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Resiliency for older people represents the capacity to return to equilibrium when difficulties arise and was found as integral predictor of their health status. This study aims to develop a theory that attempts to explain the older adults’ resiliency perspectives during crisis and how it has affected their well-being and quality of life as population group. Deductive theory generation using axiomatic approach was adopted resulting to five axioms that served as basis for the generation of three propositions such as: (1) An older person’s health needs have tendencies to develop into a health threat, (2) when the threat is perceived, older persons developed a sense of internal control and adaptation to the changes it creates known as internal resiliency, and (3) internal resiliency can influence the quality of life in old age. The evolved theory suggests that in times of crisis (e.g., COVID-19 pandemic), health needs develop into a health threat that compels older persons to develop internal resiliency in order to preserve their integrity, wellbeing and quality of life. This study widens the nursing perspectives in addressing older persons’ resiliency by the unique condition at which older clients are placed affecting both the pathological nature of the illness as well as the preventive interventions which the society is forced to implement.

Axiomatic systems can be understood as subsets of syntactic systems. By a process of increasing abstraction, the notion of a syntactic system can become useful for understanding Meillassoux's concept of hyper-Chaos. Fil: Orensanz, Martín. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Departamento de Biología. Laboratorio de Zoonosis Parasitarias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata; Argentina

Formal verification technology has been widely applied in the fields of mathematics and computer science. The formalization of fundamental mathematical theories is particularly essential. Axiomatic set theory is a foundational system of mathematics and has important applications in computer science. Most of the basic concepts and theories in computer science are described and demonstrated in terms of set theory. In this paper, we present a formal system of axiomatic set theory based on the Coq proof assistant. The axiomatic system used in the formal system refers to Morse-Kelley set theory which is a relatively complete and concise axiomatic set theory. In this formal system, we complete the formalization of the basic definitions of sets, functions, ordinal numbers, and cardinal numbers and prove the most commonly used theorems in Coq. Moreover, the non-negative integers are defined, and Peano's postulates are proved as theorems. According to the axiom of choice, we also present formal proofs of the Hausdorff maximal principle and Schröeder-Bernstein theorem. The whole formalization of the system includes eight axioms, one axiom schema, 62 definitions, and 148 corollaries or theorems. The “axiomatic set theory” formal system is free from the more apparent paradoxes, and a complete axiomatic system is constructed through it. It is designed to give a foundation for mathematics quickly and naturally. On the basis of the system, we can prove many famous mathematical theorems and quickly formalize the theories of topology, modern algebra, data structure, database, artificial intelligence, and so on. It will become an essential theoretical basis for mathematics, computer science, philosophy, and other disciplines.

In this thesis, I develop and investigate various novel semantic frameworks for deontic logic. Deontic logic concerns the logical aspects of normative reasoning. In particular, it concerns reasoning about what is required, allowed and forbidden. I focus on two main issues: free-choice reasoning and the role of norms in deontic logic. Free-choice reasoning concerns permissions and obligations that offer choices between different actions. Such permissions and obligations are typically expressed by a disjunctive clause in the scope of a deontic operator. For instance, the sentence "Jane may take an apple or a pear" intuitively offers Jane a choice between two permitted courses of action: she may take an apple, and she may take a pear. In the first part of the thesis, I develop semantic frameworks for deontic logic that account for free-choice reasoning. I show that the resulting logics avoid problems that arise for other logical accounts of free-choice reasoning. The main technical contributions are completeness results for axiomatizations of the different logics. Semantic frameworks for deontic logic typically only talk about norms implicitly. In the second part of the thesis, I clarify the role of norms in deontic logic. I develop a formal model of normative systems where norms are represented explicitly. The model takes into account both the hierarchical structure of normative systems, and the function of norms to regulate agent behavior. I show how the model can be used to clarify issues in the study of normative systems. I also develop a norm-based semantics for deontic action logic based on the model.

I present the proof of Gödel's First Incompleteness theorem in an intuitive manner assuming college-level computational background, while covering all technically challenging steps. I also discuss Gödel's Second Incompleteness theorem, their connection to Gödel's Completeness theorem, and conclude with brief remarks on implications for mathematics, computation, theory of mind and AI.

