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Starting from the notion of a multidistance, we formalize, through a suitable system of axioms, the concept of an inequality measure defined on a nonempty set with no additional structure implemented a priori. Among inequality measures, apart from multidistances we pay special attention to dispersions, and study their main features. Classical concepts will be generalized to this abstract setting. Multidistances are then revisited, and some new methods to generate them are implemented. A wide spectrum of interdisciplinary applications is outlined in the final section. This work is partially supported by the research projects ECO2015-65031-R , MTM2015-63608-P , TIN2016-77356-P , TIN2017-87600-P (MINECO/AEI-FEDER, UE) and PID2019-108392GB-I00 (AEI/10.13039/501100011033)

This paper introduces a logical characterization of multi-adjoint algebras with a twofold contribution. On the one hand, the study of multi-adjoint algebras, from a logical perspective, will allow us to discover both the core and new features of these algebras. On the other hand, the axiomatization of multi-adjoint algebras will be useful to take advantage of the properties of the logical connectives considered in the corresponding deductive system. The mechanism considered to carry out the mentioned axiomatization follows the one given by Petr Hajek for residuated lattices. Specifically, the paper presents the bounded poset logic (BPL) as an axiomatization of a bounded poset, since this algebraic structure is the most simple structure from which a multi-adjoint algebra is defined. In the following, the language of BPL is enriched with a family of pairs, composed of a conjunctor and an implication, and its axiomatic system is endowed with new axioms, giving rise to the multi-adjoint logic (ML). The soundness and completeness of BPL and ML are proven. Finally, a comparison between the axiomatization of the multi-adjoint logic and the one given for the BL logic is introduced.

Fuzzy relations R : A 2 → [ 0 , 1 ] are widely used in decision-making in order to represent degrees of preference between alternatives in A . Examining fuzzy relations, this paper first reflects on an inherent incompatibility inside the domain of reciprocal fuzzy preference relations and preference structures. Aiming at clarifying the semantics of reciprocal fuzzy relations by means of that of weak fuzzy relations, a general frame is then proposed for studying the transformation between them, based on the notion of reciprocal functions. Next, we show that reciprocal relations can be naturally endowed with a preference structure of their own, containing two relations respectively showing a semantics of strict preference and lack of preference. Finally, we study the compatibility between these reciprocal preference structures and the standard preference structures of weak fuzzy preference relations by means of an axiomatic approach, leading to a system of functional equations that is shown to admit different solutions, allowing to explicitly assign one of the available semantics for reciprocal preference relations.

Considering a continuous lattice L as the lattice background, an axiomatic approach to bases and subbases in the framework of L-convex spaces is provided. Firstly, the concepts of bases and subbases in L-convex spaces are introduced and then L-convexity base axioms and L-convexity subbase axioms are proposed by abstracting the properties of bases and subbases, respectively. Secondly, some applications of L-convexity base axioms and L-convexity subbase axioms are investigated. It is shown that L-convexity base axioms can be used to demonstrate some relationship between spatial structures with respect to L-convex structures and L-convexity subbase axioms can be applied to define the join space and the product space of L-convex spaces.

This paper introduces granular fuzzy rough sets in the view of fuzzy implicators and fuzzy coimplicators, and discusses the constructive and axiomatic approach to fuzzy granules based on fuzzy implicators and coimplicators. Moreover, we study the connection between fuzzy granules and fuzzy relations and discuss the relationship between existing granular fuzzy rough set models and that proposed in this paper. Considering the absolute error limit, we introduce the concept of the granular variable precision fuzzy rough sets based on fuzzy implicators and coimplicators. Then we present four propositions to ensure that the approximation operators can be efficiently calculated.

International audience; Necessity (resp. possibility) measures are very simple min-decomposable (resp. max-decomposable) representations of epistemic uncertainty due to incomplete knowledge. They can be used in both quantitative and qualitative settings. In the present work, we revisit Choquet and Sugeno integrals as criteria for decision under uncertainty and propose new axioms when uncertainty is representable in possibility theory. First, a characterization of Choquet integral with respect to a possibility or a necessity measure is proposed. We respectively add an optimism or a pessimism axiom to the axioms of the Choquet integral with respect to a general capacity. This new axiom enforces the maxitivity or the minitivity of the capacity without requiring the same property for the functional. It essentially assumes that the decision-maker preferences only reflect the plausibility ordering between states of nature. The obtained pessimistic (resp. optimistic) criterion is an average maximin (resp. maximax) criterion of Wald across cuts of a possibility distribution on the state space. The additional axiom can be also used in the axiomatic approach to Sugeno integral and generalized forms thereof to justify possibility and necessity measures. The axiomatization of these criteria for decision under uncertainty in the setting of preference relations among acts is also discussed. We show that the new axiom justifying possibilistic Choquet integrals can be expressed in this setting. In the case of Sugeno integral, we correct a characterization proof for an existing set of axioms on acts, and study an alternative set of axioms based on the idea of non-compensation.

