The IndetermSoft Set is as an extension of the Soft Set, because the data, or the function, or the sets involved in the definition of the soft set have indeterminacy - as in our everyday life, and we still need to deal with such situations. And similarly, IndetermHyperSoft Set as extension of the HyperSoft Set, when there is indeterminate data, or indeterminate functions, or indeterminate sets. Herein, ‘Indeterm’ stands for ‘Indeterminate’ (uncertain, conflicting, incomplete, not unique outcome). We now introduce (...) for the first time the TreeSoft Set as extension of the MultiSoft Set. Several applications are presented for each type of soft set. (shrink)

In this paper one introduces for the first time the IndetermSoft Set, as extension of the classical (determinate) Soft Set, that deals with indeterminate data, and similarly the HyperSoft Set extended to IndetermHyperSoft Set, where ‘Indeterm’ stands for ‘Indeterminate’ (uncertain, conflicting, not unique outcome). They are built on an IndetermSoft Algebra that is an algebra dealing with IndetermSoft Operators resulted from our real world. Afterwards, the corresponding Fuzzy / Intuitionistic Fuzzy / Neutrosophic / and other fuzzy-extension (...) class='Hi'>IndetermSoft Set & IndetermHyperSoft Set are presented together with their applications. (shrink)

In this paper we define the Soft Set Product as a product of many soft sets and afterwards we extend it to the HyperSoft Set. Similarly, the IndetermSoft Product is extended to the IndetermHyperSoft Set. We also present several applications of the Soft Set Product to Fuzzy (and fuzzy-extensions) Soft Set Product and to IndetermSoft Set and IndetermHyperSoft Set.

This is a review paper, where we recall the definitions together with practical applications of the Soft Set and its extensions to HyperSoft Set, IndetermSoft Set, IndetermHyperSoft Set, and TreeSoft Set.

This paper presents for the first time the IndetermSoft Set, as an extension of the classical (determinate) Soft Set, which operates on indeterminate data, and similarly the HyperSoft Set extended to the IndetermHyperSoft Set, where 'Indeterm' means 'Indeterminate' (uncertain, conflicting, non-unique result). They are built on an IndetermSoft Algebra which is an algebra dealing with IndetermSoft Operators resulting from our real world. Subsequently, the IndetermSoft and IndetermHyperSoft Sets and their Fuzzy/Fuzzy Intuitionistic/Neutrosophic and other fuzzy extensions and (...) their applications are presented. (shrink)

Set-theoretic pluralism is an increasingly influential position in the philosophy of set theory (Balaguer [1998], Linksy and Zalta [1995], Hamkins [2012]). There is considerable room for debate about how best to formulate set-theoretic pluralism, and even about whether the view is coherent. But there is widespread agreement as to what there is to recommend the view (given that it can be formulated coherently). Unlike set-theoretic universalism, set-theoretic pluralism affords an answer to Benacerraf’s epistemological challenge. The purpose of this paper is (...) to determine what Benacerraf’s challenge could be such that this view is warranted. I argue that it could not be any of the challenges with which it has been traditionally identified by its advocates, like of Benacerraf and Field. Not only are none of the challenges easier for the pluralist to meet. None satisfies a key constraint that has been placed on Benacerraf’s challenge. However, I argue that Benacerraf’s challenge could be the challenge to show that our set-theoretic beliefs are safe – i.e., to show that we could not have easily had false ones. Whether the pluralist is, in fact, better positioned to show that our set-theoretic beliefs are safe turns on a broadly empirical conjecture which is outstanding. If this conjecture proves to be false, then it is unclear what the epistemological argument for set-theoretic pluralism is supposed to be. (shrink)

A Cantorian argument that there is no set of all truths. There is, for the same reason, no possible world as a maximal set of propositions. And omniscience is logically impossible.

