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Starting from the notion of a multidistance, we formalize, through a suitable system of axioms, the concept of an inequality measure defined on a nonempty set with no additional structure implemented a priori. Among inequality measures, apart from multidistances we pay special attention to dispersions, and study their main features. Classical concepts will be generalized to this abstract setting. Multidistances are then revisited, and some new methods to generate them are implemented. A wide spectrum of interdisciplinary applications is outlined in the final section. This work is partially supported by the research projects ECO2015-65031-R , MTM2015-63608-P , TIN2016-77356-P , TIN2017-87600-P (MINECO/AEI-FEDER, UE) and PID2019-108392GB-I00 (AEI/10.13039/501100011033)

This paper introduces a logical characterization of multi-adjoint algebras with a twofold contribution. On the one hand, the study of multi-adjoint algebras, from a logical perspective, will allow us to discover both the core and new features of these algebras. On the other hand, the axiomatization of multi-adjoint algebras will be useful to take advantage of the properties of the logical connectives considered in the corresponding deductive system. The mechanism considered to carry out the mentioned axiomatization follows the one given by Petr Hajek for residuated lattices. Specifically, the paper presents the bounded poset logic (BPL) as an axiomatization of a bounded poset, since this algebraic structure is the most simple structure from which a multi-adjoint algebra is defined. In the following, the language of BPL is enriched with a family of pairs, composed of a conjunctor and an implication, and its axiomatic system is endowed with new axioms, giving rise to the multi-adjoint logic (ML). The soundness and completeness of BPL and ML are proven. Finally, a comparison between the axiomatization of the multi-adjoint logic and the one given for the BL logic is introduced.

In this paper we introduce a new set of axioms that characterize uniform random dictatorship (URD) as a randomized impartial nomination rule. Unlike earlier work we use a variable population model which allows us to employ axioms that reflect consistency and proportionality–axioms that ensure that the rule behaves well with respect to the combinatorial structure of the nomination profile. Earlier work characterizing URD employed strong symmetry axioms and so it is surprising that our axioms characterize the same rule.

Fuzzy rough approximation operators are the underlying concepts in fuzzy rough set theory . There are at least two approaches to develop these primary concepts, i.e., the constructive approach and the axiomatic approach . Single axiomatic characterizations of fuzzy rough approximation operators have got tons of attention. In this paper, considering L being a GL-quantale, we will develop the theory of L -valued rough sets with an L -set as the basic universe of defining L -valued rough approximation operators. Adopting the idea of single axiomatic characterizations of fuzzy rough sets , we will present the axiomatic characterizations of L -valued upper and lower rough approximation operators on an L -set with respect to reflexive, symmetric, transitive L -valued relations on an L -set as well as their compositions. Choosing an L -set as the universe will break the rules that adopting Zadeh’s fuzzy sets as the universe. By these results, we will further provide a new framework of axiomatic research of fuzzy rough set theory.

The task’s slack is the key issue to reduce the energy consumed by DVFS-enabled computing platforms. Despite the large number of scheduling algorithms that are presented in the literature, only a unique scaling axiomatic approach (SAA) is utilized in the scaling phase of the algorithms. SAA simply extends the execution of the task within its slack if a suitable scaling frequency is available. Unfortunately, when dependent-tasks applications are scheduled on such platforms, scheduling algorithms minimize the tasks’ slacks to reduce the overall completion time of the application tasks. This paper presents a mechanism that can be applied to any schedule produced by a dependent-task scheduling algorithm for both homogeneous and heterogeneous DVFS-enabled computing platforms. The proposed mechanism is called BlackLight. BlackLight attempts to extend the tasks’ slacks via rescheduling the application tasks without violating the overall completion time of the application tasks. The proposed mechanism is applied to a large number of dependent-tasks schedules of both random generated application graphs and two real-world application graphs. The experimental results based on a computer simulation show that the proposed mechanism significantly extends the tasks’ slacks compared with SAA , which leads to more reduction in the consumed energy.

We introduce an axiomatic approach to entropies and relative entropies that relies only on minimal information-theoretic axioms, namely monotonicity under mixing and data-processing as well as additivity for product distributions. We find that these axioms induce sufficient structure to establish continuity in the interior of the probability simplex and meaningful upper and lower bounds, e.g., we find that every relative entropy satisfying these axioms must lie between the Renyi divergences of order 0 and $\infty $ . We further show simple conditions for positive definiteness of such relative entropies and a characterisation in terms of a variant of relative trumping. Our main result is a one-to-one correspondence between entropies and relative entropies.

Fuzzy relations R : A 2 → [ 0 , 1 ] are widely used in decision-making in order to represent degrees of preference between alternatives in A . Examining fuzzy relations, this paper first reflects on an inherent incompatibility inside the domain of reciprocal fuzzy preference relations and preference structures. Aiming at clarifying the semantics of reciprocal fuzzy relations by means of that of weak fuzzy relations, a general frame is then proposed for studying the transformation between them, based on the notion of reciprocal functions. Next, we show that reciprocal relations can be naturally endowed with a preference structure of their own, containing two relations respectively showing a semantics of strict preference and lack of preference. Finally, we study the compatibility between these reciprocal preference structures and the standard preference structures of weak fuzzy preference relations by means of an axiomatic approach, leading to a system of functional equations that is shown to admit different solutions, allowing to explicitly assign one of the available semantics for reciprocal preference relations.