Under the assumption of rational economics, the consistency of judgments is one of the important issues in multiple criteria decision making (MCDM) methods. We propose three axiomatic properties of the consistent judgments in relative measurements being considered from the perspective of strict logic and rationality: (A1) the reciprocal property of pairwise comparisons, (A2) the invariance of consistency with respect to permutations of alternatives, and (A3) the robustness of the ranking of alternatives. These properties are further applied to analyze the consistency of the judgments expressed by positive real numbers and triangular fuzzy numbers, respectively. The inconsistency of comparison matrices encountered in the Analytic Hierarchy Process (AHP) can be considered to be the case of weakening the axiomatic property (A3). Fuzzifying the judgments is due to the weakening of axiomatic property (A1). A method of weakening the axiomatic properties for triangular fuzzy reciprocal matrices is proposed and a new concept of approximation-consistency is put forward to distinguish from the typical consistency. By considering the permutations of alternatives, an illustrative example is presented to show the use of the proposed procedures for solving the decision making problems with triangular fuzzy reciprocal preference relations.

Firstly, the disjunctive elimination rule is proved valid in Esteva and Godo's MTL and Hajek's BL logics. Secondly, from this rule a general method is presented for finitely axiomatizing the intersection of the sets of theorems of two given schematic extensions of MTL or BL. Thirdly, the axiomatic system of NM is simplified by replacing the axiom schemata (NM) with another which is built up with one variable. Fourthly, the intersection of the sets of tautologies of NM and G, NM and @P, NM and L are finitely axiomatized by our method. Finally, new axiom systems for LG, [email protected], @PG, [email protected] are presented by our method and some comments are made on the relations between these systems and those given by Cignoli et al.

In the framework of multicriteria decision making based on fuzzy measures or cooperative game theory, an axiomatization of the concept of amount of interaction index is given. It is based on four axioms: a natural property called ''absence of interaction'', a boundary condition for which the index reduces to the Shapley or Banzhaf interaction index, a pseudo-additivity axiom, and a recursivity property. The resulting index, which can be interpreted as a measure of the average degree of independence among criteria or players, takes a particularly simple and natural form for supermodular or submodular measures. The approach to the notion of interaction adopted in this paper can be seen as complementary to that recently adopted by Grabisch and Roubens.

This paper has two main goals. The first goal is to show a different axiomatic system of the Ł Π and Ł Π 1 2 propositional logics. These propositional logics were introduced in Esteva et al. (Arch. Math. Logic, to appear) and they are the combinations of the Łukasiewicz and the product logic (together with the constant 1 2 in case of the Ł Π 1 2 logic). The second goal is to show an axiomatic system and the completeness theorem of a predicate version of the Ł Π and Ł Π 1 2 propositional logics. It will be shown that Godel, product and Łukasiewicz predicate logics are contained in the Ł Π ∀ logic.

A Bernays-like axiomatic theory of intuitionistic fuzzy sets involving five primitives and seven axioms is presented.

The notion of independence is of great importance in any formalism for managing uncertainty, for both theoretical and practical reasons. In this paper we study the concept of independence in the framework of possibility theory. Our approach to defining conditional independence relationships is based on comparing conditional possibility measures. Different comparison criteria are presented, based on the ideas of ‘not to modify’, ‘not to gain’, and ‘to obtain similar’ information after conditioning. For each definition of independence considered, an axiomatic study has been carried out. Moreover, there are different operators to define conditional possibility measures, which are related to different views of possibility theory. Particularly, in the first part of the paper, we use Hisdal conditioning (whereas Dempster conditioning will be used in the second part). Finally, we study the marginal problem for possibility measures and, as an application, we show that it is possible to store large n-dimensional possibility distributions efficiently, using independence relationships among variables.

This paper provides an axiomatic characterization of the Orlovsky choice procedure for a decision maker whose preference are fuzzy. In contrast to the result in Banerjee (1993). our characterization does not require that in a pairwise choice, the choice coincides with the Orlovsky set. Moreover, our characterization result holds for a larger domain of preferences than that in Banerjee (1993).

t-norms and t-cornorms are considered as the suitable axiomatic basis for the aggregate functions used for the definition of fuzzy set intersection and union operations. When the domain of the function is infinite, as in most cases, continuity has also to be included as an axiom. This paper proves that the identity law axioms in t-norm and t-cornorm are implied by continuity and the other axioms. A simple example is given to show that in the absence of continuity, the identity law condition is necessary for a plausible axiomatic basis of fuzzy set operations. We also proved that the only idempotent t-norm (t-cornorm) functions is min (max).

All major theories of uncertainty permit defining numerical uncertainty measures . Formulae can originate by axiomatic approach or by modifying either Shannon or related entropy model. The latter has a very natural semantics as the complexity of transmission of specific information about the uncertainty assignment. Such interpretation has been lacking in the possibility model, based on fuzzy set theory. This paper presents such information semantics for this theory. It is accomplished by demonstrating that possibilistic uncertainty can be viewed as the expected value of information identifying the acceptability of choices from the fuzzy domain.