Critical-set views avoid the Repugnant Conclusion by subtracting some constant from the welfare score of each life in a population. These views are thus sensitive to facts about biographical identity: identity between lives. In this paper, I argue that questions of biographical identity give us reason to reject critical-set views and embrace the total view. I end with a practical implication. If we shift our credences towards the total view, we should also shift our efforts towards ensuring that humanity survives (...) for the long term. (shrink)

The Affordable Care Act (ACA) may be the most important health law statute in American history, yet much of the most prominent legal scholarship examining it has focused on the merits of the court challenges it has faced rather than delving into the details of its priority-setting provisions. In addition to providing an overview of the ACA’s provisions concerning priority setting and their developing interpretations, this Article attempts to defend three substantive propositions. First, I argue that the ACA is neither (...) uniformly hostile nor uniformly friendly to efforts to set priorities in ways that promote cost and quality. Second, I argue that the ACA does not take a single, unified approach to priority setting; rather, its guidance varies depending on the aspect of the health care system at issue (Patient Centered Outcomes Research Institute, Medicare, essential health benefits) and the factors being excluded from priority setting (age, disability, life expectancy). Third, I argue that cost-effectiveness can be achieved within the ACA's constraints, but that doing so will require adopting new approaches to cost-effectiveness and priority setting. By limiting the use of standard cost-effectiveness analysis, the ACA makes the need for workable rivals to cost-effectiveness analysis a pressing practical concern rather than a mere theoretical worry. (shrink)

In this contribution, I explore the possibility of characterizing the emergence of time in causal set theory (CST) in terms of the growing block universe (GBU) metaphysics. I show that although GBU seems to be the most intuitive time metaphysics for CST, it leaves us with a number of interpretation problems, independently of which dynamics we choose to favor for the theory —here I shall consider the Classical Sequential Growth and the Covariant model. Discrete general covariance of the CSG dynamics (...) does not allow us to individuate a single history of the universe (defined by a causal history of different causal sets), thereby making the claim that ‘the past exists’ at best problematic. In addition, because the evolution of the universe in CSG dynamics leads to an outward branching causal tree, it becomes impossible to determine a proper ‘line of becoming’, thereby blurring the presentists’ claim that only the present exists. Similarly, the covariant approach runs into the same, if not even more severe problems, since each configuration of the universe would amount to a set of possible causal sets, thereby making the individuation of a single configuration of the universe —and thus the physical interpretation of the theory— implausible. (shrink)

The iterative conception of set is typically considered to provide the intuitive underpinnings for ZFCU (ZFC+Urelements). It is an easy theorem of ZFCU that all sets have a definite cardinality. But the iterative conception seems to be entirely consistent with the existence of “wide” sets, sets (of, in particular, urelements) that are larger than any cardinal. This paper diagnoses the source of the apparent disconnect here and proposes modifications of the Replacement and Powerset axioms so as to allow for the (...) existence of wide sets. Drawing upon Cantor’s notion of the absolute infinite, the paper argues that the modifications are warranted and preserve a robust iterative conception of set. The resulting theory is proved consistent relative to ZFC + “there exists an inaccessible cardinal number.”. (shrink)

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense (...) of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences. (shrink)

Consider a variant of the usual story about the iterative conception of sets. As usual, at every stage, you find all the (bland) sets of objects which you found earlier. But you also find the result of tapping any earlier-found object with any magic wand (from a given stock of magic wands). -/- By varying the number and behaviour of the wands, we can flesh out this idea in many different ways. This paper's main Theorem is that any loosely constructive (...) way of fleshing out this idea is synonymous with a ZF-like theory. -/- This Theorem has rich applications; it realizes John Conway's (1976) Mathematicians' Liberation Movement; and it connects with a lovely idea due to Alonzo Church (1974). (shrink)

In this short note we show that the so-called Ambiguous Set (2019) is a subclass of the Double Refined Indeterminacy Neutrosophic Set (2017) and is a particular case of the Refined Neutrosophic Set (2013). Also, the Ambiguous Set is similar to the Quadripartitioned Neutrosophic Set (2016), and Belnap’s Four-Valued Logic (1975).

