Axiomatic Approach Research Articles

The Axiomatic Approach to Non-Classical Model Theory

Institution theory represents the fully axiomatic approach to model theory in which all components of logical systems are treated fully abstractly by reliance on category theory. Here, we survey some developments over the last decade or so concerning the institution theoretic approach to non-classical aspects of model theory. Our focus will be on many-valued truth and on models with states, which are addressed by the two extensions of ordinary institution theory known as L-institutions and stratified institutions, respectively. The discussion will include relevant concepts, techniques, and results from these two areas.

On fuzzy Sheffer strokes: New results and the ordinal sums

Recently, Baczyński et al. introduced the axiomatic definition of fuzzy Sheffer strokes, which is the generalization of the Sheffer stroke operators in classical logic. On the basis of this work, we point out that fuzzy Sheffer strokes can be transformed into common fuzzy logical operators through a fuzzy negation and fuzzy Sheffer strokes can be obtained from two increasing unary operators acting on a given fuzzy Sheffer stroke. We further discuss the relationship between fuzzy Sheffer strokes and overlap functions as well as grouping functions. The main contribution is to present the concept of ordinal sums of fuzzy Sheffer strokes which provides constructing methods of fuzzy Sheffer strokes and analyze some related properties. We argue under which condition a fuzzy Sheffer stroke can be represented as the ordinal sum of a family of fuzzy Sheffer strokes. We relate ordinal sums of fuzzy Sheffer strokes with its induced fuzzy conjunctions and fuzzy disjunctions as well as overlap functions and grouping functions under certain conditions. In the end, we give real-life application examples of fuzzy Sheffer strokes.

A systematic review of uncertainty theory with the use of scientometrical method

Uncertainty theory is an area in axiomatic mathematics recently proposed by Professor Baoding Liu and aiming to deal with belief degrees. Retrieving 1004 journal articles from the Web of Science database between 2008 and 2019, and utilizing CiteSpace and Pajek software, we analyze the publications per year and by geographical distribution, productive scholars and their cooperation, key journals, highly cited articles and main paths of the field. In this way, seven key sub-fields of uncertainty theory and their research potential are derived. The results show the following: (1) The literature on uncertainty theory follows a linear growth trend, involves an extensive network of 1000 scholars worldwide and is published in 300 journals, indicating thus that uncertainty theory has become increasingly attractive, and its academic influence is gradually expanding. (2) Seven key sub-fields of uncertainty theory have clearly been identified, including the axiomatic system, uncertain programming, uncertain sets, uncertain logic, uncertain differential equations, uncertain risk analysis, and uncertain processes. Among them, uncertain differential equations and programming are the two main sub-fields with the largest numbers of published papers. Furthermore, for evaluating the research potential of sub-fields, maturity and recent attention indicators are calculated using the citations, total number of publications, quantity of most cited literature and half-life. Based on these indicators, uncertain processes shows the greatest development potential, and has remained a hot topic in recent years, being mainly concentrated on the uncertain renewal reward process, optimal control of discrete-time uncertain systems, and uncertain linear quadratic optimal control. Additionally, uncertain risk analysis is ranked second, and focuses on the analysis of expected losses, investment risk, and structural reliability of uncertain systems.

Mechanical equipment health management method based on improved intuitionistic fuzzy entropy and case reasoning technology

