Learning preferences from paired opposite-based semantics

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Learning preferences from paired opposite-based semantics

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Camilo Franco a,∗ , J. Tinguaro Rodríguez b , Javier Montero b,c a

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IFRO, Faculty of Science, University of Copenhagen, Frederiksberg, 1870, Denmark Faculty of Mathematics, Complutense University, Madrid, 28040, Spain Institute IGEO (CISC-UCM), Complutense University, Madrid, 28040, Spain

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Article history: Received 22 February 2016 Received in revised form 21 October 2016 Accepted 30 April 2017 Available online xxxx

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Keywords: Paired concepts Fuzzy logic Preference structures Semantic opposition Fuzzy reinforcement Significance

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Preference semantics examine the meaning of the preference predicate, according to the way that alternatives can be understood and organized for decision making purposes. Through opposite-based semantics, preference structures can be characterized by their paired decomposition of preference into opposite poles, and their respective valuation of binary preference relations. Extending paired semantics by fuzzy sets, preference relations can be represented in a gradual functional form, under an enhanced representational frame for examining the meaning of preference. Following a semantic argument on the character of opposition, the compound meaning of preference emerges from the fuzzy reinforcement of paired opposite concepts, searching for significant evidence for affirming dominance among the decision objects. Here we propose a general model for the paired decomposition of preference, examining its characteristic semantics under a binary and fuzzy logical frame, and identifying solutions with different values of significance for preference learning. © 2017 Published by Elsevier Inc.

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1. Introduction The meaning of the preference concept refers to the way decision objects or alternatives can be ordered for decision making purposes. This concept can be examined through the preference predicate, regarding the paired opposite decomposition of the positive and negative perceptions of preference. In psychology (see e.g. [3,11,17]), the meaning of concepts has been studied in relation to its valuation as being either positive or negative, eliciting a subjective measurement from the individual, but at the same time, requiring that the individual somehow solves the natural ambivalence involved in understanding opposite perceptions. For doing this, a semantic scale has been commonly used (initially proposed in [17]), measuring the meaning of concepts according to a given pair of opposite poles. In decision theory (see e.g. [4,10,22]), such a bi-polarity has been studied from two perspectives. The first one can be referred as the univariate model, introducing a one-dimensional scale with opposite references as endpoints, in such a way that one of these references is taken to be (or is understood) as positive, in opposition to the other which is considered as its negative counterpart. In this setting, it is possible to further introduce a reciprocity assumption, so that the verification status of one pole entails by complementation a particular status of the other. For example, if we assume reciprocal preferences and we have absolute preference for watching a fiction movie over a documentary, then it is already assumed

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This paper is part of the virtual special issue on Advances in Weighted Logics for Artificial Intelligence, edited by Marcelo Finger, Lluis Godo, Henri Prade and Guilin Qi. Corresponding author. E-mail addresses: [email protected] (C. Franco), [email protected] (J. Tinguaro Rodríguez), [email protected] (J. Montero).

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http://dx.doi.org/10.1016/j.ijar.2017.04.010 0888-613X/© 2017 Published by Elsevier Inc.

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that we have absolutely no preference for watching a documentary. In this (restricted) univariate reciprocal model, where opposite preference is directly associated with the inverse preference, it is then possible to prefer one or the other, and if no reciprocity is imposed, we may even prefer nothing at all, such that the value for preference is either positive, negative, or neither positive nor negative. On the other hand, the second perspective on bipolarity can be referred as the unipolar bivariate model (see e.g. [10, 11]), where a concept can be positive, negative, neither positive nor negative, or both positive and negative, thus allowing preference (and aversion) for watching both a fiction and a documentary. Then the relation among poles is not simple, but rather complex and depends on the particular semantic relation holding among the opposite concepts. Even more, taking into consideration multi-dimensional concepts, like multiple viewpoints describing the properties of objects, the semantic relation holding among opposites may require a more complex analysis. Focusing on the different neutral states holding in between the opposite poles, and stressing the determinant role that those states have for representing the meaning of concepts, logical paired structures [13] provide an adequate framework to address such complexity. Particularly, in previous works we have studied how the different neutral states in between opposite poles that are postulated in the context of paired structures may be useful tools to represent and understand the complexity of preference concepts (see [8]), allowing to configure a pertinent valuation preference structure for ordering the decision objects or alternatives. Furthermore, paired structures [8,13] explicitly represent the different and non-reciprocal sources of information building up the complex meaning of preference. In this respect, and on a neurological level, it is observed that the meaning of concepts emerges from the multiple positive and aversive stimuli composing perceptions and emotions [3,9]. In this sense, different pleasant and unpleasant affective components of the same sensory stimulus, processed separately in different physical areas of the brain (see e.g. [16,23]), may provide the inputs of human behavior and decision making [1,9]. Therefore, as neurological observation suggests, the positive and negative counterparts are formed and evaluated separately, in an independent manner, configuring a significant decision space with respect to the available (positive and negative) evidence. In this way, opposite sets of evidence can be simultaneously evaluated as separate entities, coming together under the fuzzy reinforcement of their intensities. That is, given the separate nature of positive and negative aspects, they can be jointly examined as they reinforce each other under an appropriate aggregation process based on opposition operators [13,18]. As it will be examined in detail in the later sections of this paper (Sections 5 and 6), opposite pieces of evidence, coming from different sources, can be used to reinforce each other in order to find greater significance to their inferred meaning. Here we propose a general setting where the performance of the different preference models can be formally assessed according to their significance. For this purpose, a measure of relative significance is introduced for evaluating the amount of evidence for affirming preference. As a result, the aggregation process unravels while searching for significant evidence on pairwise dominance for preference learning and intelligent (automatic) decision support, where the reliance on the emotional meaning associated to the alternatives allows explaining and identifying satisfactory (descriptive) viewpoints for decision-making (see e.g. [12,22]). In order to examine opposite-based preference semantics for preference learning, Section 2 introduces standard preference structures, where the inverse preference relation represents the negative perception on preference. Section 3 extends the analysis to more complex preference structures, where positive and negative preferences are independently represented and measured. Then, in Section 4, fuzzy preference structures are introduced, examining a general frame for fuzzy paired preference semantics. In Section 5, a proposal for the significance of preference orders is given, and in Section 6, the general methodology for learning preferences is proposed, based on fuzzy reinforcement and maximal significance. Finally, Section 7 offers a numerical example illustrating the proposed methodology, ending with some open challenges for future research.

