Structural Axioms Research Articles

Material calculation and its unconscious: approaching computerization with Heidegger and Lacan

The following paper focuses on discerning a specific epistemic effect that modern computers and especially the technology of artificial intelligence (AI) have. To discern this effect, it is necessary to reflect on the use of mathematics, that is its practice, and its ontological underpinnings. To do this, we combine Heideggerian and Lacanian concepts to approach the theoretical problem that AI and computers pose to the practice of calculation. The paper discusses that the computer as a material calculator has limiting factors that make it unable to utilize important uses of formalization. Central to this is the forced absence of virtual voids, which compels computers to act as if the symbolic would behold to the same structural axioms as the imaginary. Far from being a simple inability of computers, this proximate failure allows us instead to understand AI and modern computation in terms of their use and misuse as an epistemic tool.

Axiomatizing the Education System: Analysis, Taxonomy or Professional Development?

The author starts by attempting to Axiomatize the Education System and by clearly outlining what is meant by Axioms and the Axiomatic Method. Essentially. he takes an Axiomatic Approach to defining Axiomatization. He then describes how a facilitated workshop was used to agree the key features of the education system: – a list of short statements (say 100) about education; – cognate groups (say 10) that are used to sort the statements; and – longer statements that describe those groups. The longer statements are initially given the status First Order Axioms, while short statements are given the status of Second Order Axioms. The First Order Axioms were then grouped into: – Fundamental Axioms; – Structural Axioms; and – Functional Axioms. The main body of the paper then clearly articulates: – those First Order Axioms: Pupils; Education; The State; Subsidiarity; The School; Leadership and Management; The Curriculum; Teachers and Teaching; Assessment and Accreditation and Monitoring and Evaluation; along with – the accompanying Second Order Axioms that were generated when the author participated in their own facilitated workshop. The author also provides the diagrammatic overview of the education system that resulted from the facilitated workshop. He concludes that although the approach can be useful, it has its limitations: a number of the key requirements of Axiomatic Systems cannot be met. The education system is so complex that the Axiomatic System derived from an Axiomatic Approach cannot be Consistent and Independent; nor can all the statements be shown to follow logically from statements that are, in turn, the logical result of other statements. The author finally concludes that, although the approach is useful, the terms Axiomatic System and First and Second Order Axioms should be replaced by Taxonomy and First and Second Order Statements. However, he does suggest that developing a Taxonomy of Educational Objectives could be very useful for: – providing an agreed high-level description of the education system; and – in-service education and staff development purposes.

(L,⁎)-valued coarse structures on powersets

The main purpose of this paper is to establish a fundamental framework of L-valued coarse structures. At first an approach to fuzzification of the notion of coarse structure is proposed. An L-valued coarse structure on a set X as an L-fuzzy subset U of the powerset 2X×X satisfying L-valued analogues of the axioms of a coarse structure is given. Some basic properties of L-valued coarse structure spaces are studied. Then we introduce the notion of asymptotic dimension in [0,1]-valued coarse spaces and show that the degree of asymptotic dimension is invariant under fuzzy coarse equivalences.

A Study on Sylow Theorems for Finding out Possible Subgroups of a Group in Different Types of Order

This paper aims at treating a study on Sylow theorem of different algebraic structures as groups, order of a group, subgroups, along with the associated notions of automorphisms group of the dihedral groups, split extensions of groups and vector spaces arises from the varying properties of real and complex numbers. We must have used the Sylow theorems of this work when it's generalized. Here we discuss possible subgroups of a group in different types of order which will give us a practical knowledge to see the applications of the Sylow theorems. In algebraic structures, we deal with operations of addition and multiplication and in order structures, those of greater than, less than and so on. It is through the study of Sylow theorems that we realize the importance of some definitions as like as the exact sequences and split extensions of groups, Sylow p-subgroup and semi-direct product. Thus it has been found necessary and convenient to study these structures in detail. In situations, where it was found that a given situation satisfies the basic axioms of structure and having already known the properties of that structure. Finally, we find out possible subgroups of a group in different types of order for abelian and non-abelian cases.

Entitlements to continued life and the evaluation of population health

We analyze the implications of axioms formalizing entitlements to continued life for the evaluation of population health, when combined with basic structural axioms. A straightforward implication of our analysis is that if the scope of equal entitlements to continued life is not limited, concerns for morbidity are dismissed in the evaluation of population health. Nevertheless, with axioms formalizing a more limited scope of equal entitlement to continued life, we provide several characterization results of focal population health evaluation functions, ranging from lifetime utilitarianism to generalized healthy years equivalent utilitarianism.

