Intermediate Quantifiers Research Articles

Logical syllogisms with “Almost all, Most, Many, A few” and “Several”

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)”.

Intermediate quantifiers and valid syllogisms on EQ-algebras

Intermediate quantifiers are expressions of natural language, for example “most, almost all, many, a few” using which we quantify a number of some objects in a given universe. We have shown in [23] that all valid syllogisms with intermediate quantifiers are a consequence of only two algebraic inequalities and one equality. The result was obtained in the formalism of Lukasiewicz fuzzy type theory whose truth values form a linearly ordered complete MV-algebra. In this paper we will prove that the same holds if we replace MV-algebra by a much more general IEQ-algebra (involutive EQ-algebra).

Verification of validity of syllogisms with intermediate quantifiers is equivalent with checking Peterson's rules

This paper is a continuation of the previous work in which we analyzed six Peterson's rules for checking the validity of Aristotle's classical syllogisms and also their extension for checking the validity of syllogisms with intermediate quantifiers. We formalized the rules and proved that only four rules are sufficient. We also proved that if a syllogism is valid, then it satisfies all four (and consequently all six) rules. In this paper, we return to this topic and prove much stronger result, namely that a syllogism with intermediate quantifiers is valid if and only if it satisfies all four (and consequently all six) extended Peterson's rules.

Formal analysis of Peterson's rules for checking validity of syllogisms with intermediate quantifiers

In this paper, we follow up on previous publications in which we studied generalized Peterson's syllogisms with intermediate quantifiers. We present results of two kinds. First we show that on semantic level all the valid syllogisms follow from two inequalities and one equality. Furthermore, we focus on six rules suggested by Peterson in his book using which he was able to verify validity of all syllogisms. The problem is that the rules are formulated in free natural language and so, they do not provide formal means using which it would be possible to explain why the rules do their job. Therefore, we suggested formal reformulation of them and showed that all the valid syllogisms with intermediate quantifiers indeed satisfy Peterson's rules.

Partial Fuzzy Quantifiers and their Computation

Background: In computer science, one often needs to deal with undefined values. For example, they naturally increase when a mistake such as the square root of a negative number or division by zero occurs. A similar problem occurs in the logical analysis of natural language. For example, the expression “Czech president in the 18th century” has no denotation because there was no Czech president before 1918. Such a situation in mathematics is characterized by partial functions, i.e., functions that may be undefined for specific arguments. Method: In this paper, we will extend the theory of intermediate quantifiers (i.e., expressions such as “most, almost all, many, a few,” etc.) to deal with partially defined fuzzy sets. First, we will extend algebraic operations that are used in fuzzy logic by the additional value “undefined.” Then we will introduce intermediate quantifiers using the former. The theory of intermediate quantifiers has been developed as a special theory of higher-order fuzzy logic. Results: In this paper, we introduce the quantifiers semantically and show how they can be computed. The latter is also demonstrated in three illustrative examples. Conclusion: The paper contributes to the development of fuzzy quantifier theory and its extension by undefined values and suggests methods for computation of truth values.

A Formal Analysis of Generalized Peterson’s Syllogisms Related to Graded Peterson’s Cube

This publication builds on previous publications in which we constructed syntactic proofs of fuzzy Peterson’s syllogisms related to the graded square of opposition. The aim of the publication is to be formally able to find syntactic proofs of fuzzy Peterson’s logical syllogisms with forms of fuzzy intermediate quantifiers that design the graded Peterson’s cube of opposition.

Graded polygons of opposition in fuzzy formal concept analysis

Quantifier-based operators are fuzzy quantifiers that are, similarly to the intermediate quantifiers, based on the evaluative linguistic expressions not small, very big and extremely big. This article focuses mainly on achieving two goals. Firstly, quantifier-based operators are introduced to create extended fuzzy concept lattices that capture more detailed information from datasets compared to the existing ones. Then, graded extensions of Aristotle's square, named polygons of opposition, are constructed using quantifier-based operators. Our results highlight new connections between three different research areas: the theory of evaluative linguistic expressions, fuzzy formal concept analysis, and the studies on Aristotle's square.1

Graded Structures of Opposition in Fuzzy Natural Logic

The main objective of this paper is devoted to two main parts. First, the paper introduces logical interpretations of classical structures of opposition that are constructed as extensions of the square of opposition. Blanche’s hexagon as well as two cubes of opposition proposed by Morreti and pairs Keynes–Johnson will be introduced. The second part of this paper is dedicated to a graded extension of the Aristotle’s square and Peterson’s square of opposition with intermediate quantifiers. These quantifiers are linguistic expressions such as “most”, “many”, “a few”, and “almost all”, and they correspond to what are often called “fuzzy quantifiers” in the literature. The graded Peterson’s cube of opposition, which describes properties between two graded squares, will be discussed at the end of this paper.