Priority setting in health care is ubiquitous and health authorities are increasingly recognising the need for priority setting guidelines to ensure efficient, fair, and equitable resource allocation. While cost-effectiveness concerns seem to dominate many policies, the tension between utilitarian and deontological concerns is salient to many, and various severity criteria appear to fill this gap. Severity, then, must be subjected to rigorous ethical and philosophical analysis. Here we first give a brief history of the path to today’s severity criteria in (...) Norway and Sweden. The Scandinavian perspective on severity might be conducive to the international discussion, given its long-standing use as a priority setting criterion, despite having reached rather different conclusions so far. We then argue that severity can be viewed as a multidimensional concept, drawing on accounts of need, urgency, fairness, duty to save lives, and human dignity. Such concerns will often be relative to local mores, and the weighting placed on the various dimensions cannot be expected to be fixed. Thirdly, we present what we think are the most pertinent questions to answer about severity in order to facilitate decision making in the coming years of increased scarcity, and to further the understanding of underlying assumptions and values that go into these decisions. We conclude that severity is poorly understood, and that the topic needs substantial further inquiry; thus we hope this article may set a challenging and important research agenda. (shrink)

This article presents an overview of the basic philosophical motivations for, and some recent work in, modal set theory.

Suppose that the members of a group each hold a rational set of judgments on some interconnected questions, and imagine that the group itself has to form a collective, rational set of judgments on those questions. How should it go about dealing with this task? We argue that the question raised is subject to a difficulty that has recently been noticed in discussion of the doctrinal paradox in jurisprudence. And we show that there is a general impossibility theorem that that (...) difficulty illustrates. Our paper describes this impossibility result and provides an exploration of its significance. The result naturally invites comparison with Kenneth Arrow's famous theorem (Arrow, 1963 and 1984; Sen, 1970) and we elaborate that comparison in a companion paper (List and Pettit, 2002). The paper is in four sections. The first section documents the need for various groups to aggregate its members' judgments; the second presents the discursive paradox; the third gives an informal statement of the more general impossibility result; the formal proof is presented in an appendix. The fourth section, finally, discusses some escape routes from that impossibility. (shrink)

Recently theorists have demonstrated a growing interest in the ethical aspects of resource allocation in international non-governmental humanitarian, development and human rights organizations (INGOs). This article provides an analysis of Thomas Pogge's proposal for how international human rights organizations ought to choose which projects to fund. Pogge's allocation principle states that an INGO should govern its decision making about candidate projects by such rules and procedures as are expected to maximize its long-run cost-effectiveness, defined as the expected aggregate moral value (...) of the projects it undertakes divided by the expected aggregate cost of these projects? I critique Pogge's argument on two fronts: (1) I demonstrate that his view is problematic on his own terms, even if we accept the cost-effectiveness framework he employs. (2) I take issue with his overall approach because it generates results which can undermine the integrity of INGOs. Further, his approach mis-characterizes the nature of INGOs, and this mistake is at the root of his problematic view of INGO priority-setting. Ultimately, I argue for a conception of INGOs in which they are understood as ?organizations of principle?, in the sense that they are independent moral agents and so should be permitted a fairly wide sphere of autonomy within reasonable moral constraints. (shrink)

Proposals for allocating scarce lifesaving resources in the face of the Covid-19 pandemic have aligned in some ways and conflicted in others. This paper attempts a kind of priority setting in addressing these conflicts. In the first part, we identify points on which we do not believe that reasonable people should differ—even if they do. These are (i) the inadequacy of traditional clinical ethics to address priority-setting in a pandemic; (ii) the relevance of saving lives; (iii) the flaws of first-come, (...) first-served allocation; (iv) the relevance of post-episode survival; (v) the difference between age and other factors that affect life-expectancy; and (vi) the need to avoid quality-of-life judgments. In the second part, we lay out some positions on which reasonable people can and do differ. These include (i) conflicts between maximizing benefits and priority to the worst off; (ii) role-based priority; and (iii) whether patients’ existing lifesaving resources should be subject to redistribution. (shrink)

Multiverse Views in set theory advocate the claim that there are many universes of sets, no-one of which is canonical, and have risen to prominence over the last few years. One motivating factor is that such positions are often argued to account very elegantly for technical practice. While there is much discussion of the technical aspects of these views, in this paper I analyse a radical form of Multiversism on largely philosophical grounds. Of particular importance will be an account of (...) reference on the Multiversist conception, and the relativism that it implies. I argue that analysis of this central issue in the Philosophy of Mathematics indicates that Radical Multiversism must be algebraic, and cannot be viewed as an attempt to provide an account of reference without a softening of the position. (shrink)

