A set theory within fuzzy logic

Soft set theory was first introduced by D. A. Molodtsov [D. Molodtsov, Soft set theory first results, Computers & Mathematics with Applications, 37(4–5) (1999) 19–31] in 1999 as an effective approach to deal with uncertainty. Over time, fuzzy soft set theory has evolved significantly, integrating concepts from both fuzzy set theory and soft set theory to enhance decision-making processes. Acharjee and Medhi [S. Acharjee and S. Medhi, Logical connectives of fuzzy soft set theory, New Mathematics and Natural Computation, 21(1) (2025) 339–352] introduced several advanced concepts related to logical connectives in fuzzy soft set theory. Building on Molodtsov’s foundational work, this paper explores new developments in fuzzy soft logic, extending the findings of Acharjee and Medhi. Specifically, we introduce the concepts of [Formula: see text]-fuzzy soft implication and [Formula: see text]-fuzzy soft bi-implication, along with their related results within the framework of fuzzy soft logic.

PurposeThis paper aims to present the development of a holistic campus facility management (CFM) performance assessment framework that incorporates a fuzzy logic approach and integrates a comprehensive set of key factors for successful management of campus facilities. The devised framework aims to cater to the needs of campus facilities management firms and departments for the purpose of gauging and assessing their performance across different management domains. Through this approach, facility management organizations can detect potential areas of enhancement and adopt preemptive steps to evade issues, foster progress and ensure success.Design/methodology/approachAfter a comprehensive analysis of the literature, conducting in-depth interviews with industry experts and employing the Delphi technique in two rounds, a total of 45 indicators critical to CFM success were identified and subsequently sorted into seven distinct groups. Through an online questionnaire, 402 subject-matter experts proficiently assessed the significance of the critical success indicators and their groups. A fuzzy logic framework was developed to evaluate and quantify a firm's compliance with the critical success indicators and groups of indicators. The framework was subsequently weighted using computations of the relative importance index (RII) based on the responses received from the questionnaire participants. The initial section of the framework involved a comprehensive analysis of the firm's performance vis-à-vis the indicators, while the latter part sought to evaluate the impact of the indicators groups on the overall firm's performance.FindingsThe utilization of fuzzy logic has uncovered the significant effects each effective CFM key indicator on indicators groups, as well as the distinct effects of each CFM indicators group on the overall performance of CFM. The results reveal that financial management, communications management, sustainability and environment management and workforce management are the most impactful indicators groups on the CFM performance. This suggests that it is imperative for management to allocate increased attention to these specific areas.Originality/valueThis study contributes to the advancement of current knowledge by revealing vital indicators of effective CFM and utilizing them to construct a thorough fuzzy logic framework that can assist in evaluating the effectiveness of CFM firms worldwide. This has the potential to provide crucial assistance to facility management organizations, facility managers and policymakers in their quest for informed decision-making.

This paper investigates the applicability of the Vaganov–Shashkin–Lite (VSL) forward model for tree-ring-width chronologies as observation operator within a proxy data assimilation (DA) setting. Based on the principle of limiting factors, VSL combines temperature and moisture time series in a nonlinear fashion to obtain simulated TRW chronologies. When used as observation operator, this modelling approach implies three compounding, challenging features: (1) time averaging, (2) “switching recording” of 2 variables and (3) bounded response windows leading to “thresholded response”. We generate pseudo-TRW observations from a chaotic 2-scale dynamical system, used as a cartoon of the atmosphere-land system, and attempt to assimilate them via ensemble Kalman filtering techniques. Results within our simplified setting reveal that VSL’s nonlinearities may lead to considerable loss of assimilation skill, as compared to the utilization of a time-averaged (TA) linear observation operator. In order to understand this undesired effect, we embed VSL’s formulation into the framework of fuzzy logic (FL) theory, which thereby exposes multiple representations of the principle of limiting factors. DA experiments employing three alternative growth rate functions disclose a strong link between the lack of smoothness of the growth rate function and the loss of optimality in the estimate of the TA state. Accordingly, VSL’s performance as observation operator can be enhanced by resorting to smoother FL representations of the principle of limiting factors. This finding fosters new interpretations of tree-ring-growth limitation processes.

