A Mathematical Setting for Fuzzy Logics

1997 ◽  
Vol 05 (03) ◽  
pp. 223-238 ◽  
Author(s):  
Mai Gehrke ◽  
Carol Walker ◽  
Elbert Walker
Keyword(s):  
Fuzzy Logic ◽  
Propositional Logic ◽  
Normal Forms ◽  
Unit Interval ◽  
Fuzzy Logics ◽  
Real Numbers ◽  
Truth Values ◽  
Finite Algebras ◽  
Finite Algorithms ◽  
Interval Valued

The setup of a mathematical propositional logic is given in algebraic terms, describing exactly when two choices of truth value algebras give the same logic. The propositional logic obtained when the algebra of truth values is the real numbers in the unit interval equipped with minimum, maximum and -x=1-x for conjunction, disjunction and negation, respectively, is the standard propositional fuzzy logic. This is shown to be the same as three-valued logic. The propositional logic obtained when the algebra of truth values is the set {(a, b)|a≤ b and a,b∈[0,1]} of subintervals of the unit interval with component-wise operations, is propositional interval-valued fuzzy logic. This is shown to be the same as the logic given by a certain four element lattice of truth values. Since both of these logics are equivalent to ones given by finite algebras, it follows that there are finite algorithms for determining when two statements are logically equivalent within either of these logics. On this topic, normal forms are discussed for both of these logics.

SUBSTRUCTURAL INQUISITIVE LOGICS

2019 ◽  
Vol 12 (2) ◽  
pp. 296-330 ◽  
Author(s):  
VÍT PUNČOCHÁŘ
Keyword(s):  
Fuzzy Logic ◽  
General Theory ◽  
Propositional Logic ◽  
Substructural Logics ◽  
Substructural Logic ◽  
Fuzzy Logics ◽  
Inquisitive Semantics ◽  
Semantic Framework ◽  
Noncommutative Version ◽  
Image Position

AbstractThis paper shows that any propositional logic that extends a basic substructural logic BSL (a weak, nondistributive, nonassociative, and noncommutative version of Full Lambek logic with a paraconsistent negation) can be enriched with questions in the style of inquisitive semantics and logic. We introduce a relational semantic framework for substructural logics that enables us to define the notion of an inquisitive extension of λ, denoted as ${\lambda ^?}$, for any logic λ that is at least as strong as BSL. A general theory of these “inquisitive extensions” is worked out. In particular, it is shown how to axiomatize ${\lambda ^?}$, given the axiomatization of λ. Furthermore, the general theory is applied to some prominent logical systems in the class: classical logic Cl, intuitionistic logic Int, and t-norm based fuzzy logics, including for example Łukasiewicz fuzzy logic Ł. For the inquisitive extensions of these logics, axiomatization is provided and a suitable semantics found.

scholarly journals A FAMILY OF FINITE DE MORGAN AND KLEENE ALGEBRAS

2012 ◽  
Vol 20 (05) ◽  
pp. 631-653 ◽  
Author(s):  
CAROL L. WALKER ◽  
ELBERT A. WALKER
Keyword(s):  
Fuzzy Sets ◽  
Unit Interval ◽  
Point Of View ◽  
Truth Values ◽  
Special Cases ◽  
Per Se ◽  
Special Circumstances ◽  
Interval Valued

The algebra of truth values for fuzzy sets of type-2 consists of all mappings from the unit interval into itself, with operations certain convolutions of these mappings with respect to pointwise max and min. This algebra generalizes the truth-value algebras of both type-1 and of interval-valued fuzzy sets, and has been studied rather extensively both from a theoretical and applied point of view. This paper addresses the situation when the unit interval is replaced by two finite chains. Most of the basic theory goes through, but there are several special circumstances of interest. These algebras are of interest on two counts, both as special cases of bases for fuzzy theories, and as mathematical entities per se.

Very true on CBA fuzzy logic

2010 ◽  
Vol 60 (4) ◽  
Author(s):  
Michal Botur ◽  
Filip Švrček
Keyword(s):  
Fuzzy Logic ◽  
Propositional Logic ◽  
Fuzzy Logics ◽  
Algebraic Semantic

AbstractCBA logic was introduced as a non-associative generalization of the Łukasiewicz many-valued propositional logic. Its algebraic semantic is just the variety of commutative basic algebras. Petr Hájek introduced vt-operators as models for the “very true” connective on fuzzy logics. The aim of the paper is to show possibilities of using vt-operators on commutative basic algebras, especially we show that CBA logic endowed with very true connective is still fuzzy.

