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This study derives space travel pricing by Walrasian Equilibrium, which is logical reasoning from the general relativity theory (GRT), the accounting equation, and economic supply and demand functions. The Cobb–Douglas functions embed the endogenous space factor as new capital to form the space travel firm's production function, which is also transformed into the consumer's utility function. Thus, the market equilibrium occurs at the equivalence of supply and demand functions, like the GRT, which presents the equivalence between the spatial geometric tensor and the energy–momentum tensor, explaining the principles of gravity and the motion of space matter in the spacetime framework. The mathematical axiomatic set theory of the accounting equation explains the equity premium effect that causes a short-term accounting equation inequality, then reaches the equivalence by suppliers' incremental equity through the closing accounts process of the accounting cycle. On the demand side, the consumption of space travel can be assumed as a value at risk (VaR) investment to attain the specific spacetime curvature in an expected orbit. Spacetime market equilibrium is then achieved to construct the space travel pricing model. The methodology of econophysics and the analogy method was applied to infer space travel pricing with the model of profit maximization, single-mindedness, and envy-free pricing in unit-demand markets. A case study with simulation was conducted for empirical verification of the mathematical models and algorithm. The results showed that space travel pricing remains associated with the principle of market equilibrium, but needs to be extended to the spacetime tensor of GRT. [ABSTRACT FROM AUTHOR]

"Joel D. Hamkins.Lectures on the Philosophy of Mathematics" is an introductory textbook written by a prominent set theorist. The book covers both mathematical and philosophical material, combining interesting mathematical material with discussions of philosophical issues. While the book may cover too much mathematical material for some readers, it could be of great interest to students studying a combination of math and philosophy. The book is divided into chapters on Number, Rigor, Infinity, Geometry, Proof, Computability, Incompleteness, and Set theory, with the chapters on Number and Set theory being the longest. The author often adopts a lecturing attitude, introducing key ideas but leaving the issues open for further exploration. The book also discusses the ontological controversy between the universe and multiverse views in set theory, with the author presenting arguments for the multiverse view. Overall, the book provides a wide and ambitious coverage of the philosophy of mathematics, although some readers may find certain topics could have been presented in a different light or more carefully. [Extracted from the article]

In this paper, fuzzy stress strengths R F = P (Y ≺ X) and traditional stress strengths R = P (Y < X) are considered and compared when X and Y are independently inverse Weibull random variables. When axiomatic fuzzy set theory is taken into account in the stress–strength inference, it enables the generation of more precise studies on the underlying systems. We discuss estimating both conventional and fuzzy models of stress strength utilizing a maximum product of spacing, maximum likelihood, and Bayesian approaches. Simulations based on the Markov Chain Monte Carlo method are used to produce various estimators of conventional and fuzzy dependability of stress strength for the inverse Weibull model. To generate both conventional and fuzzy models of dependability, we use the Metropolis–Hastings method while performing Bayesian estimation. In conclusion, we will examine a scenario taken from actual life and apply a real-world data application to validate the accuracy of the provided estimators. [ABSTRACT FROM AUTHOR]

Before we dive into the book, we are warned that this is not a typical math popularization book; it won't gush about genius mathematicians, nor will it simplify any fashionable math into meaningless drivel. Another possible issue some may raise about the book is its identification of mathematics with proof and its identification of proof with Greek mathematics. Lolli here aims to convince us that even a formalist perspective on mathematics can coexist with the conception of mathematics as a narrative. What do you think of when you read the phrase "mathematics as storytelling?". [Extracted from the article]

"The Influence of Quantum Physics on Philosophy" continues the discussion on quantum physics. The aim of this special issue is broad in scope, since its contributions range over different topics in logic and philosophy of science. Connecting Things in the Setting of Foundations and Philosophy of Science: Introduction to the special issue. [Extracted from the article]

