Axiomatic Approach Research Articles

Logics for reasoning about degrees of confirmation

Abstract In this paper, we present a first-order and a propositional logic for reasoning about degrees of confirmation. We define the appropriate formal languages and describe the corresponding classes of models. We provide infinitary axiomatizations for both logics and we prove that the axiomatizations are sound and strongly complete. We also show that our propositional logic is decidable. For some restrictions of the logics, we provide finitary axiomatic systems.

A decomposition of general premium principles into risk and deviation

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

Trembling-Hand Perfection and Correlation in Sequential Games

We initiate the study of trembling-hand perfection in sequential (i.e., extensive-form) games with correlation. We introduce the extensive-form perfect correlated equilibrium (EFPCE) as a refinement of the classical extensive-form correlated equilibrium (EFCE) that amends its weaknesses off the equilibrium path. This is achieved by accounting for the possibility that the players may make mistakes while following recommendations independently at each information set of the game. After providing an axiomatic definition of EFPCE, we show that one always exists since any perfect (Nash) equilibrium constitutes an EFPCE, and that it is a refinement of EFCE, as any EFPCE is also an EFCE. Then, we prove that, surprisingly, computing an EFPCE is not harder than finding an EFCE, since the problem can be solved in polynomial time for general n-player extensive-form games (also with chance). This is achieved by formulating the problem as that of finding a limit solution (as $\epsilon \rightarrow 0$) to a suitably defined trembling LP parametrized by $\epsilon$, featuring exponentially-many variables and polynomially-many constraints. To this end, we show how a recently developed polynomial-time algorithm for trembling LPs can be adapted to deal with problems having an exponential number of variables. This calls for the solution of a sequence of (non-trembling) LPs with exponentially-many variables and polynomially-many constraints, which is possible in polynomial time by applying an ellipsoid against hope approach.

On the Methods of Constructing Hilbert-type Axiom Systems for Finite-valued Propositional Logics of Łukasiewicz

The article explores the following question: which among the most often examined in the literature method of constructing Hilbert-type axiom systems for finite-valued propositional logics of Łukasiewicz – the Rosser-Turquette method, the Tuziak method or the Grigolia system of axiom schemata – is in fact applicable in the researches on many-valued logics? Although the method offered by Rosser and Turquette is considered to be a solution of the problem of axiomatizability of finite-valued logics of Łukasiewicz, it has a serious limitation in producing adequate axiom systems. If the Rosser-Turquette axiom schemata are expressed just in terms of the standard propositional connectives which are definable in terms of the propositional connectives of Łukasiewicz’s logics, then every Rosser-Turquette axiom system for a Łukasiewicz’s logic is semantically incomplete. The article also examines the Tuziak axiom systems that actually axiomatize Łukasiewicz’s finite-valued logics and can be applied in the logical researches. The article compares the Tuziak axiom systems with the Grigolia set of axiom schemata, and it demonstrates that since a crucial in the Grigolia method definition of connectives is in fact invalid in Łukasiewicz’s logics, the Grigolia method does not provide sound axiom systems for Łukasiewicz’s logics.

Evaluating radial efficiency considering environmental factors: A generalization of classical DEA

Environmental issues are becoming a significant and serious worldwide problem. Undesirable outputs like waste and pollution are by-products of desirable outputs in a production system. To motivate environment-friendly production, undesirable outputs should be included in the performance assessment of firms, organizations and countries. Data Envelopment Analysis (DEA) is one of the most successful methods of efficiency evaluation. Unfortunately, there is no reliable radial efficiency measure for environmental assessment in DEA. In this paper, the conventional modeling in DEA is extended to develop a new radial model for efficiency assessment in the presence of undesirable outputs. This method enables more accurate modeling of the problem. It opens a new door to the issue, and shows more reliabilities than the existing methods. The new efficiency measure is more rigorous, and provides higher discriminating power. Evaluating divisional efficiency is another advantage of the model. It also provides credible targets with greater practical accessibility for each inefficient decision making unit. The model is studied by an axiomatic approach and employed to evaluate the performance of 28 countries in economic, social and environmental dimensions. The results show that only one country achieves efficiency, and the average environmental efficiency is considerably lower than the operational efficiency among the studied countries.

Restrictions on shareability of classical correlations for random multipartite quantum states

Unlike quantum correlations, the shareability of classical correlations (CCs) between two-parties of a multipartite state is assumed to be free since there exist states for which CCs for each of the reduced states can simultaneously reach their algebraic maximum value. However, when one randomly picks out states from the state space, we find that the probability of obtaining those states possessing the algebraic maximum value is vanishingly small. We explore the possibility of nontrivial upper bound by Haar uniformly generating random multipartite states and computing the frequency distribution for various CC measures, conventional classical correlators, and two axiomatic measures of classical correlations, namely the classical part of quantum discord and local work of work-deficit. We find that the distributions are typically Gaussian-like and their standard deviations decrease with the increase in number of parties. It also reveals that among the multiqubit random states, most of the reduced density matrices possess a low amount of CCs which can also be confirmed by the mean of the distributions, thereby showing a kind of restrictions on the sharability of classical correlations for random states. Furthermore, we also notice that the maximal value for random states is much lower than the algebraic maxima obtained for a set of states, and the gap between the two increases further for states with a higher number of parties. We report that for a higher number of parties, the classical part of quantum discord and local work can follow monogamy-based upper bound on sharability while classical correlators have a different upper bound. The trends of sharability for classical correlation measures in random states clearly demarcate between the axiomatic definition of classical correlations and the conventional ones.

