Axiomatic Approach Research Articles

Axiomatization of Boolean Connexive Logics with syncategorematic negation and modalities

Abstract In the article we investigate three classes of extended Boolean Connexive Logics. Two of them are extensions of Modal and non-Modal Boolean Connexive Logics with a property of closure under an arbitrary number of negations. The remaining one is an extension of Modal Boolean Connexive Logic with a property of closure under the function of demodalization. In our work we provide a formal presentation of mentioned properties and axiom schemata that allow us to incorporate them into Hilbert-style calculi. The presented axiomatic systems are provided with proofs of soundness, completeness and decidability. The properties of closure under negation and demodalization are motivated by the syncategorematic view on the connectives of negation and modalities, which is discussed in the paper.

A generalized TODIM approach based on information measurement and consensus building model in probabilistic hesitant fuzzy environment

Information measurement and decision-making methods are two important aspects in the application of probabilistic hesitant fuzzy elements (PHFEs), and these issues are crucial for improving decision quality and enhancing the applicability of PHFSs. This paper focuses on the entropy measure and score function of PHFEs, and then proposes a generalized TOIDM method based on information measurement and consensus building. In this contribution, we develop a novel non-probabilistic entropy measure by two non-fuzzy elements and a novel score function based on sigmoid function. The novel non-probabilistic entropy of PHFEs can comprehensively reflect the dynamic changes of fuzzy uncertainty and hesitant uncertainty, and provide a new axiomatic framework. The novel score function of PHFEs can synthesize multiple information effectively with strong integrity. In the sequel, the two proposed information measurement methods are compared with existing methods to further verify the validity of the developed methods. Eventually a consensus building model is constructed based on the proposed information measurement method, and then a generalized TODIM approach based on information measurement and consensus building is proposed for probabilistic hesitant fuzzy environment. The developed approach is demonstrated with an emergency rescue planning decision-making problem. Sensitivity analysis and comparative analysis are employed to verify the validity and stability of the approach and information measures proposed in this paper. Based on the findings of this study, we conclude that the proposed methods effectively enrich the existing information measurement methods and decision-making methods in the field of probabilistic hesitant fuzzy.

Why Kahneman matters

Abstract Daniel Kahneman's legacy is best understood in light of developments in economic theory in the early-mid-20th century, when economists were eager to put utility functions on a firm mathematical foundation. The axiomatic system that provided this foundation was not originally intended to be normative in a prescriptive sense but later came to be seen that way. Kahneman took the axioms seriously, tested them for descriptive accuracy, and found them wanting. He did not view the axioms as necessarily prescriptive. Nevertheless, in the research program he conceived, factual discoveries about real decision-making were stated as deviations from the axioms and thus deemed ‘errors’. This was an unfortunate turn that needs to be corrected for the psychological enrichment of economics to proceed in a productive direction.

Towards a probabilistic foundation of relativistic quantum theory: the one-body Born rule in curved spacetime

In this work, we establish a novel approach to the foundations of relativistic quantum theory, which is based on generalizing the quantum-mechanical Born rule for determining particle position probabilities to curved spacetime. A principal motivator for this research has been to overcome internal mathematical problems of relativistic quantum field theory (QFT) such as the ‘problem of infinities’ (renormalization), which axiomatic approaches to QFT have shown to be not only of mathematical but also of conceptual nature. The approach presented here is probabilistic by construction, can accommodate a wide array of dynamical models, does not rely on the symmetries of Minkowski spacetime, and respects the general principle of relativity. In the analytical part of this work, we consider the 1-body case under the assumption of smoothness of the mathematical quantities involved. This is identified as a special case of the theory of the general-relativistic continuity equation. While related approaches to the relativistic generalization of the Born rule assume the hypersurfaces of interest to be spacelike and the spacetime to be globally hyperbolic, we employ prior contributions by C. Eckart and J. Ehlers to show that the former condition is naturally replaced by a transversality condition and that the latter one is obsolete. We discuss two distinct formulations of the 1-body case, which, borrowing terminology from the non-relativistic analog, we term the Lagrangian and Eulerian pictures. We provide a comprehensive treatment of both. The main contribution of this work to the mathematical physics literature is the development of the Lagrangian picture. The Langrangian picture shows how one can address the ‘problem of time’ in this approach and, therefore, serves as a blueprint for the generalization to many bodies and the case that the number of bodies is not conserved. We also provide an example to illustrate how this approach can in principle be employed to model particle creation and annihilation.