In the epistemological tradition, there are two main interpretations of the semantic relation that an empirical theory may bear to the real world. According to realism, the theory-world relationship should be conceived as truth; according to instrumentalism, instead, it should be limited to empirical adequacy. Then, depending on how empirical theories are conceived, either syntactically as a class of sentences, or semantically as a class of models, the concepts of truth and empirical adequacy assume different and specific forms. In this paper, we review two main conceptions of truth (one sentence-based and one model-based) and two of empirical adequacy (one sentence-based and one model-based), we point out their respective difficulties, and we give a first formulation of a new general view of the theory-world relationship, which we call Methodological Constructive Realism (MCR). We then show how the content of MCR can be further specified and expressed in a definite and precise form. The bulk of the paper shows in detail how it is possible to accomplish this goal for the special case of deterministic dynamical phenomena and their correlated deterministic models. This special version of MCR is formulated as an axiomatic extension of set theory, whose specific axioms constitute a formal ontology that provides an adequate framework for analyzing the two semantic relations of truth and empirical correctness, as well as their connections. [ABSTRACT FROM AUTHOR]

Geographic information systems have undergone rapid growth for decades. Topology has provided valuable modeling tools in the development of this field. Formal verification ofthe model of topological spatial relations can provide a reliable guarantee for the correctness of geographic information systems. We present a proof of the topological spatial relations model that has been formally verified in the Coq proof assistant. After an introduction to the formalization of the axiomatic set theory of Morse–Kelley, the formal description of the elementary concepts and properties of general topology is developed. The topological spatial relations between two sets are described by using the concept of the intersection value. Finally, we formally proved the topological spatial relations between two sets which are restricted to the regularly closed and the planar spatial regions. All the proof details are strictly completed in Coq, which shows that the correctness of the theoretical model for geographic information systems can be checked by a computer. This paper provides a novel method to verify the correctness of the topological spatial relations model. This work can also contribute to the creation and validation of various geological models and software. [ABSTRACT FROM AUTHOR]

Deflationism about truth describes truth as a logical notion. In the present paper, I explore the implication of the alleged logicality of truth from the perspective of axiomatic theories of truth, and argue that the deflationist doctrine of the logicality of truth gives rise to two types of self-undermining arguments against deflationism, which I call the conservativeness argument from logicality and the topic-neutrality argument. [ABSTRACT FROM AUTHOR]

Axiomatic fuzzy sets (AFS) theory has seen extensive application in classification tasks in recent years. Yet, efficiently generating precise fuzzy descriptions for each target class from a multitude of complex concepts is challenging. This study aims to bridge this gap by transforming the challenge into a combinatorial optimization problem. We adopt an advanced model-based evolutionary optimization method, Randomized Coordinate Shrinking Classification (RACOS), to create an interpretable classifier within the AFS framework. By refining the definition of the complex concept set in AFS theory, we define a feasible search space for the optimization method. Innovative fitness functions have been developed focusing on semantic discrimination and prediction accuracy. Concurrently, we establish an encoding–decoding mechanism to link the solution vector with the complex concept for each fitness function. Ultimately, the complex concepts, guided by various fitness functions, are integrated into the class's fuzzy description using AFS logical operations. Our method demonstrates competitive classification performance and superior interpretability compared to other evolutionary fuzzy rule-based classifiers. • Develop encoding–decoding mechanism to bridge complex concept and solution vector. • Design various fitness function to balance prediction accuracy and interpretability. • Covert training AFS-based classifier to solving combinatorial optimization problem. • Generate semantic description for prediction result in form of natural language. [ABSTRACT FROM AUTHOR]

There are several decision-making based situations in which it is necessary to categorize the evaluating parameters into their respective sub-parametric values based non-overlapping sets. The existing soft set model is not compatible with such situations therefore hypersoft set (Hs-set) is developed which manages such situations by utilizing a novel mapping called multi-argument approximate mapping which broadens the domain of soft approximate mapping. This research presents the characterization of several essential axiomatic properties and set-based operations of Hs-set which will help the researchers to implement this emerging theory to other fields of study. The brief discussion on some hybridized structures of Hs-set with fuzzy set-like models is also provided. [ABSTRACT FROM AUTHOR]

I Mathematics and Its Logics i is a collection of Geoffrey Hellman's papers. According to Hellman, there are principles statable in the language of his modal stage theory - essentially, that of modal, plural logic with vocabulary for stages and sets - that are at once significantly more self-evident than the potentialist's axiom of infinity and that nevertheless imply it as a consequence. Saliently, Boolos abandons the axiom of replacement altogether; more subtly, in order to secure infinity, he has to write in the existence of an infinite stage as a new axiom in the stage theory. On potentialist theories, in general, sets are possibilia, and so the relevant "axiom of infinity" does not assert that infinite sets I exist i , but rather only the "potentialist translation" of the standard axiom, that infinite sets I are possible i . [Extracted from the article]

