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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.

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

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

This paper introduces the inquisitive extension of R, denoted as InqR, which is a relevant logic of questions based on the logic R as the background logic of declaratives. A semantics for InqR is developed, and it is shown that this semantics is, in a precisely defined sense, dual to Routley-Meyer semantics for R. Moreover, InqR is axiomatized and completeness of the axiomatic system is established. The philosophical interpretation of the duality between Routley-Meyer semantics and the semantics for InqR is also discussed.

Considering L being a complete Heyting algebra, this paper mainly proposes a general framework of L-rough approximate operators in which constructive and axiomatic approaches are used. In the constructive approach, upper and lower L-rough approximate operators are introduced and their connections with L-relations are investigated. In the axiomatic approach, various types of set-theoretic L-operators are defined. It is shown that each type of L-rough approximate operators corresponding to special kind of L-relations, including serial, reflexive, symmetric, transitive, mediate, Euclidean and adjoint L-relations as well as their compositions, can be characterized by single axioms.

In the design process of cognitive human-machine systems, focus is on novel engineered components while the human component is generally considered as known, at least at large: as the cognitive ability of a person expresses in communication, verbal and behavioral, within his or her life–sphere surroundings, whether kindergarten, school, neighborhood, work place, leisure activities, or elderly home, its characteristics are persistent patterns that emerge with biological and cognitive development during childhood and adolescence, are shaped by social interaction in adult life, and finally modified by shrinking vitality in biological aging. However, such phenomenalistic categorization is certainly inadequate as communication is actually an expression of in–system functional dynamics and their controls. It is then their high degree of interweavement across several scale levels, that also interlaces the controls of reproductive subsystems with all other functional subsystems within the human body and thus mandates a fundamental distinction between male and female cognition beyond pregnancy and maternity. As the structure of physiological couplings of functional dynamics within the human body is not understood, impact from outside on within-dynamics is analytically unforeseeable for the machine component. A recent axiomatic theory of multi–scale holistic functional biodynamics for human–body system suggests a concept for functionally equivalent virtual machines, “human–similar AI robots”, that “live” through a human life–cycle from childhood to old age, as appropriate for each sex. Connected to it is an understanding of human social life’s driving forces as anxiety about loosing (access to) life resources.

In this paper, we refer to the axiomatic theory of risk and investigate the problem of formal verification of the expected shortfall (ES) model based on a sample ES. Recognizing the infeasibility of parametric methods, we explore the bootstrap technique, which, unlike the current value-at-risk model-based (VaR model-based) Basel III testing framework, permits the creation of more powerful sample ES-based procedures. Our contribution to the debate on the possibilities of sample ES-based testing is twofold. First, we introduce a bootstrap test based on the idea of ES prediction corrected variables. In this way, we obtain a procedure that makes no distributional assumptions about the underlying returns process, and whose p-value computation does not assume any asymptotic convergence. Second, we provide a unifying framework for ES value verification, in which we compare alternative sample ES-based approaches: the residual-based procedures versus the ES prediction corrected tests as well as the VaR model-dependent approach versus the fixed failure rate tests. By examining its statistical properties and practical applicability, we find evidence that the proposed bootstrap procedure, based on ES prediction corrected variables, is superior to other methods. This provides important guidance for developing international standards of market risk management.

Profit allocation plays an important role in the decision-making field. In this paper, we study an allocation method on restricted coalition cooperation with intuitionistic fuzzy coalitions, in which the partners have some hesitation degree and different risk preferences when they participate in a limited communication structure game. In order to sufficiently analyze the profit allocation strategy, an average tree solution (A-T solution) with Choquet integrals and hesitation degrees is studied. In particular, a simple solving method for the A-T solution is proposed by proving that the characteristic functions of the cooperative game satisfy the monotonicity condition. Using this method, the upper and lower bounds of the A-T solution can be calculated directly from the upper and lower bounds of the interval characteristic functions. This method avoids the subtraction of interval numbers. Furthermore, the properties of the A-T solution according to an axiomatic system are proved in this paper. Finally, the applicability and superiority of the proposed approach are demonstrated through comparison with other methods.

