into applied policy questions. These questions span the challenges of jobless growth to the unanswered questions posed by the role of artificial intelligence in a wide range of daily life tasks and decisions. While not all readers will agree with the communitarian solutions that he proposes, many will welcome an approach that is, at its core, inclusive and accepting of the increasingly global nature of all societies at the same time. It is a must read for all readers concerned about the future of Western liberal democracy. Carol Graham, Leo Pasvolsky Senior Fellow, The Brookings Institution and College Park Professor/University of Maryland In characteristically lively, engaging, and provocative style Etzioni tackles many of the great public policy dilemmas that afflict us today. Arguing that we are trapped into a spiral of slavish consumerism, he proposes a form of liberal communitarian that, he suggests, will allow human beings to flourish in changing circumstances. Jonathan Wolff, Blavatnik Chair of Public Policy, Blavatnik School of Government, University of Oxford. (shrink)

Contemporary sentencing practices treat time served as the primary metric of punishment severity. The appeal of temporal sentencing is familiar: it promises legality, proportionality, transparency, and uniformity by making punishment publicly specifiable and comparable across cases. This paper argues that time alone cannot plausibly function as a legitimate metric of penal severity under conditions modern penal systems knowingly tolerate. The core claim is that stable and institutionally known variation in conditions of confinement—differences in safety, medical access, isolation, overcrowding, (...) and deprivation—systematically alters the burden of imprisonment. As a result, identical sentences often authorize substantially different punitive experiences, producing predictable proportionality failures: some offenders are punished beyond what their desert warrants, while others are punished less than they deserve. The paper develops this diagnosis by (i) reconstructing the philosophical rationales for temporal sentencing, (ii) explaining “metric failure” as the breakdown of duration’s ability to track punitive severity, and (iii) addressing the objection that confinement conditions are merely administrative matters outside sentencing theory. The conclusion is not that time is irrelevant, nor that retributive punishment must be abandoned. It is that duration cannot bear the normative weight punishment theory places on it when confinement severity varies in foreseeable, structurally tolerated ways. (shrink)

In the anti-de Sitter/conformal field theory correspondence, one often finds claims along the lines that “spacetime emerges from entanglement.” This paper argues that behind these general statements hide three distinct emergence claims about, respectively, metric, gravitational dynamics, and topological connectivity. Thus, despite being advertised with the same terminology, these results are not about the same spatiotemporal aspects. They can therefore not just be grouped as evidence for one unanimous conclusion, though they do point in similar directions. The paper also (...) investigates whether the three emergence claims satisfy two of the necessary conditions for emergence: Determination and novelty. The paper argues that none of the emergence claims satisfies the determination condition: More than entanglement is needed to furnish the emergence basis. Besides entanglement, the emergence basis for the bulk metric must include the induced metric on the boundary. Thus, this claim might not satisfy the novelty condition for emergence that the emergent should be novel as compared to the emergence basis. Likewise, the emergence basis for topological connectivity seems to include connectivity whereby this emergence claim is also questionable. The paper concludes that only gravitational dynamics is novel compared to the fully furnished emergence basis. (shrink)

This study analyzes tokenization in Natural Language Processing (NLP) at the intersection of language, technology, and political economy. Recent studies show that subword tokenization in multilingual Large Language Models (LLMs) distributes processing costs unevenly across languages, as non-Latin scripts are disproportionately fragmented by statistical segmentation. Taking this disparity as a point of departure, the study reconceptualizes tokenization beyond a purely linguistic or computational problem. The analysis approaches tokenization through the epistemological problems of translation and situates it within a longer history (...) of socio-technical infrastructures from telegraphy to machine learning that mediate between linguistic abstraction and regimes of social productivity. Rather than treating tokens as neutral technical units, it shows that they function as operational measures of linguistic value beyond meaning, enabling the quantification and automation of linguistic labor. Drawing on Marxist and structuralist accounts of linguistic value, and postcolonial critiques of semiotic standardization, the study frames tokenization as both a technical abstraction and a metric in the social division of labor. (shrink)