Recent discussions of how axioms are extrinsically justified have appealed to abductive considerations: on such accounts, axioms are adopted on the basis that they constitute the best explanation of some mathematical data, or phenomena. In the first part of this paper, I set out a potential problem caused by the appeal made to the notion of mathematical explanation and suggest that it can be remedied once it is noted that all the justificatory work is done by appeal to the theoretical (...) virtues. In the second part of the paper, I appeal to the theoretical virtues account of axiom justification to provide an argument that judgements of theoretical virtuousness, and therefore of extrinsic justification, are subjective in a substantive sense. This tells against a recent claim by Penelope Maddy that such justification is “wholly objective”. (shrink)

The incompleteness of set theory ZF C leads one to look for natural nonconservative extensions of ZF C in which one can prove statements independent of ZF C which appear to be “true”. One approach has been to add large cardinal axioms.Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski-Grothendieck set theory T G or It is a nonconservative extension of ZF C and is obtained from other axiomatic set theories by the inclusion of Tarski’s axiom (...) which implies the existence of inaccessible cardinals. See also related set theory with a filter quantifier ZF (aa). In this paper we look at a set theory NC# ∞# , based on bivalent gyper infinitary logic with restricted Modus Ponens Rule In this paper we deal with set theory NC# ∞# based on bivalent gyper infinitary logic with Restricted Modus Ponens Rule. Nonconservative extensions of the canonical internal set theories IST and HST are proposed. (shrink)

A decision maker (DM) selects a project from a set of alternatives with uncertain productivity. After the choice, she observes a signal about productivity and decides how much effort to put in. This paper analyzes the optimal decision problem of the DM who rationally filters information to deal with her post-decision cognitive dissonance. It is shown that the optimal effort level for a project can be affected by unchosen projects in her choice set, and the nature of the choice set-dependence (...) is determined by the signal structure. Some comparative statics of choice set-dependence is also provided. Finally, based on the results, the optimal choice set design is also explored. This paper offers a simple framework to explain the experimental finding in psychology that people’s effort level for a project can be enhanced when the project is chosen by themselves rather than by others. (shrink)

In this paper we introduce the notion of filter on the neutrosophic crisp set, then we consider a generalization of the filter’s studies. Afterwards, we present the important neutrosophic crisp filters. We also study several relations between different neutrosophic crisp filters and neutrosophic topologies. Possible applications to database systems are touched upon.

In the real word, in most cases, everything (an attribute, event, proposition, theory, idea, person, object, action, etc.) is evaluated in general by many sources (called experts), not only one. The more sources evaluate a subject, the better accurate result (after fusioning all evaluations). That’s why, in this paper, we straightforwardly extend the Refined Neutrosophic Set to the MultiNeutrosophic Set, and we show that the last two are isomorphic. A MultiNeutrosophic Set is a Neutrosophic Set whose all elements’ degrees of (...) truth/indeterminacy/falsehood are evaluated by many (Multi) sources. Afterwards, we introduce a total order on the set of n-plets of the form (p, r, s), we build the operators on the (p, r, s)-plets, and show several applications of the MultiNeutrosophic Sets. Several particular cases of the MultiNeutrosophic Sets are presented: such as MultiFuzzy Set, MultiIntuitionistic Fuzzy Set, MultiPicture Fuzzy Set, and other Multi(Fuzzy Extension) Set. (shrink)

In this book set theory INC# based on intuitionistic logic with restricted modus ponens rule is proposed. It proved that intuitionistic logic with restricted modus ponens rule can to safe Cantor naive set theory from a triviality. Similar results for paraconsistent set theories were obtained in author papers [13]-[16].

This work introduces the framework for selecting architecture in 5G networks, considering various technological, performance, economic, and operational factors. With the emergence of 5G technology, the architecture selection process has become pivotal in meeting diverse requirements for ultra-high-speed connectivity, low latency, scalability, and diverse service demands. The evaluation comprehensively analyses different architecture options, including centralized, distributed, cloud-based, and virtualized architectures. Factors such as network performance, scalability, cost-effectiveness, security, and compatibility are considered within a multi-criteria decision-making framework. Findings reveal each architecture (...) option's diverse strengths and limitations, emphasizing the importance of aligning architectural choices with specific use cases, deployment scenarios, and business objectives. We used the concept of multi-criteria decision-making (MCDM) methodology to control the various criteria. We used the single-valued neutrosophic set (SVNS) to deal with vague and inconsistent data. The SVNS is integrated with the hyperSoft set. The proposed methodology uses a single-valued neutrosophic hyperSoft set (SVNHSS) to select the best 5G architecture. We used the VIKOR method as an MCDM method to rank the alternatives. The proposed framework provides stakeholders with a structured methodology to evaluate and prioritize architecture options, facilitating informed decision-making in the complex landscape of 5G network deployments. (shrink)

