The purpose of this paper is to develop an approach to multiattribute group decision making under interval-valued hesitant fuzzy environment. To do this, this paper defines some new operations on interval-valued hesitant fuzzy elements, which eliminate the disadvantages of the existing operations. Considering the fact that elements in a set may be interdependent, two generalized interval-valued hesitant fuzzy operators based on the generalized Shapley function and the Choquet integral are defined. Then, some models for calculating the optimal fuzzy measures (...) on the expert set and the ordered position set are established. Because fuzzy measures are defined on the power set, it makes the problem exponentially complex. To simplify the complexity of solving a fuzzy measure, models for the optimal 2-additive measures are constructed. Finally, an investment problem is offered to show the practicality and efficiency of the new method. (shrink)

This paper generalizes the simple regret model by Bell in Operations Research 30(5), 961-981 and Loomes and Sugden in The Economic Journal 92(368), 805-824 to cope with the situation in which decision outcomes are multi-attributed. We propose a model that combines the simple regret model for ex ante preferences and the additive difference representation for ex post preferences. We first present a necessary and sufficient axiomatization of our model in Savage’s framework. The proposed model is composed of three types of (...) functions. One is a value function for each attribute. The others are attribute-dependent and holistic regret-rejoicing functions that capture trade-offs among value-differences of chosen and forgone outcomes for each attribute and among all attributes, respectively. We then provide constructive procedures for those functions and methods for consistency checks of the procedures. We, finally, conduct an experiment to estimate those functions in the two attributes case by means of the constructive procedure and give an application of the model and our findings. (shrink)

We present a method to characterize the preferences of a decision maker in decisions with multiple attributes. The approach modifies the outcomes of a multivariate lottery with a multivariate transformation and observes the change in the decision maker’s certain equivalent. If the certain equivalent follows this multivariate transformation, we refer to this situation as multiattribute transformation invariance, and we derive the functional form of the utility function. We then show that any additive or multiplicative utility function that is formed (...) of continuous and strictly monotonic utility functions of the individual attributes must satisfy transformation invariance with a multivariate transformation. This result provides a new interpretation for multiattribute utility functions with mutual utility independence as well as a necessary and sufficient condition that must be satisfied when assuming these widely used functional forms. We work through several examples to illustrate the approach. (shrink)

A theoretical framework for the reinforcement learning of aesthetic biases was recently proposed based on brain circuitries revealed by neuroimaging. A model grounded on that framework accounted for interesting features of human aesthetic biases. These features included individuality, cultural predispositions, stochastic dynamics of learning and aesthetic biases, and the peak-shift effect. However, despite the success in explaining these features, a potential weakness was the linearity of the value function used to predict reward. This linearity meant that the learning process (...) employed a value function that assumed a linear relationship between reward and sensory stimuli. Linearity is common in reinforcement learning in neuroscience. However, linearity can be problematic because neural mechanisms and the dependence of reward on sensory stimuli were typically nonlinear. Here, we analyze the learning performance with models including optimal nonlinear value functions. We also compare updating the free parameters of the value functions with the delta rule, which neuroscience models use frequently, vs. updating with a new Phi rule that considers the structure of the nonlinearities. Our computer simulations showed that optimal nonlinear value functions resulted in improvements of learning errors when the reward models were nonlinear. Similarly, the new Phi rule led to improvements in these errors. These improvements were accompanied by the straightening of the trajectories of the vector of free parameters in its phase space. This straightening meant that the process became more efficient in learning the prediction of reward. Surprisingly, however, this improved efficiency had a complex relationship with the rate of learning. Finally, the stochasticity arising from the probabilistic sampling of sensory stimuli, rewards, and motivations helped the learning process narrow the range of free parameters to nearly optimal outcomes. Therefore, we suggest that value functions and update rules optimized for social and ecological constraints are ideal for learning aesthetic biases. (shrink)

