modelling a linguistic variable as a hierarchical family of partitions induced by an indistinguishability operator this work shows a method to obtain a hierarchy of partitions on the universe 0 1 in such a way that each of them is compatible with a refinement of lukasiewicz indistinguishability operator the classes of the partition at a given level present a relation of antonymy between them moreover the partition at a certain level can be seen as the refinement of a previous level by means of a class of linguistic modifiers due to this fact they seem appropriate for modeling linguistic labels of a linguistic variable the associated indistinguishability operators show the increasing granularity when the number of classes rises up

reasoning with graded information in medicine one frequently deals with vague information as a tool for reasoning in this area fuzzy logic suggests itself in this paper we explore the applicability of the basic ideas of fuzzy set theory in the context of medical assessment questionnaires which are commonly used for instance to support the diagnosis of psychological disorders the items of a questionnaire are answered in a graded form patients are asked to choose an element on a linear scale the derived diagnostic hypotheses are graded as well this leads to the question whether there is a logical formalism that is suitable to capture the score calculation of medical assessment questionnaires and thereby provides a mathematical justification of the way in which the calculation is typically done we elaborate two alternative approaches to this problem first we follow the lines of mathematical fuzzy logic for the proposed logic which can deal with the formation of mean values we present a hilbert style deduction system in addition we consider a variant of the prototype approach to vagueness in this case we are led to a framework for which to obtain a logical calculus turns out to be difficult yet our gain is a model that is conceptually comparably well justifiable

fuzzy sets and sheaves part ii this essay shows that large parts of fuzzy set theory are actually subfields of sheaf theory resp of the theory of complete w valued sets hence fuzzy set theory is closer to the mainstream in mathematics than many people would expect part ii of this series of two papers explains the sheaf theoretic basis of zadeh s operations on fuzzy sets with the exception of the complementation further the quotient problem w r t similarity relations and the quotient problem of fuzzy groups w r t fuzzy congruence relations are solved finally stratified fuzzy topologies are identified with internal topologies on constant sheaves

on fuzzy type theory this paper is a generalization of classical simple type theory we have developed a formal system of fuzzy type theory which differs from the classical one essentially in extension of truth values from two to infinitely many the structure of truth values is assumed to be an imtl algebra residuated lattice with prelinearity and double negation extended by the baaz delta operation various properties of fuzzy type theory are proved including its completeness

similarity preserving linear maps on b h we determine kernels of similarity preserving bounded linear maps on b h and give characterizations for elementary operators of length 1 to be similarity preserving

evaluative linguistic expressions vs fuzzy categories in this paper we discuss the distinction between categories characterized by verbal labels taken from a fuzzy rating scale and special class of linguistic expressions called evaluative the latter form a general class of expressions that includes gradable and evaluative adjectives and their hedges first we will provide a brief linguistic analysis of them then we outline basic principles for construction of the mathematical model of semantics of evaluative expressions in section 3 we will analyze the concepts of rating scale with verbal labels fuzzy rating scale their semantics and demonstrate that the latter cannot be identified with the semantics of evaluative expressions finally we suggest a method for construction of fuzzy rating scales on the basis of evaluative linguistic expressions the main goal of this paper is to demonstrate the power of fuzzy set theory and to show that it is capable of providing tools for advanced modeling of linguistic semantics

continuity issues of the implicational interpretation of fuzzy rules the implicational interpretation of fuzzy rules has received little attention in real world applications so far this is largely due to the fact that ensuring continuity of the resulting function is not a straightforward task this paper targets this subject departing from consistent linguistic descriptions rule bases we introduce sufficient conditions for the continuity of the implicational interpretation with mean of maximum defuzzification we demonstrate that continuity can be achieved under practically feasible conditions regardless of the dimensionality of the input

nonlinear maps preserving similarity on b h abstract let x be a real or complex banach space with dimension at least 3 n 1 x be the family of all rank one nilpotent operators we give the concrete form of every bijective map n 1 x n 1 x such that t s n 1 x t s n 1 x based on this result we characterize the surjective map b h b h witht s r t s r for all t s r b h where h is a hilbert space real or complex with dimension at least 3