We take the axiomatic approach to uncover the structure of the revenue-sharing problem from broadcasting sports leagues. We formalize two notions of impartiality, depending on the stance one takes with respect to the revenue generated in the games involving each pair of teams. We show that the resulting two axioms lead towards two broad categories of rules, when combined with additivity and some other basic axioms. We complement those results strengthening the impartiality notions to consider axioms of order preservation., Universidad Pablo de Olavide, Department of Economics, Carretera de Utrera, Km. 1, 41013, Seville, Spain

This paper contributes to the theory of average rate of change (ARC) measurement. We use an axiomatic approach to generalize the conventional ARC measures (such as the difference quotient and the continuously compounded growth rate) in several directions: to outcome variables with arbitrary connected domains, to not necessarily time-shift invariant dependence on time, to more general (than an interval) time sets, to a path-dependent setting, and to a benchmark-based evaluation. We also revisit and generalize the relationship between the ARC measurement and intertemporal choice models.

Hesitant fuzzy set is a natural generalization of the classical fuzzy set. A hesitant fuzzy set on a universe of discourse is in terms of a function that when applied to the universe returns a finite subset of [0, 1]. Since the axiomatic method of approximation operator is of great significance in the research of the mathematical structure of rough set theory, it is a fundamental problem in axiomatic method to find the minimum set of abstract axioms. This paper first introduces the basic concepts, properties and related operations of hesitant fuzzy set, hesitant fuzzy rough set and hesitant fuzzy rough approximation operator. Secondly, by defining inner product, outer product and by exploring their related properties, the single axiomatization problem of the classical hesitant fuzzy rough approximation operator is solved. Furthermore, we study the single axiomatization of hesitant fuzzy rough approximation operators derived from serial, reflexive, symmetric and transitive hesitant fuzzy relations, respectively. Finally, we compare and analyze the advantages and disadvantages of hesitant fuzzy set, fuzzy rough set and hesitant fuzzy rough set through some cases.

In this paper, we present constructions of Riecan, Bosbach and internal and general states on Sheffer stroke basic algebra $${\mathcal {A}}.$$ We deal with some characteristic features and independency of axiomatic system of forementioned states. We prove that a Riecan state corresponds to a Bosbach state or vice versa. In the sequel, we introduce faithful and fixed set generated by internal states on $${\mathcal {A}}.$$ We demonstrate that each Sheffer stroke basic algebra is also a Sheffer stroke basic algebra under internal state. In addition, we mention about internal state-(filter, compatible filter and prime filter) on $${\mathcal {A}}$$ and their related results. We touch on internal state congruence and investigate its relation with filters. We are interested in finding whether general states have implied Riecan and Bosbach states or not. We handle the notion of general state-morphism and some related results on $${\mathcal {A}}.$$ In order to support mentioned concepts, we give place to some attractive examples.

Purpose of the study. Analysis of the literature shows that the ordinal theory of utility is widespread in the theory of consumer behavior. To analyze consumer preferences, a utility function is used, which characterizes the value of the utility of the consumed goods and services on a scale of order. Moreover, to find the marginal utility of a product, arithmetic operations are used, which are impossible on a scale of order. To allow arithmetic operations, a quantitative analysis of the utility function is required. Consequently, the problem of quantitative measurement of the utility function is relevant.The measurement problem also arises in decision theory. For example, the hierarchy analysis method is a popular method for solving multicriteria problems, but contains an erroneous model of subjective measurement. For this reason, other methods appear in decision-making theory that should replace the method of analyzing hierarchies. The theory of the importance of criteria is being actively developed. However, the theory of the importance of criteria also does not solve the problem of quantitative measurement.For a long time, the problem of measurement has also existed in psychophysics. The existence of two mismatched psychophysical laws contradicts the classical theory of measurements. Recently, a rating solution has been proposed. The equivalence of the basic laws of psychophysics has been proved. In this paper, it is proposed to use the rating method to measure preferences in utility theory and in decision theory.Materials and methods. The domain of the rating is the set of ordered pairs of objects. Moreover, the composition (operation of addition) of objects is defined on the set of ordered pairs. A rating is a number that is assigned during a measurement to an ordered pair of objects.The rating is assumed to preserve the operation of composition of ordered pairs.An arithmetic operation is selected to carry out the measurement. The measurement result must match the result of the arithmetic operation. The result of an arithmetic operation is the difference or ratio of the values of the quantity. The rating values coincide with the result of the arithmetic operation (up to isomorphism).The additivity of the rating is used to check the adequacy of the measurement results. For this, it is assumed that the rating is independent of the measurement method. The theoretical justification for independence is the isomorphism condition. The empirical confirmation of independence is the equivalence of the basic psychophysical laws.Results. The paper presents an axiomatic approach to the measurement problem. Measurement can be carried out in both objective and subjective ways. The axiomatic and classical definition of the rating has been formulated. The axiomatic definition implies the classical definition for a special set of objects. The classic definition is constructive. To check the adequacy of the measurement results, it is enough to compare the ratings obtained by different measurement methods (method of alternatives).Conclusion. The rating method is a quantitative measurement method. The rating method can be used to construct a model of consumer behavior and in decision-making theory.