Russell had two theories of definite descriptions: one for singular descriptions, another for plural descriptions. We chart its development, in which ‘On Denoting’ plays a part but not the part one might expect, before explaining why it eventually fails. We go on to consider many-valued functions, since they too bring in plural terms—terms such as ‘4’ or the descriptive ‘the inhabitants of London’ which, like plain plural descriptions, stand for more than one thing. Logicians need to take plural reference (...) seriously if only because mathematicians take many-valued functions seriously. We assess the objection (by Russell, Frege and others) that many-valued functions are illegitimate because the corresponding terms are ambiguous. We also assess the various methods proposed for getting rid of them. Finding the objection ill-founded and the methods ineffective, we introduce a logical framework that admits plural reference, and use it to answer some earlier questions and to raise some more. (shrink)

In this paper we study intrinsic notions of “computability” for open and closed subsets of Euclidean space. Here we combine together the two concepts, computability on abstract metric spaces and computability for continuous functions, and delineate the basic properties of computable open and closed sets. The paper concludes with a comprehensive examination of the Effective Riemann Mapping Theorem and related questions.

We provide conditions under which an incomplete strongly independent preorder on a convex set X can be represented by a set of mixture preserving real-valued functions. We allow X to be infi nite dimensional. The main continuity condition we focus on is mixture continuity. This is sufficient for such a representation provided X has countable dimension or satisfi es a condition that we call Polarization.

Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valued field [Formula: see text]. Given a definable set [Formula: see text] let [Formula: see text] be the ring of continuous definable functions from [Formula: see text] to [Formula: see text]. Under very mild assumptions on the geometry of [Formula: see text] and on the structure [Formula: see text], in particular when [Formula: see text] is [Formula: see text]-minimal or [Formula: see text]-minimal, (...) or an expansion of a local field, we prove that the ring of integers [Formula: see text] is interpretable in [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] is definably connected of pure dimension [Formula: see text], then [Formula: see text] defines the subring [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] has no isolated points, then there is a discrete ring [Formula: see text] contained in [Formula: see text] and naturally isomorphic to [Formula: see text], such that the ring of functions [Formula: see text] which take values in [Formula: see text] is definable in [Formula: see text]. (shrink)

We show that without assuming completeness or continuity, a strongly independent preorder on a possibly infinite dimensional convex set can always be given a vector-valued representation that naturally generalizes the standard expected utility representation. More precisely, it can be represented by a mixture-preserving function to a product of lexicographic function spaces.

Prospect theory emerged as one of the leading descriptive decision theories that can rationalize a large body of behavioral regularities. The methods for eliciting prospect theory parameters, such as its value function and probability weighting, are invaluable tools in decision analysis. This paper presents a new simple method for eliciting prospect theory’s value function without any auxiliary/simplifying parametric assumptions. The method is applicable both to choice under ambiguity (Knightian uncertainty) and risk (when events are characterized by objective probabilities). (...) Our new elicitation method is implemented in a simple paper-and-pencil experiment (via an iterative multiple price list format). This is one of the first experiments that elicits non-parametric prospect theory’s value function with salient rewards. The collected data generally confirm findings in the existing literature: the value function is S-shaped (concave in the gain domain and convex in the loss domain) though there is a weaker loss aversion on the aggregate level and a substantial heterogeneity in loss aversion on the individual level (41% loss averse, 6% loss neutral and 53% gain seeking). (shrink)

In this paper we consider the algebra of differential operators with polynomial coefficients acting on Clifford algebra-valued functions from both sides. We characterize the subalgebra of SO(m)-invariant differential operators, which itself contains the subalgebra of GL(m)-invariant differential operators.

Two sets are said to be almost disjoint if their intersection is finite. Almost disjoint subsets of [ω]ω and ωω have been studied for quite some time. In particular, the cardinal invariants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{a}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{a}_e}$$\end{document}, defined to be the minimum cardinality of a maximal infinite almost disjoint family of [ω]ω and ωω respectively, are known to be consistently less than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} (...) \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{c}}$$\end{document}. Here we examine analogs for functions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^\omega}$$\end{document} and projections on l2, showing that they too can be consistently less than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{c}}$$\end{document}. (shrink)

The notion of an instrument in the quantum theory of measurement is studied in the context of transformation valued linear maps on von Neumann algebras and their *-subalgebras. An extension theorem is proved which yields among other things characterizations of the Fourier transforms of instruments and their noncommutative analogues. As an application, an ergodic type theorem for a general class of transformation valued functions on a locally compact group is obtained.