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Given a set of alternatives A, a crisp preference structure can be defined ∀(a, b) ∈ A2 , by the decomposition of the weak preference predicate R (a, b) = a is at least as desired as b, into three basic binary relations P , I , J , such that (see e.g. [5]),

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P: The pair of alternatives (a, b) ∈ A2 belongs to the strict preference relation P , if and only if a is more desired than b, expressed by P (a, b). The inverse strict preference predicate, b is more desired than a, is expressed by P (b, a) = P −1 (a, b). I: The pair of alternatives (a, b) ∈ A2 belongs to the indifference relation I , if and only if a is as much as desired as b, expressed by I (a, b). J: The pair of alternatives (a, b) ∈ A2 belongs to the incomparability relation J , if and only if a cannot be compared with b, expressed by J (a, b).

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2. Standard preference semantics

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Under standard preference modeling (see again [5] but also [19] and [14]), it is assumed that ∀a, b ∈ A, the relations I and J are symmetrical, such that I (a, b) = I (b, a) and J (a, b) = J (b, a) hold, I is reflexive, such that I (a, a) holds, J is irreflexive, such that J (a, a) does not hold, and P is asymmetrical, such that P (a, b) and P (b, a) cannot hold simultaneously. Then, the preference structure ( P , I , J ), is such that only one situation holds, as in,

P ∩ I = ∅,

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fulfilling properties,

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where R d = N ( R −1 ) = ( N ( R ))−1 , for the logical complement N, and being complete in the sense of,

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P = Q ∩ N (V ) = R ∩ N (R

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but notice here that it could also be, e.g., that Q = P and V = P −1 . In consequence, the meaning assigned to the poles Q and V may respectively correspond with R and R −1 , or with P and P −1 .

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Proof. Under the hypothesis that N ( V ) = Q , where it holds that Q = R and V = R −1 = N ( R ), it is true that N ( V ) = N ( N ( R )) = R = Q . According to (4), it holds that R = P ∪ I , and hence that N ( R ) = P −1 . Therefore, it follows that R −1 = N ( R ) ⇐⇒ R −1 = P −1 , being true that ( R −1 )−1 = ( P −1 )−1 = R = P . 2

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As a result, based on Proposition 1, there is a single concept Q and its complementation V = N ( Q ), being there no place for indifference as a neutral category [13] if the poles of preference are complementary. In this sense, reciprocity entails a binary unipolar model where either P or P −1 holds (see [8]).

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Example 1. Consider the decision problem of choosing a place to live, and a recommendation system offering feedback on suitable alternatives. Let us have two alternatives a, b ∈ A, assessing their desirability according to the price. After eliciting the preferences for a over b, a given user states that R (a, b) = 1, and at the same time, that R (b, a) = 1. Then, if the system takes both pieces of evidence, the result is that I (a, b) = 1 holds, but if the system assumes (additive) reciprocity, after verifying that R (a, b) = 1, it then obtains that R (b, a) = 0. In this way, under the reciprocal condition, it results that P (a, b) = 1, and consequently, the recommendation system would identify a as the most desired alternative for the user. Otherwise, allowing the separate elicitation of the inverse preference, the system would identify both alternatives as being just as desirable. In consequence, the reciprocity condition entails some important loss of information, as the elicitation of the inverse preference reveals that the user is indifferent between both alternatives.

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Considering the process for the formation of preferences, candidate alternatives are gradually understood and brought forward regarding their specific attributes, which are perceived under a paired decomposition that allows representing the

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Proposition 1. If R = P ∪ I is a reciprocal preference relation, i.e., such that its semantic poles are complementary, then R = P .