Pairwise comparisons are used in a wide variety of decision situations where the importance of alternatives should be measured on a numerical scale. One popular method to derive the priorities is based on the right eigenvector of a multiplicative pairwise comparison matrix. We consider two monotonicity axioms in this setting. First, increasing an arbitrary entry of a pairwise comparison matrix is not allowed to result in a counter-intuitive rank reversal, that is, the favoured alternative in the corresponding row cannot be ranked lower than any other alternative if this was not the case before the change (rank monotonicity). Second, the same modification should not decrease the normalised weight of the favoured alternative (weight monotonicity). Both properties are satisfied by the geometric mean method but violated by the eigenvector method. The axioms do not uniquely determine the geometric mean. The relationship between the two monotonicity properties and the Saaty inconsistency index are investigated for the eigenvector method via simulations. Even though their violation turns out not to be a usual problem even for heavily inconsistent matrices, all decision-makers should be informed about the possible occurrence of such unexpected consequences of increasing a matrix entry., Comment: 16 pages, 2 figures, 2 tables

Given the steady growth in demand for Internet resources (due to specific benefits for users) a significant issue is the governing regulation of relations that develop during the interaction of subjects of legal relations on this sphere. Intellectual property law expands the scope of regulation of legal relations via the Internet. In this regard, there is an essential need for a comprehensive study of theoretical and practical aspects of the protection of intellectual property rights on the Internet. During the study, a number of general scientific methods were used: method of observation, method of comparison, method of experiment, method of convergence from abstract to concrete, method of formalization, axiomatic method, method of analysis, and historical method. As a result of the study, the peculiarities of legal regulation of relations arising in the process of using objects of intellectual property rights on the Internet, which are caused by the difficulty in identifying the infringer of intellectual property rights, determining the place of the crime, the speed of spread of intellectual property on the Internet, low level of legal culture of the population. The ways to improve the system of protection of intellectual property rights on the Internet are the following were suggested.

Nonmodularity is a prominent property of capacity that deeply links to the internal interaction phenomenon of multiple decision criteria. Following the common architectures of the simultaneous interaction indices as well as of the bipartition interaction indices, in this paper, we construct and study the notion of probabilistic nonmodularity index and also its particular cases, such as Shapely and Banzhaf nonmodularity indices, which can be used to describe the comprehensive interaction situations of decision criteria. The connections and differences among three categories of interaction indices are also investigated and compared theoretically and empirically. It is shown that three types of interaction indices have the same roots in their first and second orders, but meanwhile the nonmodularity indices have involved less amount of subsets and can be adopted to describe the interaction phenomenon in decision analysis.

How to apply allocation in an life cycle assessment (LCA) is a long-running and controversial debate. Consensus seems to exist on the fact that the allocation procedure should follow logically from the LCA goal definition. This paper proposes to use an axiomatic method to (1) identify an allocation procedure for co-production, joint treatment, and recycling, that best responds to a specific LCA goal and (2) communicate the rationale applied by the LCA practitioner transparently. The method is illustrated via a case study. The specific goal definition for which a suitable allocation procedure is identified is to evaluate what impacts can be attributed to a product, which could inform a company about potential sources of reputation damage. Subjective assumptions that reflect our vision of “what impacts can be attributed to a product” are described in definitions and axioms. Axioms are formulated that describe the system boundaries of the product system and the partitioning criterion. The derived allocation procedure corresponds to “Allocation at the Point of Substitution,” which is applied in one of the system models of ecoinvent. Partitioning is based on market information, which corresponds to a cause-oriented perspective on “what impacts can be attributed to a product.” Other LCA goal definitions and rationales could require different system boundaries or a different partitioning criterion. The axiomatic method presented in this paper supports the identification of a suitable allocation procedure for a defined LCA goal and the transparent communication of the rationale that backs up this procedure. Building forth on the approach of this paper, a collection of axioms and corresponding allocation procedures could be developed on which consensus might exist within the LCA community. Such goal-dependent allocation procedures could form the basis of future guidance on LCA.