The Role of Philosophy of Science in Quantitative Linguistics

Abstract The paper aims to evaluate the role of the philosophy of science in contemporary quanti-tative linguistics. The primary goal is the reflection of the scientific methods and models of scientific explanations (Köhler 1986, 2012) used in quantitative linguistics. The paper shows that the current philosophy of science has adopted some new approaches to ex-planation that are conducive to quantitative linguistics (Craver, Darden 2013, Eck, Mennes 2016). Above all, the philosophy of science offers new models of non-causal explanations (Reutlinger, Saatsi 2018; Lange 2017). The paper’s primary goal is to propose a new topo-logical model of explanation in system-theoretical linguistics, following Kostić’s (2019a) and Huneman’s (2018) approaches. This model of explanation overcomes various difficul-ties of functional explanation, such as the nature of the structural axiom and the func-tional equivalents. Simultaneously, it establishes a non-causal form of linguistic explana-tion which builds on the previous analysis of quantitative linguists (Andres 2009; Ferrer-i-Cancho, Solé 2003; Hřebíček 2002).

The Unique Characterization of the Shapley Value for Bi-cooperative Games

The aim of the present paper is to study the unique characterization for bicooperative games. Shapley value is the expected marginal contribution of the alliance. We will introduce some properties for bicooperative games. Our first characterization is based on the classical axioms determining the Shapley value with the symmetry axiom replaced excluded null axiom. In our second axiomatization we use structural axiom and a zero excluded axiom instead of effective axiom in classical cooperative games. Finally, We provide here linearity, anonymity, dummy and efficiency and structural axioms to study a one-point solution concept for Bi-cooperative games.

A Complete Axiomatisation for Quantifier-Free Separation Logic

We present the first complete axiomatisation for quantifier-free separation logic. The logic is equipped with the standard concrete heaplet semantics and the proof system has no external feature such as nominals/labels. It is not possible to rely completely on proof systems for Boolean BI as the concrete semantics needs to be taken into account. Therefore, we present the first internal Hilbert-style axiomatisation for quantifier-free separation logic. The calculus is divided in three parts: the axiomatisation of core formulae where Boolean combinations of core formulae capture the expressivity of the whole logic, axioms and inference rules to simulate a bottom-up elimination of separating connectives, and finally structural axioms and inference rules from propositional calculus and Boolean BI with the magic wand.

D-Complete Posets: Local Structural Axioms, Properties, and Equivalent Definitions

Although d-complete posets arose along the interface between algebraic combinatorics and Lie theory, they are defined using only requirements on their local structure. These posets are a mutual generalization of rooted trees, shapes, and shifted shapes. They possess Stanley’s hook product property for their P-partition generating functions and Schutzenberger’s well-defined jeu de taquin rectification property. The original definition of d-complete poset was lengthy, but more succinct definitions were later developed. Here several definitions are shown to be equivalent. The basic properties of d-complete posets are summarized. Background and a partial bibliography for these posets is given.

Eight Inference Rules for Implication

Utilizing an idea that has its first appearance in Gerhard Gentzen’s unpublished manuscripts, we generate an exhaustive repertoire of all the possible inference rules that are related to the left implication inference rule of the sequent calculus from a ground sequent, that is, a logical axiom. We discuss the similarities and differences of these derived rules as well as their interaction with the implication right rule under cut and the structural axiom. We further consider the question of analyticity of cuts in calculi using one of the new rules instead of the standard left implication rule.

Yoneda structures and KZ doctrines

In this paper we strengthen the relationship between Yoneda structures and KZ doctrines by showing that for any locally fully faithful KZ doctrine, with the notion of admissibility as defined by Bunge and Funk, all of the Yoneda structure axioms apart from the right ideal property are automatic.

A new family of information content models with an experimental survey on WordNet

This paper introduces a new family of intrinsic and corpus-based Information Content (IC) models for ontology-based similarity measures based on the IC theory, a detailed state of the art, an experimental survey of IC models and IC-based similarity measures on WordNet, and a comparison between intrinsic and corpus-based IC models. The family of IC models is made up of five intrinsic IC models, called CondProbHypo, CondProbUniform, CondProbLeaves, CondProbLogistic, and CondProbCosine, and one corpus-based IC model called CondProbCorpus which completes the family. The proposed IC models rely on two previously unconsidered notions: (1) the preservation of the probabilistic structure of the taxonomy associated to the conditional probabilities between child and parent concepts, and (2) the explicit consideration of a cognitive similarity notion in the definition of the IC model. The family of IC models defines a new method for the proposal of new intrinsic IC models based on the exploration of other alternatives for the intrinsic estimation of the conditional probabilities between child and parent concepts. Our work is inspired by an unexplored relationship between the Jiang–Conrath distance and a shortest path on an IC-based weighted graph, derived from the conditional probabilities between concepts, as well as certain cognitive evidence about the perception distance between concepts. The new IC models obtain results comparable to the state of the art and satisfy a set of well-founded structure axioms. In addition, we prove that most of intrinsic IC models and IC-based similarity measures do not show a significant statistical difference as regards a baseline corpus-based IC model and the Jiang–Conrath similarity, with the exception of the overall outperformance shown by the Sánchez et al. (2012) IC model and the cosJ&C similarity measure, which has recently been introduced by the authors.