Existential Import, Aristotelian Logic, and its Generalizations

The paper uses the theory of generalized quantifiers to discuss existential import and its implications for Aristotelian logic, namely the square of opposition, conversions and the assertoric syllogistic, as well as for more recent generalizations to intermediate quantifiers like “most”. While this is a systematic discussion of the semantic background one should assume in order to obtain the inferences and oppositions Aristotle proposed, it also sheds some light on the interpretation of his writings. Moreover by applying tools from modern formal semantics to the investigation of classical Aristotelian logic and its extensions, we combine different approaches to the logic of quantification. We will present variants of quantifiers that are associated to the four corners of the square of opposition with and without existential import and discuss their role for the logical square, conversions and the syllogistic. It will turn out that there is no way to ascribe existential import that validates all inferences and relations which one is willing to hold in Aristotelian logic. Two options, however, provide reasonable results. Existential import should either be ascribed only to affirmative statements or only be ascribed to universal quantification. The former option is preferable for a mere reconstruction of the classical Aristotelian logic while the latter option is more attractive if Aristotelian logic is generalized to intermediate quantifiers.

The theory of intermediate quantifiers in fuzzy natural logic revisited and the model of “Many”

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.

Linguistic Characterization of Natural Data by Applying Intermediate Quantifiers on Fuzzy Association Rules

The paper aims at applying fuzzy natural logic together with the Fuzzy GUHA method to analyse and linguistically characterise scientific data. Fuzzy GUHA is a tool for extracting linguistic association rules from data. Obtained associations are IF-THEN rules composed of evaluative linguistic expressions, which allow the quantities to be characterized with vague linguistic terms such as very small, big, medium etc. Originally, fuzzy GUHA provides several numerical indices of rule quality, which may not be easily understandable for domain experts that are not familiar with GUHA association rules. Therefore, we show in this paper that the theory of intermediate quantifiers (a constituent of fuzzy natural logic) can be applied to the results in an automatic manner in order to obtain natural linguistic summarization. We also present an idea of how the theory of generalized Aristotles's syllogisms can be used for a detailed data analysis. We also open the possibility to use fuzzy partial logic for cases where some data is missing or undefined.

Mining information from time series in the form of sentences of natural language

The goal of this paper is to provide a more detailed explanation of the principles how special formulas that characterize properties of trend of time series can be formed and how they are interpreted. Then we show how these formulas can be used in a tectogrammatical tree that construes special sentences of natural language, using which information on behavior of time series is provided. We also outline the principles of mining this information. The last part is devoted to application of the theory of intermediate quantifiers to mining summarized information on time series also in sentences of natural language.

Syllogisms and 5-Square of Opposition with Intermediate Quantifiers in Fuzzy Natural Logic

In this paper, we provide an overview of some of the results obtained in the mathematical theory of intermediate quantifiers that is part of fuzzy natural logic (FNL). We briefly introduce the mathematical formal system used, the general definition of intermediate quantifiers and define three specific ones, namely, “Almost all”, “Most” and “Many”. Using tools developed in FNL, we present a list of valid intermediate syllogisms and analyze a generalized 5-square of opposition.

Intermediate Syllogisms in Fuzzy Natural Logic

The goal of this paper is twofold: first, we give overview of the results of one part of the Fuzzy Natural Logic - the formal theory of intermediate quantifiers and syllogisms based on them. We present 105 valid basic syllogisms and show that for the proof of validity of all of them, we need to prove validity of only few of them so that validity of the other ones immediately follows. In the second part of the paper, we focused on conditions under which non-trivial intermediate syllogisms are valid. The latter are syllogisms which contain two or even three general intermediate quantifiers (not equal to any of $\forall$ or $\exists$).

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.

The structure of generalized intermediate syllogisms

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.

An algebraic study of Peterson’s Intermediate Syllogisms

Peterson's Intermediate Syllogisms, generalizing Aristotelian syllogisms by intermediate quantifiers `Many', `Most' and `Almost all', are studied. It is demonstrated that, by associating certain values V, W and U on standard ?ukasiewicz MV-algebra with the first and second premise and the conclusion, respectively, the validity of a corresponding intermediate syllogism is determined by a simple MV-algebra (in-)equation. Possible conservative extensions of Peterson's system are discussed. Finally it is shown that Peterson's bivalued intermediate syllogisms can be viewed as fuzzy theories in Pavelka's fuzzy propositional logic, i.e. a fuzzy version of Peterson's Intermediate Syllogisms is introduced.

Analysis of generalized square of opposition with intermediate quantifiers

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, Łukasiewicz fuzzy type theory, and is very general, with a high potential for applications.

On modelling with words

In this paper, we discuss a class of methods encapsulated under the term modelling with words. The theoretical frame is mathematical fuzzy logic in broader sense, namely its constituents: formal logical theory of evaluative linguistic expressions, intermediate quantifiers, and the related concepts of linguistic description and perception-based logical deduction. We present various kinds of applications based on this theory: control of complex processes, managerial decision making, analysis, forecasting and linguistic evaluation of time series, mining linguistic associations, and also linguistic summarization and deduction based on intermediate quantifier theory.

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.