In this article, we will show how relations forming the graded Peterson's square of opposition (contrary, contradictory, sub-contrary and sub-alterns) are connected with intermediate logical syllogisms, i.e., syllogisms that contain intermediate quantifiers. Our results are related to the theory of non-monotonic reasoning.

In this article, we will form a graded hexagon of opposition as an extension of the graded Peterson's square, incorporating new variations of intermediate quantifiers. We will start with the classical Aristotle's square of opposition and its extension to hexagon. Then we introduce the graded square of opposition that was suggested by Peterson and fully formalized by the authors. We also introduce quantifiers A few ( A little ) and Several and include them in the latter. Then we introduce new forms of quantifiers and study relations of contrary, sub-contrary, contradictory and sub-(sup-)altern among them. On the basis of the knowledge of the relations among all the intermediate quantifiers we construct the graded Peterson's hexagon of opposition.

This paper delves into logical syllogisms featuring intermediate quantifiers. In our previous works, we established the validity of logical syllogisms involving fundamental intermediate quantifiers “Almost all”, “Most”, and “Many”. In this paper, we focus on syllogisms incorporating also the quantifiers “Several” and “A few (A little)”.

We recently introduced special fuzzy quantifiers named quantifier-based operators to form extended fuzzy concept lattices and to construct graded squares, hexagons, octagons and decagons of oppositions. This article aims to extend our previous works by organizing quantifier-based operators in more general structures of oppositions: the so-called graded cubes of opposition and 5-graded cubes of opposition.

In this paper, we describe the software LFL Forecaster for analysis, forecasting and mining information from time series using special methods of fuzzy modeling. The main theoretical tools are fuzzy transform and methods of fuzzy natural logic. The software also provides information in sentences of natural language mined from time series. We demonstrate functioning of the software on several real time series from the area of economy and other sectors.

This contribution is devoted to a method for forecasting of the trend-cycle of time series based on the application of a combination of the higher degree fuzzy transform and fuzzy natural logic (FNL) techniques. The main idea is to predict future components of the direct fuzzy transform of a time series. The forecast of trend-cycle is then computed using the inverse fuzzy transform applied to the predicted components.

One of the proclaims often emphasized in papers on fuzzy sets and fuzzy logic is their ability to model semantics of certain linguistic expressions because their inherent vagueness can be captured by fuzzy sets. This direction of research was initiated by L. A. Zadeh already in his early papers and since then, most of the applications of fuzzy sets emphasize presence of natural language, at least in hidden form. Still, this ability is not generally accepted by linguists. In this paper we try to show that capturing linguistic semantics requires more sophisticated models. One possibility has been elaborated in the concept of fuzzy natural logic (FNL) that is a mathematical theory whose roots lay in the concept of natural logic developed by linguists and logicians. We also argue that it is reasonable to develop a simplified model that would capture the main features of the semantics of natural language and thus make it possible to realize sophisticated technical applications. In the paper, we outline how model of the meaning of basic constituents of natural language (nouns, adjectives, adverbs, verbs) has been elaborated in FNL.

This paper is a contribution to the study of a special kind of syllogisms with intermediate quantifiers. We stem from our previous papers where a formal theory of the intermediate quantifiers was introduced. Besides other results, we syntactically proved validity of 105 basic syllogisms with them. We also demonstrated how our theory works in the semantic interpretation. In this paper, we will address some special kinds of syllogisms that are non-trivial in the sense that both premises as well as conclusion contain general intermediate quantifiers.

In this paper, we apply the F-transform technique with respect to raised cosine generalized uniform fuzzy partition to analysis of time series. We assume that the time series can be additively decomposed into trend-cycle, seasonal component and noise. We will prove that when constructing reasonably the fuzzy partition then the inverse F-transform of the time series provides a good approximation of its trend-cycle. Namely, the F-transform with respect to raised cosine generalized uniform fuzzy partition is able to eliminate completely the seasonal component and significantly reduce noise.

In this paper, three special soft computing software systems are presented. The systems are based on the original results in two areas: fuzzy natural logic and fuzzy transform. The first software is LFL Controller that is a universal SW system that can be used in fuzzy or linguistic control, and in decision-making. The system implements results of fuzzy natural logic, namely the theory of evaluative linguistic expressions and perception-based logical deduction. This means the control or decision strategy are defined directly in natural language. Additionally, it can also realize classical fuzzy control on the basis of relational interpretation of fuzzy IF-THEN rules. This system has wide range of applications and was already applied in control of real plants. The second system is LFL Forecaster that is a specialized SW for analysis and forecasting of time series. The analysis is realized using F-transform and forecasting using results of fuzzy natural logic. The third system is FT-Studio that is a specialized SW for computation of fuzzy transform of functions that can be defined either using a formula, or given by data.