On axioms of frobenius like structure in the theory of arrangements

A Frobenius manifold is a manifold with a flat metric and a Frobenius algebra structure on tangent spaces at points of the manifold such that the structure constants of multiplication are given by third derivatives of a potential function on the manifold with respect to flat coordinates. In this paper we present a modification of that notion coming from the theory of arrangements of hyperplanes. Namely, given natural numbers $n>k$, we have a flat $n$-dimensional manifold and a vector space $V$ with a non-degenerate symmetric bilinear form and an algebra structure on $V$, depending on points of the manifold, such that the structure constants of multiplication are given by $2k+1$-st derivatives of a potential function on the manifold with respect to flat coordinates. We call such a structure a Frobenius like structure . Such a structure arises when one has a family of arrangements of $n$ affine hyperplanes in ${\mathbb C}^k$ depending on parameters so that the hyperplanes move parallely to themselves when the parameters change. In that case a Frobenius like structure arises on the base ${\mathbb C}^n$ of the family.

Dynamic Programming on Nominal Graphs

Many optimization problems can be naturally represented as (hyper) graphs, where vertices correspond to variables and edges to tasks, whose cost depends on the values of the adjacent variables. Capitalizing on the structure of the graph, suitable dynamic programming strategies can select certain orders of evaluation of the variables which guarantee to reach both an optimal solution and a minimal size of the tables computed in the optimization process. In this paper we introduce a simple algebraic specification with parallel composition and restriction whose terms up to structural axioms are the graphs mentioned above. In addition, free (unrestricted) vertices are labelled with variables, and the specification includes operations of name permutation with finite support. We show a correspondence between the well-known tree decompositions of graphs and our terms. If an axiom of scope extension is dropped, several (hierarchical) terms actually correspond to the same graph. A suitable graphical structure can be found, corresponding to every hierarchical term. Evaluating such a graphical structure in some target algebra yields a dynamic programming strategy. If the target algebra satisfies the scope extension axiom, then the result does not depend on the particular structure, but only on the original graph. We apply our approach to the parking optimization problem developed in the ASCENS e-mobility case study, in collaboration with Volkswagen. Dynamic programming evaluations are particularly interesting for autonomic systems, where actual behavior often consists of propagating local knowledge to obtain global knowledge and getting it back for local decisions.

Economics for Economists

The characteristic capability of science – to turn whatever it might touch into knowledge – seems to have eluded economics. Currently, economists do not understand how the economy works. To get out of the cul-de-sac requires a paradigm shift. It consists in replacing behavioral axioms by structural axioms. The subject matter of theoretical economics is not human behavior but systemic behavior. From the structural analysis follows a new Law of Supply and Demand and a new Profit Law for the economy as a whole. The conventional supply-demand-equilibrium approach is refuted. This implies that the reliance on the spontaneous order metaphor is unfounded.

The Synthesis of Economic Law, Evolution, and History

It has long been criticized that history is almost entirely absent from orthodox economics. This deficiency is due to the fact that equilibrium and time make an odd couple. Because equilibrium is one of the crucial hard-core propositions of the research program it cannot be abandoned. This impedes the treatment of time in a methodologically acceptable manner. The orthodox approach is based on indefensible axioms which are in this paper replaced by objective structural axioms. This enables the synthesis of timeless economic laws, randomness, and goal-oriented human action, which are the essential elements of a formally consistent historical account.

The Ideal Economy: A Prototype

Standard economics starts with behavioral assumptions that are formally expressed as axioms. This approach met with little scientific success but still enjoys some popularity for lack of a convincing alternative. To replace the subjective formal foundations by objective structural axioms is the first task of this paper. To give a correct account of how the monetary economy works is the second. This entails an explanation of the continuous clearing of both the product and the labor market in the random consumption economy, that is, of how the economy could establish ongoing full employment and price stability in principle.

Redemption and Depression

According to prevailing methodological criteria, standard economics is definitively refuted. Joan Robinson’s wake-up call “Scrap the lot and start again” has therefore lost nothing of its original freshness and urgency. Yet, how can the restart succeed? This inquiry builds on structural axioms. First, conceptual consistency is assured and the confusion about profit and income is dissolved. The question of interest is then how a recession or depression develops as the result of the normal functioning of the monetary economy. This involves the identification of positive feedback. A very effective mechanism consists of the circular interaction of profit and distributed profit.

Understanding Profit and the Markets: The Canonical Model

Neither Walrasians nor Keynesians have a clear idea of the fundamental economic concepts income and profit, nor of the interdependence of qualitatively different markets. Critique of these approaches is necessary but not overly productive. A real breakthrough requires a new set of premises because no way leads from the accustomed behavioral assumptions to the understanding of how the economy works. More precisely, the hitherto accepted behavioral axioms have to be replaced by structural axioms. Starting from new formal foundations, this paper gives a comprehensive and consistent account of the objective interrelations of the monetary economy’s elementary building blocks.

How to Get Rid of Demand – Supply – Equilibrium for Good

This paper provides a substantial reconceptualization of the serial clearing of the product market on the basis of structural axioms. The change of premises is required simply because from the accustomed premises only the accustomed conclusions can be derived and these are known to be inapplicable in the real world. This holds in particular for the still popular idea that the working of a market can be described in terms of the triad supply-function – demand-function – equilibrium. Structural axiomatization provides the complete and consistent picture of interrelated product market events.