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In this way, in the standard structure, poles Q and V are allowed to overlap, under the specific semantic condition that Q = R and V = R −1 . Otherwise, if poles are assigned the strict preference values, such that Q = P and V = P −1 , then the neutral situation of indifference (I ) can only be defined by exclusion, as the state holding if no pole is verified. Thus, it is a general attitude towards preference, following a semantic argument, which allows distinguishing between indifference, either as a situation defined by exclusion among poles (case of the univariate model), or as a situation defined by the overlap among poles (case of the bivariate model). In order to grasp the relevance of this semantic argument, consider the following result, taking the valuation set {0, 1}, such that ∀(a, b) ∈ A2 , R (a, b) = 1 if R (a, b) holds, otherwise R (a, b) = 0, an involutive negation operator N, such that ∀x ∈ {0, 1}, N ( N (x)) = x, and a reciprocal preference relation R, such that Q = R and V = R −1 = N ( R ), defined for all pairs {(a, b) ∈ A2 |a = b}.1

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Exploring preference semantics under the standard structure, the meaning of R is captured between the two predicates of R and R −1 , which stand as opposite (inverse) poles evaluating the meaning of preference. Opposite poles offer a set of valuation references for positive and negative judgments, hereby denoted by Q and V , respectively. Hence, poles can be assigned any pair of relations having a specific type of opposite semantics between them, being here the case that Q = R and V = R −1 , such that,

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Notice that being R a reciprocal relation, indifference (I ) is defined by exclusion, such that pairs (a, a) ∈ A2 are always indifferent, i.e., ∀a ∈ A, I (a, a) = 1.

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emotional character of decisions (see e.g. [1,3,12,23]). Hence, following on Example 1, the meaning of price can refer not only to the negative condition of establishing a restriction on the user’s budget, but also, to the positive perception on the social condition that a high price entails for the individual. Therefore, an extended frame for preference representation is needed to properly take into account both the positive and the negative dimensions of decisions, aiming at capturing the relevant aspects for identifying and explaining different viewpoints for decision-making.

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The standard preference structure can be extended according to a deeper conceptual decomposition of the preference predicate, based on four different sources of information (see e.g. [6,8]). On the one hand, for every a, b ∈ A, take the positive aspects of alternatives to evaluate the preference predicate R + (a, b) = R (a, b) = a is at least as desired as b and its inverse R + (b, a), and on the other, take their negative aspects for evaluating the negative preference predicate R − (a, b) = a is at least as non-desired/rejected as b and its inverse R − (b, a). These paired references are associated to logically independent poles ( Q , V ), such that R + (a, b) = Q (a, b), R + (b, a) = Q (b, a) = Q −1 (a, b), R − (a, b) = V (a, b) and R − (b, a) = V (b, a) = V −1 (a, b). After some neurological evidence on how the brain deals with the multiple positive and aversive stimuli composing perceptions and emotions (processing separately the pleasant and the unpleasant affective components in different physical areas of the brain [9,16,23]), it can be examined how opposite-based semantics may provide the inputs for preference-based decision making. Therefore, preference can be understood under a more complex evaluation space [3], generalizing the one of the standard model (as shown in [8]), representing the activation channels for positive and negative affections by a pair of opposite conceptual poles Q = “desire” and V = “rejection”. Thus, each pole is firstly, and separately, decomposed into their respective standard structures for Preference and Aversion (P–A) [6], where positive aspects are measured with respect to ( Q , Q −1 ), obtaining the standard structure ( P , I , J ) through expressions (1)–(7). Analogously, the negative aspects are measured with respect to ( V , V −1 ), obtaining the aversion structure ( Z , G , H ). In consequence, the aversion structure is defined ∀(a, b) ∈ A2 , by the decomposition of the weak aversion predicate R − = ( V , V −1 ), into three basic binary relations Z , G , H , such that,

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Z: The pair of alternatives (a, b) ∈ A belongs to the strict aversion relation Z , if and only if a is more rejected than b, denoted by Z (a, b). G: The pair of alternatives (a, b) ∈ A2 belongs to the aversion indifference relation I , if and only if a is as much as rejected as b, denoted by G (a, b). H: The pair of alternatives (a, b) ∈ A2 belongs to the aversion incomparability relation J , if and only if a cannot be compared with b on their negative aspects, denoted by H (a, b). 2

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Hence, ∀a, b ∈ A, the one-dimensional relations G and H are symmetrical, G is reflexive, H is irreflexive, and Z is asymmetrical, and the three relations are linked together as in,

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Z ∩ G = ∅,

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such that,

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Z∪H=V ,

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Z ∪ Z −1 ∪ G ∪ H = A2 .

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As a result, every pair a, b ∈ A can be simultaneously valued as belonging to one relation in ( P , I , J ), and one in ( Z , G , H ), building the meaning of preference on the separate and simultaneous verification of the positive and negative aspects of decisions. Extending this binary model to a continuous one, fuzzy logic allows a gradual valuation of preference, exploring how are preferences weighed according to the intensity in which they are perceived (see e.g. [5,6,14]). Next, fuzzy preference structures are examined together with paired-opposite semantics.

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4. Fuzzy paired semantics

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Fuzzy preference structures [5,14,24] represent preference relations as gradual predicates which may be verified up to a certain degree of intensity, fulfilling as much as possible conditions (4)–(7) and (11)–(14). In this way, the decision outcome is valued by different preference states and their degrees of verification, instead of only one preference/aversion situation as in the classical-binary setting of (1)–(3) and (8)–(10). From this standpoint, the characterization of a fuzzy preference relation is given by

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I = i (μ Q , μ Q −1 ) = T (μ Q , μ Q −1 ),

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J = j (μ Q , μ Q −1 ) = T ( N (μ Q ), N (μ Q −1 )),

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Z = z(μ V , μ V −1 ) = T (μ V , N (μ V −1 )),

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where T is a continuous t-norm (a standard fuzzy conjunctive operator), N is a strict negation (in this fuzzy setting, given by a strictly decreasing, continuous function), and i , j , g and h are symmetrical functions. Therefore, the standard properties (4)–(7) and (11)–(14), can be respectively formulated in fuzzy logic by means of a continuous t-conorm S,2 such that (see again [5,6]),

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S ( p , j ) = N (μ Q ),

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S ( z, g , z−1 ) = S (μ V , μ V −1 ),

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Taking the paired decomposition of preference according to the P–A framework,

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is the membership function of R, measuring the degree up to which any pair of decision objects verify the preference predicate. In this way, there are a pair of functions representing the predicates R + and R − , that under the hypothesis that Q = R + and V = R − , are given by,

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Table 1 Fuzzy preference/aversion structures.