Determining food safety in canned food using fuzzy logic based on sulphur dioxide, benzoic acid and sorbic acid concentration

Labyrinth of Thought discusses the emergence and development of set theory and the set-theoretic approach to mathematics during the period 1850-1940. Rather than focusing on the pivotal figure of Georg Cantor, it analyzes his work and the emergence of transfinite set theory within the broader context of the rise of modern mathematics. The text has a tripartite structure. Part 1, The Emergence of Sets within Mathematics, surveys the initial motivations for a mathematical notion of a set within several branches of the discipline (geometry, algebra, algebraic number theory, real and complex analysis), emphasizing the role played by Riemann in fostering acceptance of the set-theoretic approach. In Part 2, Entering the Labyrinth, attention turns to the earliest theories of sets, their evolution, and their reception by the mathematical community; prominent are the epoch-making contributions of Cantor and Dedekind, and the complex interactions between them. Part 3, In Search of an Axiom System, studies the four-decade period from the discovery of set-theoretic paradoxes to Godel’s independence results, an era during which set theory gradually became assimilated into mainstream mathematics; particular attention is given to the interactions between axiomatic set theory and modern systems of formal logic, especially the interplay between set theory and type theory. A new Epilogue for this second edition offers further reflections on the foundations of set theory, including the dichotomy conception and the well-known iterative conception. The author paints on a grand scale. He sees clearly, and he sees whole. The result is a spacious canvas full of intriguing scenes and portraits from the history of set theory, seamlessly juxtaposed to form a fascinating and accurate picture of a vast area of modern mathematics. It is a must-have book for anyone who wishes to gain a balanced picture of this history. It is written in clear and elegant language for the learner, while experts in the area will enjoy seeing this beautiful presentation of what they already know, perhaps arguing about some of the author’s conclusions and choices of material. (Roger Cooke, University of Vermont)

Rough function is one of interesting new directions of the classical Pawlak rough set theory. This paper presents a generalized rough function with one variable under the framework of classical Pawlak rough set theory. In the definition of generalized rough function, we extend the domain of the classical rough function from the general natural number set to integer set, and then the rough function with one variable defined on an integer set is obtained. After that, we investigate the basic properties of the proposed rough function with one variable base on the classical function analysis theory in detail. Furthermore, some interesting results about the rough function with one variable over integer set are given and the relationship between the existing literatures is discussed. The contribution of this is twofold: one is present a new concept of rough function with one variable over the integer set. Another is of enriches the basis of theory of classical Pawlak rough set and also extends the areas of the applications.

Assuming the Axiom of Constructibility, necessary and sufficient conditions are given for the vanishing of Ext\ for rings A of global dimension 1.Using Martin's Axiom, the necessity of these conditions is shown not to be a theorem of ZFC.Applications are given to abelian group theory, including a partial solution (assuming V = L) to a problem of Baer on the splitting of abelian groups.Some independence results in abelian group theory are also proved.Introduction.In a remarkable recent paper [22], S. Shelah proved that Whitehead's problem is unsolvable in ordinary set theory (Zermelo-Frankel set theory with the Axiom of Choice).More precisely, he proved that two different answers to the problem are obtained when set theory is extended by adding two different axioms: the Axiom of Constructibility and Martin's Axiom.(See Theorem 4.1.)Shelah's arguments were specific to Whitehead's problem and nonhomological in character.In this paper we extend Shelah's techniques in order to obtain some general results about the vanishing of ExtA (for modules over a ring A of global dimension 1) in different models of set theory, and we apply them to some specific problems in abelian group theory.Assuming the Axiom of Constructibility we obtain necessary and sufficient conditions for the vanishing of ExtA (Theorems 1.2 and 1.5).To state the result in.a special case: if C and A are abelian groups of cardinality N,, Ext(C, A) -0 if and only if C is the union of an increasing chain of countable subgroups [Cv\v < «,} such that the chain is smooth (i.e., C0 -U"<oC, for aii iimit ordinals a) and satisfies Ext(C0, A) = 0 and E\i(Cv+x/Cv,A) = 0 for all v < K,.Although the sufficiency of this condition is an easy consequence of classical homological algebra (see Theorem 1.2), the proof of necessity makes use of the Axiom of Constructibility (Theorem 1.5).The necessity of this condition is not provable in ordinary set theory; in particular the condition is not necessary when Martin's Axiom is assumed ( §3).