Totalisation of operations and functions is a subject of great importance in the precise sciences at large, and in particular, in computer science, where computed functions must be totalised to account for edge cases and prevent system failures. In this paper, we investigate the totalisation of computable functions in the general and abstract setting of the axiomatic theory of computing systems, such as automata, computer networks, and algorithms, allowing us to define characteristics and discover relations which resonate true through many computational models – from individual automata through complex distributed systems with concurrent functioning. We explore the possible types of totalisations, the complexes of computing systems that do (or do not) have totalisations, and the behaviour of totalisation in various transformations and compositions. In addition, we point to a few interesting directions for future research. [ABSTRACT FROM AUTHOR]

In this paper, we reflect on the ethico-political and epistemological implications of a critical trans pedagogy that takes as its focus the generative stance of refusal. Our purpose is to identify and explain the significance of key axiomatic principles at the heart of our conception of such a pedagogical endeavour, which entails an interrogative stance vis-à-vis cisgenderism, antinormativity and trans necropolitics. These principles define a governing logics and rationality for enacting a trans pedagogy of refusal in its potential to create curricular spaces of recognition and intelligibility in educational institutions that are committed to addressing the erasure of trans and non-binary people. They also illuminate a necessary pedagogical commitment to centring desubjugated and submerged knowledges of transness and the blackness of transness that defy the limits of antinormativity and necropolitics. [ABSTRACT FROM AUTHOR]

This paper proposes a novel classification technology—fuzzy rule-based oblique decision tree (FRODT). The neighborhood rough sets-based FAST feature selection (NRS_FS_FAST) is first introduced to reduce attributes. In the axiomatic fuzzy set theory framework, the fuzzy rule extraction algorithm is then proposed to dynamically extract fuzzy rules. And these rules are regarded as the decision function during the tree construction. The FRODT is developed by expanding the unique non-leaf node in each layer of the tree, which results in a new tree structure with linguistic interpretation. Moreover, the genetic algorithm is implemented on σ to obtain the balanced results between classification accuracy and tree size. A series of comparative experiments are carried out with five classical classification algorithms (C4.5, BFT, LAD, SC and NBT), and recently proposed decision tree HHCART on 20 UCI data sets. Experiment results show that the FRODT exhibits better classification performance on accuracy and tree size than those of the rival algorithms. [ABSTRACT FROM AUTHOR]

We explore the relationships between Description Logics and Set Theory. The study is carried on using, on the set-theoretic side, a very rudimentary axiomatic set theory Ω, consisting of only four axioms characterizing binary union, set difference, inclusion, and the power-set. An extension of ALC , dubbed ALC Ω , is defined in which concepts are naturally interpreted as sets living in Ω-models. In ALC Ω not only membership between concepts is allowed—even admitting membership circularity—but also the power-set construct is exploited to add metamodelling capabilities. We investigate translations of ALC Ω into standard description logics as well as a set-theoretic translation. A polynomial encoding of ALC Ω in ALCOI proves the validity of the finite model property as well as an ExpTime upper bound on the complexity of concept satisfiability. We develop a set-theoretic translation of ALC Ω in the theory Ω, exploiting a technique proposed for translating normal modal and polymodal logics into Ω. Finally, we show that the fragment LC Ω of ALC Ω not admitting roles and individual names, is as expressive as ALC Ω. [ABSTRACT FROM AUTHOR]