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M

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where N is a strong (involutive) negation, such that ∀x ∈ [0, 1], N ( N (x)) = x, and ∀x, y ∈ [0, 1], T L (x, y ) = max(x + y − 1, 0), T M = min(x, y ), and S = S L = min(x + y , 1). That is, ( T L , S L , N ) is the Lukasiewicz De Morgan triple along with its respective residual T M . Under this solution, i and j are mutually exclusive in ( p , i , j ), as well as g and h are mutually exclusive in ( z, g , h), expressing the classical logical impossibility of finding any pair of objects that are equally desired (rejected) but at the same time incomparable on their desirable (rejectable) attributes. Another solution (not fully complying with the standard properties) can be identified for fuzzy preference structures [5], if p (z) is defined as a strongly asymmetrical relation, such that ∀x, y ∈ [0, 1], p (x, y ) > 0 ⇒ p −1 (x, y ) = 0. Then, defining p , z by means of the t-norm T L , and i , j , g , h by means of T M , both the preference (16), (18)–(19), and the aversion conditions (20), (22)–(23), are satisfied, but not (17) nor (21). Lastly (see [24]), there is another limit solution that allows the simultaneous verification of all basic relations in ( p , i , j ) and ( z, g , h), but that does not fulfill (16)–(18) nor (20)–(22). This solution (which is complete in the sense of (19) and (23)), is given by the multiplicative or probabilistic De Morgan triple ( T p , S p , N ), such that ∀x, y ∈ [0, 1], T p (x, y ) = x · y and S = S p = x + y − x · y.

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Example 2. Following Example 1, the initial two alternatives a, b ∈ A can be assessed regarding their desirability as well as their rejectability regarding relevant attributes such as price, location and area. Consider here the case where the user reveals the overall weak preference and aversion values, given by μ Q (a, b) = 0.5, μ Q −1 (a, b) = 0.9, μ V (a, b) = 0.6 and

μ V −1 (a, b) = 0.7. All the different solutions for the fuzzy preference structures can be seen in Table 1, where p = z = T

M

stands for the solution complying with all conditions (16)–(23), p = z = T L agrees with the asymmetrical modeling of p and z, and p = z = T p takes the multiplicative De Morgan triple. As a result, the first solution obtains that b is preferred to a, although both alternatives seem to be similar both on the positive and negative aspects. The other solutions confirm their P–A indifference, with a slightly greater intensity for preference on b over a, suggesting a weak recommendation for choosing b.

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These solutions for the multi-valued characterization of R P − A (15), through fuzzy preference structures R + , and R − , illustrate the fuzzy-paired semantics of P–A. This approach proposes an analytical neurological foundation for preference learning (as it will be examined in the following sections), stressing the separate conception of both positive and negative components, being accordingly treated for understanding the meaning of human perceptions and stimuli. Then, in order to address the P–A meaning, both opposite dimensions have to be aggregated. The paired networks of P–A may interact in different ways, according to the semantic requirements on their aggregation. Thus, they can be either conjunctively or disjunctively aggregated, or under a fair aggregation, they can be allowed to interact by compensating the positive and the negative aspects, through the reinforcement of desirable and non-rejectable intensities. For this purpose, different types of aggregation operators can be examined for representing the P–A meaning, learning preference orders with different degrees of (evidence-supported) significance.

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5. Significance of preference orders

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The meaning of preference responds to the perception of opposite stimuli, which, absorbed through the proposed structure R P − A (15), offer a pair of complex relational networks that may interact together by reinforcing the intensity for desire through the corroboration of non-aversion.3 In this sense, the significance of verifying both classes of desirable and non-rejectable evidence, is greater than verifying only one, or either one of them.4 Consider a preference (weak) order  on A, such that θ a stands for the position of a ∈ A in the order . Then, ∀a, b ∈ A, it holds one of the following: θ a > θ b , θ b > θ a , or θ b = θ a . In case that θ b > θ a (θ a > θ b ), it holds that a dominates b (b dominates a), otherwise a and b are equivalent in order . In this way, it is proposed that by learning a significant order , a position is assigned to every alternative a ∈ A, such that the significance of  increases with the amount of evidence supporting it. For every a, b ∈ A, let Q = { Q , Q −1 }∀a,b∈A stand for the set of evidence on the positive aspects of preference, and let V = { V , V −1 }∀a,b∈A stand for the set of evidence on the negative aspects of aversion. Then significance can be defined as follows.

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

33 34 35 36 37 38 39 40 41 42 43 44

preference for a is measured by an evidence counter ea , initially set up to ea = 0. In this way, whenever it holds that Q (a, b) > Q (b, a), then ea = ea + 1 (i.e., there is one more unit of evidence for affirming that a is a desirable alternative), and in the same way, if it holds that V (b, a) > V (a, b), then ea = ea + 1 (i.e., there is one more unit of evidence for affirming that a is a non-rejectable alternative).