In the topological semantics for modal logic, S4 is well-known to be complete for the rational line, for the real line, and for Cantor space: these are special cases of S4’s completeness for any dense-in-itself metric space. The construction used to prove completeness can be slightly amended to show that S4 is not only complete, but also strongly complete, for the rational line. But no similarly easy amendment is available for the real line or for Cantor space and the (...) question of strong completeness for these spaces has remained open, together with the more general question of strong completeness for any dense-in-itself metric space. In this paper, we prove that S4 is strongly complete for any dense-in-itself metric space. (shrink)

In this paper, we focus on digital images with $\kappa$ adjacency relation. We define rectangular metric on digital images. By using the differences related to convergency, Cauchy sequence, and $\kappa$ continuity in digital images and by using rectangular property we obtain and prove Kannan-type fixed-point theorems.

This paper develops a non-metric model of causal structure based on the directed variation of a scalar field F(x)F. Departing from traditional frameworks that tie causality to spacetime metrics, we propose that causal precedence arises from internal asymmetries in F, independently of any geometric or temporal background. We formalize this idea via acyclic graphs, gradient structures, and threshold actualization, and show how regions of local stabilization in F give rise to emergent metric properties. The model supports a new (...) type of modal logic based on functional flow, with possibility and necessity defined relative to the direction of variation. We discuss the philosophical implications for structural realism, the direction of time, and the ontology of causation, and suggest that this framework offers a viable foundation for causal reasoning in background-independent physical theories such as quantum gravity. (shrink)

We show that the Axiom of Countable Choice is necessary and sufficient to prove that the existence of a Borel measure on a pseudometric space such that the measure of open balls is positive and finite implies separability of the space. In this way a negative answer to an open problem formulated in Górka (Am Math Mon 128:84–86, 2020) is given. Moreover, we study existence of maximal $$\delta $$ δ -separated sets in metric and pseudometric spaces from the point (...) of view the Axiom of Choice and its weaker forms. (shrink)

An attempt is made to remove singularities arising in general relativity by modifying it so as to take into account the existence of a fundamental rest frame in the universe. This is done by introducing a background metric γμν (in addition to gμν) describing a spacetime of constant curvature with positive spatial curvature. The additional terms in the field equations are negligible for the solar system but important for intense fields. Cosmological models are obtained without singular states but simulating (...) the “big bang.” The field of a particle differs from the Schwarzschild field only very close to, and inside, the Schwarzschild sphere. The interior of this sphere is unphysical and impenetrable. A star undergoing gravitational collapse reaches a state in which it fills the Schwarzschild sphere with uniform density (and pressure) and has the geometry of a closed Einstein universe. Any charge present is on the surface of the sphere. Elementary particles may have similar structures. (shrink)

We develop the general theory of topometric spaces, i.e., topological spaces equipped with a well-behaved lower semi-continuous metric. Spaces of global and local types in continuous logic are the motivating examples for the study of such spaces. In particular, we develop Cantor-Bendixson analysis of topometric spaces, which can serve as a basis for the study of local stability (extending the ad hoc development in Ben Yaacov I and Usvyatsov A, Continuous first order logic and local stability. Trans Am Math (...) Soc, in press), as well as of global ${\aleph_0}$ -stability. We conclude with a study of perturbation systems (see Ben Yaacov I, On perturbations of continuous structures, submitted) in the formalism of topometric spaces. In particular, we show how the abstract development applies to ${\aleph_0}$ -stability up to perturbation. (shrink)

