In this short communication, we introduce a class of functions defined via quasi-arithmetic means on the unit interval. We demonstrate that when these means are induced by strictly convex and differentiable additive generators of strict triangular norms, the resulting functions are distance functions that satisfy the properties of continuous and symmetric monometrics as well.

Approximate reasoning Threshold-based Modus PonensIn fuzzy logic, most of the implication operators are based on generalizations of the classical, material implication.That is, these implications are defined as the disjunction of the negated value of the first argument and the value of the second argument, while the underlying disjunction operators are associative triangular conorms.In our study, we concentrate on how a class of implication operators, called the preference implication operators, can be used in approximate reasoning.Using this implication operator family, we present a novel, Modus Ponens-like approximate reasoning method, in which we have two premises: (1) a statement and (2) a preference implication with an antecedent of this statement.Here, we show how the continuous logical value of the consequent of the preference implication can be derived from the continuous logical values of the premises.We point out that this novel approximate reasoning method is strongly connected with the so-called aggregative operator, which is a representable uninorm.Next, we also present a threshold value-based generalization of the Modus Ponens syllogism and demonstrate that the Modus Tollens syllogism can be generalized in the same way.Lastly, we provide an illustrative example.

Abstract This research explores individual investors’ preferences for cryptocurrencies in Hungary using a novel analytical method with flexible fuzzy numbers, which accounts for uncertainty in responses, unlike the traditional Likert scale-based analysis. An online questionnaire yielded 116 responses, which were analyzed using descriptive and inferential statistical methods. The flexible fuzzy number-based questions assessed respondents’ knowledge of cryptocurrencies, blockchain, and trust levels. Responses were aggregated mathematically, and demographic factors such as age, gender, education, marital status, income, and cryptocurrency ownership were analyzed. Median tests were used for hypothesis testing between cryptocurrency owners and non-owners based on defuzzified values. Relationships among variables were visualized using heatmaps. Additionally, comparative analysis of cryptocurrency preferences was conducted between investors in Hungary and Germany. Insights from this study can help financial institutions tailor investment portfolios by understanding individual cryptocurrency preferences in Hungary.

In this short communication, we will show that a quasi-arithmetic mean induced by an additive generator of a strict t-norm (strict t-conorm, respectively) can be represented by the composition of a unary operator (called the tau function) and the strict t-norm (strict t-conorm, respectively), both induced by the same generator function as the quasi arithmetic mean. Here, we will also state a connection between the idempotency of a transformed strict t-norm (strict t-conorm, respectively) and the tau function.

Abstract Following tfhe seminal paper of Offerman et al. (2009), in this study, adaptations of constructions of continuous-valued logic to prospect theory are presented. Here, we demonstrate that the so-called kappa function and its special cases are viable alternatives to some elements of quadratic scoring rule prospects theory. After, we present the tau-eta scoring rule prospect and show that it may be treated as a generalization of the quadratic scoring rule prospect. Furthermore, we prove that if this new prospect for an uncertain event is evaluated using specific kappa functions as utility functions, then (1) the weighting measure of the event is a function of the optimal value of its reported probability, (2) the inverse of the latter function, and (3) the (risk-) corrected reported probability of the event, also as a function of the optimal value of its reported probability, all have a common formula. The parameters of the common formula are unambiguously determined by four tuning parameters. Lastly, we show that with our approach, by fitting one of the abovementioned functions to corresponding empirical data, we can immediately obtain the other two functions as well.

In the application of fuzzy sets, the greatest uncertainty appears when the membership grade in a fuzzy set is equal to the membership grade in the complement of this fuzzy set. That is, the membership grade is equal to the value of the fixed point ($\nu$) of the underlying complement (negation) operator. The fixed point of the standard Zadeh negation is 0.5. In fuzzy set theory, the complement of a fuzzy set can be defined using a strong complement (negation) operator that differs from the standard Zadeh negation. In this case, the fixed point of the strong complement (negation) operator is not necessarily 0.5. In this short paper, we present the concept of a parametric fuzziness measure called the$\nu$-maximal vagueness measure. This new measure may be regarded as a generalized fuzziness measure. Here, we present an operator system-dependent kernel, which we call the$\nu$-maximal vagueness entropy, and using this, we construct a$\nu$-maximal vagueness measure. The proposed vagueness entropy is based on a common generator function of a strict triangular norm and a strong negation operator. The$\nu$-maximal vagueness entropy is flexible and operator-dependent, and it can be readily adapted to a continuous-valued logical system. Furthermore, the parameter$\nu$has a clear semantic meaning.

In this short communication, we will present a connection between two well-known representations of the strong negations (involutive negations). Namely, we will provide a necessary and sufficient condition for the equality of Trillas and Dombi forms of negations. We will also show a connection between the additive generators of nilpotent and strict triangular norms.