• High school students possess general knowledge about road safety. • This general knowledge is not sufficient for decision-making in the road environment. • Students fail to recognize specific situations in which the cause of the accident is linked to the pedestrian behavior. • In practical situations, students engage in risky behavior that they justify by citing a variety of reasons. • Students displayed an inclination to support compliance with traffic rules and proposed strategies to reduce road risks. This study aims to comprehend traffic safety competencies in high school students from two specific regions of Peru, as well as to analyze the differences between both groups. For this purpose, a qualitative study was carried out which consisted of eight focus groups in high schools, three of them in the metropolitan area of Lima (Lima region) and five in the province of Rodríguez de Mendoza (the Amazonas region). The focus groups were constituted by 73 students enrolled in the last three years of high school, named 3rd, 4th and 5th grade of secondary education, whose ages ranged from 14 to 18 years. The collected data were analyzed employing elements of Grounded Theory and Axiomatic Set Theory. The main findings reveal that most participants have a general understanding of traffic rules and the proper use of the components of traffic safety. Thus, they understand the traffic rules as road signs that regulate the behavior of drivers and pedestrians. In addition, participants know the proper procedure to follow at traffic lights, pedestrian walkways and how to use seat belts. They also identify the driver's tiredness/drowsiness, and the consumption of alcohol and drugs as potential factors that cause traffic accidents. No causes that are linked to pedestrian behavior were identified, and only participants from the Amazonas region mentioned speeding as a contributing factor. On the other hand, participants report that in practical situations they engage in risky behavior and fail to comply with transit rules. They justify this behavior by citing poor infrastructure, vehicle malfunctions, and the need get around quickly in order to get things done. Finally, the implications of the results are discussed and compared to the content and implementation of the Peruvian Road Safety Educational Program (PENSV, for its acronym in Spanish), providing recommendations that can aid evidence‐based policy making in Peru. [ABSTRACT FROM AUTHOR]

• Examines the question of objectivity for latent variable modelling. • Focuses on conditional independence, circularity, and factor indeterminacy. • Proposes a conceptual framework to address problems for objectivity. The aim of this paper is raise and address questions regarding the status of objectivity for the generalized latent variable model (GLVM) in psychometric research, given the conceptual, logical and mathematical problems of circularity, conditional independence, and factor indeterminacy, respectively. The question of objectivity for the model is examined with respect to measurement and realist perspectives. Drawing on insights from measurement and systems dynamics literature, a proposal for a conceptual framework is presented, that integrates: i) inference from the best systematisation; and ii) axiomatic set theory. This conceptual framework, which addresses the whole of a research project, invites specification of the expected relations, conditions, and assumptions which are relevant to the implementation of the GLVM. While this does not eliminate the problems for the GLVM, it provides future researchers with maximal objective information in standardized form, supporting minimization of definitional and instrumental uncertainty, in psychological modelling practices. [ABSTRACT FROM AUTHOR]

We may also note that the axiomatic system consisting of axioms (1)-(5) admits only models that are isomorphic to the natural numbers, and since the natural numbers are well-ordered, in this system well-ordering ( To the human mind, the natural numbers may be a more primitive intuitive concept than sets, but that notwithstanding, the more common reduction in texts on the foundations of mathematics is usually that the concept of natural number is reduced to the concept of sets. Note that the natural numbers were introduced by means of Peano's axioms, so if one removes the principle of induction from the axiom set and replaces it with the well-ordering property, the proof of Theorem 1 is incorrect. Bachelor's thesis in mathematics, Dept. of Mathematics and Mathematical Statistics, Umeå University, 2016. [Extracted from the article]

In fuzzy time series, a way of representing their original numeric data through a collection of fuzzy sets plays a pivotal role and impacts the prediction performance of the constructed forecasting models. An evident shortcoming of most existing models is that fuzzy sets (their membership functions) are developed in an intuitive manner so that not all aspects of the time series could be fully captured. In this study, using an idea of axiomatic fuzzy set clustering we take the distribution of data into account to position time series in the framework of fuzzy sets. The obtained clusters exhibit well-defined semantics. To produce numeric results of forecasting, we develop a method to determine the prototypes based on the corresponding fuzzy description of the clusters. The commonly used enrollment time series is applied to demonstrate how the proposed method works. The experimental results exploiting the Taiwan Stock Exchange Capitalization Weighted Stock Index demonstrate that the proposed method can effectively improve forecasting accuracy. Furthermore, the proposed approach is of a general form and as such can be easily integrated with various fuzzy time series models. [ABSTRACT FROM AUTHOR]

Examples of nontrivial (that is, non-Cartesian product) multidimensional discrete variable representation (DVR) basis sets are presented that are generalizations of sinc functions in one dimension. Their use in solving quantum problems in two dimensions is illustrated. Unlike all standard (one-dimensional) examples of DVR bases, these bases cannot be created by dividing out the roots of a generating function. It is argued that the difficulty of constructing nontrivial, multidimensional DVR bases is due to the restrictive nature of the DVR conditions, which cannot be satisfied on most subspaces of wave functions. The bases considered in this paper, however, are invariant under translations on a lattice in n-dimensional space, and the properties of the Abelian group of lattice translations allow the DVR conditions to be satisfied. More generally, the question of the relation between group theory and the conditions necessary to qualify a set of basis functions as a DVR basis is considered. It is shown how to construct orthonormal states that are related by the action of some group, and, in the case of Abelian groups, the extra conditions required to qualify the basis as a DVR set are discussed. © 2002 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The next chapter broadens the scope with a comprehensive outline of the history of the philosophy of mathematics. Our aim is to introduce mathematicians and teachers of mathematics as well as students of mathematics into the philosophy of mathematics (pp. xviii-xiv).This book is quite rich, and at over 450 pages, it delivers much of what the authors promise. The first chapter, "On the Way to the Reals", begins with the question, What is HT ht ?. [Extracted from the article]