Constantin Carathéodory axiomatic approach and Grigory Perelman thermodynamics for geometric flows and cosmological solitonic solutions

We elaborate on statistical thermodynamics models of relativistic geometric flows as generalisations of G. Perelman and R. Hamilton theory centred around C. Carath\'{e}odory axiomatic approach to thermodynamics with Pfaffian differential equations. The anholonomic frame deformation method, AFDM, for constructing generic off--diagonal and locally anisotropic cosmological solitonic solutions in the theory of relativistic geometric flows and general relativity is developed. We conclude that such solutions can not be described in terms of the Hawking--Bekenstein thermodynamics for hypersurface, holographic, (anti) de Sitter and similar configurations. The geometric thermodynamic values are defined and computed for nonholonomic Ricci flows, (modified) Einstein equations, and new classes of locally anisotropic cosmological solutions encoding solitonic hierarchies.

An axiomatic and non-cooperative approach to the multi-step Shapley value

Inspired by the two-step Shapley value, in this paper we introduce and axiomatize the multi-step Shapley value for cooperative games with levels structures. Moreover, we design a multi-step bidding mechanism, which implements the value strategically in subgame perfect Nash equilibrium for superadditve games.

Spur Gear Design using Axiomatic Design Approach

Spur Gear Design using Axiomatic Design Approach

Entropy in Thermodynamics: from Foliation to Categorization

AbstractWe overview the notion of entropy in thermodynamics. We start from the smooth case using differential forms on the manifold, which is the natural language for thermodynamics. Then the axiomatic definition of entropy as ordering on a set that is induced by adiabatic processes will be outlined. Finally, the viewpoint of category theory is provided, which reinterprets the ordering structure as a category of pre-ordered sets.

Real-Valued Systemic Risk Measures

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

The development of Need-Threat Internal Resiliency Theory in COVID-19 crisis utilizing deductive axiomatic approach.

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

Entry Mode Strategy and Firm Performance in Emerging Economy: Moderating Role of Organisational Structure and Environmental Turbulence

Research question: This study assessed the combined moderating effect of organizational structure and environmental turbulence on entry mode strategy - performance link focusing on baby care industry in Lagos State, Nigeria. Motivation: The manufacturers of baby care products in Nigeria have struggled to understand the complexities of entry mode strategy and how to use firm-specific capabilities to contain the threats from new entrants given the idiosyncrasies of the business environment in Nigeria. Also, considering the fast changing business environment and the need for firms to align internal organisational structure to manage the external environmental challenges, this study via the supposition of Hage (1965) axiomatic theory of organisations examined the joint moderating effect of organisational structure and environmental turbulence on entry mode strategy - performance linkage focusing on the baby care industry in Lagos State, Nigeria. Idea: Empirical submissions on the combined moderating effect of organizational structure and environmental turbulence on the interactions of entry mode strategy and organizational performance are sparse. Hence, this study addressed this gap in literature. Data: The survey research design and a sample of 518 employees engaged in FMCG manufacturers in Lagos State, Nigeria were adopted for this study. Tools: A validated structured questionnaire was the instrument of data collection for this study and the hierarchical regression analysis was adopted to test the hypotheses formulated. Findings: The results showed that the interaction between entry mode strategy and firm performance was positive and significant. Further analysis revealed that the interaction term of organizational structure and environmental turbulence accounted for the rise in firm performance to suggest that organizational structure and environmental turbulence are joint significant moderators. This suggests that entry mode strategy appropriateness is key to firm performance and that the fit between organisational structure and the macro-environment is a precondition to higher performance. Contribution: This study adds to recent empirical literature on the link between entry mode strategy, organizational structure, environmental turbulence, and firm performance within emerging economy context, and it provides additional support for the assumptions of the eclectic theory and Hage’s axiomatic theory of organization.

Measures of conflict, basic axioms and their application to the clusterization of a body of evidence

There are several approaches for evaluating conflict within belief functions. In this paper, we develop one of them based on axioms and show its connections to the decomposition approach. We describe a class of conflict measures satisfying this system of axioms and show that measuring conflict can be realized through the clusterization of a body of evidence. We also show that well-known conflict measures like the auto-conflict measure and the measure of dissonance do not satisfy the proposed system of axioms. We also tackle the problem of simplifying a body of evidence based on clusterization and show the application of developed theoretical constructions for data processing.