This article presents two related challenges to the idea that, to ensure policy evaluation is comprehensive, all costs and benefits should be aggregated into a single, equity-weighted wellbeing metric. The first is to point out how, even allowing for equity-weighting, the use of a single metric limits the extent to which we can take distributional concerns into account. The second challenge starts from the observation that in this and many other ways, aggregating diverse effects into a single (...) class='Hi'>metric of evaluation necessarily involves settling many moral questions that reasonable people disagree about. This raises serious questions as to what role such a method of policy evaluation can and should play in informing policy-making in liberal democracies. Ultimately, to ensure comprehensiveness of policy evaluation in a wider sense, namely, that all the diverse effects that reasonable people might think matter are kept score of, we need multiple metrics as inputs to public deliberation. (shrink)

In “Happiness Is the Wrong Metric,” Amitai Etzioni largely argues that human beings are motivated by more than just their own happiness, whether conceived in terms of pleasant experiences or fulfilled preferences, and that the state should attend to more than merely people’s happiness. He contends that people are often disposed to seek out, and that public policy ought to promote, what is morally right and good. While not disagreeing with this thrust of Etzioni’s position, I maintain in my (...) contribution that it is too narrow. There is a large range of goods that people tend to pursue, and that social and political institutions should plausibly foster, which are reducible to neither happiness nor morality. They are values that are instead well captured by the concept of what makes a life meaningful. If Etzioni is correct that the state ought to enable people to live morally upright lives, then it has no less reason to enable them to live meaningful ones, too. (shrink)

We study the isomorphism relation on Borel classes of locally compact Polish metric structures. We prove that isomorphism on such classes is always classifiable by countable structures (equivalently: Borel reducible to graph isomorphism), which implies, in particular, that isometry of locally compact Polish metric spaces is Borel reducible to graph isomorphism. We show that potentially $\boldsymbol {\Pi }^{0}_{\alpha + 1}$ isomorphism relations are Borel reducible to equality on hereditarily countable sets of rank $\alpha $, $\alpha \geq 2$. We (...) also study approximations of the Hjorth-isomorphism game, and formulate a condition ruling out classifiability by countable structures. (shrink)

Much of Grünbaum's work may be regarded as a careful development and systematic elaboration of the Riemann-Poincaré thesis of the conventionality of congruence, the thesis that the continuous manifolds of space, time, and space-time are intrinsically metrically amorphous, i.e. are devoid of intrinsic metrics. Therefore, to appreciate Grünbaum's philosophical contributions, one must have a clear understanding of what he means by an intrinsic metric. The second and fourth sections of this paper are exegetical; in them we try to piece (...) together, from his sundry remarks about intrinsic metrics, what Grünbaum means by the term 'intrinsic metric.' We shall argue that, the customary carefulness and precision of Grünbaum's writings notwithstanding, there are residual unclarities and difficulties which beset his conception of an intrinsic metric. In the fifth section we shall propose an explication of Grünbaum's notion of an intrinsic metric which seems on the whole faithful to Grünbaum's intuitions and insights and which also seems capable of performing the philosophical services which his work demands of that notion. The third section is a digression on Zeno's metrical paradox of extension. (shrink)

This paper explores the circumstances that influence whether managers in the public services manipulate the measurement information that is used to assess performance; and if they do, what level of deception they might use. The realistic evaluation approach is adopted. A Delphi survey and the collection of critical incidents through interviews are used to identify possible configurations of contexts–mechanisms–outcomes that provide possible explanations of information manipulation. A number of these configurations are discussed. In a later stage of the project these (...) configurations will be further tested through another Delphi survey, with the intention of developing proposals for improved governance of performance measurement systems in the public services. (shrink)

We present an adaptation of continuous first order logic to unbounded metric structures. This has the advantage of being closer in spirit to C. Ward Henson's logic for Banach space structures than the unit ball approach, as well as of applying in situations where the unit ball approach does not apply. We also introduce the process of single point emph{emboundment}, allowing to bring unbounded structures back into the setting of bounded continuous first order logic. Together with results from cite{BenYaacov:Perturbations} (...) regarding perturbations of bounded metric structures, we prove a Ryll-Nardzewski style characterisation of theories of Banach spaces which are separably categorical up to small perturbation of the norm. This last result is motivated by an unpublished result of Henson. (shrink)