Based on axiomatic fuzzy set (AFS) theory and fuzzy information entropy, a novel fuzzy oblique decision tree (FODT) algorithm is proposed in this paper. Traditional axis-parallel decision trees only consider a single feature at each non-leaf node, while oblique decision trees partition the feature space with an oblique hyperplane. By contrast, the FODT takes dynamic mining fuzzy rules as a decision function. The main idea of the FODT is to use these fuzzy rules to construct leaf nodes for each class in each layer of the tree; the samples that cannot be covered by the fuzzy rules are then put into an additional node – the only non-leaf node in this layer. Construction of the FODT consists of four major steps: (a) generation of fuzzy membership functions automatically by AFS theory according to the raw data distribution; (b) extraction of dynamically fuzzy rules in each non-leaf node by the fuzzy rule extraction algorithm (FREA); (c) construction of the FODT by the fuzzy rules obtained from step (b); and (d) determination of the optimal threshold δ to generate a final tree. Compared with five traditional decision trees (C4.5, LADtree (LAD), Best-first tree (BFT), SimpleCart (SC) and NBTree (NBT)) and a recently obtained fuzzy rules decision tree (FRDT) on eight UCI machine learning data sets and one biomedical data set (ALLAML), the experimental results demonstrate that the proposed algorithm outperforms the other decision trees in both classification accuracy and tree size. [ABSTRACT FROM AUTHOR]

In this paper we provide an axiomatic characterization of a total preorder for assessing the mobility associated with any pair of rankings relative to a same group by means of a new class of rank-mobility measures. These measures constitute a strong generalization of Spearman's ρ index. The problem of dealing with mobility of subgroups, which is a pressing issue in several and various applicative contexts ranging from the use of socio-economic indicators to information retrieval, is addressed in terms of partial permutations which include standard permutations as a special case. Literature is lacking in a theoretical discussion for comparing mobility of variable-size subgroups: we show that our axiomatic approach fills this gap. [ABSTRACT FROM AUTHOR]

A social dichotomy function maps a collection of weak orders to a set of dichotomous weak orders. Every dichotomous weak order partitions the set of alternatives into approved alternatives and disapproved alternatives. The Borda mean rule returns all dichotomous weak orders that approve all alternatives with above-average Borda score and disapprove alternatives with below-average Borda score. We show that the Borda mean rule is the unique social dichotomy function satisfying neutrality, reinforcement, faithfulness, and the quasi-Condorcet property. Our result holds for all domains of weak orders that are sufficiently rich, including the domain of all linear orders and the domain of all weak orders. [ABSTRACT FROM AUTHOR]

Linking systems are crucial for studying the homotopy theory of fusion systems, but are also of interest from an algebraic point of view. We propose a definition of a linking system associated to a saturated fusion system which is more general than the one currently in the literature and thus allows a more flexible choice of objects of linking systems. More precisely, we define subcentric subgroups of fusion systems in a way that every quasicentric subgroup of a saturated fusion system is subcentric. Whereas the objects of linking systems in the current definition are always quasicentric, the objects of our linking systems only need to be subcentric. We prove that, associated to each saturated fusion system F, there is a unique linking system whose objects are the subcentric subgroups of F. Furthermore, the nerve of such a subcentric linking system is homotopy equivalent to the nerve of the centric linking system associated to F. We believe that the existence of subcentric linking systems opens a new way for a classification of fusion systems of characteristic p-type. The various results we prove about subcentric subgroups give furthermore some evidence that the concept is of interest for studying extensions of linking systems and fusion systems. [ABSTRACT FROM AUTHOR]