ТЕОРЕТИКО-МЕТОДОЛОГІЧНИЙ БАЗИС СОЦІАЛЬНО-ЕКОНОМІЧНОЇ ВЗАЄМОДІЇ СУБ’ЄКТІВ ГОСПОДАРЮВАННЯ В УМОВАХ ГЛОБАЛЬНИХ ВИКЛИКІВ

Formulation of the problem. Modern global challenges and imbalances necessitate cooperation in new realities, change the interaction nature to counter global challenges by activating the most effective forms, methods, types and mechanisms of interaction and creating a new values philosophy and motivational guidelines for socio-economic interaction. The processes of socio-economic uncertainty of economic prospects in the context of global challenges require the formation of theoretical and methodological basis for the economic entities interaction. The aim of the research is formation of theoretical and methodological basis for economic entities socio-economic interaction in the context of modern global challenges. The subject of the research is theoretical aspect and methodological aspect of business entities socio-economic interaction in global challenges context. The methods of the research: historical method, methods of verification of theoretical positions (morphological analysis of the content and interrelation of categories, principles and laws, assessment of historical facts to theoretical hypotheses), methods of analysis and synthesis, induction and deduction, logical method (hypothetical and axiomatic approach), comparison method. The hypothesis of the research. In the context of global challenges, it is necessary to change the nature of interaction, for which it is necessary to form a theoretical and methodological basis for such interaction and determine the directions of necessary changes in interaction. The statement of basic materials. Within the epistemological field of research, the definitions of "interaction" as social, economic, philosophical, psychological and managerial categories are analyzed. The providing subsystems of business entities social and economic interaction in the conditions of global challenges are determined, among which the normative and legal, financial, logistical, personnel, informational, logistic, infrastructural, psychological, scientific, institutional, organizational supporting subsystems are singled out and characterized. Within the content (paradigmatic and methodological) basis of the research, it is proposed to distinguish the paradigmatic provisions of socio-economic interaction and the methodological basis of business entities socio-economic interaction research in the context of global challenges. The originality and practical significance of the research. The research of business entities socio-economic interaction in the context of global challenges includes four interrelated blocks – theoretical and explanatory basis of interaction, content (paradigmatic and methodological), applied and managerial basis of interaction, which complex covers possible interaction aspects. Conclusions and perspectives of further research. The article considers the theoretical and methodological basis of business entities interaction during global challenges, within which the implementation of the relevant logical blocks is proposed. Within the framework of further research it is planned to develop an applied basis of interaction, which will provide a diagnostic and prognostic field of socio-economic interaction in the context of global challenges and modeling of of business entities socio-economic interaction, as well as management basis of interaction of business entities, which will provide a sequence of management stages of the socio-economic interaction process and determine the interaction effectiveness in global challenges.

Exploring the Jungle of Intuitionistic Temporal Logics

AbstractThe 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 programming languages with new features via type theory, while from the semantics perspective, several logics for reasoning about dynamical systems and several semantics for logic programming have their roots in this framework. 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. We provide two distinct interpretations of “henceforth”, both of which are natural intuitionistic variants of the classical one. We completely establish the order relation between the semantically defined logics based on both interpretations of “henceforth” and, using our soundness results, show that the axiomatically defined logics enjoy the same order relations.

How to measure the average rate of change?

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.

An axiomatic approach to existence and liveness for differential equations

This article presents an axiomatic approach for deductiveverification of existence and liveness for ordinary differentialequations (ODEs) with differential dynamic logic (dL). The approachyields proofs that the solution of a given ODE exists long enough toreach a given target region without leaving a given evolutiondomain. Numerous subtleties complicate the generalization ofdiscrete liveness verification techniques, such as loop variants, tothe continuous setting. For example, ODE solutions may blow up infinite time or their progress towards the goal may converge to zero.These subtleties are handled in dL by successively refining ODEliveness properties using ODE invariance properties which have acomplete axiomatization. This approach is widely applicable: severalliveness arguments from the literature are surveyed and derived asspecial instances of axiomatic refinement in dL. These derivationsalso correct several soundness errors in the surveyed literature,which further highlights the subtlety of ODE liveness reasoning andthe utility of an axiomatic approach. An important special case ofthis approach deduces (global) existence properties of ODEs, whichare a fundamental part of every ODE liveness argument. Thus, allgeneralizations of existence properties and their proofs immediatelylead to corresponding generalizations of ODE liveness arguments.Overall, the resulting library of common refinement steps enablesboth the sound development and justification of new ODE existenceand of liveness proof rules from dL axioms. These insights are putinto practice through an implementation of ODE liveness proofs inthe KeYmaera X theorem prover for hybrid systems.

LEONARD SAVAGE, THE ELLSBERG PARADOX, AND THE DEBATE ON SUBJECTIVE PROBABILITIES: EVIDENCE FROM THE ARCHIVES

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

Open system dynamics from thermodynamic compatibility

Thermodynamics entails a set of mathematical conditions on quantum Markovian dynamics. In particular, strict energy conservation between the system and environment implies that the dissipative dynamical map commutes with the unitary system propagator. Employing spectral analysis we prove the general form of the ensuing master equation. We compare this result to master equations obtained from standard microscopic derivations. The obtained formal structure can be employed to test the compatibility of approximate derivations with thermodynamics. For example, it designates that global master equations are the compatible choice. The axiomatic approach sheds light on the validity of the secular approximation in microscopic derivations, the form of the steady state in heat transport phenomena, and indicates the lack of exceptional points in the dynamics of open quantum systems.