We analyse three soil horizons in each square kilometre of certain local area. The goal of the work is to determine the degree of damage to soil horizons by cluster analysis. From these analyses we constructed RGB maps that show areas with the same or similar damage of soil horizons in the whole area and we also tried to determine the degree of soil damage. We applied cluster analysis and fuzzy cluster analysis.

In this paper, we will focus on the application of fuzzy transform (F-transform) in the analysis of time series. We assume that the time series is decomposed into two constituents: the trend-cycle and random noise. We will demonstrate that using the F-transform we can reduce the variability of random noise which consequence is an extraction of the trend-cycle.

In many practical situations, e.g., in signal processing, image processing, analysis of temporal data, it is very useful to use fuzzy (F-) transforms. In an F-transform, we first replace a function x(t) by a few local averages (this is called forward F-transform), and then reconstruct the original function from these averages (this is called inverse F-transform). While the formula for the forward F-transform makes perfect intuitive sense, the formula for the inverse F-transform seems, at first glance, somewhat counter-intuitive. On the other hand, its empirical success shows that this formula must have a good justification. In this paper, we provide such a justification - a justification which is based on formulating a reasonable compression-based criterion.

A new methodology for forecasting of time series is proposed. It is based on combination of two techniques: fuzzy transform and perception-based logical deduction on the basis of learned linguistic description.

This contribution is intended as a position paper that favors the viewpoint that inference based on deductive rules (i.e., the rules are interpreted using fuzzy implication) can indeed be considered as a valuable inference scheme in real-world applications. For this purpose, we highlight the basic concepts behind the most common fuzzy inference schemes and demonstrate their interpretation by means of illustrative examples. We conclude that, under some reasonable conditions, deductive inference is able to compete with or even outperform the well-known Mamdani-Assilian inference.

This paper is a contribution to the approximate reasoning theory. We present an alternative to the well known Mamdani rule, which we call the perception-based logical deduction. The leading assumption is that fuzzy IF-THEN rules are interpreted as genuine linguistic sentences consisting of the, so called, evaluating linguistic expressions. Evaluating expressions determined on the basis of an observation in a concrete context are called perceptions. The latter and sets of fuzzy IF-THEN rules, called linguistic descriptions, are used in derivation of a conclusion. The perception-based deduction is demonstrated to behave in accordance with human way of reasoning. A lot of simulations as well as real applications corroborate that it works very well and is easy for the user because of natural language comprehension

In this article, we return to the problem of the derivation of a conclusion on the basis of fuzzy IF–THEN rules. The so-called Mamdani method is well elaborated and widely applied. In this article, we present an alternative to it. The fuzzy IF–THEN rules are here interpreted as genuine linguistic sentences consisting of the so-called evaluating linguistic expressions. Sets of fuzzy IF–THEN rules are called linguistic descriptions. Linguistic expressions derived on the basis of an observation in a concrete context are called perceptions. Together with the linguistic description, they can be used in logical deduction, which we will call a perception-based logical deduction. We focus on semantics only and confine ourselves to one specific model. If the perception-based deduction is repeated and the result interpreted in an appropriate model, we obtain a piecewise continuous and monotonous function. Though the method has already proved to work well in many applications, the nonsmoothness of the output may sometimes lead to problems. We propose in this article a method for how the resulting function can be made smooth so that the output preserves its good properties. The idea consists of postprocessing the output using a special fuzzy approximation method called F-transform. © 2004 Wiley Periodicals, Inc. Int J Int Syst 19: 1007–1031, 2004.

An algorithm for the extraction of linguistic knowledge from data records is presented. General outline of fuzzy logic deduction is introduced. The behavior of the algorithm is illustrated on “difficult order” example.

In this paper a simple procedure for pattern pre-recognition is presented. The task is to recognize whether a given pattern should be interesting for further analysis or not. The description of the procedure presented in this paper is based on formal theory of fuzzy logic in narrow sense so that the procedure – being general – can be used for many other purposes where patterns are to be recognized.