The standard elementary number theory is not a finite axiomatic system due to the presence of the induction axiom scheme. Absence of a finite axiomatic system is not an obstacle for most tasks, but may be considered as imperfect since the induction is strongly associated with the presence of set theory external to the axiomatic system. Also in the case of logic approach to the artificial intelligence problems presence of a finite number of basic axioms and states is important. Axiomatic hyperrational analysis is the axiomatic system of hyperrational number field. The properties of hyperrational numbers and functions allow them to be used to model real numbers and functions of classical elementary mathematical analysis. However hyperrational analysis is based on well-known non-finite hyperarithmetic axiomatics. In the article we present a new finite first-order arithmetic theory designed to be the basis of the axiomatic hyperrational analysis and, as a consequence, mathematical analysis in general as a basis for all mathematical application including AI problems. It is shown that this axiomatics meet the requirements, i.e., it could be used as the basis of an axiomatic hyperrational analysis. The article in effect completes the foundation of axiomatic hyperrational analysis without calling in an arithmetic extension, since in the framework of the presented theory infinite numbers arise without invoking any new constants. The proposed system describes a class of numbers in which infinite numbers exist as natural objects of the theory itself. We also do not appeal to any “enveloping” set theory.

Analogical proportions are statements of the form "A is to B as C is to D" that are used for several reasoning and classification tasks in artificial intelligence and natural language processing (NLP). For instance, there are analogy based approaches to semantics as well as to morphology. In fact, symbolic approaches were developed to solve or to detect analogies between character strings, e.g., the axiomatic approach as well as that based on Kolmogorov complexity. In this paper, we propose a deep learning approach to detect morphological analogies, for instance, with reinflexion or conjugation. We present empirical results that show that our framework is competitive with the above-mentioned state of the art symbolic approaches. We also explore empirically its transferability capacity across languages, which highlights interesting similarities between them., Comment: Submitted and accepted by the 8th IEEE International Conference on Data Science and Advanced Analytics (DSAA)

An axiomatic theory is proposed that reconciles the existence of an absolute scale for time (Planck time) and special relativity. According to this theory speed of light c becomes a variable which is proposed to be taken as the fifth dimension.

En este trabajo se exploran los principales planteamientos filosóficos de la teoría de la prueba de David Hilbert. Específicamente, el trabajo se centra en sus ideas respecto de la lógica, el concepto de prueba, la axiomática, el concepto de verdad, la metamatemática, el conocimiento a priori y la naturaleza general del conocimiento científico. El objetivo principal es mostrar y caracterizar su enfoque epistemológico acerca de la fundamentación del conocimiento, donde la lógica aparece como garantía de dicha fundamentación. Hilbert supone que el apriorismo proposicional, que propone para sustentar las matemáticas, sustenta —a su vez— un método general para el tratamiento del problema en otros ámbitos como lo es el de las ciencias naturales. Dicho método es la axiomática. Se trata entonces de contribuir a recuperar y actualizar el pensamiento filosófico de Hilbert acerca del papel que juega la lógica para la totalidad del conocimiento científico.

Set theory faces two difficulties: formal definitions of sets/subsets are incapable of assessing biophysical issues; formal axiomatic systems are complete/inconsistent or incomplete/consistent. To overtake these problems reminiscent of the old-fashioned principle of individuation, we provide formal treatment/validation/operationalization of a methodological weapon termed “outer approach” (OA). The observer’s attention shifts from the system under evaluation to its surroundings, so that objects are investigated from outside. Subsets become just “holes” devoid of information inside larger sets. Sets are no longer passive containers, rather active structures enabling their content’s examination. Consequences/applications of OA include: a) operationalization of paraconsistent logics, anticipated by unexpected forerunners, in terms of advanced truth theories of natural language, anthropic principle and quantum dynamics; b) assessment of embryonic craniocaudal migration in terms of Turing’s spots; c) evaluation of hominids’ social behaviors in terms of evolutionary modifications of facial expression’s musculature; d) treatment of cortical action potentials in terms of collective movements of extracellular currents, leaving apart what happens inside the neurons; e) a critique of Shannon’s information in terms of the Arabic thinkers’ active/potential intellects. Also, OA provides an outer view of a) humanistic issues such as the enigmatic Celestino of Verona’s letter, Dante Alighieri’s “Hell” and the puzzling Voynich manuscript; b) historical issues such as Aldo Moro’s death and the Liston/Clay boxing fight. Summarizing, the safest methodology to quantify phenomena is to remove them from our observation and tackle an outer view, since mathematical/logical issues such as selective information deletion and set complement rescue incompleteness/inconsistency of biophysical systems.