Exploring the foundations of Islamic moral economy: a return to substantive morality

Purpose This paper aims to explore the general flaws in the theorization of Islamic economics, particularly its failure to incorporate substantive morality into its framework. Since a considerable number of Islamic economic studies remain within the bounds of instrumental morality, this paper seeks to propose an alternative understanding of Islamic economics by emphasizing the essential role of Islamic moral economy (IME) through substantive morality. Design/methodology/approach The study critically examines the theorization of Islamic economics through a discursive methodology. The adopted method and strategy allow for the discursive evaluation of conceptual developments within the field of modern Islamic economics. It proposes an authentic definition of IME by outlining its paradigmatic framework of political economy, value system, axiomatic approach and operational principles. Findings Islamic economics has been theorized in a reactive and reflexive manner, aiming to provide an alternative to the neoclassical hegemonic discourse. However, it has largely neglected its inherent moral dimension, focusing instead on instrumental morality and failed to embed substantive morality into its system. By introducing an alternative reading of Islamic economics, this study presents IME as a distinct economic model that centers substantive morality in its theorization. Originality/value This paper contributes to the existing literature by offering a novel perspective on Islamic economics that challenges the prevalent instrumental approach and emphasizes the importance of substantive morality. It provides a comprehensive framework for understanding IME as a distinct economic model, thereby enriching the discourse on Islamic economics with an authentic and morally grounded paradigm.

Free vibration and buckling characteristics of temperature dependent FGCNT reinforced plates in finite element framework of a non-polynomial axiomatic approach

In the present work, a generalized finite element formulation is established and implemented for the free vibration and buckling characteristics for temperature dependent carbon nanotube (CNT) reinforced plate. The uniform and functionally graded distribution of CNT is assumed in the plate and the plate is modeled in non-polynomial axiomatic framework. The extended rule of mixture is employed for computing the equivalent material parameters which are temperature dependent. Further, an eight-noded C0 continuous element with seven degree of freedom is adopted for discretization of the plate domain. The convergence and comparison studies are carried out to ensure the accuracy and effectiveness of the proposed formulation. As temperature conditions significantly influence structures integrity, on that account, thorough parametric studies are performed to examine the influence of volume fraction (VCNT ), span to thickness ratio ( a / h ), CNT distribution and boundary conditions on the FG-CNTR plate for the different temperature conditions. The results demonstrate that the efficient use of CNT distribution and boundary conditions can consequently improve free vibration and buckling characteristics of the temperature dependent FG-CNTR plates.

Logics for contact and measure

Abstract We enrich contact algebras with a new binary relation that compares the size of regions, and provide axiom systems for various logics of contact and measure. Our contribution is three-fold: (1) we characterize the relations on a Boolean algebra that derive from a measure, thereby improving an old result of Kraft, Pratt and Seidenberg; (2) for all $n \geq 1$, we axiomatize the logic of regular closed sets of ${\mathbb{R}}^{n}$ with null boundary; (3) considering a broad class of equational theories that contains all logics of contact, we prove that they all have unitary or finitary unification, and that unification and admissibility are decidable.

Some advances in cooperative game theory: Indivisibilities, externalities and axiomatic approach

Some advances in cooperative game theory: Indivisibilities, externalities and axiomatic approach

View-based axiomatic reasoning for the weak memory models PSO and SRA

Weak memory models describe the semantics of concurrent programs in modern multicore architectures. As these semantics deviate from the commonly assumed model of sequential consistency, reasoning techniques like Owicki-Gries-style proof calculi need to be adapted to specific memory models. To avoid having to design a new proof calculus for every new memory model, a uniform approach for axiomatic reasoning has recently been proposed. This approach bases reasoning on memory-model independent axioms about thread views and how they are changed by program actions like reads and writes. It allows to prove program correctness based on axioms only. Such proofs are valid for all memory models instantiating the axioms.In this paper, we study instantiations of the axioms for two memory models, the Partial Store Order (PSO) and the Strong Release Acquire (SRA) model. We see that both models fulfil all but one axiom, a different one though. For PSO, the missing axiom refers to message-passing abilities of memory models; for SRA, the missing axiom refers to the independence of actions on executing threads. We discuss the consequences of these missing axioms and illustrate the reasoning technique on a specific litmus test.