Management becomes more challenging as machinery becomes more widely used. From the health management history case records of mechanical devices, we can find that there are many health problems of the same type occurring repeatedly, but when similar problems occur again, solutions are not found in a short period. Case-based reasoning technology aimed at solving the above problems are widely used in equipment health management, but there are problems such as inadequate data utilization and failure to achieve the required accuracy and reliability. To uncover important information from historical failure case data and help reduce equipment downtime due to failure, this paper proposes a machinery equipment health management method based on improved intuitionistic fuzzy entropy and case reasoning technology. Firstly, the axiomatic definition of traditional intuitionistic fuzzy entropy is optimized and a new intuitionistic fuzzy entropy formula in the framework of case-based reasoning decision matrix is proposed. Secondly, the weight value of each attribute is obtained by combining the formula of the entropy weight method. Finally, the feature attribute weight values are fused into the case-based reasoning, and the historical cases that are most similar to the target cases are obtained by combining the existing case base. The effectiveness of the method was verified by comparing and analyzing the example calculation results with the traditional method. The method improves the management of machinery equipment operation and maintenance data. Furthermore, it helps enterprises to carry out the management and analysis of machinery equipment health problems.

Preservation in many-valued truth institutions

The theory of L-institutions represents an axiomatic category-based approach to many-valued truth model theory. Technically, L-institutions are many-valued truth extensions of institution theory of Goguen and Burstall. In this paper, in the general framework of L-institutions we study preservation properties of the satisfaction of sentences by filtered products of models and by submodels, and apply them to Horn sentences. General consequences include semantic compactness and initial semantics results. Our general concepts and results are applied on several concrete many-valued logical systems.

The extent of partially resolving uncertainty in assessing coherent conditional plausibilities

Handling uncertainty and reasoning under partial knowledge are challenging tasks that require to deal with coherent assessments and their extensions. Plausibility theory is shown to rest upon the principle of partially resolving uncertainty due to Jaffray, together with a systematically optimistic behavior. This means that we allow situations in which the agent may only acquire the information that a non-impossible event occurs, without knowing which is the true state of the world. This leads to assume that a target event is plausibly true if it is compatible with the acquired piece of information. The aim of the paper is to provide coherence conditions for a conditional plausibility assessment (namely, Pl-coherence), by referring to a suitable axiomatic definition based on the Dempster's rule of conditioning. We provide different equivalent notions of Pl-coherence in terms of consistency, betting scheme, and penalization that, as a by-product, highlight different interpretations. We then specialize the Pl-coherence conditions to the subclasses of (finitely additive) conditional probabilities and (finitely maxitive) conditional possibilities.

Dynamic capital allocation rules via BSDEs: an axiomatic approach

In this paper, we study capital allocation for dynamic risk measures, with an axiomatic approach but also by exploiting the relation between risk measures and BSDEs. Although there is a wide literature on capital allocation rules in a static setting and on dynamic risk measures, only a few recent papers on capital allocation work in a dynamic setting and, moreover, those papers mainly focus on the gradient approach. To fill this gap, we then discuss new perspectives to the capital allocation problem going beyond those already existing in the literature. In particular, we introduce and investigate a general axiomatic approach to dynamic capital allocations as well as an approach suitable for risk measures induced by g-expectations under weaker assumptions than Gateaux differentiability.

Quine’s Underdetermination Thesis

In On Empirically Equivalent Systems of the World from 1975, Quine formulated a thesis of underdetermination roughly to the effect that every scientific theory has an empirically equivalent but logically incompatible rival, one that cannot be discarded merely as a terminological variant of the former. For Quine, the truth of this thesis was an open question. If true, some would argue that it undermines any belief in scientific theories that is based purely on their empirical success. But despite its potential significance, surprisingly little has been done by way of establishing or refuting it. My aim is to establish the thesis for as large a class of theories as possible. Under various precisifications of the concepts involved, I show that it holds for all consistent and recursively axiomatizable theories that postulate infinitely many theoretical entities.

Decompositions of stratified institutions

Abstract The theory of stratified institutions is a general axiomatic approach to model theories where the satisfaction is parameterized by states of the models. In this paper we further develop this theory by introducing a new technique for representing stratified institutions, which is based on projecting to such simpler structures. On the one hand this can be used for developing general results applicable to a wide variety of already existing model theories with states, such as those based on some form of Kripke semantics. On the other hand this may serve as a template for defining new such model theories. In this paper we emphasize the former application of this technique by developing general results on model amalgamation and on the existence diagrams for stratified institutions. These are two most useful properties to have in institution theoretic model theory.