Consider a language SL having as its primitive signs one or more atomic statements, the two connectives ‘∼’ and ‘&,’ and the two parentheses ‘’; and presume the extra connectives ‘V’ and ‘≡’ defined in the customary manner. With the statements of SL substituting for sets, and the three connectives ‘∼,’ ‘&,’and ‘V’ substituting for the complementation, intersection, and union signs, the constraints that Kolmogorov places in [1] on probability functions come to read:K1. 0 ≤ P,K2. P) = 1,K3. (...) If ⊦ ∼, then P = P + P,K4. If ⊦ A ≡ B, then P = P.2. (shrink)

Abstract: In this article, I offer a proposal to clarify what I believe is the proper relation between value maximization and stakeholder theory, which I call enlightened value maximization. Enlightened value maximization utilizes much of the structure of stakeholder theory but accepts maximization of the long-run value of the firm as the criterion for making the requisite tradeoffs among its stakeholders, and specifies long-term value maximization or value seeking as the firm’s objective. This proposal (...) therefore solves the problems that arise from the multiple objectives that accompany traditional stakeholder theory. I also discuss the Balanced Scorecard, which is the managerial equivalent of stakeholder theory, explaining how this theory is flawed because it presents managers with a scorecard that gives no score—that is, no single-valued measure of how they have performed. Thus managers evaluated with such a system (which can easily have two dozen measures and provides no information on the tradeoffs between them) have no way to make principled or purposeful decisions. The solution is to define a true (single dimensional) score for measuring performance for the organization or division (and it must be consistent with the organization’s strategy), and as long as their score is defined properly, (and for lower levels in the organization it will generally not be value) this will enhance their contribution to the firm. (shrink)

In this paper we study 12 four-valued logics arisen from Belnap's truth and/or knowledge four-valued lattices, with or without constants, by adding one or both or none of two new non-regular operations—classical negation and natural implication. We prove that the secondary connectives of the bilattice four-valued logic with bilattice constants are exactly the regular four-valued operations. Moreover, we prove that its expansion by any non-regular connective (such as, e.g., classical negation or natural implication) is strictly functionally complete. Further, finding axiomatizations (...) of the quasi varieties generated by the 12 logics involved (that prove to be varieties), we find naturell equational axiomatizations of these logics. Finally, applying Pynko's general theory of algebraizable sequential consequence operations, we also find equivalent natural sequentiell axiomatizations of the logics under consideration that expand either of two Pynko's sequential calculi for the constant-free truth-lattice four-valued logic. (shrink)

The philosophy of medicine and psychiatry has considered the defining of disease, illness, and disorder an important project for over three decades. Within this literature, accounts based on adaptive "functions" have been prominent, particularly in the DSM nosology. In response to this trend, Jerome Wakefield has presented a view of mental disorder as "harmful dysfunction." In this view, "harm" contributes the value-element to disorder concepts, while "dysfunction" implies a value-free foundation as long as the latter is grounded (...) in evolutionary biology. In a critical review of Wakefield's and others' functionalist arguments for disorder concepts, we make four major points. We recommend a shift away from the definition of value-free disorder concepts, instead emphasizing the analysis of the logical features and the value commitments in nosological categories. (shrink)

Is the concept of a work of art an evaluative concept: does its application imply a positive evaluation? I shall discuss this question by considering two opposing attempts at defining art, namely the Institutional Theory and the view that art is a functional concept. I shall argue that the concept of art does not imply an unconditionally positive evaluation, but that art is a prestige concept. Moreover, it will be shown that functional definitions of art are flawed.

