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"Possibility"
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Metaphysical and absolute possibility
It is widely alleged that metaphysical possibility is “absolute” possibility (Kripke in Naming and necessity, Harvard University Press, Cambridge, 1980; Lewis in On the plurality of worlds, Blackwell, Oxford, 1986; van Inwagen in Philos Stud 92:68–84, 1997; Rosen, in: Gendler and Hawthorne (eds) Conceivability and possibility, Clarendon, Oxford, 2002, p 16; Stalnaker, in: Stalnaker (ed) Ways a world might be: metaphysical and anti-metaphysical essays, Oxford University Press, Oxford, 2003, pp 201–215;Williamson in Can J Philos 46:453–492, 2016). Kripke calls metaphysical necessity “necessity in the highest degree” (1980, p. 99). Van Inwagen claims that if P ismetaphysically possible, then it is possible “tout court. Possible simpliciter. Possible period... possib(le) without qualification (1997, p. 72).” And Stalnaker writes, “we can agree with Frank Jackson, David Chalmers, Saul Kripke, David Lewis, and most others who allow themselves to talk about possible worlds at all, that metaphysical necessity is necessity in the widest sense (2003, p. 203).” What exactly does the thesis that metaphysical possibility is absolute amount to? Is it true? In this article, I argue that, assuming that the thesis is not merely terminological, and lacking in any metaphysical interest, it is an article of faith. I conclude with the suggestion that metaphysical possibility may lack the metaphysical significance that is widely attributed to it.
Journal Article
Possibility mean, variance and standard deviation of single-valued neutrosophic numbers and its applications to multi-attribute decision-making problems
Single-valued neutrosophic numbers (SVN-numbers) are a special kind of neutrosophic set on the real number set. The concept of a SVN-number is important for quantifying an ill-known quantity and ranking of SVN-number is a very difficult situation in decision-making problems. The main aim of this paper is to present a new ranking methodology of SVN-numbers for solving multi-attribute decision-making problems. Therefore, we firstly define the possibility mean, variance and standard deviation of single-valued neutrosophic numbers. Using the ratio of possibility mean and standard deviation, we have developed the proposed ranking approach and applied to MADM problems. Finally, a numerical example is examined to show the applicability and embodiment of the proposed method.
Journal Article
Exploring the Structure of Possibility Multi-Fuzzy Soft Ordered Semigroups Through Interior Ideals
This paper aims to introduce a novel idea of possibility multi-fuzzy soft ordered semigroups for ideals and interior ideals. Various results, formulated as theorems based on these concepts, are presented and further validated with suitable examples. This paper also explores the broad applicability of possibility multi-fuzzy soft ordered semigroups in solving modern decision-making problems. Furthermore, this paper explores various classes of ordered semigroups, such as simple, regular, and intra-regular, using this innovative method. Based on these concepts, some important conclusions are drawn with supporting examples. Moreover, it defines the possibility of multi-fuzzy soft ideals for semiprime ordered semigroups.
Journal Article
Type-2 fuzzy variables and their arithmetic
This paper proposes an axiomatic framework from which we develop the theory of type-2 (T2) fuzziness, called
fuzzy possibility theory
. First, we introduce the concept of a fuzzy possibility measure in a fuzzy possibility space (FPS). The fuzzy possibility measure takes on regular fuzzy variable (RFV) values, so it generalizes the scalar possibility measure in the literature. One of the interesting consequences of the FPS is that it leads to a new definition of T2 fuzzy set on the Euclidean space
which we call T2 fuzzy vector, as a map to the space instead of on the space. More precisely, we define a T2 fuzzy vector as a measurable map from an FPS to the space
of real vectors. In the current development, we are suggesting that T2 fuzzy vector is a more appropriate definition for a T2 fuzzy set on
In the literature, a T2 fuzzy set is usually defined via its T2 membership function, whereas in this paper, we obtain the T2 possibility distribution function as the transformation of a fuzzy possibility measure from a universe to the space
via T2 fuzzy vector. Second, we develop the product fuzzy possibility theory. In this part, we give a general extension theorem about product fuzzy possibility measure from a class of measurable atom-rectangles to a product ample field, and discuss the relationship between a T2 fuzzy vector and T2 fuzzy variables. We also prove two useful theorems about the existence of an FPS and a T2 fuzzy vector based on the information from a finite number of RFV-valued maps. The two results provide the possible interpretations for the concepts of the FPS and the T2 fuzzy vector, and thus reinforce the credibility of the approach developed in this paper. Finally, we deal with the arithmetic of T2 fuzzy variables in fuzzy possibility theory. We divide our discussion into two cases according to whether T2 fuzzy variables are defined on single FPS or on different FPSs, and obtain two theorems about T2 fuzzy arithmetic.
Journal Article
Hyperspectral Image Classification with Capsule Network Using Limited Training Samples
Deep learning techniques have boosted the performance of hyperspectral image (HSI) classification. In particular, convolutional neural networks (CNNs) have shown superior performance to that of the conventional machine learning algorithms. Recently, a novel type of neural networks called capsule networks (CapsNets) was presented to improve the most advanced CNNs. In this paper, we present a modified two-layer CapsNet with limited training samples for HSI classification, which is inspired by the comparability and simplicity of the shallower deep learning models. The presented CapsNet is trained using two real HSI datasets, i.e., the PaviaU (PU) and SalinasA datasets, representing complex and simple datasets, respectively, and which are used to investigate the robustness or representation of every model or classifier. In addition, a comparable paradigm of network architecture design has been proposed for the comparison of CNN and CapsNet. Experiments demonstrate that CapsNet shows better accuracy and convergence behavior for the complex data than the state-of-the-art CNN. For CapsNet using the PU dataset, the Kappa coefficient, overall accuracy, and average accuracy are 0.9456, 95.90%, and 96.27%, respectively, compared to the corresponding values yielded by CNN of 0.9345, 95.11%, and 95.63%. Moreover, we observed that CapsNet has much higher confidence for the predicted probabilities. Subsequently, this finding was analyzed and discussed with probability maps and uncertainty analysis. In terms of the existing literature, CapsNet provides promising results and explicit merits in comparison with CNN and two baseline classifiers, i.e., random forests (RFs) and support vector machines (SVMs).
Journal Article