In this study, we present a continuous-valued logical approach to generalize the sigmoid function. Our starting point is the kappa function, which is known as a unary operator in continuous-valued logic. First we extend the kappa function to the (a,b) interval, and then we interpret the generalized sigmoid function as the limit of the extended kappa function when a and b tend to the negative and positive infinity, respectively. Since the extended kappa function is induced by an additive generator of a strict t-norm or strict t-conorm, the generalized sigmoid function is operator dependent. Based on the properties of this new function, we show that it can be viewed as the generalization of the classical sigmoid function. Also, we demonstrate that the classical sigmoid function is a special case of the generalized sigmoid function. Next, we provide a sufficient condition for the equality of two generalized sigmoid functions. It is well known that the classical sigmoid function can be utilized in logistic regression and in preference modeling. Here, we demonstrate how the logistic regression can be generalized using the generator function-based sigmoid function. Also, we show that the generalized sigmoid function can be viewed as a preference measure.

Here, we present an operator-dependent, analytic membership function family that is derived from two soft inequalities by using the so-called middle hedge operator. This operator is defined over a pair of a strictly increasing and a strictly decreasing parametric membership functions that represent two soft inequalities. The middle hedge operator can also be treated as a membership function of an interval on a bounded or unbounded domain. Owing to its construction, the values of this membership function at the end points of the interval, which it represents, are equal to a given parameter value. This parameter may be interpreted as an intersection parameter. That is, if a fuzzy partition of a bounded or unbounded interval is created by using middle hedge operators with a fixed value of the intersection parameter, then each two successive membership functions in the fuzzy partition intersect at the same level. Here, we demonstrate that this new membership function family is very flexible and it can be readily used for constructing a membership function system in a fuzzy control system.

In this paper, we study monotone set functions defined as the composition of an additive measure with a strictly increasing function. This function is a unary operator in continuous-valued logic, called the tau function, and it is a generator function-based parametric mapping. We provide a necessary and sufficient condition for the equality of two tau functions that are induced by different generator functions. Using the tau function and its properties, we introduce a new monotone measure that we call the tau-additive measure. This measure is computationally simple and it can be viewed as an upper or lower probability depending on the parameter settings of the tau function. We present the parameter-dependent submodularity and supermodularity of the tau-additive measure and show how this measure can be constructed from a set function on a finite set. This procedure is analogous to how the well-known lambda-additive measure can be constructed, but our method is computationally simpler. We demonstrate that the tau-additive measure can be used to approximate the lambda-additive measure. Lastly, exploiting these theoretical results, we present an application in the area of human resource management.

In this study, we present the operator-dependent sigmoid function, which is derived from a universal unary operator called the kappa function. Here, we describe how the generalized sigmoid function is related to representable uninorms (i.e., Dombi's aggregative operator). Namely, we show that the inverse of a generalized sigmoid function is an additive generator of the aggregative operator. We provide the necessary and sufficient conditions for the form of the function that transforms the aggregative operator into a conjunctive or disjunctive logical operator. This transformation is also based on the generalized sigmoid function. Here, we show how conjunctive and disjunctive operators, which form a De Morgan system with a negation, can be derived from the aggregative operator. We point out that, under certain conditions, a set of generalized sigmoid functions is closed under the negation and modifier operators. Lastly, we demonstrate an important connection between the weighted aggregative operator and the generalized sigmoid function. Based on this connection, we provide a new interpretation of the feed-forward neural networks. We show that a perceptron-based neural network can be modeled using the aggregative operator and the generalized sigmoid function.

In this study, the notion of a dual pair of modal operators is interpreted according to the algebraic criteria for necessity and possibility operators on De Morgan lattices presented by Cattaneo, Ciucci and Dubois, 2011. Here, a representation theorem is introduced which demonstrates that, in this algebraic model, a dual pair of modal operators can be represented by compositions of two strong negations, where one of them is stricter than the other. Then, the Pliant negation operator is utilized to derive dual modal operators. It is demonstrated that using the generator function of Dombi operators, the composition of two Pliant negations results in modal operators that have simple forms and easy-to-use characteristics. Next, we examine how the proposed modal operators are connected with the drastic necessity and possibility operators. Also, the necessary and sufficient condition for the distributivity of modal operators induced by compositions of strong negations over strict t-norms and strict t-conorms is presented. Lastly, a connection between the modal operators and hedges is highlighted.

In this paper, we study operator dependent modifiers and we interpret the dual pair of modal operators based on an algebraic definition. It is a known fact that the substantiating and weakening modifier operators can be induced by repeating the arguments of conjunctive and disjunctive operators. We provide the conditions for which these modifier operators satisfy the requirements for a dual pair of necessity and possibility operators. Next, the necessary and sufficient condition for the distributivity of unary operators over conjunctive and disjunctive operators is presented. This also means that setting the distributivity as a requirement results in a unary operator that is identical to the modal operators mentioned above. Using this property, we establish an important connection between modal operators and linguistic hedges. Previously, we demonstrated that the unary operators induced by compositions of two strong negations satisfy the requirements for a dual pair of modal operators. Here, we view the negation operator as a modifier operator. Then, it is shown that (1) the strong negations, (2) the substantiating and weakening modifier operators, modal operators and linguistic hedges mentioned above, and (3) the unary operators, which are distributive over conjunctive and disjunctive operators, may be viewed as special cases of a unified unary operator class.