This paper proposes a functional approach to characterize epistemic and nonepistemic values. The paper argues that epistemic values are functionally homogeneous since they act as criteria to evaluate the epistemic virtues a hypothesis ought to possess, and they validate scientific knowledge claims objectively. Conversely, non-epistemic values are functionally heterogeneous since they may promote multiple and sometimes conflicting aims in different research contexts. An incentive of espousing the functional approach is that it helps us understand how values can operate in appropriate (...) and inappropriate ways in scientific research and inappropriate influences can eventually be prevented. The idea is to argue that since non-epistemic values are functionally heterogeneous, they cannot provide objective reasons for the acceptance of a hypothesis. However, their involvement is necessary in certain research contexts and the problem is the involvement of these need not be always legitimate. By analyzing a case from chemical research, I demonstrate that how non-epistemic values might influence scientific research and, then, I go on to demonstrate that how a proper understanding of the functions of different kinds of values might promote the attainment of multiple goals of a particular research in a legitimate and socially relevant way. (shrink)

We develop a bottom-up approach to truth-value semantics for classical logic of partial terms based on equality and apply it to prove the conservativity of the addition of partial description and selection functions, independently of any strictness assumption.

Functional words are of substantial interest in moral philosophy because they appear to lie at the juncture of description and evaluation. This is no doubt the reason that they have played a significant part in much recent discussion of the relation between facts and values. Yet, in spite of the many discussions in which functional words have made an appearance, their significance for an understanding of the relation between facts and values remains unclear. A thorough-going examination of the nature of (...) functional words would appear to be in order. And while such a study is beyond the scope of a single article it should nevertheless be possible to make a beginning. That, at any rate, is my objective in what follows. (shrink)

This thesis (110 pages) was submitted in March 1970 in partial fulfillment of the requirements for the degree of Master of Arts. The work was supervised by Professor Raymond Klibansky, McGill University.

The purpose of this paper is modelling and controlling the spread of COVID-19 disease in Morocco. A nonlinear mathematical model with two subclasses of infectious individuals is proposed. The population is divided into five classes, namely, susceptible, exposed, undiagnosed infectious, diagnosed patients, and removed individuals. To reflect the real dynamic of the COVID-19 transmission in Morocco, the real reported data are used for estimating model parameters. Two controls representing screening effort and limited treatment are considered. Based on viability theory and (...) set-valued analysis, a Lyapunov function is constructed such that both exposed and infected populations are decreased to zero asymptotically. The corresponding controls are derived via a continuous selection of adequately designed feedback map. Numerical simulations are presented with three scenarios. Our results show that when only one control is to be applied, screening is the most effective in decreasing the number of people in the three infected compartments, whereas combining both controls is found to be highly effective and leads to a significant improvement in the epidemiological situation of Morocco. To the best of our knowledge, this work is the first one that applies the set-valued approach to a controlled COVID-19 model which agrees with the observed cases in Morocco. (shrink)

A theoretical structure for multiattribute decision making is presented, based on a dynamical system for interactions in a neural network incorporating affective and rational variables. This enables modeling of problems that elude two prevailing economic decision theories: subjective expected utility theory and prospect theory. The network is unlike some that fit economic data by choosing optimal weights or coefficients within a predetermined mathematical framework. Rather, the framework itself is based on principles used elsewhere to model many other cognitive and (...) behavioral data, in a manner approximating how humans perform behavioral functions. Different, interconnected modules within the network encode (a) attributes of objects among which choices are made, (b) object categories, (c) and goals of the decision maker. An example is utilized to simulate the actual consumer choice between old and new versions of Coca-Cola. Potential applications are also discussed to market decisions involving negotiations between participants, such as international petroleum traders. (shrink)

The concept of disease remains hotly contested. In light of problems with existing accounts, some theorists argue that the disease concept ought to be eliminated. We answer this skeptical challenge by reframing the discussion in terms of the role that the disease concept plays in the complex network of health-care institutions in which it is deployed. We argue that while prevailing accounts do not suffer from the particular defects that critics have identified, they do suffer from other deficits, and this (...) leads us to propose a new account that satisfies the desiderata for a concept of disease in human medicine. (shrink)