Impossibility of universal work extraction from coherence: reconciling axiomatic and resource-theory approaches

Abstract We compare how the impossibility of a universal work extractor from coherence arises from different approaches to quantum thermodynamics: an explicit protocol accounting for all relevant quantum resources, and axiomatic, information-theoretic constraints imposed by constructor theory. We first explain how the impossibility of a universal work extractor from coherence is directly implied by a recently proposed constructor-theoretic theorem based on distinguishability, which is scale- and dynamics- independent. Then we give an explicit demonstration of this result within quantum theory, by proving the impossibility of generalising a proposed quantum protocol for deterministically extracting work from coherence. We demonstrate a new connection between the impossibility of universal work extractors and constructor-based irreversibility, which was recently shown using the quantum homogenizer. Finally we discuss additional avenues for applying the constructor-theoretic formulation of work extraction to quantum thermodynamics, including the irreversibility of quantum computation and thermodynamics of multiple conserved quantities.

Two novel pure-state coherence measures in quantifying coherence

Quantification is one of the primary goals of quantum coherence resource theory. Here, we put forward two coherence measures, analytically prove their validity in the pure-state regime following the axiomatic definition of a legitimate pure-state coherence measure (PSCM), and provide their general normalized forms. We further discuss their extensions in the domain of mixed states with the assistance of the convex roof of coherence, and top coherence (quantification in terms of pure state coherence) and define their classes (coherence \emph{monotone} or \emph{measure}); in this regard, we thoroughly investigate the top coherence class. For the quantitative demonstration, we pick up different laser pulse-two-qubit interaction scenarios and compare the evolutions of these two coherence measures along with \(l_{1}\)-norm of coherence and relative entropy of coherence; this study also signifies the importance of coherence in probing an atomic or molecular system.

Incompleteness of boundedly axiomatizable theories

Our main result (Theorem A) shows the incompleteness of any consistent sequential theory T T formulated in a finite language such that T T is axiomatized by a collection of sentences of bounded quantifier-alternation-depth. Our proof employs an appropriate reduction mechanism to rule out the possibility of completeness by simply invoking Tarski’s Undefinability of Truth theorem. We also use the proof strategy of Theorem A to obtain other incompleteness results.

An Axiomatic Approach in the Development of the Theory of Early Childhood Teachers’ Work Engagement

An Axiomatic Approach in the Development of the Theory of Early Childhood Teachers’ Work Engagement

A semantic axiomatic design for integrity in IoT

AbstractIn the complex, critical, and rapidly evolving era of Industry 4.0 manufacturing, mature engineering disciplines can be developed to validate system designs at an abstract level, enabling effective fault‐free environments. However, designing integrated IoT platforms presents significant challenges due to their inherent complexity and interoperability issues. To address this challenge, the proposed framework integrates axiomatic design principles with considerations for data integrity and domain ontology. Furthermore, this paper introduces a multi‐level integrity rule repository for IoT application to provides a robust foundation for designing and maintaining IoT systems. To validate the effectiveness of this approach, we conducted a study across four domains of smart cities. The design's quality was assessed using model quality parameters, including completeness, consistency and validity. Our analysis shows that the semantic‐aware axiomatic design framework can effectively address the problem of inconsistencies and low‐quality design in scientific IoT environments, providing a more reliable and efficient solution.

Properties of Group Fairness Measures for Rankings

In recent years, several measures have been developed for evaluating group fairness of rankings. Given that these measures were developed with different application contexts and ranking algorithms in mind, it is not straightforward which measure to choose for a given scenario. Previous work has already laid out some categorizations of measures and explored relationships between these, however, there has not yet been a thorough mathematical analysis of practically grounded properties of individual measures. In this paper, we therefore apply an axiomatic approach to perform a comprehensive analysis of existing group fairness measures that have been developed in the context of fair ranking. To this end, we propose a set of fourteen properties for group fairness measures that consider different ranking settings. These properties specifically provide information about how fairness scores of ranked outputs can be interpreted and contextualized. For a given use case, one can then identify which properties are of interest, and select a measure based on whether it satisfies these properties. We further apply our properties on twelve existing group fairness measures, and through both empirical and theoretical results demonstrate that most of these measures only satisfy a small subset of the proposed properties. These findings highlight limitations of existing measures, and provide insights into how to evaluate and interpret different fairness measures in practical deployment. Overall, our work can assist practitioners in selecting appropriate group fairness measures for a specific application, and also aid researchers in designing and evaluating such measures.