Following our previous paper Dombi and Jónás (2019) [16], we will now present new inequalities using the general Poincaré formula for λ-additive measures. These inequalities represent bounds for the well-known Poincaré formula of probability theory.

Huynh et al. introduced a probability-based fuzzy relation for comparing fuzzy numbers (see V. Huynh, Y. Nakamori and J. Lawry (2008) [40]), but they did not detail how to compute it. Here, we will consider this fuzzy relation as a probability-based preference intensity index and present closed formulas for the integrals needed to compute this index for fuzzy sets that have trapezoidal membership functions. Also, we will propose an algorithm to compute this index and a numerical method to approximate it. The comparison of two fuzzy numbers should also be able to capture the situation where the order of the fuzzy numbers cannot be judged; and so, their order may be considered as being indifferent. Here, using the probability-based preference intensity index, we will introduce two crisp relations, which have a common parameter, over a collection of fuzzy sets that have trapezoidal membership functions. Next, we will show that - depending on the parameter value - one of them is a strict order relation and the other one may be interpreted as a relation that expresses the order indifference of fuzzy numbers. We will call this latter one the order indifference relation. Lastly, we will demonstrate how these two relations can be utilized to rank a collection of fuzzy sets that have trapezoidal membership functions.

In this study, a new regression method called Kappa regression is introduced to model conditional probabilities. The regression function is based on Dombi’s Kappa function, which is well known in fuzzy theory. Here, we discuss how the Kappa function relates to the Logistic function as well as how it can be used to approximate the Logistic function. We introduce the so-called Generalized Kappa Differential Equation and show that both the Kappa and the Logistic functions can be derived from it. Kappa regression, like binary Logistic regression, models the conditional probability of the event that a dichotomous random variable takes a particular value at a given value of an explanatory variable. This new regression method may be viewed as an alternative to binary Logistic regression, but while in binary Logistic regression the explanatory variable is defined over the entire Euclidean space, in the Kappa regression model the predictor variable is defined over a bounded subset of the Euclidean space. We will also show that asymptotic Kappa regression is Logistic regression. The advantages of this novel method are demonstrated by means of an example, and afterwards some implications are discussed.

The aim of this paper is twofold. On the one hand, the \(\lambda \) -additive measure (Sugeno \(\lambda \) -measure) is revisited, and a state-of-the-art summary of its most important properties is provided. On the other hand, the so-called \(\nu \) -additive measure as an alternatively parameterized \(\lambda \) -additive measure is introduced. Here, the advantages of the \(\nu \) -additive measure are discussed, and it is demonstrated that these two measures are closely related to various areas of science. The motivation for introducing the \(\nu \) -additive measure lies in the fact that its parameter \(\nu \in (0,1)\) has an important semantic meaning as it is the fix point of the complement operation. Here, by utilizing the \(\nu \) -additive measure, some well-known results concerning the \(\lambda \) -additive measure are put into a new light and rephrased in more advantageous forms. It is discussed here how the \(\nu \) -additive measure is connected with the belief-, probability- and plausibility measures. Next, it is also shown that two \(\nu \) -additive measures, with the parameters \(\nu _1\) and \(\nu _2\) , are a dual pair of belief- and plausibility measures if and only if \(\nu _1+\nu _2 = 1\) . Furthermore, it is demonstrated how a \(\nu \) -additive measure (or a \(\lambda \) -additive measure) can be transformed to a probability measure and vice versa. Lastly, it is discussed here how the \(\nu \) -additive measures are connected with rough sets, multi-attribute utility functions and certain operators of fuzzy logic.

Abstract In this study, we present a novel methodology that can be used to generate parametric probability weighting functions, which play an important role in behavioral economics, by making use of the Dombi modifier operator of continuous-valued logic. Namely, we will show that the modifier operator satisfies the requirements for a probability weighting function. Next, we will demonstrate that the application of the modifier operator can be treated as a general approach to create parametric probability weighting functions including the most important ones such as the Prelec and the Ostaszewski, Green and Myerson (Lattimore, Baker and Witte) probability weighting function families. Also, we will show that the asymptotic probability weighting function induced by the inverse of the so-called epsilon function is none other than the Prelec probability weighting function. Furthermore, we will prove that, by using the modifier operator, other probability weighting functions can be generated from the dual generator functions and from transformed generator functions. Finally, we will show how the modifier operator can be used to generate strictly convex (or concave) probability weighting functions and introduce a method for fitting a generated probability weighting function to empirical data.