linguistic characterization of time series the goal of this paper is to provide an overview of applications of special soft computing theories the fuzzy transform and fuzzy natural logic to analysis forecasting and mining information from time series the focus is especially placed on the ability of methods of fuzzy natural logic to provide information in sentences of natural language our approach is based on the decomposition of time series into three components the trend cycle seasonal component and noise the trend cycle is extracted using the f transform and its course is characterized by automatically generated linguistic description the latter is then used to forecast the trend cycle the trend cycle can be furthermore decomposed into trend general tendency and cycle the former is computed again using the f transform moreover the f1 transform makes it possible to estimate the direction of the trend which can then be characterized by expressions of natural language stagnating slightly increasing sharply decreasing etc finally we focus on selected problems of mining information from time series first we suggest an algorithm for finding intervals of monotonous behavior and then show how the theory of intermediate quantifiers a constituent of fuzzy natural logic and generalized aristotle s syllogisms can be applied to automatic summarization of information on time series

a comprehensive theory of trichotomous evaluative linguistic expressions in this paper a logical theory of the so called trichotomous evaluative linguistic expressions tev expressions is presented these are frequent expressions of natural language such as small very small roughly medium extremely big etc the theory is developed using the formal system of higher order fuzzy logic namely the fuzzy type theory generalization of classical type theory first we discuss informally what are properties of the meaning of tev expressions then we construct step by step axioms of a formal logical theory t e v of tev expressions and prove various properties of t e v all the proofs are syntactical and so our theory is very general we also outline construction of a canonical model of t e v the main elegancy of our theory consists in the fact that semantics of all kinds of evaluative expressions is modeled in a unified way we also prove theorems demonstrating that essential properties of the vagueness phenomenon can be captured within our theory

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

adjacency preserving maps on matrices and operators we characterize injective continuous maps on the space of real or complex rectangular matrices preserving adjacent pairs of matrices we also extend hua s fundamental theorem of the geometry of rectangular matrices to the infinite dimensional case an application in the theory of local automorphisms is presented

a formal theory of generalized intermediate syllogisms in this paper we continue developing the formal theory of intermediate quantifiers expressions such as most few almost all a lot of many a great deal of a large part of a small part of the theory is a fuzzy logic formalization of the concept introduced by peterson in his book we will syntactically prove that 105 generalized aristotle s syllogisms introduced in this book are valid in our theory at the same time we will also prove that syllogisms listed there as invalid are invalid also in our theory therefore we believe that our theory provides a reasonable mathematical model of the generalized syllogistics

triangular norm based predicate fuzzy logics the paper surveys the present state of knowledge on t norm based predicate fuzzy logics with their double semantics standard the set of truth values being the real interval 0 1 and general with abstract algebras of truth functions

similarity preserving linear maps on matrices abstract we determine all similarity preserving linear maps on the space of n x n complex matrices and all unitary equivalence preserving linear maps on the space of n x n hermitian matrices sub majorization preserving linear maps on hermitian matrices are also determined

antonyms and linguistic quantifiers in fuzzy logic abstract the paper is a contribution to the theory of fuzzy logic in broader sense flb namely the discussion of linguistic expressions fundamental for it the evaluating linguistic predications the pairs nominal syntagm antonym and the theory of linguistic quantifiers the aim is to develop a theory of natural human reasoning whose characteristic feature is the use of natural language formalism of flb is based on the theory of fuzzy logic in narrow sense with evaluated syntax which provides us means for modelling of the concepts of intension possible world and extension characterization of some of the main properties of the above expressions is provided we also propose a modified definition of the linguistic variable

design of natural classification kernels using prior knowledge a new class of kernels has been designed to enhance the usability of prior knowledge prior knowledge is shown to improve the generalization ability of kernel algorithms for binary classification problems the prior knowledge is expressed in natural language via fuzzy rules first the concepts of a fuzzy rule base and prior confidence region are proposed to formulate the prior knowledge then the new kernels which are referred to as natural classification kernels ncks are represented by fuzzy equivalence relations based on the formulation of prior knowledge an nck is interpreted as a measure of similarities between samples it is proven that ncks have two desired properties 1 transitivity with respect to the triangular norms and 2 the ability to provide higher similarities to spatially closer samples from the same class using transitivity a large number of ncks may be directly obtained by means of triangular norms additionally the theoretical results show that the second property makes it possible for the support vector machine svm and convex hull separation algorithm to generalize from training samples to test samples in the prior confidence region finally some synthetic datasets and a benchmark dataset are employed to validate the proposed approach