We describe the axiomatic approach to real-valued Systemic Risk Measures, which is a natural counterpart to the nowadays classical univariate theory initiated by Artzner et al. in the seminal paper “Coherent measures of risk”, Math. Finance, (1999). In particular, we direct our attention towards Systemic Risk Measures of shortfall type with random allocations, which consider as eligible, for securing the system, those positions whose aggregated expected utility is above a given threshold. We present duality results, which allow us to motivate why this particular risk measurement regime is fair for both the single agents and the whole system at the same time. We relate Systemic Risk Measures of shortfall type to an equilibrium concept, namely a Systemic Optimal Risk Transfer Equilibrium, which conjugates Bühlmann’s Risk Exchange Equilibrium with a capital allocation problem at an initial time. We conclude by presenting extensions to the conditional, dynamic framework. The latter is the suitable setup when additional information is available at an initial time.

This paper explores archival material concerning the reception of Leonard J. Savage’s foundational work of rational choice theory in its subjective-Bayesian form. The focus is on the criticism raised in the early 1960s by Daniel Ellsberg, William Fellner, and Cedric Smith, who were supporters of the newly developed subjective approach but could not understand Savage’s insistence on the strict version he shared with Bruno de Finetti. The episode is well known, thanks to the so-called Ellsberg Paradox and the extensive reference made to it in current decision theory. But Savage’s reaction to his critics has never been examined. Although Savage never really engaged with the issue in his published writings, the private exchange with Ellsberg and Fellner, and with de Finetti about how to deal with Smith, shows that Savage’s attention to the generalization advocated by his correspondents was substantive. In particular, Savage’s defense of the normative value of rational choice theory against counterexamples such as Ellsberg’s did not prevent him from admitting that he would give careful consideration to a more realistic axiomatic system, should the critics be able to provide one.

We provide an axiomatic approach to general premium principles in a probability-free setting that allows for Knightian uncertainty. Every premium principle is the sum of a risk measure, as a generalization of the expected value, and a deviation measure, as a generalization of the variance. One can uniquely identify a maximal risk measure and a minimal deviation measure in such decompositions. We show how previous axiomatizations of premium principles can be embedded into our more general framework. We discuss dual representations of convex premium principles, and study the consistency of premium principles with a financial market in which insurance contracts are traded. (C) 2021 Elsevier B.V. All rights reserved.

We study the interpretation of the observed/expected ratio as an indicator of international scientific collaboration. This indicator cannot be considered as representing the affinity for collaboration, in the sense of 'the more the better', between two countries.

Complex black-box machine learning models are regularly used in critical decision-making domains. This has given rise to several calls for algorithmic explainability. Many explanation algorithms proposed in literature assign importance to each feature individually. However, such explanations fail to capture the joint effects of sets of features. Indeed, few works so far formally analyze high-dimensional model explanations. In this paper, we propose a novel high dimension model explanation method that captures the joint effect of feature subsets. We propose a new axiomatization for a generalization of the Banzhaf index; our method can also be thought of as an approximation of a black-box model by a higher-order polynomial. In other words, this work justifies the use of the generalized Banzhaf index as a model explanation by showing that it uniquely satisfies a set of natural desiderata and that it is the optimal local approximation of a black-box model. Our empirical evaluation of our measure highlights how it manages to capture desirable behavior, whereas other measures that do not satisfy our axioms behave in an unpredictable manner., 31 pages, 10 Figures, 2 Tables

In this paper we develop a combinatorial abstraction of tropical linear programming. This generalizes the search for a feasible point of a system of min-plus-inequalities. It is based on the polyhedral properties of triangulations of the product of two simplices and the combinatorics of the associated set of bipartite graphs with an additional sign information which we call a signed tropical matroid. We demonstrate the connections with the classical simplex method, mean payoff games and scheduling., Comment: 56 pages, 25 figures; essential improvements in results and exposition

Conference postponed to 2021 due to covid-19; International audience; Undoing computations of a concurrent system is beneficial in many situations, e.g., in reversible debugging of multi-threaded programs and in recovery from errors due to optimistic execution in parallel discrete event simulation. A number of approaches have been proposed for how to reverse formal models of concurrent computation including process calculi such as CCS, languages like Erlang, prime event structures and occurrence nets. However it has not been settled what properties a reversible system should enjoy, nor how the various properties that have been suggested, such as the parabolic lemma and the causal-consistency property, are related. We contribute to a solution to these issues by using a generic labelled transition system equipped with a relation capturing whether transitions are independent to explore the implications between these properties. In particular, we show how they are derivable from a set of axioms. Our intention is that when establishing properties of some formalism it will be easier to verify the axioms rather than proving properties such as the parabolic lemma directly. We also introduce two new notions related to causal consistent reversibility, namely causal safety and causal liveness, and show that they are derivable from our axioms.