Privacy, in the sense of control over access to one’s personal information, is a central concern in the context of online decision making, both in general and in relation to online platforms in particular. For at least some agents, a belief that one online platform jeopardizes users’ privacy more than another may tip the scales in favor of the latter. Thus, understanding how privacy considerations come into play is central for any economic or social analysis. To this end, we study how agents rank online platforms (or mechanisms, as we call them) from a privacy perspective. We propose a very simple model of privacy-jeopardizing mechanisms, along with a normative methodology for understanding how these mechanisms are ranked. Similarly to classic work in decision theory, we postulate several axioms that we believe a privacy order should satisfy, and then characterize the set of orders that comply with these axioms. These orders turn out to be related to the notion of f -divergence from information theory, one example of which is KL divergence. We test the usefulness of our theoretical result by using it to rank clustering models based on data provided by the Recommendation Team at Microsoft Research.

This paper looks into the writings of Jurgen Rodig (1942–1975) on legal logic with the aim of identifying those results which still appear to be essential contributions to legal theory. Of particular relevance are (i) the reasons why propositional and predicate logic in its classsical form should be considered adequate and by and large sufficient to represent formal legal reasoning. Other issues discussed in this paper are (ii) Rodig’s views concerning specifically legal variants of logical arguments, (iii) the axiomatic method in legal theory, (iv) ways of substantiating general norms as a central step in what constitutes the legal syllogism, (v) the correct method of interpreting written law, (vi) the question as to whether there are formal characteristics of general norms, and (vii) structural properties of legal definitions and of similar legislative drafting tools.

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

This paper aims to characterize the relationship between information as defined in the information-theoretic approach and linguistic meaning by way of formulation of computations over the lexicon of a natural language. Information in its information theoretic sense is supposed not to be equivalent to linguistic meaning, whereas linguistic meaning has an intrinsic connection to information as far as the form and structure of the lexicon of a language (in a non-lexicological sense) is concerned. We argue that these two apparently conflicting aspects of the relationship between information and linguistic meaning can be unified by showing that information conserves linguistic meaning, only insofar as computations of a certain kind are defined on the symbolic elements of a lexicon. This has consequences not merely for the nature of lexical learning – natural or artificial or otherwise – but also for the conservation of information in axiomatic systems.

In this article, we wish to explore the influence of the figure and the drawing on students’ argumentation and proof during the solving problem. Our research is based on both Toulmin’s model and Vinner’s concept image and concept definition. After analysing the arguments with the Toulmin model, we analyse the personnel concept definition, concept image evoked about the figure and the effect of drawing which intervene in students’ arguments. Our data suggest that wrong student’s arguments seem to be based on their concept image evoked on the figure manipulated which is in contradiction with the formal axiomatic system. Moreover, the data of the arguments seem to come from an abusive interpretation of the drawings. Observations that emerge from students’ oral and written speeches reveal continuities and gaps between their argumentation and proof.

In a classical 1976 paper, Solovay proved the arithmetical completeness of the modal logic GL; provability of a formula in GL coincides with provability of its arithmetical interpretations of it in Peano Arithmetic. In that paper, he also provided an axiomatic system GLS and proved arithmetical completeness for GLS; provability of a formula in GLS coincides with truth of its arithmetical interpretations in the standard model of arithmetic. Proof theory for GL has been studied intensively up to the present day. However, it might sound somewhat strange that no proof theory for GLS was ever developed nor even suggested thus far, except for the axiomatic system offered by Solovay. In this paper, we develop a proof theory for GLS based on the sequent calculus method. We provide a sequent calculus for GLS and prove the cut-elimination and some standard consequences of it: the interpolation and definability theorems. As another consequence of cut-elimination, we also prove the equivalence of GL and GLS with respect to a special form of formulas which we call Godel sentences, using a purely proof-theoretical method.

Introduction. The article is devoted to identification of interrelations between the modeling of legal development of public relations and global issues. The necessity of using modeling methods in the development of regulatory impact mechanisms and the importance of taking into account modern megatrends in this process are substantiated. Although the history of researching legal development modeling accounts for more than half a century, the establishment of the relationship of modern problems with the methods of legal forecasting is a relatively new area of research.Materials and methods. The methodological basis of the work consists of general scientific and special methods of knowledge of legal phenomena and processes, such as the method of historical and logical analysis, the structural method, the axiomatic method and the method of comparative law. The UN data on demographic dynamics of the population are used.Research results. The article presents a structural and legal model for achieving the UN sustainable development goals, which comprises two levels: elaborative and integral. In the course of the analysis significant interrelations between modern global problems and legal development modeling have been established.Discussion and conclusions. The discussion of solutions to global problems in the scientific literature highlighted the importance of joint efforts to achieve the desired result. According to the author, legal development modeling can become one of the effective tools for solving current global problems of our time.