Thus, the significance of a preference order can be understood as a monotone, increasing function of the amount of evidence supporting each pairwise ordering of alternatives. In this sense, if both Q (a, b) > Q (b, a) and V (b, a) > V (a, b) hold, then ea = ea + 2, and if it holds ∀b = a ∈ A, that Q (a, b) > Q (b, a) and V (b, a) > V (a, b), then ea = 2(n − 1). Proposition 2. Given two separate sets of evidence Q = { Q , Q −1 }∀a,b∈A and V = { V , V −1 }∀a,b∈A , and two preference orders on A, 1 and 2 , such that 1 is supported by both sets of evidence Q and V, and 2 is supported by only one set of evidence from Q or V. If there exists some a, b ∈ A where both Q (a, b) > Q (b, a) and V (b, a) > V (a, b) hold, then the significance of 1 is greater than the significance of 2 .

Based on Proposition 2, a measure of relative significance can be proposed, given two separate sets of evidence, one for the desired attributes Q, and another for the rejected attributes V, and given a reference order 1 , built from both bodies of evidence Q and V. In this way, the degree of significance of another order 2 , with respect to the reference order 1 , is measured by

σ (1 , 2 ) = 1 −

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1

|A|2



dist (θ1a , θ2a ),

∀a∈A

dist (θ1a , θ2a ) = |θ1a − θ2a |.

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The relative significance measure (24) computes the significance of the order 2 with respect to a reference, significant order 1 , coming from the joint verification of desire and non-aversion. Thus, it holds that σ (1 , 2 ) = 1, only if 1 = 2 , being equivalently maximal significant orders. Otherwise, 2 is a significant order up to a degree, given by 0 < σ (1 , 2 ) < 1. Notice that on a strictly formal level, there is a certain similarity between (24) and Spearman’s rank correlation coefficient. Nonetheless, they belong to different contexts, as the degree of significance (24) measures if there

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(24)

where dist can be taken e.g. as a 1-norm distance, such that

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Proof. For every a ∈ A, count the evidence for affirming preference for a under index ea , such that if Q (a, b) > Q (b, a) holds, then ea = ea + 1, and in the same way, if V (b, a) > V (a, b) holds, then ea = ea + 1. It then follows that ∀a ∈ A,  the value of ea under 2 , ea 2 , counts the evidence on either Q (a, b) > Q (b, a) or V (b, a) > V (a, b), such that if both    Q (a, b) > Q (b, a) and V (b, a) > V (a, b) hold, then ea 2 = ea 2 + 1. But under 1 , ea 1 counts the evidence on both Q (a, b) >   Q (b, a) and V (b, a) > V (a, b), such that if both Q (a, b) > Q (b, a) and V (b, a) > V (a, b) hold, then ea 1 = ea 1 + 2, such that 1 2 ea > ea . Thus, if there exists some a, b ∈ A where both Q (a, b) > Q (b, a) and V (b, a) > V (a, b) hold, then the significance of 1 is greater than the significance of 2 . 2

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Definition 1. Given two separate sets of evidence Q = { Q , Q −1 }∀a,b∈A and V = { V , V −1 }∀a,b∈A , the significance for affirming

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Notice that this approach is different from the one that represents the meaning of R P − A by aggregating its opposite conceptual structures as in

( P , I , J ) × ( Z , G , H ) , generating 16 different relational situations (see e.g. [6,8]). 4

This observation on the greater significance of verifying the evidence on both the desirable and non-rejectable aspects is grounded on a dialectical decision principle (see e.g. [7]), stating that needs are more important to satisfy than desires, where needs refer to both desired and not rejected alternatives.

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is any difference between two rankings (where one of them is fixed as a reference, significant order); and the Spearman correlation measures the statistical dependence or correlation between the ranking of two random variables. Now, the procedure for building significant orders from the available evidence requires representing the way that both opposite paired concepts of P–A are aggregated, as two separate complex perceptions, together with how they are weighed along the aggregation process. Therefore, the meaning of R P − A is examined next, modeling the weighted compound predicate for Q and not-V , as it organizes the positive and negative stimuli under a neurological frame for preference learning.

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Focusing on the separate nature of positive and negative perceptions, and how they are weighed and transformed into meaningful knowledge, it has been observed that opposite pieces of evidence, coming from different sources, can be used to reinforce each other in order to find greater significance to their inferred meaning. Therefore, the meaning of Q can be reinforced by not-V , once the predicate V has been transformed by some involutive operation into representing that which needs to be avoided (see e.g. [6,7]). Here, opposition operators [13,18] allow modeling in a general and flexible way the meaning of the predicate not-V , which is independent from the predicate Q , such that Q = not-V and not-Q = V . In this way, ∀x, y ∈ [0, 1] and a strong negation N, an opposition operator A : [0, 1] → [0, 1], is such that [18],

A ( A (x)) = x,

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x ≤ y ⇒ A ( y ) ≤ A (x).

20 21

Then, A is an antonym [21] if it holds that,

22

A ≤ N,

23 24

and A is an antagonism [18], also called sub-antonym [13], if it holds that,

25 26

A ≥ N.