The importance of intuitionistic temporal logics in Computer Science and Artificial Intelligence has become increasingly clear in the last few years. From the proof-theory point of view, intuitionistic temporal logics have made it possible to extend functional languages with new features via type theory, while from its semantical perspective several logics for reasoning about dynamical systems and several semantics for logic programming have their roots in this framework. In this paper we consider several axiomatic systems for intuitionistic linear temporal logic and show that each of these systems is sound for a class of structures based either on Kripke frames or on dynamic topological systems. Our topological semantics features a new interpretation for the `henceforth' modality that is a natural intuitionistic variant of the classical one. Using the soundness results, we show that the seven logics obtained from the axiomatic systems are distinct., Comment: Under consideration in Theory and Practice of Logic Programming (TPLP). arXiv admin note: text overlap with arXiv:1803.05077

The purpose of this paper is to propose an axiomatic characterization of ecological footprint indices. Using an axiomatic approach, we define a set of axioms representing the properties considered ...

Axiomatic Theories and Improving the Relevance of Information Systems Research This paper examines the fact that a significant number of empirical information systems (IS) studies engage in confirmative testing of self-evident axiomatic theories without yielding highly relevant knowledge for the IS community. The authors conduct both a horizontal analysis of 72 representative IS theories and an in-depth vertical analysis of 3 well-known theories (i.e., technology acceptance model, diffusion of innovation theory, and institutional theory) in order to measure how pervasive such testing of axiomatic theories is. The authors discovered that more than 60% of 666 hypotheses from the horizontal analysis could be regarded as axiomatic theory elements. In the vertical analysis, 68.1% of 1,301 hypotheses from 148 articles were axiomatic. Based on these findings, the authors propose four complementary IS research approaches: (1) identifying disconfirming boundary conditions, (2) measuring the relative importance of axiomatic causal factors, (3) measuring the stage of progression toward visionary goals when the nature of the axiomatic theory can be extended to future visions, and (4) engaging in the conceptual design of visionary axiomatic goals. They argue that these complementary IS research approaches can enhance the relevance of IS research outcomes without sacrificing methodological rigor.

Genuine Paraconsistent logics $$\mathbf {L3A}$$ and $$\mathbf {L3B}$$ were defined in 2016 by Beziau et al, including only three logical connectives, namely, negation disjunction and conjunction. Afterwards in 2017 Hernandez-Tello et al, provide implications for both logics and define the logics $$\mathbf {L3A_G}$$ and $$\mathbf {L3B_G}$$ . In this work we continue the study of these logics, providing sound and complete Hilbert-type axiomatic systems for each logic. We prove among other properties that $$\mathbf {L3A_G}$$ and $$\mathbf {L3B_G}$$ satisfy a restricted version of the Substitution Theorem, and that both of them are maximal with respect to Classical Propositional Logic. To conclude we make some comparisons between $$\mathbf {L3A_G}$$ and $$\mathbf {L3B_G}$$ and among other logics, for instance $${\mathbf {Int}}$$ and some $${\mathbf {LFI}}$$ s.

In this paper, we analyse the various meanings of probability and its different applications, and we focus especially on the classical, the frequentist, and the subjectivist view. We describe the different problems of how probability can be measured in each of the approaches, and how each of them can be well justified by a mathematical theory. We analyse the foundations of probability, where the scientific analysis of the theory that allows for a frequentist interpretation leads to unsolvable problems. Kolmogorov’s axiomatic theory does not suffice to establish statistical inference without further definitions and principles. Finally, we show how statistical inference essentially determines the meaning of probability and a shift emerges from purely objectivist views to a complementary conception of probability with frequentist and subjectivist constituents. For didactical purpose, the result of the present analyses explains basic problems of teaching, originating from a biased focus on frequentist aspects of probability. It also indicates a high priority for the design of suitable learning paths to a complementary conception of probability. In the applications, modellers use information in a pragmatic way processing this information regardless of its connotation into formal mathematical models, which are always thought as essentially wrong but useful.

A model is proposed to allocate Formula One World Championship prize money among the constructors. The methodology is based on pairwise comparison matrices, allows for the use of any weighting method, and makes possible to tune the level of inequality. We introduce an axiom called scale invariance, which requires the ranking of the teams to be independent of the parameter controlling inequality. The eigenvector method is revealed to violate this condition in our dataset, while the row geometric mean method always satisfies it. The revenue allocation is not influenced by the arbitrary valuation given to the race prizes in the official points scoring system of Formula One and takes the intensity of pairwise preferences into account, contrary to the standard Condorcet method. Our approach can be used to share revenues among groups when group members are ranked several times., Comment: 19 pages, 3 figures, 6 tables

We propose an axiomatic approach which economically underpins the representation of dynamic intertemporal decisions in terms of a utility function, which randomly reacts to the information available to the decision maker throughout time. Our construction is iterative and based on time dependent preference connections, whose characterization is inspired by the original intuition given by Debreu’s State Dependent Utilities (1960).