This Paper combines interval- valued neutrouphic sets and rough sets. It studies roughness in interval- valued neutrosophic sets and some of its properties. Finally we propose a Hamming distance between lower and upper approximations of interval valued neutrosophic sets.

Both neutrosophic sets theory and rough sets theory are emerging as powerful tool for managing uncertainty, indeterminate, incomplete and imprecise information .In this paper we develop an hybrid structure called “ rough neutrosophic sets” and studied their properties.

In this paper, we generalize the soft set to the hypersoft set by transforming the function F into a multi-attribute function. Then we introduce the hybrids of Crisp, Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Hypersoft Set.

Set-theoretic and category-theoretic foundations represent different perspectives on mathematical subject matter. In particular, category-theoretic language focusses on properties that can be determined up to isomorphism within a category, whereas set theory admits of properties determined by the internal structure of the membership relation. Various objections have been raised against this aspect of set theory in the category-theoretic literature. In this article, we advocate a methodological pluralism concerning the two foundational languages, and provide a theory that fruitfully interrelates a `structural' perspective (...) to a set-theoretic one. We present a set-theoretic system that is able to talk about structures more naturally, and argue that it provides an important perspective on plausibly structural properties such as cardinality. We conclude the language of set theory can provide useful information about the notion of mathematical structure. (shrink)

In this paper, we introduce the plithogenic set (as generalization of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets), which is a set whose elements are characterized by many attributes (parameters)’ values. An attribute value v has a corresponding (fuzzy, intuitionistic fuzzy, or neutrosophic) degree of appurtenance d(x,v) of the element x, to the set P, with respect to some given criteria. In order to obtain a better accuracy for the plithogenic aggregation operators in the plithogenic set, and for a more (...) exact inclusion (partial order), a (fuzzy, intuitionistic fuzzy, or neutrosophic) contradiction (dissimilarity) degree is defined be-tween each attribute value and the dominant (most important) attribute value. The plithogenic intersection and union are linear combinations of the fuzzy operators tnorm and tconorm, while the plithogenic complement, inclusion (inequality), equality are influenced by the attribute values contradiction (dissimilarity) degrees. This article offers some examples and applications of these new concepts in our everyday life. (shrink)

It is a striking fact from reverse mathematics that almost all theorems of countable and countably representable mathematics are equivalent to just five subsystems of second order arithmetic. The standard view is that the significance of these equivalences lies in the set existence principles that are necessary and sufficient to prove those theorems. In this article I analyse the role of set existence principles in reverse mathematics, and argue that they are best understood as closure conditions on the powerset of (...) the natural numbers. (shrink)

In this paper we have defined neutrosophic ideals, neutrosophic interior ideals, netrosophic quasi-ideals and neutrosophic bi-ideals (neutrosophic generalized bi-ideals) and proved some results related to them. Furthermore, we have done some characterization of a neutrosophic LA-semigroup by the properties of its neutrosophic ideals. It has been proved that in a neutrosophic intra-regular LA-semigroup neutrosophic left, right, two-sided, interior, bi-ideal, generalized bi-ideal and quasi-ideals coincide and we have also proved that the set of neutrosophic ideals of a neutrosophic intra-regular LA-semigroup forms (...) a semilattice structure. (shrink)

Neutrosophy has been introduced by Smarandache [7, 8] as a new branch of philosophy. The purpose of this paper is to construct a new set theory called the neutrosophic set. After given the fundamental definitions of neutrosophic set operations, we obtain several properties, and discussed the relationship between neutrosophic sets and others. Finally, we extend the concepts of fuzzy topological space [4], and intuitionistic fuzzy topological space [5, 6] to the case of neutrosophic sets. Possible application to superstrings and space–time (...) are touched upon. (shrink)