Contact recommendation is an important functionality in many social network scenarios including Twitter and Facebook, since they can help grow the social networks of users by suggesting, to a given user, people they might wish to follow. Recently, it has been shown that classical information retrieval (IR) weighting models – such as BM25 – can be adapted to effectively recommend new social contacts to a given user. However, the exact properties that make such adapted contact recommendation models effective at the task are as yet unknown. In this paper, inspired by new advances in the axiomatic theory of IR, we study the existing IR axioms for the contact recommendation task. Our theoretical analysis and empirical findings show that while the classical axioms related to term frequencies and term discrimination seem to have a positive impact on the recommendation effectiveness, those related to length normalization tend to be not desirable for the task.

Expected Utility Theory (EUT), the first axiomatic theory of risky choice, describes choices as a utility maximization process: decision makers assign a subjective value (utility) to each choice option and choose the one with the highest utility. The continuity axiom, central to EUT and its modifications, is a necessary and sufficient condition for the definition of numerical utilities. The axiom requires decision makers to be indifferent between a gamble and a specific probabilistic combination of a more preferred and a less preferred gamble. While previous studies demonstrated that monkeys choose according to combinations of objective reward magnitude and probability, a concept-driven experimental approach for assessing the axiomatically defined conditions for maximizing subjective utility by animals is missing. We experimentally tested the continuity axiom for a broad class of gamble types in four male rhesus macaque monkeys, showing that their choice behavior complied with the existence of a numerical utility measure as defined by the economic theory. We used the numerical quantity specified in the continuity axiom to characterize subjective preferences in a magnitude-probability space. This mapping highlighted a trade-off relation between reward magnitudes and probabilities, compatible with the existence of a utility function underlying subjective value computation. These results support the existence of a numerical utility function able to describe choices, allowing for the investigation of the neuronal substrates responsible for coding such rigorously defined quantity.

We analyze an environmental agreement as a cooperative differential game between two regions. The agreement is jointly profitable, since the current contributions in terms of emissions reduction is overcompensated by the benefits of a cleaner environment now and in the future. The regions are asymmetric in two respects: the valuation of a cleaner environment and the responsibility for the state of the environment at the beginning of the agreement. Unlike standard dynamic distribution schemes, the proposed mechanism distributes the efforts of lowering current emissions, not the benefits of a cleaner environment. It is built following an axiomatic approach. It must be time consistent: at any intermediate time, no country can do better by deviating from cooperation. A benefits pay principle is also required: the greater one region's relative benefit from cooperation, the greater must be its relative contribution. A novelty of our approach is the addition of a responsibility or polluter pay principle: a region's relative contribution increases with its responsibility. We characterize a family of dynamic distribution schemes which satisfy the three desired axioms. Interestingly, the proposed scheme could equivalently arise from the Nash bargaining solution considering an asymmetric bargaining power., Ministerio de Economía y Competitividad (projects ECO2014-52343-P and ECO2017-82227-P), Junta de Castilla y León (projects VA024P17 and VA105G18), co-financed by FEDER funds., Project GREEN-Econ (ANR-16-CE03-0005) and the LabEx Entreprendre (ANR-10-LABX-11-01)

Blockchain data structures maintained via the longest-chain rule have emerged as a powerful algorithmic tool for consensus algorithms. The technique—popularized by the Bitcoin protocol—has proven to be remarkably flexible and now supports consensus algorithms in a wide variety of settings. Despite such broad applicability and adoption, current analytic understanding of the technique is highly dependent on details of the protocol’s leader election scheme. A particular challenge appears in the proof-of-stake setting, where existing analyses suffer from quadratic dependence on suffix length. We describe an axiomatic theory of blockchain dynamics that permits rigorous reasoning about the longestchain rule in quite general circumstances and establish bounds—optimal to within a constant—on the probability of a consistency violation. This settles a critical open question in the proof-of-stake setting where we achieve linear consistency for the first time. Operationally, blockchain consensus protocols achieve consistency by instructing parties to remove a suffx of a certain length from their local blockchain. While the analysis of Bitcoin guarantees consistency with error 2-k by removing O(k) blocks, recent work on proof-of-stake (PoS) blockchains has suffered from quadratic dependence: (PoS) blockchain protocols, exemplified by Ouroboros (Crypto 2017), Ouroboros Praos (Eurocrypt 2018) and Sleepy Consensus (Asiacrypt 2017), can only establish that the length of this suffx should be Θ(k2). This consistency guarantee is a fundamental design parameter for these systems, as the length of the suffix is a lower bound for the time required to wait for transactions to settle. Whether this gap is an intrinsic limitation of PoS—due to issues such as the “nothing-at-stake” problem—has been an urgent open question, as deployed PoS blockchains further rely on consistency for protocol correctness: in particular, security of the protocol itself relies on this parameter. Our general theory directly improves the required suffix length from Θ(k2) to Θ(k). Thus we show, for the first time, how PoS protocols can match proof-of-work blockchain protocols for exponentially decreasing consistency error. Our analysis focuses on the articulation of a two-dimensional stochastic process that captures the features of interest, an exact recursive closed form for the critical functional of the process, and tail bounds established for associated generating functions that dominate the failure events. Finally, the analysis provides an explicit polynomial-time algorithm for exactly computing the exponentially-decaying error function which can directly inform practice.