For a completely distributive De Morgan algebra L, we develop a general framework of L-fuzzy rough sets. Said precisely, we introduce a pair of L-fuzzy approximation operators, called upper and lower L-fuzzifying approximation operators derived from general L-fuzzifying neighborhood systems. It is shown that the proposed approximation operators are a common extension of the L-fuzzifying approximation operators derived from L-fuzzy relations (INS 2019) and the approximation operators derived from general neighborhood systems (KBS 2014). Furthermore, we investigate the unary, serial, reflexive, transitive and symmetric conditions in general L-fuzzifying neighborhood systems, and then study the associated approximation operators from both a constructive method and an axiomatic method. Particularly, for transitivity (resp., symmetry), we give two interpretations, one is an appropriate generalization of transitivity (resp., symmetry) for L-fuzzy relations, and the other is a suitable extension of transitivity (resp., symmetry) for general neighborhood systems. In addition, for some special L-fuzzifying approximation operators, we use single axiom to characterize them, respectively. At last, the proposed approximation operators are applied in the research of incomplete information system, and a three-way decision model based on them is established. To exhibit the effectiveness of the model, a practical example is presented.

This is an expository paper on tensor products where the standard approaches for constructing concrete instances of algebraic tensor products of linear spaces, via quotient spaces or via linear maps of bilinear maps, are reviewed by reducing them to different but isomorphic interpretations of an abstract notion, viz. the universal property, which is based on a pair of axioms.

The research purpose is invention (construction) of a formal logical inference of the Law of Conservation of Energy within a logically formalized axiomatic epistemology-and-axiology theory Sigma from a precisely defined assumption of a-priori-ness of knowledge. For realizing this aim, the following work has been done: 1) a two-valued algebraic system of formal axiology has been defined precisely and applied to proper-philosophy of physics, namely, to an almost unknown (not-recognized) formal-axiological aspect of the physical law of conservation of energy; 2) the formal axiomatic epistemology-and-axiology theory Sigma has been defined precisely and applied to proper-physics for realizing the above-indicated purpose. Thus, a discrete mathematical model of relationship between philosophy of physics and universal epistemology united with formal axiology has been constructed. Results: 1) By accurate computing relevant compositions of evaluation-functions within the discrete mathematical model, it is demonstrated that a formal-axiological analog of the great conservation law of proper physics is a formal-axiological law of two-valued algebra of metaphysics. (A precise algorithmic definition of the unhabitual (not-well-known) notion “formal-axiological law of algebra of metaphysics” is given.) 2) The hitherto never published significantly new nontrivial scientific result of investigation presented in this article is a formal logical inference of the law of conservation of energy within the formal axiomatic theory Sigma from conjunction of the formal-axiological analog of the law of conservation of energy and the assumption of a-priori-ness of knowledge.

We provide a complete taxonomic characterization of robust hierarchical clustering methods for directed networks following an axiomatic approach. We begin by introducing three practical properties associated with the notion of robustness in hierarchical clustering: linear scale preservation, stability, and excisiveness. Linear scale preservation enforces imperviousness to change in units of measure whereas stability ensures that a bounded perturbation in the input network entails a bounded perturbation in the clustering output. Excisiveness refers to the local consistency of the clustering outcome. Algorithmically, excisiveness implies that we can reduce computational complexity by only clustering a subset of our data while theoretically guaranteeing that the same hierarchical outcome would be observed when clustering the whole dataset. In parallel to these three properties, we introduce the concept of representability, a generative model for describing clustering methods through the specification of their action on a collection of networks. Our main result is to leverage this generative model to give a precise characterization of all robust -- i.e., excisive, linear scale preserving, and stable -- hierarchical clustering methods for directed networks. We also address the implementation of our methods and describe an application to real data., arXiv admin note: substantial text overlap with arXiv:1607.06339

The general purpose of the research—systematical clarifying and explicating the too vague proper philosophical concepts of space, void, matter, motion, inertia, for making a logical harmony between them and the corresponding notions of proper physics. The special purpose of the research—invention (construction) of a formal inference of the well-known Newton’s first law of mechanics within a logically formalized axiomatic epistemology system from a set of precisely defined presumptions. For realizing this aim the following work has been done: a two-valued algebraic system of metaphysics as formal axiology has been applied to philosophical epistemology and philosophy of nature; a formal axiomatic theory called Sigma has been applied to physics for realizing the above-indicated special purpose of the research. Thus, constructing a discrete mathematical model of relationship between universal epistemology and philosophy of physics has been done. Research results: The main hitherto not published significantly new nontrivial scientific result of applied investigations presented in this article is a formal inference of the well-known Newton’s first law of mechanics within the formal axiomatic epistemology system Sigma from conjunction of the formal-axiological analog of the proper-law-of-mechanics (which analog is the formal-axiological law of two-valued algebra of metaphysics) and the assumption of a-priori-ness of knowledge. For obtaining this main research result, a set of accessory nontrivial novelties has been used, for instance; a precise algorithmic definition is given for the notion “law of metaphysics” in the algebraic system of metaphysics as formal axiology; a formal-axiological equivalence in the algebraic system is defined precisely. Precise tabular definitions are given for relevant evaluation-functions determined by evaluation-arguments, for example; “movement of (what, whom) x”; “speed of x”; “vector of x”; “velocity of x”; “magnitude of x”; “finiteness (definiteness) of x”; “dynamical closed-ness (isolated-ness) of x”; “constant-ness, immutability, conservation of x”.