Recent work has defended “Euclidean” theories of set size, in which Cantor’s Principle (two sets have equally many elements if and only if there is a one-to-one correspondence between them) is abandoned in favor of the Part-Whole Principle (if A is a proper subset of B then A is smaller than B). It has also been suggested that Gödel’s argument for the unique correctness of Cantor’s Principle is inadequate. Here we see from simple examples, not that Euclidean theories of set (...) size are wrong, but that they must be either very weak and narrow or largely arbitrary and misleading. (shrink)

This paper explores a middle way between realism and eliminativism about grounding. Grounding-talk is intelligible and useful, but it fails to pick out grounding relations that exist or obtain in reality. Instead, grounding-talk allows us to convey facts about what metaphysically explains what, and about the worldly dependence relations that give rise to those explanations.

This is a short piece of humor (I hope) on nonstandard set theories. An earlier version appeared in Bull. EATCS 45: 146-147 (1991).

Quasi-set theory $\cal Q$ allows us to cope with certain collections of objects where the usual notion of identity is not applicable, in the sense that $x = x$ is not a formula, if $x$ is an arbitrary term. $\cal Q$ was partially motivated by the problem of non-individuality in quantum mechanics. In this paper I discuss the range of explanatory power of $\cal Q$ for quantum phenomena which demand some notion of indistinguishability among quantum objects. My main focus is (...) on the double-slit experiment, a major physical phenomenon which was never modeled from a quasi-set-theoretic point of view. The double-slit experiment strongly motivates the concept of degrees of indistinguishability within a field-theoretic approach, and that notion is simply missing in $\cal Q$. Nevertheless, other physical situations may eventually demand a revision on quasi-set theory axioms, if someone intends to use it in the quantum realm for the purpose of a clear discussion about non-individuality. I use this opportunity to suggest another way to cope with identity in quantum theories. (shrink)

Hylomorphically complex objects are things that change their parts or matter or that might have, or have had, different parts or matter. Often ontologists analyze such objects in terms of sets (or functions, understood set-theoretically) or other extensional entities such as mereological fusions or quantities of matter. I urge two reasons for being wary of any such analyses. First, being extensional, such things as sets are ill-suited to capture the characteristic modal and temporal flexibility of hylomorphically complex objects. Secondly, sets (...) are often appealed to because they seem to contain their members. But the idea that sets do contain their members, in the ordinary sense of containment, is a substantive metaphysical position that makes analyses that rely on that idea for their plausibility much more metaphysically committing than is generally thought. (shrink)

In this book the authors introduce and study the following notions: Neutrosophic Crisp Points, Neutrosophic Crisp Relations, Neutrosophic Crisp Sets, Neutrosophic Set Generated by (Characteristic Function), alpha-cut Level for Neutrosophic Sets, Neutrosophic Crisp Continuous Function, Neutrosophic Crisp Compact Spaces, Neutrosophic Crisp Nearly Open Sets, Neutrosophic Crisp Ideals, Neutrosophic Crisp Filter, Neutrosophic Crisp Local Functions, Neutrosophic Crisp Sets via Neutrosophic Crisp Ideals, Neutrosophic Crisp L-Openness and Neutrosophic Crisp L-Continuity, Neutrosophic Topological Region, Neutrosophic Closed Set and Neutrosophic Continuous Function, etc. They compute (...) the distances between neutrosophic sets and extend it to Neutrosophic Hesitancy Degree. The authors also generalize the Crisp Topological Space and Intuitionistic Topological Space to the notion of Neutrosophic Crisp Topological Space. At the end, they present applications to Neutrosophic Database, and show a security scheme based on Public Key Infrastructure (PKI) using Neutrosophic Logic Manipulation. The authors utilize neutrosophic sets in order to analyze social networks data conducted through learning activities, and for the Geographical Information Systems (GIS) they employ fundamental concepts and properties of a Neutrosophic Spatial Region. Keywords: Neutrosophic Crisp Points; Neutrosophic Crisp Relations; Neutrosophic Crisp Sets; Neutrosophic Crisp Continuous Function; Neutrosophic Crisp Compact Spaces; Neutrosophic Crisp Nearly Open Sets; Neutrosophic Crisp Ideals; Neutrosophic Crisp Filter; Neutrosophic Crisp Local Functions; Neutrosophic Crisp Sets via Neutrosophic Crisp Ideals; Neutrosophic Crisp L-Openness and Neutrosophic Crisp L-Continuity; Neutrosophic Topological Region; Neutrosophic Closed Set and Neutrosophic Continuous Function; Neutrosophic Hesitancy Degree; Neutrosophic Crisp Topological Space; Neutrosophic Database; Neutrosophic Logic Manipulation; Neutrosophic Spatial Region. (shrink)