Nowadays, the methods of program-targeted management for the development of various socio-economic systems of complex structure, such as agricultural areas, have become ubiquitous. Therefore, the current tasks at hand are the verification of already created development programs and the development of “proper” programs for the development of such systems, by analogy with the verification and development of proper computer programs through advanced disciplines in theoretical programming. In this paper, in order to solve the problem of the verification of development programs for agricultural territories, a structural scheme of the program is first constructed, through which an axiomatic theory is created using the Hoare’s algorithmic logic system. The main problem in the construction of the axiomatic theory is the development of the axioms of the theory that reflect preconditions and effects of the implementation of meaningful actions indicated in the text of the development program. The verification of the development program corresponds to the provability check of some Hoare triplet, according to the initial and target conditions of the program. For the task of elaborating proper development programs, we describe the mechanism for constructing a domain model using the PDDL family description languages. The description of a specific model is purely declarative in nature and consists of descriptions of predicates and actions of the chosen subject area. In this paper, using the described model with the help of intelligent planners, including temporal planners such as OPTIC, we show how to automatically build solutions to the targets of development programs. Based on expert knowledge and industry standards, a model of an agricultural territory is constructed, a brief description of which is given in this work. The conducted experiments showed the effectiveness of the proposed approach for the development of proper development programs.

New approaches that allow to expand the logical capabilities of control algorithms based on a thorough analysis of the logic of the decision making process for formalizing the process of object management is considered. Based on the analysis of decision-making methods, it is established that the automation of decision-making processes in the event of emergencies of an anthropogenic nature in the management of forces and facilities should be based on the semiotic principle. To solve the problem of developing solutions in emergency situations of anthropogenic nature, the axiomatic approach is the most acceptable. The development of a method for the synthesis of solution options allows the use of adaptive control algorithms with the modeling of human mental activity. The use of an axiomatic approach to formalizing and solving problems of developing solutions for managing the risks of an industrial accident can only be formalized on the basis of quantitative mathematical methods.

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

Behavior-related neuronal signals often vary between neurons. Despite the unreliability of individual neurons, brains are able to accurately represent and drive behavior. The notion may also apply to economic (‘value-based’) choices and the underlying reward signals. Reward value is subjective and can be defined by nonlinear weighting of magnitude (utility) and probability. Using a wide variety of reward magnitude and probability, we assessed subjective reward value at choice indifference between safe and risky rewards as prescribed by the continuity axiom that provides stringent criteria for meaningful choice. We found that individual neurons in the orbitofrontal cortex (OFC) of monkeys carry unreliable and heterogeneous neuronal signals for subjective value that largely fails to match the animal’s choice. However, the averaged neuronal signals matched well the animals’ choices, suggesting reliable subjective economic value encoding by the observed population of unreliable neurons.HighlightsDifferent from widely held views, reliable neuronal information processing may not require reliable processors.Neurons in monkey orbitofrontal cortex (OFC) process reward magnitude and probability heterogeneously and unreliably.Despite unreliable neuronal processing, OFC population activity codes choices reliably.Reliability systems performance from unreliable elements seems to be a broad feature of neuronal reward coding.In briefUsing stringent concepts of behavioral choice, Ferrari-Toniolo and Schultz describe unreliable individual reward neurons in monkey orbitofrontal cortex whose activity combines to a reliable population code for economic choice.

Intercultural Communication Competence is essential in creating healthy relations and prevent conflict across numerous cultures. It also emphasizes the essence of accurate and proper information transmission among people. This study aimed to analyze existing literature and derive the perfect formula for learners to develop their Intercultural Communication Competence. A theory on Intercultural Communication Competence was generated using the deductive axiomatic approach. After the analysis and synthesis of related literature and studies, twelve axioms were formulated. From there, four propositions were identified: (1) Intercultural Communication Competence entails cultural actualization; (2) Intercultural Communication Competence calls for a comprehensive curriculum and assessment; (3) Digitalization boosts Intercultural Communication Competence; (4) Upskilled instructors are confident in refining learners’ Intercultural Communication Competence. With these propositions, the Interdependent Theory of Intercultural Communication Competence was generated. This theory highlights the importance of the interdependence among cultural actualization, comprehensive curriculum, digitalization, and upskilled instructors in realizing the full potential of Intercultural Communication Competence of learners. Article visualizations