The Borda rule, originally defined on profiles of individual preferences modelled as linear orders over the set of alternatives, is one of the most important voting rules. But voting rules often need to be used on preferences of a different format as well, such as top-truncated orders, where agents rank just their most preferred alternatives. What is the right generalisation of the Borda rule to such richer models of preference? Several suggestions have been made in the literature, typically considering specific contexts where the rule is to be applied. In this work, taking an axiomatic perspective, we conduct a principled analysis of the different options for defining the Borda rule on top-truncated preferences.

This paper proposes a new method for solving the problem of multicriteria optimization of a numerical vector function on a fuzzy set. The membership function of a fuzzy feasible set is joined to the original set of criteria that allows the original problem of multi-criteria optimization to be treated as the task of finding a suitable compromise (Pareto-optimal) solution with respect to an extended set of criteria. It is assumed that in a search for the “best” compromise solution there is only fuzzy information about the preferences of decision maker in the form of information quanta. At the first stage of the proposed method, the search for a compromise is made on the basis of an axiomatic approach, with which the Pareto set is reduced. The result of the reduction is a fuzzy set with a membership function, which is determined on the basis of the used fuzzy information. At the second stage, the obtained membership function is added to the extended set of criteria, after which the scalarization procedure based on the idea of goal programming is used to solve the formed multicriteria problem.

Historically, our theme is situated within the triangle of three of Kant's students: Hegel (1770-1831), Bolzano (1781-1848) and Peirce (1839-1914). All three wanted to change Kant's strict separation of philosophy and science by developing a new conception of logic. Bolzano inaugurated the so-called linguistic turn of philosophy which became the guiding principle of all analytical philosophy (Dummett, 2014) and he opposed Hegel’s unity of concept and object of knowledge. Charles Peirce took a middle position, a position that is expressed in his so-called Pragmatic Maxim (Peirce, CP 5.3). Taken together we might say that a universal principle of complementarity of meaning and reference, or of meaning and information (in the sense of Shannon) finds its origin in Post-Kantian philosophy. We encounter here the very same approach of principled thinking endorsed by Einstein in physics (special theory of relativity) or by the formal axiomatic approach in mathematics (Hilbert)! Key Words: Bolzano, Hegel, Peirce; Complementarity of sense and reference; Geometry from Euclid to Einstein; Hilbert. Historicamente, nosso tema se situa dentro do triângulo de três alunos de Kant: Hegel (1770-1831), Bolzano (1781-1848) e Peirce (1839-1914). Todos os três queriam mudar a separação estrita de Kant entre filosofia e ciência, desenvolvendo uma nova concepção de lógica. Bolzano inaugurou a chamada virada linguística da filosofia, que se tornou o princípio orientador de toda filosofia analítica (Dummett, 2014) e se opôs à unidade de conceito e objeto de conhecimento de Hegel. Charles Peirce assumiu uma posição intermediária, posição que se expressa em sua assim chamada Máxima Pragmática (Peirce, CP 5.3). Tomados em conjunto, podemos dizer que um princípio universal de complementaridade de significado e referência, ou de significado e informação (no sentido de Shannon) encontra sua origem na filosofia pós-kantiana. Encontramos aqui a mesma abordagem do pensamento baseado em princípios endossada por Einstein na física (teoria da relatividade especial) ou pela abordagem axiomática formal na matemática (Hilbert)! Palavras-chave: Bolzano, Hegel, Peirce; Complementaridade de sentido e referência; Geometria de Euclides a Einstein; Hilbert.

This paper shows the importance of applying a certain approach to the history and philosophy of mathematical practice to the study of Zygmunt Janiszewski's contribution to the topological foundations of Continuum theory. In the first part, a biography of Janiszewski is presented. It emphasizes his role as one of the founders of the Polish School of Mathematics, and the social, political and military facets in which his intellectual character was revealed, as well as the values that guided his academic and scientific life. Kitcher's view of mathematical practice is then adopted to examine the philosophical conceptions and epistemological style of Janiszewski in relation to the construction of the formal axiomatic system of knowledge about the continua. Finally, it is shown the convenience of differentiating in Kitcher's approach, the methods, procedures, techniques and strategies of practice, and the aesthetic values of mathematics. Keywords: Zygmunt Janiszewski; Continuum theory; Philosophy of mathematical practice; Polish school of mathematics. Este artigo mostra a importância de aplicar uma certa abordagem da história e da filosofia da prática matemática ao estudo da contribuição de Zygmunt Janiszewski para os fundamentos topológicos da Teoria do continuum. Na primeira parte, é apresentada uma biografia de Janiszewski. Ela enfatiza seu papel como um dos fundadores da Escola de Matemática da Polônia, e as facetas sociais, políticas e militares nas quais seu caráter intelectual foi revelado, assim como os valores que guiaram sua vida acadêmica e científica. A visão de Kitcher sobre a prática matemática é então adotada para examinar as concepções filosóficas e o estilo epistemológico de Janiszewski em relação à construção do sistema axiomático formal de conhecimento sobre o contínuo. Finalmente, mostra-se a conveniência de diferenciar na abordagem de Kitcher, os métodos, procedimentos, técnicas e estratégias da prática, e os valores estéticos da matemática. Palavras-chave: Zygmunt Janiszewski; Teoria do Continuum; Filosofia da prática matemática; Escola de Matemática da Polônia.

