Petra Murinová

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 Aristotles 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.

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

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.

Omitting types in fuzzy logic with evaluated syntax

This paper is a contribution to the development of model theory of fuzzy logic in narrow sense. We consider a formal system EvŁ of fuzzy logic that has evaluated syntax, i. e. axioms need not be fully convincing and so, they form a fuzzy set only. Consequently, formulas are provable in some general degree. A generalization of Godels completeness theorem does hold in EvŁ. The truth values form an MV-algebra that is either finite or Łukasiewicz algebra on [0, 1]. The classical omitting types theorem states that given a formal theory T and a set Σ(x1, … , xn ) of formulas with the same free variables, we can construct a model of T which omits Σ, i. e. there is always a formula from Σ not true in it. In this paper, we generalize this theorem for EvŁ, that is, we prove that if T is a fuzzy theory and Σ(x1, … , xn ) forms a fuzzy set , then a model omitting Σ also exists. We will prove this theorem for two essential cases of EvŁ: either EvŁ has logical (truth) constants for all truth values, or it has these constants for truth values from [0, 1] ∩ ℚ only. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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.

Undefined Values in Fuzzy Logic

The main objective of this paper is to propose an extended algebra of truth values by special truth values which may have several interpretations, such as “undefined”, “non-applicative” “overdetermined”, “undetermined”, etc. In this paper, we will analyze several situations, where the non-existent data may come from, and show within a fuzzy sets framework that different cases of non-existence have to be carefully treated and interpreted in a different way.

Graded Generalized Hexagon in Fuzzy Natural Logic

In our previous papers, we formally analyzed the generalized Aristotle’s square of opposition using tools of fuzzy natural logic. Namely, we introduced general definitions of selected intermediate quantifiers, constructed a generalized square of opposition consisting of them and syntactically analyzed the emerged properties. The main goal of this paper is to extend the generalized square of opposition to graded generalized hexagon.

The analysis of the generalized square of opposition

In this paper, we continue the development of a formal theory of intermediate quantifiers (linguistic expressions such as “most”, “many”, “few”, “almost all”, etc.). In previous work, we demonstrated that 105 generalized syllogisms are valid in our theory. We turn our attention to another problem which is the analysis of the generalized Aristotelian square of opposition which, besides the classical quantifiers, is extended also by several selected intermediate quantifiers. We show that the expected relations can be well modeled in our theory. The formal theory of intermediate quantifiers is developed within a special higher-order fuzzy logic — Eukasiewicz fuzzy type theory.

Fuzzy Association Rules on Data with Undefined Values

Handling of missing values is a very common in data processing. However, data values may be missing not only because of lack of information, but also because of undefinedness (such as asking for the age of non-married person’s spouse). The aim of this paper is to propose an extension of fuzzy association rules framework for data with undefined values.

An Algorithm for Intermediate Quantifiers and the Graded Square of Opposition Towards Linguistic Description of Data.

The aim of this paper is to apply main theories of fuzzy natural logic together with fuzzy GUHA method for a linguistic characterization of relationships in data. Namely, we utilize the theory of intermediate quantifiers, which provides mathematical interpretation of natural language expressions describing quantity such as “Almost all”, “Few” etc., to describe relationships in data using vague terms that are natural in human expression. We provide an algorithm for computation of truth degrees of expressions containing such quantifiers. Moreover, we discuss some basic properties of intermediate quantifiers (contraries, contradictories, sub-contraries and sub-alterns), which formulate the graded Peterson’s square of opposition, and which can be used to infer new expressions from existing ones.