27

An example for the opposition operator A, previously assuming the standard strong negation N = 1 − x, can be given by the Sugeno family of strong negations or λ-complement [20], defined by (∀x ∈ [0, 1], λ > −1),

N λ (x) =

1−x 1 + λx

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,

(25)

being an antonym whenever λ ≥ 0, and an antagonist whenever −1 < λ ≤ 0. Through the opposition operator A, the reinforced interaction can take place among the paired network formed by Q and A ( V ), denoted by R  =  Q , A ( V ) , such that ∀a, b ∈ N, the degree of reinforced preference, or dominance, is given by,

R  (a, b) = C (μ Q (a, b), A (μ V (a, b))),

(26)

where C is an aggregation function C : [0, 1]2 → [0, 1], being {0,1}-idempotent (i.e., ∀x ∈ {0, 1}, C (x, x) = x), and monotonic (i.e., ∀x, y , z, w ∈ [0, 1], C (x, y ) > C ( w , z) whenever x > w and y > z. If instead of taking the weak predicates Q = R + , V = R − , the strict predicates Q = P , V = Z are taken as opposite poles, following a strict decision attitude towards preference, the dominance relation R  , can be transformed by a semantic argument for taking the strict reinforcing predicates of P and A ( Z ). Thus, based on (26), the weak dominance relation is characterized by the weak assignation of opposites Q = R + and V = R − , while the strict dominance relation is characterized by the strict assignation of opposites Q = P and V = Z . Under a strict attitude, fuzzy preference structures hold a set of solutions which allow the joint verification of ( p , p −1 , i , j ) and (z, z−1 , g , h) under the multiplicative solution; or having mutual exclusion among p and p −1 , z and z−1 , i and j or g and h, according to any of the Lukasiewicz–De Morgan solutions. In consequence, considering the decision attitude towards preference (weak or strict), the specific semantic requirements for A (antonym or antagonist), and the type of aggregation for C , the meaning for R  (26) can be measured under three general perspectives. The first perspective develops from a disjunctive meaning, where either degree of desirability or non-rejection allows affirming a general preference order. The second one is more demanding, requiring both degrees of desirability and non-rejection to be verified, while the third one allows some fair compensation among them, allowing further interaction among the paired fuzzy reinforcing concepts of μ Q and A (μ V ).

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6.1. Disjunctive meaning

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Examining the meaning of R  from a disjunctive perspective, requires using a disjunctive aggregation operator. For this, take an aggregation operator C = C s , such that C s (0, 1) = C s (1, 0) = 1. Hence, this is the most tolerant type of aggregation, as the verification of either degree of desirability or non-rejection is enough for having some degree of dominance. Take e.g. the (smallest) t-conorm C s = S M (being also a grouping operator [2]), such that ∀x, y ∈ [0, 1], S M = max(x, y ).

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As a result, the disjunctive dominance fuzzy relation is defined ∀a, b ∈ A, such that, s R (a, b) = C s (μ Q (a, b), A (μ V (a, b))).

1

(27)

s Hence, ∀a, b ∈ A, a simple voting procedure can be applied for preference learning based on R  , such that a vote for the s s dominance of a over b is conceeded when R  (a, b) > R  (b, a). Thus, alternatives can be ranked (weakly ordered) according to their total number of votes. This order is referred to as the disjunctive order, based on (27) and the votes in favor of, or not against dominance, due to the intensity of preference on desire or non-rejection. Such a weak order is denoted by s . The order s is the result of the most tolerant ordering procedure, as the verification of just one of the intensities being aggregated is enough to affirm dominance. Hence (as pointed out in Proposition 2), this is a solution that fails to comply with the more significant one of the conjunctive procedure, which is examined next.

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From a conjunctive perspective, the meaning of R  requires a conjunctive aggregation operator. Thus, take an aggregation operator C = C t , such that C t (0, 1) = C t (1, 0) = 0. Therefore, this is the most demanding type of aggregation, as the verification of both a degree of desirability and of non-rejection is needed for having some degree of dominance. Take e.g. the t-norm C t = T M (being the largest t-norm and also an overlap operator [2]). In consequence, the conjunctive dominance fuzzy relation is defined ∀a, b ∈ A, such that,

R t (a, b) = C t (μ Q (a, b), A (μ V (a, b))).

14 15 16 17 18 19

(28)

20

Applying the same voting procedure as before, this time based on R  , ∀a, b ∈ A, a vote for the dominance of a over b is conceeded when R t (a, b) > R t (b, a). In this way, alternatives are ranked according to their total number of votes under the conjunctive order, based on (28) and the votes in favor of, and not against dominance, due to the intensity of preference on desire and non-rejection. This order is denoted by t . The order t is the result of the most demanding ordering procedure, as the verification of both the positive and non-negative intensities is necessary to affirm dominance. Hence, this is the most significant solution, which nonetheless can be difficult to obtain due to its strong requirements. In usual decision problems, in particular multi-criteria problems (see e.g. [10,19]), some trade-off or compensation among the decision attributes is often required to arrive at satisfactory solutions. For this reason, a compensative meaning is explored next, with the purpose of finding a compromise over the positive and non-negative intensities of preference.