Propositional logic

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Fuzzy Logic ◽  
Propositional Logic ◽  
Building Blocks ◽  
English Word ◽  
The Other ◽  
American Women ◽  
Truth Values ◽  
Other Hand ◽  
Truth Value ◽  
The Relationship

In propositional logic, atomic formulas are propositions. Any assertion will do. For example, . . . A = “Aristotle is dead,” B = “Barcelona is on the Seine,” and C = “Courtney Love is tall” . . . are atomic formulas. Atomic formulas are the building blocks used to construct sentences. In any logic, a sentence is regarded as a particular type of formula. In propositional logic, there is no distinction between these two terms. We use “formula” and “sentence” interchangeably. In propositional logic, as with all logics we study, each sentence is either true or false. A truth value of 1 or 0 is assigned to the sentence accordingly. In the above example, we may assign truth value 1 to formula A and truth value 0 to formula B. If we take proposition C literally, then its truth is debatable. Perhaps it would make more sense to allow truth values between 0 and 1. We could assign 0.75 to statement C if Miss Love is taller than 75% of American women. Fuzzy logic allows such truth values, but the classical logics we study do not. In fact, the content of the propositions is not relevant to propositional logic. Henceforth, atomic formulas are denoted only by the capital letters A, B, C,. . . (possibly with subscripts) without referring to what these propositions actually say. The veracity of these formulas does not concern us. Propositional logic is not the study of truth, but of the relationship between the truth of one statement and that of another. The language of propositional logic contains words for “not,” “and,” “or,” “implies,” and “if and only if.” These words are represented by symbols: . . . ￢ for “not,” ∧ for “and,” ∨ for “or,” → for “implies,” and ↔ for “if and only if.” . . . As is always the case when translating one language into another, this correspondence is not exact. Unlike their English counterparts, these symbols represent concepts that are precise and invariable. The meaning of an English word, on the other hand, always depends on the context.

scholarly journals On BL-Algebras and its Interval Counterpart

2019 ◽  
Vol 20 (2) ◽  
pp. 241
Author(s):  
Rui Paiva ◽  
Regivan Santiago ◽  
Benjamín Bedregal
Keyword(s):  
Fuzzy Logic ◽  
Fuzzy Sets ◽  
Algebraic Structures ◽  
Fuzzy Logics ◽  
Interval Valued ◽  
Algebraic Counterpart

Interval Fuzzy Logic and Interval-valued Fuzzy Sets have been widely investigated. Some Fuzzy Logics were algebraically modelled by Peter Hájek as BL-algebras. What is the algebraic counterpart for the interval setting? It is known from literature that there is a incompatibility between some algebraic structures and its interval counterpart. This paper shows that such incompatibility is also present in the level of BL-algebras. Here we show both: (1) the impossiblity of match imprecision and the correctness of the underlying BLimplication and (2) some facts about the intervalization of BL-algebras.

NORMAL FORMS FOR FUZZY LOGIC — AN APPLICATION OF KOLMOGOROV’S THEOREM

1996 ◽  
Vol 04 (04) ◽  
pp. 331-349 ◽  
Author(s):  
VLADIK KREINOVICH ◽  
HUNG T. NGUYEN ◽  
DAVID A. SPRECHER
Keyword(s):  
Neural Networks ◽  
Fuzzy Logic ◽  
Normal Form ◽  
Normal Forms ◽  
Approximation Property ◽  
Universal Approximation ◽  
Approximation Result ◽  
Logical Operations ◽  
Kolmogorov's Theorem

This paper addresses mathematical aspects of fuzzy logic. The main results obtained in this paper are: 1. the introduction of a concept of normal form in fuzzy logic using hedges; 2. using Kolmogorov’s theorem, we prove that all logical operations in fuzzy logic have normal forms; 3. for min-max operators, we obtain an approximation result similar to the universal approximation property of neural networks.

scholarly journals A Modern Syllogistic Method in Intuitionistic Fuzzy Logic with Realistic Tautology