Entropy measures of multigranular unbalanced hesitant fuzzy linguistic term sets for multiple criteria decision making

The hesitant fuzzy linguistic term set (HFLTS) is an efficient tool for modeling linguistic information in multi-criteria decision making (MCDM), and the entropy measure of HFLTS, as a substantial representation of uncertainty, merits additional investigation. This article aims to exploit a general framework to facilitate the construction of entropy measure for multigranular unbalanced HFLTS. An axiomatic definition of the entropy for HFLTSs that considers both types of uncertainty (fuzziness and hesitation) is presented, with the entropy measure subsequently derived from distance-based mapping. From this definition, several deduced results have been developed for the mapping that depicts the entropy expression in order to get such functions with ease. Whereafter, a MCDM weight-determining model for multigranular unbalanced linguistic information without preset weights is devised, and an empirical application of the suggested model in MCDM is illustrated. Ultimately, comparisons and analyses with existing studies are conducted to demonstrate the advantages of the proposed work.

Special cubic zeros and the dual variety

AbstractLet be a diagonal cubic form over in six variables. From the dual variety in the delta method of Duke–Friedlander–Iwaniec and Heath‐Brown, we unconditionally extract a weighted count of certain special integral zeros of in regions of diameter . Heath‐Brown did the same in four variables, but our analysis differs and captures some novel features. We also put forth an axiomatic framework for more general .

Descent modulus and applications

The norm of the gradient ‖∇f(x)‖ measures the maximum descent of a real-valued smooth function f at x. For (nonsmooth) convex functions, this is expressed by the distance dist(0,∂f(x)) of the subdifferential to the origin, while for general real-valued functions defined on metric spaces by the notion of metric slope |∇f|(x). In this work we propose an axiomatic definition of descent modulus T[f](x) of a real-valued function f at every point x, defined on a general (not necessarily metric) space. The definition encompasses all above instances as well as average descents for functions defined on probability spaces. We show that a large class of functions are completely determined by their descent modulus and corresponding critical values. This result is already surprising in the smooth case: a one-dimensional information (norm of the gradient) turns out to be almost as powerful as the knowledge of the full gradient mapping. In the nonsmooth case, the key element for this determination result is the break of symmetry induced by a downhill orientation, in the spirit of the definition of the metric slope. The particular case of functions defined on finite spaces is studied in the last section. In this case, we obtain an explicit classification of descent operators that are, in some sense, typical.

Sequent calculi and an efficient theorem prover for conditional logics with selection function semantics

Abstract In this paper we present our final solution to the problem of designing an efficient theorem prover for Conditional Logics with the selection function semantics. Conditional Logics recently have received a renewed attention and have found several applications in knowledge representation and artificial intelligence. In order to provide an efficient theorem prover for Conditional Logics, we introduce labelled sequent calculi for the logics characterized by well-established axioms systems including the axiom of strong centering CS, the axiom of conditional identity ID, the axiom of conditional modus ponens MP, as well as the conditional third excluded middle CEM, rejected by Lewis but endorsed by Stalnaker, as well as for the whole cube of extensions. The proposed calculi revise and improve the calculi SeqS introduced in Olivetti et al. (2007, ACM Trans. Comput. Logics, 8). We also present an implementation of these calculi in SWI Prolog, including a graphical interface in Python as well as standard heuristics and refinements that allow us to obtain an efficient theorem prover for the logics under consideration. Moreover, we present some statistics about the performances of the theorem prover, which are promising and significantly better than those of its predecessor CondLean, an implementation of the calculi SeqS.

An axiom system for absolute plane geometry in terms of collinearity and congruence

AbstractWe present an axiom system in terms of collinearity and congruence for plane geometry independent of assumptions about parallels. The line reflections and motions of corresponding geometries are studied, and it is shown that the axiom system characterizes the non-elliptic metric planes in the sense of Bachmann with midpoints. If the hyperbolic axiom is added, we show that the familiar algebraic models of hyperbolic planes result.