The neoclassical synthesis has been defined as a bridge between Keynes-ian theory and Walrasian general equilibrium theory. The aim of this article is to show that founders of the neoclassical synthesis were not homogenous in their appraisal of the importance of Walrasian theory. To do so, we focus on Robert Solow’s contributions as a case study and examine the history of his lifelong criticism of what he called “axiomatics.” According to Solow, the axiomatic approach aims at founding economics on one general and complex model based on first principles or axioms. In contrast, Solow advocated the use of a diversity of simple and partial models, which have practical utility, are realistic in their crucial assumptions, consider institutions and the evolving nature of the economy, and rely on common sense microfoundations. We conclude by suggesting that Solow can be characterized as Cournotian.

"Systems neuroergonomics" (Mau, J. In: R. Wang and X. Pan, editors, Advances in Cognitive Neurodynamics (V), chapter 59, pages 431-437, Springer Science+Business Media, Singapore, 2016) showed a separation of human-body system's functional organization from its cellular material in order to open a holistic perspective that can comprise all body functions. This was achieved with a strictly hierarchical structure in drill-down from system's functional whole to base functional elements. Implied multi-scale functional dynamics with coherent up-scaling were mathematically described in an axiomatic way, but the daily wake/sleep dynamics cover mainly anticipated functional challenges. Here, the theory is expanded to permit "functional learning" in the sense of adjustment of functional capacities to meet higher demand. "Functional aging" as a whole-body frailty is derived from biodynamic entropy that eventually drives the living to meet its inevitable destiny, death. The ages at which frailty begins to dominate can currently be obtained only from official life tables as population averages and are different for men and women, accordingly. The objective is to develop a mathematical theory in medicine from an axiomatic approach to phenomenological biodynamics, in which cognition is viewed as an embodied activity.

We revisit a problem faced by all programmers. Can one write a program that determines whether any given program is the shortest program? How does one prove that a given program is the shortest? After answering these questions, we discuss very briefly the Kolmogorov complexity of a string of zero and one, which leads to a barrier on any axiomatic system, known as Chaitin’s barrier.

This work presents an axiomatization for entropy based on an extension of concavity called core-concavity. We show that core-concavity characterizes the largest class of functions for which the data-processing inequality holds, under the assumption that conditional entropy is defined as a generalized average. Also, under the same assumption, we show that data-processing and “conditioning reduces entropy” properties are equivalent. We prove several properties of core-concave functions, including generalization of perfect secrecy and of Fano’s inequality. We also show that definitions of conditional entropy based on worst-case can be retrieved as limit cases of generalized averages. A connection between statistical decision making and this axiomatic approach is also presented.

Bitcoin’s main innovation lies in allowing a decentralized system that relies on anonymous, profit-driven miners who can freely join the system. We formalize these properties in three axioms: anonymity of miners, no incentives for miners to consolidate, and no incentive to assuming multiple fake identities. This novel axiomatic formalization allows us to characterize what other protocols are feasible: every protocol with these properties must have the same reward scheme as Bitcoin. This implies an impossibility result for risk-averse miners. Furthermore, any protocol either gives up on some degree of decentralization or its reward scheme is equivalent to Bitcoin’s. (JEL D82, E42, O33)

In memoriam José Arrazola Ramírez (1962–2018) The logic $\textbf{G}^{\prime}_3$ was introduced by Osorio et al. in 2008; it is a three-valued logic, closely related to the paraconsistent logic $\textbf{CG}^{\prime}_3$ introduced by Osorio et al. in 2014. The logic $\textbf{CG}^{\prime}_3$ is defined in terms of a multi-valued semantics and has the property that each theorem in $\textbf{G}^{\prime}_3$ is a theorem in $\textbf{CG}^{\prime}_3$. Kripke-type semantics has been given to $\textbf{CG}^{\prime}_3$ in two different ways by Borja et al. in 2016. In this work, we continue the study of $\textbf{CG}^{\prime}_3$, obtaining a Hilbert-type axiomatic system and proving a soundness and completeness theorem for this logic.

We investigate the existence of solutions satisfying converse consistency together with other well-known axioms in the context of bargaining. Our results are negative. First, we show that there is no solution satisfying converse consistency together with Pareto optimality and contraction independence. Second, we show that there is no solution satisfying converse consistency together with Pareto optimality and individual monotonicity.