Vagueness in scientific studies poses a challenge. Intuitionistic Fuzzy Set (IFS) theory has emerged as a powerful and flexible tool to counter such challenge. So, there is a need to develop such measures which can not only measure the vagueness but also quantify the differences in underlying IFSs. The aim of this communication is to introduce one such divergence measure called Intuitionistic Fuzzy Jensen-Tsalli Divergence measure in the settings of IFS theory. The presence of parameters makes the proposed divergence measure flexible and competent for applications. Besides discussing some of its major properties, the findings are applied in pattern recognition problem and in medical diagnosis of some diseases with same set of symptoms. The performance of the proposed measure is genuinely compared with some other existing measures in literature through numerical examples based on medical diagnosis and pattern recognition. [ABSTRACT FROM AUTHOR]

In exploratory data mining, most classifiers pay more attention on the accuracy and speed of learned models, but they are lacking of the interpretability. In this paper, an interpretable and comprehensible classifier is proposed based on Linear Discriminant Analysis (LDA) and Axiomatic Fuzzy Sets (AFS). The algorithm utilizes LDA to extract features with the largest inter-class variance. Besides, the proposed approach aims to explore a transformation from the selected feature space to a semantic space where the samples in the same class are made as close as possible to one another, whereas the samples in the different class are as far as possible from one another. Moreover, the descriptions of each class can be obtained by the proposed approach. When compared with well-known classifiers such as LogisticR, C4.5Tree, SVM and KNN, the proposed method not only can achieve better performance in terms of accuracy but also has the capability of interpretability and comprehension. [ABSTRACT FROM AUTHOR]

The authors of [13] formulated a soft topological ordered spaces concept and then they established and studied some ordered mappings [14]. In the present work, we define new ordered mappings via soft topological ordered spaces based on soft β-open sets, namely soft xβ-continuous, soft xβ-open, soft xβ-closed and soft xβ-homeomorphism mappings, for x ∈ {I;D;B}. We give various characterizations of each one of the introduced soft mappings. One of the most important obtained results is that an extended soft topologies notion guarantees the equivalent between the soft mappings initiated herein and their counterparts of mappings on topological ordered spaces. We provide several interesting examples to examine the relationships among these soft mappings. [ABSTRACT FROM AUTHOR]

Highlights • An image color correction hierarchical association rule classifier based on axiomatic fuzzy is proposed. • Efficient hierarchical structure association rules are used to construct a color correction classifier. • Results show that the proposed algorithm has high computational accuracy and efficiency. Abstract In order to improve the performance of color correction classifier for color constancy finite dimensional linear model under complex illumination, an image color correction hierarchical association rule classifier based on axiomatic fuzzy sets under complex illumination with finite dimensional linear model of color constancy is proposed. Firstly, in order to improve the accuracy of the algorithm, based on the feature extraction of image data sets, the AFS attribute representation of image set fuzzy concept is constructed by using axiomatic theory (AFS) to improve the recognition degree of image set attributes. Secondly, in order to improve the computational efficiency of the algorithm, the hierarchical structure association rules are used to construct a color correction classifier for color constancy finite dimensional linear model under complex illumination. The ontology information between concepts is used to improve the ability of parallel classification. Finally, the experimental results show that the proposed algorithm has high computational accuracy and efficiency. [ABSTRACT FROM AUTHOR]

Validation is critical to the success of software trustworthiness measurement. A large number of software trustworthiness measures are proposed; however, most of them are not validated from a theory perspective. Therefore, they lack theoretical foundation and will induce unnecessary cost and useless calculation. In this paper, we bring measurement theory into software trustworthiness measurement, construct a source codes oriented software trustworthiness measure based on extensive structure in the measurement theory, and validate the developed measure by use of axiomatic approaches. Compared with some software trustworthiness measures that are already presented, this measure can evaluate software trustworthiness better from a theory perspective. [ABSTRACT FROM AUTHOR]

A Levi class L(ℳ) generated by a class ℳ of groups is a class of all groups in which the normal closure of each element belongs to ℳ. It is stated that there exist finite groups G such that a Levi class L(qG), where qG is a quasivariety generated by a group G, has infinite axiomatic rank. This is a solution for [The Kourovka Notebook, Quest. 15.36]. Moreover, it is proved that a Levi class L(ℳ), where ℳ is a quasivariety generated by a relatively free 2-step nilpotent group of exponent ps with a commutator subgroup of order p, p is a prime, p ≠ 2, s ≥ 2, is finitely axiomatizable. [ABSTRACT FROM AUTHOR]