Contemporary analytic philosophy is dominated by metaphilosophical naturalism, the view that philosophy ought to be continuous with science. This naturalistic turn is for a significant part due to the work of W. V. Quine. Yet, the development and the reception of Quine’s naturalism have never been systematically studied. In this paper, I examine Quine’s evolving naturalism as well as the reception of his views. Scrutinizing a large set of unpublished notes, correspondence, drafts, papers, and lectures as well as published responses (...) to Quine’s work, I show how both internal tensions and external criticisms forced him to continuously develop, rebrand, and refine his metaphilosophy before he explicitly decided to label his view ‘naturalism’ in the late 1960s. (shrink)

We study active set methods for optimization problems in Block Angular Form (BAF). We begin by reviewing some standard basis factorizations, including Saunders' orthogonal factorization and updates for the simplex method that do not impose any restriction on the pivot sequence and maintain the basis factorization structured in BAF throughout the algorithm. We then suggest orthogonal factorization and updating procedures that allow coarse grain parallelization, pivot updates local to the affected blocks, and independent block reinversion. A simple parallel environment appropriate (...) to the description and complexity analysis of test procedures is defined in Section 5. The factorization and updating procedures are presented in Sections 6 and 7. Our update procedure outperforms conventional Updating procedures even in a purely sequential environment. (shrink)

A rough fuzzy set is the result of the approximation of a fuzzy set with respect to a crisp approximation space. It is a mathematical tool for the knowledge discovery in the fuzzy information systems. In this paper, we introduce the concepts of rough standard neutrosophic sets and standard neutrosophic information system, and give some results of the knowledge discovery on standard neutrosophic information system based on rough standard neutrosophic sets.

The purpose of this paper is to introduce new types of neutrosophic crisp sets with three types 1, 2, 3. After given the fundamental definitions and operations, we obtain several properties, and discussed the relationship between neutrosophic crisp sets and others. Also, we introduce and study the neutrosophic crisp point and neutrosophic crisp relations. Possible applications to database are touched upon.

I prove that the Boolean Prime Ideal Theorem is equivalent, under some weak set-theoretic assumptions, to what I will call the Cut-for-Formulas to Cut-for-Sets Theorem: for a set F and a binary relation |- on Power(F), if |- is finitary, monotonic, and satisfies cut for formulas, then it also satisfies cut for sets. I deduce the CF/CS Theorem from the Ultrafilter Theorem twice; each proof uses a different order-theoretic variant of the Tukey- Teichmüller Lemma. I then discuss relationships between various (...) cut-conditions in the absence of finitariness or of monotonicity. (shrink)

An introductory textbook on metalogic. It covers naive set theory, first-order logic, sequent calculus and natural deduction, the completeness, compactness, and Löwenheim-Skolem theorems, Turing machines, and the undecidability of the halting problem and of first-order logic. The audience is undergraduate students with some background in formal logic.

Set-theoretic and category-theoretic foundations represent different perspectives on mathematical subject matter. In particular, category-theoretic language focusses on properties that can be determined up to isomorphism within a category, whereas set theory admits of properties determined by the internal structure of the membership relation. Various objections have been raised against this aspect of set theory in the category-theoretic literature. In this article, we advocate a methodological pluralism concerning the two foundational languages, and provide a theory that fruitfully interrelates a ‘structural’ perspective (...) to a set-theoretic one. We present a set-theoretic system that is able to talk about structures more naturally, and argue that it provides an important perspective on plausibly structural properties such as cardinality. We conclude the language of set theory can provide useful information about the notion of mathematical structure. (shrink)

In this paper, we first propose the concept of divergence measure on neutrosophic sets. We also provide some formulas for the divergence measure for neutrosophic sets. After that, we investigate the properties of proposed neutrosophic divergence measure. Finally, we also apply these formulas in medical problem and the classification problem.