In this paper we consider the modal logic with both $$\Box $$ and $$\Diamond $$ arising from Kripke models with a crisp accessibility and whose propositions are valued over the standard Godel algebra $$[0,1]_G$$ . We provide an axiomatic system extending the one from Caicedo and Rodriguez (J Logic Comput 25(1):37–55, 2015) for models with a valued accessibility with Dunn axiom from positive modal logics, and show it is strongly complete with respect to the intended semantics. The axiomatizations of the most usual frame restrictions are given too. We also prove that in the studied logic it is not possible to get $$\Diamond $$ as an abbreviation of $$\Box $$ , nor vice-versa, showing that indeed the axiomatic system we present does not coincide with any of the mono-modal fragments previously axiomatized in the literature.

To specify and verify the concurrent and reactive systems with the theorem proving approach, a complete axiomatization is formalized for first order projection temporal logic (PTL) with both finite and infinite time. To this end, PTL is restricted to a finite domain, and the syntax, semantics as well as the axiomatization of PTL are presented. Further, the techniques of labeled normal form and labeled normal form graph of PTL formulas are introduced respectively, with which a decision procedure for quantifier free PTL (QFPTL) formulas is given. Moreover, a generalized labeled normal form graph is defined and employed to transform a quantified PTL formula into its equivalent QFPTL formula. Finally, a decision procedure for PTL is formalized and the completeness of the axiomatic system is proved based on the decidability of PTL formulas.

This paper studies the form of the instantaneous impact cost function in a financial market with transaction costs via an axiomatic approach. We show that several kinds of convexity of the cost function are equivalent to the corresponding properties of the price impact functions. The results clarify the implicit assumptions made when selecting a particular form of the cost function and can be used when choosing the correct portfolio optimization framework.

A major challenge in cross-language information retrieval (CLIR) is the adoption of translation knowledge in retrieval models, as it affects term weighting which is known to highly impact the retrieval performance. Despite its importance, how different approaches for integration of translation knowledge into retrieval models relatively perform has not been analytically examined. In this paper, we present an analytical investigation of using translation knowledge in CLIR. In particular, by adopting the axiomatic analysis framework, we formulate impacts of using translation knowledge on document ranking as constraints that any cross-language retrieval model should satisfy. We then consider state-of-the-art CLIR methods and check whether they satisfy these constraints. Our study shows that none of the existing methods satisfies all constraints. Based on the defined constraints, we propose the hierarchical query modeling method for CLIR which satisfies more constraints and achieves a higher CLIR performance, compared to the existing methods.

Jeffrey conditionalization is a rule for updating degrees of belief in light of uncertain evidence. It is usually assumed that the partitions involved in Jeffrey conditionalization are finite and only contain positive-credence elements. But there are interesting examples, involving continuous quantities, in which this is not the case. Q1 Can Jeffrey conditionalization be generalized to accommodate continuous cases? Meanwhile, several authors, such as Kenny Easwaran and Michael Rescorla, have been interested in Kolmogorov’s theory of regular conditional distributions (rcds) as a possible framework for conditional probability which handles probability-zero events. However the theory faces a major shortcoming: it seems messy and ad hoc. Q2 Is there some axiomatic theory which would justify and constrain the use of rcds, thus serving as a possible foundation for conditional probability? These two questions appear unrelated, but they are not, and this paper answers both. We show that when one appropriately generalizes Jeffrey conditionalization as in Q1, one obtains a framework which necessitates the use of rcds. It is then a short step to develop a general theory which addresses Q2, which we call the theory of extensions. The theory is a formal model of conditioning which recovers Bayesian conditionalization, Jeffrey conditionalization, and conditionalization via rcds as special cases.

A way of studying the synchronization of relaxation self-oscillations is considered. The technique is based on a modified axiomatic approach that uses the properties of uniform near-periodic functions. Five regimes of synchronizing neural ensembles of 100 peripheral neurons in the shaping of attention are studied. The adequacy of the mathematical computational model used to describe the process of shaping attention is discussed.

We present a proof of the equivalence between two deductive systems for the constructive modal logic S4. On one side, an axiomatic characterization inspired by Hakli and Negri's Hilbert-style system of derivations from assumptions for modal logic K. On the other side, the judgmental reconstruction given by Pfenning and Davies by means of a so-called dual natural deduction approach that makes a distinction between valid, true and possible formulas. Both systems and the proof of their equivalence are formally verified using the Coq proof assistant.

Using schemes of Euclid's geometrical concepts in long-term memory to understand hyperbolic geometry and elliptic geometry concepts with assimilation and accommodation allows for cognitive conflict. This study aims to reduce the occurrence of cognitive conflict by understanding the mathematical content of the three of Euclidean geometries, hyperbolic and elliptic. The research was conduct used descriptive exploratory. The results indicate that Euclid's geometry representation is still used in representing hyperbolic and elliptic geometry so that cognitive conflict occurs. Cognitive conflicts that occur are related to the position of two lines, parallels, two triangles with the same that correspond angles, intersects one of two parallel lines, the number of angles in a triangle, and Sacherri's valid hypothesis. The efforts that can be made to reduce the occurrence of cognitive conflict are to change existing schemes or create new schemes so that the information obtained can be combined into existing schemes in a deductive axiomatic approach to material content through the accommodation process