First-order logic of change

Abstract We present the first-order logic of change, which is an extension of the propositional logic of change $\textsf {LC}\Box $ developed and axiomatized by Świętorzecka and Czermak (2015, Log. et Anal., 232, 511–527). $\textsf {LC}\Box $ has two primitive operators: ${\mathcal {C}}$ to be read it changes whether and $\Box $ for constant unchangeability. It implements the philosophically grounded idea that with the help of the primary concept of change it is possible to define the concept of time. One of the characteristic axioms for ${\mathcal {C}}$ is ${\mathcal {C}} A \to {\mathcal {C}}\neg A$ and one of the primitive rules is $\omega $-rule introducing $\Box $. It turns out that the next operator is definable in $\textsf {LC}\Box $. Logic $\textsf {LC}\Box $ with next is equivalent to certain fragment of $\textsf {LTL}$ extended by the appropriate definition of ${\mathcal {C}}$. Recently, $\textsf {LC}\Box $ has also been modified to the propositional logic of ‘branching’ changes $\textsf {BTC}$ by Łyczak (2022, Log. J. IGPL) and the logic of ‘parallel’ changes $\textsf {LC}\Box $ ↳ by Świętorzecka and Łyczak (2022, Log. Log. Philios.). Extended in an appropriate manner, $\textsf {BTC}$ is equivalent to a certain fragment of logic $\textsf {CTL}$ to which definitions of two kinds of changes have been added. Here we propose an extension of $\textsf {LC}\Box $ to first-order logic in which, again, the only primitive modal operators are ${\mathcal {C}}$ and $\Box $. We interpret our logic in the semantics of histories of changes. We give an axiomatic system $\textsf {LC}\Box Q$ for the considered logic, and we show selected theses about some relationships between ${\mathcal {C}}$, $\Box $ and $\forall $ (in particular, we prove two versions of the Barcan formula). Then we prove the completeness of $\textsf {LC}\Box Q$. Finally, we compare our logic with $\textsf {FOLT}$ and show the relation between $\textsf {LC}\Box Q$ and a certain fragment of the Kröger system $\varSigma $ to which the definition of ${\mathcal {C}}$ was added.

Income taxation and equity: new dominance criteria with a microsimulation application

This paper addresses the problem of the normative evaluation of income tax systems and income tax reforms. While most of the existing criteria, framed in the utilitarian tradition, are uniquely based on information about individual incomes, this paper, building upon the opportunity egalitarian theory, proposes new equity criteria which take into account also the socio-economic characteristics of individuals. Suitable dominance conditions that can be used to rank alternative tax systems are derived by means of an axiomatic approach. Moreover, the theoretical results are used to assess the redistributive effects of an hypothetical tax reform in Romania through a microsimulation analysis.

An axiomatic approach towards pandemic performance indicators

During a pandemic, each country (or region) is characterized by a status matrix indicating its positive cases, hospitalizations and deaths. A pandemic performance indicator is a real-valued mapping from the set of status matrices to the set of non-negative real numbers, whereby lower values stand for better performance. We show that four axioms together characterize a family of indicators arising from a weighted average of the incidence rate, morbidity rate and mortality rate. We use these indicators to evaluate the impact of COVID-19 in major countries worldwide.

Axiomatic (and non-axiomatic) mathematics

Axiomatizing mathematical structures and theories, or postulating them as Russell (1919) put it, is a goal of mathematical logic. Some axiomatic systems are mere definitions, such as the axioms of Group Theory; but some are much deeper, such as the axioms of complete ordered fields with which real analysis starts. Groups abound in the mathematical sciences, while by Dedekind’s theorem (1888) there exists only one complete ordered field, up to isomorphism. Cayley’s theorem (1854) in abstract algebra implies that the axioms of group theory completely axiomatize the class of permutation sets that are closed under composition and inversion. In this expository article, we survey some old and new results on the first-order axiomatizability of various mathematical structures. As we will see, axiomatizability of some structures are still unsolved questions in mathematics, and several results have been open problems in the past. We will also review identities over addition, multiplication, and exponentiation that hold in the set of positive real numbers; and will have a look at Tarski’s high school problem (1969) and its solution.