An axiomatic approach is developed concerning the existence of a basis for certain fuzzy algebraic substructures. Let the set of truth values be a complete Brouwerian lattice L . Let G be an Abelian group. We show that there exists an L -subgroup of G which does not have a p -basis even if G is finite. Let F be a field of characteristic p > 0 and let A , B be L -subfields of F such that A ⊇ B . We show there exists an intermediate L -subfield of A / B which does not have a relative p -basis over B even if F has finite relative imperfection degree over the support of B . When L = [0, 1], we show that every fuzzy subgroup of G with the sup property has a p -basis and that every intermediate fuzzy subfield of A / B with the sup property has a relative p -basis if certain compatibility conditions hold.

An axiomatic system of fuzzy set theory (for short FST) which requires fewer axioms than those of FST studies published before is developed as a Bernays-like theory based on four primitives and formal language.

We propose an axiomatic approach to the definition of fuzzy strict preference P , indifference I and incomparability J associated with a fuzzy preference relation R . Expression for P , I and J are given via solving two systems of functional equations.

Two axiomatizations of fuzzy set theory have been proposed by Chapin and Weidner. The comparison and the interpretations between these axiomatizations and ZF already begun by Weidner is carried on further; in particular it is shown that ZF can be interpreted in both theories.

Starting from the results of French [9] we discuss the notions of Fuzzy Set and of fuzziness: we show some characteristic properties that should be fulfilled by an axiomatic theory of fuzzy classes. We discuss two axiomatizations of fuzzy classes already proposed [5, 18] and we show the first axioms of a new axiomatic theory of fuzzy classes satisfying the properties previously showed.

In the sense of basic theory of mathematics, ML (Medium Logic) and MS (Medium Set Theory) have widened the foundation of logic and set theory of the precise classical mathematics and extended the research objects of mathematics from precision into fuzziness. Hence ML and MS, in a higher form, contain the whole of precise classical mathematics and its theoretical foundation. ML and MS have also resolved the problems of the revision of the comprehension principle left over historically, i.e. they have not only excluded all the paradoxes in the system, but also preserved all the reasonable content of the comprehension principle. ML and MS have provided a common theoretical foundation for both precise classical mathematics and future uncertain mathematics. The fifth generation of computers must lay its foundation on the acknowledgement of intelligent logic of medium states. Obviously ML will be this logic. ML and MS will also be the theoretical foundation of expert systems and advance the techniques of pattern recognition, graphic analysis, artificial intelligence, intelligent robots, computer vision, etc. to a new stage. In addition to the uncertain quantitative analysis, based on the theoretical foundation of ML and MS, the theory of large systems will present extraordinarily new prospects.

We develop a first-order theory MSTR for positive real-valued multisets. By restricting multiplicities of elements to the real interval (0, 1] we obtain a theory MSTF that gives a Zermelo-Fraenkel-like axiomatization for Zadeh's fuzzy sets. Moreover, MSTR and MSTF contain an exact copy of classical ZFC set theory and both are shown to be relatively consistent. MSTF is compared to other axiom systems that have been proposed for fuzzy sets.

The axiomatic approach to rational decision malting in a fuzzy environment proposed by Fung and Fu (1975) lays down a too restrictive family of aggregation rules. Because of such axioms, the independence of the size of the groups under aggregation is assumed; moveover, in a general problem, these groups are fuzzy sets.

Following the suggestion made by Klement [8], an axiomatic theory of TNF-σ-algebras is given, T being any measurable triangular norm and N any negation. Most of the results about T -fuzzy σ-algebbrs obtained in [8] are extended to the case of TNF-σ-algebras. Some other properties of TNF-σ-algebras are also discussed. Particularly, we point out: (1) there exists a large family of triangular norms, which contains the whole Yager family and almost the whole Sugeno family as subfamilies, such that for any negation N each TNF-σ-algebra is generated, and (2) given a set U with | U |⩾2 and a measurable triangular norm T , in order that for every negation N each TNF-σ-algebra on U is generated it is necessary that T is Archimedean.

The importance of fuzzy valuations in diagnostics is mentioned. The problem of aggregation of fuzzy opinions obtained from a group of experts in answer to the question: “Has an object the property labeled A and no property labeled ⊂1A?” is formulated. The set of Fung and Fu's [10] aggregation axioms is shortly described. The idea of weighting expert's opinions is formalized. Then a new set of aggregation axioms is presented and an aggregating operator is introduced. ‘A posteriori’ weighting of opinions in aggregation is examined.

This paper presents a proof of a strong completeness theorem for an extended axiomatic system of fuzzy logic BL with respect to all continuous t-norms. A finite strong standard completeness theorem for all continuous t-norms and their residua, the basic fuzzy logic, was proved across two papers Hajek (1998) and Cignoli et al. (2000). In Montagna (2007), the language of BL is extended by an additional connective and the axiomatic system includes an infinitary rule to achieve strong completeness result. In this paper we provide a proof of strong completeness for BL with a different infinitary inference rule but without extending the language of BL. We will also prove strong completeness for the Łukasiewicz and product t-norms using this extended axiomatic system.