Neutrality and Many-Valued Logics

In this book, we consider various many-valued logics: standard, linear, hyperbolic, parabolic, non-Archimedean, p-adic, interval, neutrosophic, etc. We survey also results which show the tree different proof-theoretic frameworks for many-valued logics, e.g. frameworks of the following deductive calculi: Hilbert's style, sequent, and hypersequent. We present a general way that allows to construct systematically analytic calculi for a large family of non-Archimedean many-valued logics: hyperrational-valued, hyperreal-valued, and p-adic valued logics characterized by a special format of semantics with an appropriate rejection of Archimedes' axiom. These logics are built as different extensions of standard many-valued logics (namely, Lukasiewicz's, Goedel's, Product, and Post's logics). The informal sense of Archimedes' axiom is that anything can be measured by a ruler. Also logical multiple-validity without Archimedes' axiom consists in that the set of truth values is infinite and it is not well-founded and well-ordered. On the base of non-Archimedean valued logics, we construct non-Archimedean valued interval neutrosophic logic INL by which we can describe neutrality phenomena.Comment: 119 page

Aggregation functions for typical hesitant fuzzy elements and the action of automorphisms

This work studies the aggregation operators on the set of all possible membership degrees of typical hesitant fuzzy sets, which we refer to as H, as well as the action of H-automorphisms which are defined over the set of all finite non-empty subsets of the unitary interval. In order to do so, the partial order ≤H, based on α-normalization, is introduced, leading to a comparison based on selecting the greatest membership degrees of the related fuzzy sets. Additionally, the idea of interval representation is extended to the context of typical hesitant aggregation functions named as the H-representation. As main contribution, we consider the class of finite hesitant triangular norms, studying their properties and analyzing the H-conjugate functions over such operators. © 2013 Elsevier Inc. All rights reserved.Peer Reviewe

Aggregation on bipolar scales

The paper addresses the problem of extending aggregation operators typically defined on $[0,1]$ to the symmetric interval $[-1,1]$, where the ``0'' value plays a particular role (neutral value). We distinguish the cases where aggregation operators are associative or not. In the former case, the ``0'' value may play the role of neutral or absorbant element, leading to pseudo-addition and pseudo-multiplication. We address also in this category the special case of minimum and maximum defined on some finite ordinal scale. In the latter case, we find that a general class of extended operators can be defined using an interpolation approach, supposing the value of the aggregation to be known for ternary vectors.bipolar scale; bi-capacity; aggregation

A Deep Study of Fuzzy Implications

This thesis contributes a deep study on the extensions of the IMPLY operator in classical binary logic to fuzzy logic, which are called fuzzy implications. After the introduction in Chapter 1 and basic notations about the fuzzy logic operators In Chapter 2 we first characterize In Chapter 3 S- and R- implications and then extensively investigate under which conditions QL-implications satisfy the thirteen fuzzy implication axioms. In Chapter 4 we develop the complete interrelationships between the eight supplementary axioms FI6-FI13 for fuzzy implications satisfying the five basic axioms FI1-FI15. We prove all the dependencies between the eight fuzzy implication axioms, and provide for each independent case a counter-example. The counter-examples provided in this chapter can be used in the applications that need different fuzzy implications satisfying different fuzzy implication axioms. In Chapter 5 we study proper S-, R- and QL-implications for an iterative boolean-like scheme of reasoning from classical binary logic in the frame of fuzzy logic. Namely, repeating antecedents $n$ times, the reasoning result will remain the same. To determine the proper S-, R- and QL-implications we get a full solution of the functional equation $I(x,y)=I(x,I(x,y))$, for all $x$, $y\in[0,1]$. In Chapter 6 we study for the most important t-norms, t-conorms and S-implications their robustness against different perturbations in a fuzzy rule-based system. We define and compare for these fuzzy logical operators the robustness measures against bounded unknown and uniform distributed perturbations respectively. In Chapter 7 we use a fuzzy implication $I$ to define a fuzzy $I$-adjunction in $\mathcal{F}(\mathbb{R}^{n})$. And then we study the conditions under which a fuzzy dilation which is defined from a conjunction $\mathcal{C}$ on the unit interval and a fuzzy erosion which is defined from a fuzzy implication $I^{'}$ to form a fuzzy $I$-adjunction. These conditions are essential in order that the fuzzification of the morphological operations of dilation, erosion, opening and closing obey similar properties as their algebraic counterparts. We find out that the adjointness between the conjunction $\mathcal{C}$ on the unit interval and the implication $I$ or the implication $I^{'}$ play important roles in such conditions

Fifty years of similarity relations: a survey of foundations and applications

On the occasion of the 50th anniversary of the publication of Zadeh's significant paper Similarity Relations and Fuzzy Orderings, an account of the development of similarity relations during this time will be given. Moreover, the main topics related to these fuzzy relations will be reviewed.Peer ReviewedPostprint (author's final draft

On extending generalized Bonferroni means to Atanassov orthopairs in decision making contexts

Extensions of aggregation functions to Atanassov orthopairs (often referred to as intuitionistic fuzzy sets or AIFS) usually involve replacing the standard arithmetic operations with those defined for the membership and non-membership orthopairs. One problem with such constructions is that the usual choice of operations has led to formulas which do not generalize the aggregation of ordinary fuzzy sets (where the membership and non-membership values add to 1). Previous extensions of the weighted arithmetic mean and ordered weighted averaging operator also have the absorbent element 〈1,0〉, which becomes particularly problematic in the case of the Bonferroni mean, whose generalizations are useful for modeling mandatory requirements. As well as considering the consistency and interpretability of the operations used for their construction, we hold that it is also important for aggregation functions over higher order fuzzy sets to exhibit analogous behavior to their standard definitions. After highlighting the main drawbacks of existing Bonferroni means defined for Atanassov orthopairs and interval data, we present two alternative methods for extending the generalized Bonferroni mean. Both lead to functions with properties more consistent with the original Bonferroni mean, and which coincide in the case of ordinary fuzzy values.<br /