An Axiomatic Approach to the Measurement of Comparative Female Disadvantage

An Axiomatic Approach to the Measurement of Comparative Female Disadvantage

ON THE PROOF OF THE THEOREMS OF FOUNDATIONS OF GEOMETRY USING ISABELLE/HOL

Isabelle/HOL is a generic proof assistant. Using Isabelle/HOL requires insight into procedures as well as into the concepts involved. In addition, how a computer manages procedures can affect mathematical concepts. Use of Isabelle/HOL can correct a current weakness in mathematical studies. The advantage of the theorem proving support system represented by Isabelle/HOL is that it mechanically guarantees the “correctness” of both human-written programs and mathematical proofs. It can allow us to clearly understand mathematical concepts and can minimize the burden of operation opportunities. However, in order to take advantage of its high versatility and reliability, the problem that all certification procedures must be clearly formalized when creating certification must be overcome. “Foundations of Geometry” is a book on mathematics written by Hilbert in 1899. The book is famous as the most rigorous study of the axiom system of Euclidean geometry by axioms and formalism. When we tried to implement Hilbert’s axioms in Isabelle/HOL, the proofs based on human cognition hindered the implementation. The purpose of this paper is “correctly” reconstruct the proofs as automated theorem proving. We are aiming to implement them “accurately” on Isabelle/ HOL and have done so for many of them. This is the originality of this study.

Towards an axiomatic approach to truth discovery

The problem of truth discovery, i.e., of trying to find the true facts concerning a number of objects based on reports from various information sources of unknown trustworthiness, has received increased attention recently. The problem is made interesting by the fact that the relative believability of facts depends on the trustworthiness of their sources, which in turn depends on the believability of the facts the sources report. Several algorithms for truth discovery have been proposed, but their evaluation has mainly been performed experimentally by computing accuracy against large datasets. Furthermore, it is often unclear how these algorithms behave on an intuitive level. In this paper we take steps towards a framework for truth discovery which allows comparison and evaluation of algorithms based instead on their theoretical properties. To do so we pose truth discovery as a social choice problem, and formulate various axioms that any reasonable algorithm should satisfy. Along the way we provide an axiomatic characterisation of the baseline ‘Voting’ algorithm—which leads to an impossibility result showing that a certain combination of the axioms cannot hold simultaneously—and check which axioms a particular well-known algorithm satisfies. We find that, surprisingly, our more fundamental axioms do not hold, and propose modifications to the algorithms to partially fix these problems.

Equivariant Coarse (Co-)Homology Theories

We present an Eilenberg-Steenrod-like axiomatic framework for equivariant coarse homology and cohomology theories. We also discuss a general construction of such coarse theories from topological ones and the associated transgression maps. A large part of this paper is devoted to showing how some well-established coarse (co-)homology theories, whose equivariant versions are either already known or will be introduced in this paper, fit into this setup. Furthermore, a new and more flexible notion of coarse homotopy is given which is more in the spirit of topological homotopies. Some, but not all, coarse (co-)homology theories are even invariant under these new homotopies. They also led us to a meaningful concept of topological actions of locally compact groups on coarse spaces.