In several applications of sequent calculi going beyond pure logic, an introduction of suitably defined rules seems to be more profitable than addition of extra axiomatic sequents. A program of formalization of mathematical theories via rules of special sort was developed successfully by Negri and von Plato. In this paper a general theorem on possible ways of transforming axiomatic sequents into rules in sequent calculi is proved. We discuss its possible applications and provide some case studies for illustration. [ABSTRACT FROM AUTHOR]

The article focuses on philosophers are divided on whether the proof-theoretic or truth theoretic approach to logic is more fruitful and the proof-theoretic approach its roots in Gentzen and Prawitz, and truth-theoretic or semantic approach has its roots in Tarski. Topics include the proof-theoretic approach has begun to encroach on semantics itself, and the truth-theoretic semantics is often associated with a truth-conditional theory of meaning, representational approach to mind and language.

Regarding the problem of the existing intuitionistic fuzzy entropy formulas in ordering the partial entropy, the constraint condition that is consistent with the intuitionistic facts is proposed in this paper, the axiomatic definition of entropy which fully reflects the intuition and fuzziness of intuitionistic fuzzy sets is given, and the improved intuitionistic fuzzy entropy formula is constructed according to the entropy axiomatic definition and its properties are studied. Finally, we compare the improved formula with the existing intuitionistic fuzzy entropy formulas, and the result turns out that the improved formula can solve the problem in the entropy ordering theoretically and practically. [ABSTRACT FROM AUTHOR]

Preferences that accommodate aversion to subjective uncertainty and its potential misspecification in dynamic settings are a valuable tool of analysis in many disciplines. By generalizing previous analyses, we propose a tractable approach to incorporating broadly conceived responses to uncertainty. We illustrate our approach on some stylized stochastic environments. By design, these discrete time environments have revealing continuous time limits. Drawing on these illustrations, we construct recursive representations of intertemporal preferences that allow for penalized and smooth ambiguity aversion to subjective uncertainty. These recursive representations imply continuous time limiting Hamilton-Jacobi-Bellman equations for solving control problems in the presence of uncertainty. [ABSTRACT FROM AUTHOR]

Using triangular norms, we present a new classiflication of fuzzy subalgebras, ideals and implicative ideals in BCK/BCI-algebras. [ABSTRACT FROM AUTHOR]

A natural problem in the methodology of Economics is how to model interactions in which the decisions made by agents depend on the beliefs held about both objective aspects of the relevant situation and the beliefs ascribed to other agents. There are two ways of representing such contexts. One is by means of a hierarchy of beliefs, the other as a state of the world. These two can be seen, from a Systems-theoretical perspective, as representing streams generated by a process and its fixed points, respectively. These two representations are in fact equivalent. The finer question we address here is whether they are identical. Resorting to a type-theoretical syntax, we obtain a unified interpretation in which both entities give rise to∞-groupoids. Voevodsky's univalent axiom then ensures their identity in small types universes. [ABSTRACT FROM AUTHOR]

The center function is defined on a connected graph G , where the input is any finite sequence of vertices of G and the output is the set of all vertices that minimize the maximum distance to the entries of the input. If the input is a sequence containing each vertex of G once, then the output is just the classical center of G . In the axiomatic approach, one wants to establish a set of properties or consensus axioms that characterize the function. We refer to an axiom as a universal axiom if the center function satisfies this axiom on any connected graph. In a previous paper, our focus was on classes of graphs on which we were able to characterize the center function in terms of universal axioms. In this paper, the focus is on classes of graphs on which these universal axioms do not characterize the center function. We introduce non-universal axioms that, together with some universal axioms, provide new characterizations of the center function: on cocktail party graphs, on complete bipartite graphs, and on block graphs. [ABSTRACT FROM AUTHOR]

In this paper, we study abstract Burnside rings of essentially finite categories. Under unique epi-mono factorization property and the existence of coequalizers for some kind, we prove the existence of a fundamental exact sequence for ABR. Furthermore, an ABR can be embedded into a direct product of rational character rings. [ABSTRACT FROM AUTHOR]