This chapter is about a change in geometry education that took place in the last century. We discuss the emergence of meaningful geometry in the Netherlands. Of course, this was not an isolated reform. Worldwide, mathematicians and mathematics educators came up with new ideas as an alternative for the traditional axiomatic approach to teaching geometry. Already at the end of the 19th century, Klein had made a start with this by advocating a transformation geometry, but in this approach the axiomatic structure still played a main role for ordering activities. This was not the case in the work of Fröbel and Montessori who by building on students’ intuitions and their attention for students’ development of spatial insight were important driving forces towards a meaningful approach to geometry education. In the Netherlands, the pioneers of such a geometry were Tatiana Ehrenfest and Dieke van Hiele–Geldof. Freudenthal was a great promoter of their ideas. For him, geometry is ‘grasping space’, meaning that geometrical experiences should start with the observation of phenomena in reality. Supported by Freudenthal, from the 1970s on, experiments were carried out in the Netherlands to develop a new intuitive and meaningful approach to geometry education, in which the focus was on spatial orientation. How big the change in geometry education that resulted from these experiments was, is illustrated in this chapter by comparing geometry problems from two Dutch mathematics textbooks: one from 1976 and one from 2002.

Innovation is traditionally considered to be an effective instrument of development for industrial enterprises, even at the time of crisis. However, as practical observations show, non-systematic and inconsistence innovative changes may lead to poor results. The sustainable innovative growth is gained as a result of in-depth transformation of all the system elements. The purpose of the research is to formulate an approach to evaluating the expected effect of transformation and restructuring an industrial company and to forecast the perspectives of innovative development. The methodological basis for the research refers to the ideas of the fuzzy set theory and the axiomatic methods of analysis. The axiomatic method makes it possible to construct formalized descriptions of a company and to achieve a sufficient level of detachment. Such descriptions, or texts, may be interpreted by managers as concise information for decision-making. The results of the research find a practical application as tools of evaluating the expected effect of innovation and transformation.

The authors of the article consider the features of the legal regulation of artificial intelligence that guarantee the sustainable development of society in the era of global digitalization. The artificial intelligence-induced transformation of the world-building is leading to a change in the legal landscape. In this regard, the authors investigate artificial intelligence as a subject and object of legal regulation. The article provides an overview of foreign and Russian legislation in artificial intelligence, based on which a legal model of a single codified act is proposed. The authors advert to the need for technical and public control when introducing artificial intelligence into operation, and also for a priori legal regulation of artificial intelligence. Based on theoretical research methods, such as the axiomatic method, analysis and synthesis, systematization, modeling and forecasting, the authors conclude that a comprehensive, consistent, systemic and prospective legal regulation can remove the possible risks of introducing AI technologies, the threat of human destruction and provide a guarantee for the sustainable development of society.

Extending classical algorithms for ordinary weighted or string-to-string automata to automata with underlying more general algebraic structures is of significant practical and theoretical interest. However, the generalization of classical algorithms sets certain assumptions on the underlying structure. In this respect the maximal factorization turns out to be a sufficient condition for many practical problems, e.g. minimization and canonization. Recently, an axiomatic approach on monoid structures suggested that monoids with most general equalizer (mge-monoids) provide an alternative framework to achieve similar results. In this paper, we study the fundamental relation between monoids admitting a maximal factorization and mge-monoids. We describe necessary conditions for the existence of a maximal factorization and provide sufficient conditions for an mge-monoid to admit a maximal factorization.