21

t

C v (x1 , x2 ) =

23 24 25 26 27 28 29 30 31 33

Focusing on a compromise solution, the meaning of R  requires a compensating or averaging aggregation operator. For this purpose, take an aggregation operator C = C v , such that 0 < C v (0, 1) = C v (1, 0) < 1. In a general sense, this type of aggregation allows some interaction among the reinforcing concepts, being a compromise solution in between the limit solutions of (27) and (28). Such a compensating operator can be modeled by means of the Choquet integral C v , widely used in decision literature, generalizing the weighted arithmetic mean (see e.g. [10,15,22]). Given the set X = {x1 , x2 }, ∀x1 , x2 ∈ [0, 1], the discrete Choquet integral can be defined with respect to a non-additive set function or fuzzy measure v : 2 X → [0, 1], such that, 2 

22

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6.3. Compensative meaning

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6.2. Conjunctive meaning

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v ( X (i ) )(x(i ) − x(i −1) ),

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i =1

where (·) indicates a permutation of X , such that x(0) = 0 ≤ x(1) ≤ x(2) , and X (1) = {(1), (2)} and X (2) = {(2)}. For example, |X| the uniform fuzzy measure v ( X ) = 2 , can be used to compute the standard arithmetic mean. It follows that the compensative dominance fuzzy relation is defined ∀a, b ∈ A2 , such that, v R (a, b) = C v (μ Q (a, b), A (μ V (a, b))).

(29)

s Applying the same voting procedure as with R  and R t , ∀a, b ∈ A, a vote for the dominance of a over b is here v v conceeded whenever R  (a, b) > R  (b, a). Then, alternatives are ranked according to their total number of votes under the compensative order, based on (29) and a compromise on the votes in favor of, and not against dominance, due to the intensity of preference on desire and/or non-rejection. This weak order is denoted by  v . Therefore, given an attitude towards preference (weak or strict), together with the type of semantic opposition (antonym or antagonist), ∀a, b ∈ A, the dominance fuzzy relation (26) can be computed, estimating the number of votes supporting the preference of a ∈ A over all other alternatives b = a ∈ A. Then, different orders are obtained under a disjunctive or compensating meaning, respectively given by s (27) and  v (29), whose significance (24) can be measured with respect to t (28), learning the order ∗ = t with maximal significance. As a result, the conjunctive order t stands as the most significant decision, at the expense of being too strong in its aggregation procedure. Hence, focusing on a compensating solution, decision support can be offered in the form of a

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Table 2 Weak and strict preference/aversion intensities.

(a, b) (a, c ) (a, d) (b, c ) (b, d) (c , d)

2

μQ

μ Q −1

μV

μ V −1

p

p −1

z

z−1

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0.6 0.3 0.4 0.1 0.9 0.5

0.7 0.7 0.4 0.2 0.9 0.5

0.1 0.6 0.3 0.1 0.7 0.1

0.5 0.1 0.6 0.3 0.2 0.4

0.3 0.3 0.4 0.1 0.1 0.5

0.4 0.7 0.4 0.2 0.1 0.5

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Consider the specific problem of looking for a place to live on rent (following Examples 1 and 2). The options are restricted to a given city, looking for alternative houses or apartments, studying them while thinking on the pros and cons of choosing one or the other, and trying to arrive at a satisfactory solution for an adequate and decent home. In a general sense, the decision is perceived based on its negative and the positive aspects, where the positive refers to the attributes that can be associated to the desirable aspects, and the negative refers to the rejectable ones. Think e.g. on common aspects related to the price: while they can be perceived as negative, the cheaper the better, they can also be perceived as positive, for a fair price indicates good quality and may offer even more confidence than a suspiciously low price (commonly associated to scams). After studying the market and the available alternatives, people articulate their preferences around plausible candidates homes, identifying the one that fits better to their desires and needs. This process can be aided by a recommendation system, which could be implemented as an open-access website.5 Suppose the website needplace2live.com exists, offering support on housing options by replicating the individual’s decision process and identifying a priority order with maximum significance. As input data from the users, it is required that they introduce information on their attitudes, whether their demands are weak or strict, and their specific desires and needs on the maximum price they are willing to pay, pmax, the greatest distance they are willing to live away from the city center, dmax, and the minimum area that they are looking for, mina. Then, the system offers the service of finding useful recommendations, presenting them in decreasing order of preference. Given a database containing all the available information on the housing options (such as the price, distance/location and area), the system goes through every option verifying if they fulfill the specific requirements introduced by the user. This first filtering process is intended to identify a small set of candidate alternatives, thus reducing the initial complexity of the following pairwise comparison process (which requires a number of 2(|A|2 − |A|) pairwise comparisons). In this way, for every pair (a, b) ∈ A2 , the desirability for a over b increases with a relatively lower (greater) price and distance (size), over an acceptance fuzzy area around the pmax and dmax (mina) reference values; while the rejectability for a over b increases with a relatively higher (lower) price and distance (size), over a rejectance fuzzy area around the pmax and dmax (mina) reference values.6 The process obtains the fuzzy weak P–A intensities (μ Q , μ Q −1 ) and (μ V , μ V −1 ), as shown in Table 2. Having filtered four candidate alternatives, the intensities of weak preference and aversion are computed, together with the strict preference and aversion values for the three limit solutions of t-norms T L , T M and T p (see Table 2 for the weak and strict (p , z = T M ) preference and aversion values. The other limit solutions for p , z are omitted as they all obtain the same ordering results). Then, as negative perceptions weigh more than positive ones under general conditions of uncertainty (see e.g. [22]), the opposition operator is taken as an antagonist operator, such that N = 1 − x and λ ∈ (−1, 0), and preferences are computed for every meaning of R  (27)–(29). The system identifies the recommended compromise solution with greater significance, given for the optimal λ value (see Table 3 and Table 4, respectively showing the estimations for preference orders under a weak and a strict attitude, including the maximal-significant solution together with some other λ-based estimations). In Table 3, the different λ-based results for a weak attitude are shown. Based on the number of votes (#) for every alternative, a ranking is built allowing ties among different alternatives (the position in the ranking is given by θ ). It can be seen that, for all values of λ ∈ (−1, 0), the ranking under a compensative meaning (29) obtains a tie among the most preferred alternatives b and d, while b is ranked first under a disjunctive meaning (27). On the other hand, under the more demanding conjunctive meaning (28), the ranking depends on the particular value of λ, resulting in different significance scores (RS) for both the disjunctive and compensative meanings. On the other hand, under the strict attitude of Table 4,

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negotiated decision among the P–A positive and negative perceptions. In this way, it is possible to learn the compensating solution that maximizes its significance for an optimal value of λ, by computing fuzzy reinforcement (26) over a set of values associated to the antonym/antagonist λ-complement operator (25). The overall methodology is illustrated in the following example.