The one that is considered the standard model in the area of logic of belief change was proposed by Alchourrón, Gärdenfors and Makinson in 1985 and is, nowadays, known as the AGM model. Two of the main shortcomings pointed out to the AGM model of belief change are the (impractical) use of belief sets to represent belief states and the (unrealistic) acceptance of any new information. In this paper we study a kind of operators—known as shielded base contractions—which address both those issues. Indeed, on the one hand, these operators are defined on belief bases (rather than belief sets) and, on the other hand, they are constructed with the underlying idea that not all new informations are accepted. We propose twenty different classes of shielded base contractions and obtain axiomatic characterizations for each of them. Additionally we thoroughly investigate the interrelations (in the sense of inclusion) among all those classes. More precisely, we analyse whether each of those classes is or is not (strictly) contained in each of the remaining ones. [ABSTRACT FROM AUTHOR]

In this article, we generalize a previously defined set of axioms for a closure operation that induces balanced big Cohen–Macaulay modules. While the original axioms were only defined in terms of finitely generated modules, these new ones will apply to all modules over a local domain. The new axioms will lead to a notion of phantom extensions for general modules, and we will prove that all modules that are phantom extensions can be modified into balanced big Cohen–Macaulay modules and are also solid modules. As a corollary, if R has characteristic p > 0 and is F -finite, then all solid algebras are phantom extensions. If R also has a big test element (e.g., if R is complete), then solid algebras can be modified into balanced big Cohen–Macaulay modules. (Hochster and Huneke have previously demonstrated that there exist solid algebras that cannot be modified into balanced big Cohen–Macaulay algebras.) We also point out that tight closure over local domains in characteristic p generally satisfies the new axioms and that the existence of a big Cohen–Macaulay module induces a closure operation satisfying the new axioms. [ABSTRACT FROM AUTHOR]

We show that, consistently, there can be maximal subtrees of P(ω) and P(ω)/fin of arbitrary regular uncountable size below the size of the continuum c. We also show that there are no maximal subtrees of P(ω)/fin with countable levels. Our results answer several questions of Campero-Arena et al. (Fund Math 234:73-89, 2016). [ABSTRACT FROM AUTHOR]

The fragment of the language of modal logic that consists of all implications A → B, where A and B are built from variables, the constant ⊤ (truth), and the connectives ∧ and ◊1,◊2,...,◊m. For the polymodal logic S5m (the logic of m equivalence relations) and the logic K4.3 (the logic of irreflexive linear orders), an axiomatization of such fragments is found and their algorithmic decidability in polynomial time is proved. [ABSTRACT FROM AUTHOR]

As was recently shown by the authors, quantum probability theory can be used for the modelling of the process of decision-making (e.g. probabilistic risk analysis) for macroscopic geophysical structures such as hydrocarbon reservoirs. This approach can be considered as a geophysical realization of Hilbert's programme on axiomatization of statistical models in physics (the famous sixth Hilbert problem). In this conceptual paper, we continue development of this approach to decision-making under uncertainty which is generated by complexity, variability, heterogeneity, anisotropy, as well as the restrictions to accessibility of subsurface structures. The belief state of a geological expert about the potential of exploring a hydrocarbon reservoir is continuously updated by outputs of measurements, and selection of mathematical models and scales of numerical simulation. These outputs can be treated as signals from the information environment E. The dynamics of the belief state can be modelled with the aid of the theory of open quantum systems: a quantum state (representing uncertainty in beliefs) is dynamically modified through coupling with E; stabilization to a steady state determines a decision strategy. In this paper, the process of decision-making about hydrocarbon reservoirs (e.g. 'explore or not?'; 'open new well or not?'; 'contaminated by water or not?'; 'double or triple porosity medium?') is modelled by using the Gorini- Kossakowski-Sudarshan-Lindblad equation. In our model, this equation describes the evolution of expert's predictions about a geophysical structure.We proceed with the information approach to quantum theory and the subjective interpretation of quantum probabilities (due to quantum Bayesianism). This article is part of the theme issue 'Hilbert's sixth problem'. [ABSTRACT FROM AUTHOR]

This is part one of a two-part paper, in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. This allows us to connect theories of partial ground with axiomatic theories of truth. In this part of the paper, we develop an axiomatization of the relation of partial ground over the truths of arithmetic and show that the theory is a proof-theoretically conservative extension of the theory PT of positive truth. We construct models for the theory and draw some conclusions for the semantics of conceptualist ground. [ABSTRACT FROM AUTHOR]