People are motivated not only by a quest for satisfaction but also by an ambition to live up to their sense of what is moral. This sense cannot be reduced to a form of satisfaction or pleasure maximization because, among other reasons, it often engenders pain and sacrifice. Further more, studies repeatedly show that income beyond a certain level does not make people happier, while “higher” sources, like spirituality, community involvement, intellectual activity, and family bonding, pay dividends with respect to (...) individual happiness. Analysis and public policy are enriched when we realize that people take moral commitments seriously and will sometimes eschew their own pleasure to pursue them. This conception of well-being, which crucially incorporates moral affirmation, is a step forward in a deep-rooted dialogue in the social sciences and philosophy about human motivation, behavior, ethics, and how one goes about living “the good life.”. (shrink)

Friedmann–Robertson–Walker (FRW) space–time with bulk viscosity in the context of f(R) gravity is considered. The field equations are solved for the Power and Exponential volumetric expansion. Two types of functional relationship i.e. f(R) = R + bRm and f(R)=R-λ4R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \,f(R) = R - \frac{{\lambda^{4} }}{R} $$\end{document} are investigated. The Phantom, Chaplygin gas and Tachyon fields are discussed. It is observed that the universe is open and inflationary.

This book offers a solution for the problem of structure and agency in sociological theory by developing a new pair of fundamental concepts: metric and nonmetric. Nonmetric forms, arising in a crowd made out of innumerable individuals, correspond to social groups that divide the many individuals in the crowd into insiders and outsiders. Metric forms correspond to congested zones like traffic jams on a highway: individuals are constantly entering and leaving these zones so that they continue to exist, (...) even though the individuals passing through them change. Building from these concepts, we can understand “agency” as a requirement for group identity and group membership, thus associating it with nonmetric forms, and “structure” as a building-up effect following the accumulation of metric forms. This reveals the contradiction between structure and agency to be a case of forced perspective, leaving us victim to an optical illusion. (shrink)

We study notions of independence appropriate for a stability theory of metric abstract elementary classes (for short, MAECs). We build on previous notions used in the discrete case, and adapt definitions to the metric case. In particular, we study notions that behave well under superstability‐like assumptions. Also, under uniqueness of limit models, we study domination, orthogonality and parallelism of Galois types in MAECs.

Role-based database management systems are most widely used for information storage and analysis but are known as vulnerable to insider attacks. The core of intrusion detection lies in an adaptive system, where an insider attack can be judged if it is different from the predicted role by performing classification on the user’s queries accessing the database and comparing it with the authorized role. In order to handle the high similarity of user queries for misclassified roles, this paper proposes a deep (...) metric neural network with strategic sampling algorithm that properly extracts salient features and directly learns a quantitative measure of similarity. A strategic sampling method of heuristically generating and learning training pairs through Monte Carlo search is proposed to select a training pair that can represent the entire dataset. With the TPC-E–based benchmark data trained with 11,000 queries for 11 roles, the proposed model produces the classification accuracy of 95.41%, which is the highest compared with the previous models. The results are verified through comparison of quantitative and qualitative evaluations, and the feature space modelled in the neural network is analysed by t-SNE algorithm. (shrink)

Causal analysis of multiscale systems is crucial for understanding how biological networks are organized and function across scales, including the origin of life itself. This article introduces an axiomatic theory of causation based on the physical concept of cause. Its rigorous derivation from first principles allows us to formulate the law of conservation of causation, expressed in terms of the continuity equation in fluid dynamics. This law states that the flow of causation in dynamical, endogenously coarse-grained systems is conserved across (...) scales. This means that these systems are causally renormalizable across scales; both upward and downward causation are forbidden. The theory also introduces a tool for calculating (i) the minimal linear chain necessary for a biological system to causally connect its two components at a distance, and (ii) the spatial span of the system, defined not as a metric volume of space occupied by the system but as the number of scales causally spanned by its dynamics. Finally, it raises age-old questions about reductionism and emergence, determinism and agency in living systems. (shrink)