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5 We could also imagine a mobile application, capturing information on house-searching behavior through a tracking system, and automatically gathering the user’s revealed preferences for suggesting recommendations. 6 P–A intensities could even be measured by means of a brain scanner, or some type of virtual tracking device, presenting images of the housing options to the user and estimating the positive and negative units of pleasure and aversion that each option stimulates in the brain.

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Table 3 Weak attitude preference orders, with the number of votes (#) and ranking position (θ ) of the alternatives under the conjunctive (t), disjunctive (s) and compensative (v) solutions, and their relative significance (RS).

2 3 4 5

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6

− 0.8

(#, θ)t

(#, θ)s

(#, θ) v

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a b c d

(1, 2) (2, 1) (1, 2) (2, 1)

(1, 2) (3, 1) (0, 3) (1, 2)

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RS

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λ

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(#, θ)t

(#, θ)s

(#, θ) v

a b c d

(2, 1) (0, 3) (1, 2) (2, 1)

(1, 2) (3, 1) (0, 3) (1, 2)

RS

1

0.69

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λ − 0.6

5

(#, θ) v

6

(1, 2) (2, 1) (1, 2) (2, 1)

(1, 2) (1, 2) (1, 2) (2, 1)

(1, 2) (3, 1) (0, 3) (1, 2)

(1, 2) (2, 1) (1, 2) (2, 1)

7

1

1

0.81

0.94

11

− 0.2

(#, θ)t

(#, θ)s

(#, θ) v

(1, 2) (2, 1) (1, 2) (2, 1)

(2, 1) (0, 3) (1, 2) (1, 2)

(1, 2) (3, 1) (0, 3) (1, 2)

(1, 2) (2, 1) (1, 2) (2, 1)

0.81

1

0.75

0.75

−1

(#, θ)t

(#, θ)s

(#, θ) v

28

a b c d

(1, 2) (2, 1) (1, 2) (2, 1)

(0, (0, (0, (0,

29

RS

1

0.88

25 26 27

39 40 41 42

1) 1) 1) 1)

(−1, 0]

(#, θ)t

(#, θ)s

(#, θ) v

(1, 2) (2, 1) (1, 2) (2, 1)

(1, 2) (2, 1) (1, 2) (2, 1)

(2, 1) (1, 2) (0, 3) (0, 3)

(1, 2) (2, 1) (1, 2) (2, 1)

1

1

0.69

1

the conjunctive and compensative meanings entail the same ranking for all values of λ ∈ (−1, 0), where b and d are ranked first, while the disjunctive meaning obtains different orders for λ = −1, having absolute indifference, and for λ ∈ (−1, 0], being a the most preferred one. As a result, following the compensating solution ((#, θ) v ) with maximum significance, b and d are both assigned the first position, followed by a and c. This ranking is identified under a weak attitude, for a λ value of −0.8, and it is confirmed by the results associated to the strict attitude, where the same compensative order holds for any value of an antagonistic λ ∈ (−1, 0]. In case a total order is required, a solution can be found according to some relevant argument that allows breaking the tie between b and d. Thus, the associated disjunctive solution can be examined, as it can be understood as a solution not complying with the significant one. It is then observed that under a weak attitude, b is ranked over d with significance 0.88, while for a strict attitude, b is ranked over d with a significance of 0.69. Hence, forcing a total order, d is placed in second place, followed by a and then by c.

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Acknowledgements

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Preference semantics have been applied to the problem of preference learning, building weak orders from the fuzzy reinforcement of opposite paired preferences. Based on dominance fuzzy relations and different attitudes towards preference, positive and negative perceptions are weighed according to their opposite character, and different pieces of evidence are aggregated for preference learning under different levels of significance. For future research, the formulation of different solutions for fuzzy (paired) preference structures remains to be explored under other types of aggregation operators, which may not satisfy the standard properties of associativity, commutativity or monotonicity. Besides, the construction of total orders should be explored in more detail, further refining weak orders, and interval valued preferences should be considered, extending the paired fuzzy frame for the representation of imprecise fuzzy preferences, learning from interval-valued or type-II fuzzy reinforcement. Finally, on an applied level, the proposed preference learning technique by fuzzy reinforcement should be tested on real data, measuring the performance of the results together with their respective degrees of significance.

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8. Final comments

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λ

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12

λ

Table 4 Strict attitude preference orders.

22

34

4

19

21

33

3

(#, θ)s

20

32

2

(#, θ)t

19

31

1

58 59

This research has been partially supported by the Government of Spain (grant TIN2015-66741-P), the Government of Madrid (grant S2013/ICE-2845, CASI-CAM-CM), and by the Danish Industry Foundation.

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