Descriptive Set Theory: Projective Sets

Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambda-notation and sorts or types as in type theory. This paper introduces a version of von-Neumann-Bernays-Godel set theory for reasoning about sets, proper classes, and partial functions represented as classes of ordered pairs. The underlying logic of the system is a partial first-order logic, so class-valued terms may be nondenoting. Functions can be specified using lambda-notation, and reasoning about the application of functions to arguments is facilitated using sorts similar to those employed in the logic of the IMPS Interactive Mathematical Proof System. The set theory is intended to serve as a foundation for mechanized mathematics systems.

An Ontological and Epistemological foundation of Fuzzy Set and Logic Theory is reviewed in comparison to Classical Set and Logic Theory. It is shown that basic equivalences of classical theory breakdown but are re-established as weak equivalences as a containment relation in fuzzy theory. It is also stressed that the law of conservation of information is still upheld within fuzzy theory.

This study aims to explore the application of fuzzy logic in multi-criteria decision-making (MCDM) processes, particularly in complex environments. A fuzzy logic framework was developed, integrating fuzzy sets, membership functions, and fuzzy rules to evaluate diverse criteria and enhance decision-making. The findings demonstrate that fuzzy logic effectively manages uncertainty and imprecision, leading to improved decision outcomes. A case study conducted in a private hospital in Istanbul validated the framework's applicability, showcasing its ability to handle non-homogeneous information and facilitate informed decisions. The implications of this research highlight the potential of fuzzy logic methodologies to enhance decision-making accuracy and robustness in various fields, including healthcare and engineering.

Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text attempts to present a largely balanced approach, which combines many elements of the different traditions of the subject. It includes a wide variety of examples, exercises (over 400), and applications, in order to illustrate the general concepts and results of the theory. This text provides a first basic course in classical descriptive set theory and covers material with which mathematicians interested in the subject for its own sake or those that wish to use it in their field should be familiar. Over the years, researchers in diverse areas of mathematics, such as logic and set theory, analysis, topology, probability theory, etc., have brought to the subject of descriptive set theory their own intuitions, concepts, terminology and notation.

An integrated fuzzy logic and web-based framework for active protocol support

Estimation of uncertainty in knowledge extracted from a repository of data is a challenging issue in artificial intelligence and cognitive sciences. The rough set theory is a prominent technique for data mining from a repository of domain data. A rough set is a formal approximation of a crisp set in terms of a pair of sets with respect to equivalence relations. The concept of lower and upper approximations is based on the assumption that information is associated with every object described by a set of attributes. The classical rough set theory with the equivalence class concept and static dependency cannot handle any uncertainty in attribute level which seems to be a major limitation. Several extensions on classical rough set theory have remarkable contributions in different applications. But excavation of the nature of dependency of the attributes and the estimation of uncertainty is still an unsolved issue. This paper proposes an extension in the classical rough set theory by introducing probabilistic relevancy of the attributes. Mining the correlation between attributes in terms of the probabilistic dependency coefficient is an important way out in discovering the uncertainty. The roughness in contributing knowledge by an attribute can be represented using approximations in the probabilistic dependency space. Using the approximations in the values of the probabilistic dependency coefficient, the inherent uncertainty can be estimated and handled more precisely.

Mathematical theories of uncertainty in which measures of uncertainty are now well established are fuzzy set theory, evidence theory, and possibility theory, in addition to classical set theory and probability theory. Some of these theories are connected. Classical set theory is subsumed under fuzzy set theory. Probability theory and possibility theory are branches of evidence theory, while evidence theory is, in turn, a branch of fuzzy measure theory. Although information aspects of fuzzy measure theory have not been investigated as yet, the theory is briefly introduced in this chapter because it represents a broad framework for future research.