the theory of intermediate quantifiers in fuzzy natural logic revisited and the model of many abstract the fuzzy natural logic fnl is a formal theory whose goal is to provide a mathematical model of natural human reasoning whose typical feature is the use of natural language one of its tasks is to develop a mathematical model of a special class of linguistic quantifiers called intermediate ones in our previous papers we introduced a general principle of how intermediate quantifiers can be formed and suggested generalized definitions of the relations of contrary contradictory subcontrary and subalterns between these quantifiers based on these relations and following peterson s book 62 we were able to suggest a formal model of the 5 square of opposition with intermediate quantifiers this paper picks up the threads of this work and has the following goals it contributes to the development of the theory of fuzzy natural logic first it gets back to some concepts of the theory of evaluative linguistic expressions introduces slight modifications and proves some not yet presented properties furthermore it improves the formal theory of intermediate quantifiers some definitions are made more precise and some new theorems are proved finally it contributes to the theory of the intermediate quantifier many and the analysis of its position in the mentioned 5 square we will show that many manifests an ambiguous relation towards the other quantifiers

fuzzy rule base ensemble generated from data by linguistic associations mining as there are many various methods for time series prediction developed but none of them generally outperforms all the others there always exists a danger of choosing a method that is inappropriate for a given time series to overcome such a problem distinct ensemble techniques that combine several individual forecasts are being proposed in this contribution we employ the so called fuzzy rule based ensemble this method is constructed as a linear combination of a small number of forecasting methods where the weights of the combination are determined by fuzzy rule bases based on time series features such as trend seasonality or stationarity for identification of fuzzy rule bases we use the linguistic association mining a huge experimental justification is provided

similarity invariant real linear subspaces and similarity preserving additive maps let h be a separable complex infinite dimensional hilbert space all similarity invariant real linear subspaces of b h the banach algebra of all bounded linear operators on h are characterized as an application all continuous additive surjective maps on b h which preserving similarity in both directions are completely classified

extensions of fuzzy relational compositions based on generalized quantifiers abstract the aim of this paper is to extend the fuzzy relational compositions motivated by their applications to medical diagnosis problem by bandler and kohout these compositions employ two quantifiers the universal quantifier and the existential quantifier as there exists a big gap between these two quantifiers that may be appropriately filled in by intermediate generalized quantifiers we take this as a motivation for our study in particular we introduce the concept of fuzzy relational compositions based on generalized quantifiers furthermore we show that the properties that are typically valid for standard fuzzy relational compositions are also valid for the compositions based on generalized quantifiers yet sometimes in a weaken form apart from fuzzy relational compositions the use of generalized quantifiers is also applied to images and preimages of fuzzy sets under fuzzy relations

analysis of generalized square of opposition with intermediate quantifiers abstract in this paper we continue the development of a formal theory of intermediate quantifiers these quantifiers are linguistic expressions such as most many few and almost all and they correspond to what are often called fuzzy quantifiers in the literature in a previous study we demonstrated that 105 generalized syllogisms with intermediate quantifiers are valid in our theory in this paper we turn our attention to another problem which is the analysis of the generalized aristotelian square of opposition which in addition to the classical quantifiers can be extended by several selected intermediate quantifiers we show that the expected relations can be well modeled in our theory our theory is developed within a special higher order fuzzy logic ukasiewicz fuzzy type theory and is very general with a high potential for applications

a hierarchical model of a linguistic variable in this work a theoretical hierarchical model of dichotomous linguistic variables is presented the model incorporates certain features of the approximate reasoning approach and others of the fuzzy control approach to fuzzy linguistic variables it allows sharing of the same hierarchical structure between the syntactic definition of a linguistic variable and its semantic definition given by fuzzy sets this fact makes it easier to build symbolic operations between linguistic terms with a better grounded semantic interpretation moreover the family of fuzzy sets which gives the semantics of each linguistic term constitutes a multiresolution system and thanks to that any fuzzy set can be represented in terms of the set of linguistic terms the model can also be considered a general framework for building more interpretable fuzzy linguistic variables with a high capacity of accuracy which could be used to build more interpretable fuzzy rule based systems frbs