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Ali Muhammad Rushdi ◽  
Mohamed Zarouan ◽  
Taleb Mansour Alshehri ◽  
Muhammad Ali Rushdi
Keyword(s):  
Fuzzy Logic ◽  
Propositional Logic ◽  
Compact Form ◽  
Logical Function ◽  
Intuitionistic Fuzzy ◽  
Complete Product ◽  
Single Function ◽  
Intuitionistic Fuzzy Logic

The Modern Syllogistic Method (MSM) of propositional logic ferrets out from a set of premisesallthat can be concluded from it in the most compact form. The MSM combines the premises into a single function equated to 1 and then produces the complete product of this function. Two fuzzy versions of MSM are developed in Ordinary Fuzzy Logic (OFL) and in Intuitionistic Fuzzy Logic (IFL) with these logics augmented by the concept of Realistic Fuzzy Tautology (RFT) which is a variable whose truth exceeds 0.5. The paper formally proves each of the steps needed in the conversion of the ordinary MSM into a fuzzy one. The proofs rely mainly on the successful replacement of logic 1 (or ordinary tautology) by an RFT. An improved version of Blake-Tison algorithm for generating the complete product of a logical function is also presented and shown to be applicable to both crisp and fuzzy versions of the MSM. The fuzzy MSM methodology is illustrated by three specific examples, which delineate differences with the crisp MSM, address the question of validity values of consequences, tackle the problem of inconsistency when it arises, and demonstrate the utility of the concept of Realistic Fuzzy Tautology.

scholarly journals Einstein Geometric Aggregation Operators using a Novel Complex Interval-valued Pythagorean Fuzzy Setting with Application in Green Supplier Chain Management

2021 ◽  
Vol 2 (1) ◽  
pp. 105-134
Author(s):  
Zeeshan Ali ◽  
◽  
Tahir Mahmood ◽  
Kifayat Ullah ◽  
Qaisar Khan ◽  
...  
Keyword(s):  
Fuzzy Set ◽  
Multicriteria Decision Making ◽  
Unit Interval ◽  
Pythagorean Fuzzy Set ◽  
Operational Laws ◽  
Special Cases ◽  
Supplier Chain ◽  
Valuable Procedure ◽  
Interval Valued ◽  
Fuzzy Setting

The principle of a complex interval-valued Pythagorean fuzzy set (CIVPFS) is a valuable procedure to manage inconsistent and awkward information genuine life troubles. The principle of CIVPFS is a mixture of the two separated theories such as complex fuzzy set and interval-valued Pythagorean fuzzy set which covers the truth grade (TG) and falsity grade (FG) in the form of the complex number whose real and unreal parts are the sub-interval of the unit interval. The superiority of the CIVPFS is that the sum of the square of the upper grade of the real part (also for an unreal part) of the duplet is restricted to the unit interval. The goal of this article is to explore the new principle of CIVPFS and its algebraic operational laws. By using the CIVPFSs, certain Einstein operational laws by using the t-norm and t-conorm are also developed. Additionally, we explore the complex interval-valued Pythagorean fuzzy Einstein weighted geometric (CIVPFEWG), complex interval-valued Pythagorean fuzzy Einstein ordered weighted geometric (CIVPFEOWG) operators and utilized their special cases. Moreover, a multicriteria decision-making (MCDM) technique is explored based on the elaborated operators by using the complex interval-valued Pythagorean fuzzy (CIVPF) information. To determine the consistency and reliability of the elaborated operators, we illustrated certain examples by using the explored principles. Finally, to determine the supremacy and dominance of the explored theories, the comparative analysis and graphical expressions of the developed principles are also discussed.

scholarly journals Constructions for t-conorms and t-norms on interval-valued and interval-valued intuitionistic fuzzy sets by paving

2020 ◽  
Vol 26 (3) ◽  
pp. 1-12
Author(s):  
Martin Kalina ◽  
Keyword(s):  
Fuzzy Sets ◽  
Intuitionistic Fuzzy Sets ◽  
Construction Method ◽  
Unit Interval ◽  
The Real ◽  
Intuitionistic Fuzzy ◽  
Interval Valued

Paving is a method for constructing new operations from a given one. Kalina and Kral in 2015 showed that on the real unit interval this method can be used to construct associative, commutative and monotone operations from particular given operations (from basic ‘paving stones’). In the present paper we modify the construction method for interval-valued fuzzy sets. From given (possibly representable) t-norms and t-conorms we construct new, non-representable operations. In the last section, we modify the presented construction method for interval-valued intuitionistic fuzzy sets.

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