A New Picture Fuzzy Entropy and Its Application Based on Combined Picture Fuzzy Methodology with Partial Weight Information

Picture fuzzy set (PFS) is more comprehensive tool than intuitionistic fuzzy set (IFS) for modeling the uncertain decision-making problems. In this paper, a new picture fuzzy entropy measure is proposed and proved that the proposed measure satisfies the axiomatic definition of entropy measures for picture fuzzy sets. Besides this, the useful mathematical properties of the new entropy measure are also investigated. The justification of the proposed picture fuzzy measure is established by discussing its particular cases and compares it with the existing entropy measures. Then, for the case where criteria weights are partially known, we used an entropy-based method to produce objective weights. For the uncertain environment, TODIM (portuguese acronym for interactive multicriteria decision-making) and ELECTRE methods are useful for practical problems. Based on the advantages of PFSs,TODIM, and ELECTRE, we proposed an integrated picture fuzzy TODIM-ELECTRE to combine the prominent benefits of these theories. We present the TODIM-ELECTRE model for PFS environment and express the computing steps in brief of this new established model. Thereafter, the superiority of the new model is verified by a numerical example of supplier selection and through comparative study with other existing methods.

Novel distance and entropy definitions for linear Diophantine fuzzy sets and an extension of TOPSIS (LDF‐TOPSIS)

AbstractThe literature of multiple attribute decision making (MADM) is fruitful since there are various and successful applications of different fuzzy set extensions such as intuitionistic, Pythagorean and q‐Rung orthopair fuzzy sets (IFS, PFS and q‐ROFS). Besides their powerful aspects, some definitional limitations are known. In order to eradicate these boundaries regarding the definitions of membership and non‐membership degrees, linear Diophantine fuzzy set (LDFS) concept has been recently emerged. By considering two parameters, LDFS extends the representation area of the previous fuzzy set definitions and provides more extensive human judgement coverage field. In this study, the first distance and entropy measures in the literature have been developed for LDFSs. Their axiomatic definitions are given, and the proofs are shown. Also, thanks to our extensive literature review, we became aware that there is no MADM extension dedicatedly proposed for LDFS. So, the first MADM method extension for LDFS environment has also been developed in this study. A very well‐known MADM approach, TOPSIS, has been extended into LDFS environment for the first time in the literature. The applicability is shown in a healthcare management decision problem and the validity is checked and approved by comparing the alternative rankings LDF‐TOPSIS and the aggregation operators that were obtained from the literature produced.

4-Manifold topology, Donaldson–Witten theory, Floer homology and higher gauge theory methods in the BV-BFV formalism

We study the behavior of Donaldson’s invariants of 4-manifolds based on the moduli space of anti-self-dual connections (instantons) in the perturbative field theory setting where the underlying source manifold has boundary. It is well-known that these invariants take values in the instanton Floer homology groups of the boundary 3-manifold. Gluing formulae for these constructions lead to a functorial topological field theory description according to a system of axioms developed by Atiyah, which can be also regarded in the setting of perturbative quantum field theory, as it was shown by Witten, using a version of supersymmetric Yang–Mills theory, known today as Donaldson–Witten theory. One can actually formulate an AKSZ model which recovers this theory for a certain gauge-fixing. We consider these constructions in a perturbative quantum gauge formalism for manifolds with boundary that is compatible with cutting and gluing, called the BV-BFV formalism, which was recently developed by Cattaneo, Mnev and Reshetikhin. We prove that this theory satisfies a modified Quantum Master Equation and extend the result to a global picture when perturbing around constant background fields. These methods are expected to extend to higher codimensions and thus might help getting a better understanding for fully extendable [Formula: see text]-dimensional field theories (in the sense of Baez–Dolan and Lurie) in the perturbative setting, especially when [Formula: see text]. Additionally, we relate these constructions to Nekrasov’s partition function by treating an equivariant version of Donaldson–Witten theory in the BV formalism. Moreover, we discuss the extension, as well as the relation, to higher gauge theory and enumerative geometry methods, such as Gromov–Witten and Donaldson–Thomas theory and recall their correspondence conjecture for general Calabi–Yau 3-folds. In particular, we discuss the corresponding (relative) partition functions, defined as the generating function for the given invariants, and gluing phenomena.