_Since the introduction of ChatGPT in EFL writing, most studies have evidenced its role primarily through qualitative lens, leaving its instructional effectiveness insufficiently verified in a rigorous, measurable way. Addressing this, the study uses a metric-driven framework known as _CAOS_—Content, Accuracy, Complexity, and Style. It aims to quantify both the benefits and potential trade-offs to guide balanced classroom implementation. Forty essays were sampled and revised with ChatGPT. Analysis revealed significant improvements in lexical sophistication, syntactic complexity, and grammatical accuracy, as (...) well as enhanced academic tone in the revised texts. However, some content measures showed slight declines in information density, highlighting the need for balanced use of AI tools. These findings provide empirical evidence for the pedagogical value of ChatGPT in EFL writing, while underscoring the importance of strategic, informed implementation._. (shrink)

Professor Massey's exposition and analysis [5] of Professor Grünbaum's writings on metric aspects of space seem to me both very helpful in understanding those writings and to contain a considerable original contribution to the subject. Nevertheless I would like to argue that there is an alternative to Massey's explication which seems to me more faithful to Grünbaum's remarks; it seems at least to have the virtue of not forcing Grünbaum to reject the usual mathematical definitions of the notions used.

The attempt to solve the serious universal problem of the multiplicity and arbitrariness of measurement systems arose at the beginning of the French Revolution, when the Constituent Assembly, by Decree of 8 May 1790, instructed the French Academy of Sciences to study a system of measurements that could be accepted throughout the world. In 1791, the Metric Commission (created in France two years earlier), proposed to create the Metric Decimal System, a system of measurements from which the meter (...) would be born as a unit of length. Thus, was born the Decimal Metric System, which was established in France by Decree of 7 April 1795. The value of this had to be determined from a natural pattern, so the commission in charge decided that the meter would be the ten-millionth part of the quadrant of the terrestrial meridian. Of course, the one that crossed the city of Paris. Given the impossibility of measuring the entire quarter of the terrestrial meridian, it was proposed to measure two arcs of equal length north and south of the 45º parallel. Since then, Paris and Greenwich competed for the meridian origin of the world, until, at an international conference held in Washington, in 1884, Greenwich was adopted as a definitive reference in the geography of our planet. With the Decimal Metric System, new units and their multiples and submultiples had been properly named. However, in socio-economic reality, its introduction added to the pre-existing chaos. In 1798 and 1799, the Institut de France organized the famous Commission of Weights and Measures, of which Gabriel Císcar was a member representing Spain, which was a source of pride for him. Gabriel Ciscar y Ciscar, Spanish sailor and scientist who represented the Kingdom of Spain on the Weights and Measures Commission of the French National Institute in 1798 and 1799. (shrink)

We analyse the connection between the computability and continuity of functions in the case of homomorphisms between topological algebraic structures. Inspired by the Pour-El and Richards equivalence theorem between computability and boundedness for closed linear operators on Banach spaces, we study the rather general situation of partial homomorphisms between metric partial universal algebras. First, we develop a set of basic notions and results that reveal some of the delicate algebraic, topological and effective properties of partial algebras. Our main computability (...) concepts are based on numerations and include those of effective metric partial algebras and effective partial homomorphisms. We prove a general equivalence theorem that includes a version of the Pour-El and Richards Theorem, and has other applications. Finally, the Pour-El and Richards axioms for computable sequence structures on Banach spaces are generalised to computable partial sequence structures on metric algebras, and we prove their equivalence with our computability model based on numerations. (shrink)