a note on similarity preserving linear maps on matrices abstract we give a simple proof of a result of hiai concerning similarity preserving linear maps on n n complex matrices as well as its extension to fields of characteristic zero

a logical approach to fuzzy truth hedges the starting point of this paper are the works of hajek and vychodil on the axiomatization of truth stressing and depressing hedges as expansions of hajek s bl logic by new unary connectives they showed that their logics are chain complete but standard completeness was only proved for the expansions over godel logic we propose weaker axiomatizations over an arbitrary core fuzzy logic which have two main advantages i they preserve the standard completeness properties of the original logic and ii any subdiagonal resp superdiagonal non decreasing function on 0 1 preserving 0 and 1 is a sound interpretation of the truth stresser resp depresser connectives hence these logics accommodate most of the truth hedge functions used in the literature about of fuzzy logic in a broader sense

monoidal t norm based logic towards a logic for left continuous t norms hajek s bl logic is the fuzzy logic capturing the tautologies of continuous t norms and their residua in this paper we investigate a weaker logic mtl which is intended to cope with the tautologies of left continuous t norms and their residua the corresponding algebraic structures mtl algebras are defined and completeness of mtl with respect to linearly ordered mtl algebras is proved besides several schematic extensions of mtl are also considered as well as their corresponding predicate calculi

type fuzzy quantifiers determined by fuzzy measures on residuated lattices part i basic definitions and examples abstract we study fuzzy quantifiers of type 1 1 defined using fuzzy measures and integrals residuated lattices are considered as structures of truth values we present basic notions on fuzzy measures over algebras of fuzzy subsets of a given fuzzy set and the definition and necessary properties of so called fuzzy integrals residuated lattice operations are established to derive operations between fuzzy sets describing degrees in which formulas expressing relations between fuzzy sets are true finally a general definition of type 1 1 fuzzy quantifiers determined by fuzzy measures and integrals is introduced and several examples of important natural language quantifiers are modeled using this approach

reasoning about mathematical fuzzy logic and its future this paper is devoted to reasoning about fuzzy logic which is based on various personal observations of the author our goal is to think of the state of the art in mathematical fuzzy logic mfl and to outline some of the tasks on which in the author s opinion mfl should focus in the future in our discussion we will mention not only the basic theory but also its extension called fuzzy logic in broader sense flb the paradigm of the latter is to be the logic of natural human reasoning whose most essential characteristic is the use of natural language besides brief description of flb we will also mention some of its applications on the basis of that we will ponder on other possible directions for research namely the possibility of using fln as a metatheory of fuzzy mathematics as a proper tool for modeling of the main manifestations of the phenomenon of vagueness and as a reasonable tool for developing models of linguistic semantics

subtypes in fuzzy type theory abstract in this paper we introduce extension of the fuzzy type theory ftt by subtypes this demand arises in situations when for example it is useful to distinguish rational and real numbers where the set of rational numbers is a subset of the reals in the original ftt this causes a problem because we cannot assign rational numbers a special type the solution is to introduce special types for both of them and declare that the former is a subtype of the latter the resulting fuzzy type theory is denoted by sftt when constructing the frame for sftt we must realize that though all the considered functions are total if we take them as functions between subsets then they become partial

the structure of generalized intermediate syllogisms abstract in this paper we continue development of the formal theory of intermediate quantifiers which are expressions of the natural language most many few etc in the previous paper we demonstrated that 105 generalized syllogisms are valid in our theory we introduce the generalization of all the figures and we show that for the proof of validity of all the generalized syllogisms we need to prove the validity of only few of them so that the validity of the other ones immediately follows

spectral functionals nonholonomic dirac operators and noncommutative ricci flows we formulate a noncommutative generalization of the ricci flow theory in the framework of spectral action approach to noncommutative geometry grisha perelman s functionals are generated as commutative versions of certain spectral functionals defined by nonholonomic dirac operators and corresponding spectral triples we derive the formulas for spectral averaged energy and entropy functionals and state the conditions when such values describe non holonomic riemannian configurations