Dynamical systems are abstract models of interaction between space and time. They are often used in fields such as physics and engineering to understand complex processes, but due to their general nature, they have found applications for studying computational processes, interaction in multi-agent systems, machine learning algorithms and other computer science related phenomena. In the vast majority of applications, a dynamical system consists of the action of a continuous `transition function' on a metric space. In this work, we consider (...) decidable formal systems for reasoning about such structures. Spatial logics can be traced back to the 1940's, but our work follows a more dynamic turn that these logics have taken due to two recent developments: the study of the topological mu-calculus, and the the integration of linear temporal logic with logics based on the Cantor derivative. In this paper, we combine dynamic topological logics based on the Cantor derivative and the `next point in time' operators with an expressively complete fixed point operator to produce a combination of the topological mu-calculus with linear temporal logic. We show that the resulting logics are decidable and have a natural axiomatisation. Moreover, we prove that these logics are complete for interpretations on the Cantor space, the rational numbers, and subspaces thereof. (shrink)

This paper is the first of a series of three articles that present the syntactic proof of the PA-completeness of the modal system G, by introducing suitable proof-theoretic objects, which also have an independent interest. We start from the syntactic PA-completeness of modal system GL-LIN, previously obtained in [7], [8], and so we assume to be working on modal sequents S which are GL-LIN-theorems. If S is not a G-theorem we define here a notion of syntactic metric d(S, G): (...) we calculate a canonical characteristic fomula H of S (char(S)) so that ⊢G ∼ H → (∼S) and ⊢GL-LIN ∼ H, and the complexity σ of ∼ H gives the distance d(S, G) of S from G. Then, in order to produce the whole completeness proof as an induction on this d(S, G), we introduce the tree-interpretation of a modal sequent Q into PA, that sends the letters of Q into PA-formulas describing the properties of a GL-LIN-proof P of Q: It is also a d(*, G)-metric linked interpretation, since it will be applied to a proof-tree T of ∼ H with H = char(S) and σ(∼ H) = d(S, G). (shrink)

Persistence-based accounts of identity-bearing systems — HAOS-type frameworks, the Free Energy Principle, Global Workspace Theory, and related approaches — rely on batteries of dynamical observables to characterize which substrates constitute persisting systems. We show that such batteries are insufficient, on at least one substrate where frameworks would naturally apply them, to individuate the systems they are used to identify. We construct an existence witness: two coupled harmonic oscillators embedded in a weak thermal bath, admitting two substrate partitions related by an (...) orthogonal transformation in the admissible transformation group of the dynamics. The partitions are distinct — one groups the oscillators by physical location, the other by normal-mode index — and satisfy each framework's systemhood criterion under its own definition. An eight-item persistence/coherence battery (autocorrelation, return statistics, spectral stability, transport, phase-locking, Kuramoto order parameter, correlation-matrix coherence, broadcast-access) takes invariant values on both partitions. The identity-sets they pick out differ. The battery therefore does not individuate within the transformation group, and any individuation drawn from the battery alone is incomplete without a supplementary structural condition we call a partition-warrant. We survey four families of candidate partition-warrants present (sometimes implicitly) in the existing literature — optimization-based, structural-definitional, statistical-structural, and process-based — and state a minimum requirement any warrant must satisfy: sensitivity to substrate properties that are not invariant under the admissible transformation group. Integrated Information Theory's exclusion postulate is the clearest example of an explicit partition-warrant; the paper's argument locates other frameworks relative to this structural slot. The result is methodological: it reclassifies some consciousness-adjacent disputes as partition-choice disputes, clarifies the division of labor between battery computation and individuation in empirical research, and makes tractable a comparative assessment of candidate warrants across frameworks. A companion paper develops one specific warrant — closure activation — as a positive proposal. -/- . (shrink)

This paper presents two general results of decidability concerning logics based on an indeterministic metric tense logic, which can be applied to, among others, logics combining knowledge, time and agency. We provide a general Kripke semantics based on a variation of the notion of synchronized Ockhamist frames. Our proof of the decidability is by way of the finite frame property, applying subframe transformations and a variant of the filtration technique.