scholarly journals FUZZY TYPE THEORY IN THE ANALYSIS OF ARGUMENTATION

2021 ◽  
pp. 37-47
Author(s):  
Oleg Domanov

The article deals with a fazzy variant of P. Martin-Löf ’s intuitionistic type theory. It presents the overview of fuzzy type theory rules and an example of its application to the analysis of the persuasiveness of argumentation. In the latter, the truth values of fuzzy logic are interpreted as degrees of persuasiveness of statements and arguments. The formalization is implemented in the proof assistant Agda.

Author(s):  
Ernesto Copello ◽  
Nora Szasz ◽  
Álvaro Tasistro

Abstarct We formalize in Constructive Type Theory the Lambda Calculus in its classical first-order syntax, employing only one sort of names for both bound and free variables, and with α-conversion based upon name swapping. As a fundamental part of the formalization, we introduce principles of induction and recursion on terms which provide a framework for reproducing the use of the Barendregt Variable Convention as in pen-and-paper proofs within the rigorous formal setting of a proof assistant. The principles in question are all formally derivable from the simple principle of structural induction/recursion on concrete terms. We work out applications to some fundamental meta-theoretical results, such as the Church–Rosser Theorem and Weak Normalization for the Simply Typed Lambda Calculus. The whole development has been machine checked using the system Agda.


Author(s):  
Radim Belohlavek ◽  
Joseph W. Dauben ◽  
George J. Klir

The term “fuzzy logic” (FL) is a generic one, which stands for a broad variety of logical systems. Their common ground is the rejection of the most fundamental principle of classical logic—the principle of bivalence—according to which each declarative sentence has exactly two possible truth values—true and false. Each logical system subsumed under FL allows for additional, intermediary truth values, which are interpreted as degrees of truth. These systems are distinguished from one another by the set of truth degrees employed, its algebraic structure, truth functions chosen for logical connectives, and other properties. The book examines from the historical perspective two areas of research on fuzzy logic known as fuzzy logic in the narrow sense (FLN) and fuzzy logic in the broad sense (FLB), which have distinct research agendas. The agenda of FLN is the development of propositional, predicate, and other fuzzy logic calculi. The agenda of FLB is to emulate commonsense human reasoning in natural language and other unique capabilities of human beings. In addition to FL, the book also examines mathematics based on FL. One chapter in the book is devoted to overviewing successful applications of FL and the associated mathematics in various areas of human affairs. The principal aim of the book is to assess the significance of FL and especially its significance for mathematics. For this purpose, the notions of paradigms and paradigm shifts in science, mathematics, and other areas are introduced and employed as useful metaphors.


Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 432 ◽  
Author(s):  
Vilém Novák

In this paper, we will visit Rough Set Theory and the Alternative Set Theory (AST) and elaborate a few selected concepts of them using the means of higher-order fuzzy logic (this is usually called Fuzzy Type Theory). We will show that the basic notions of rough set theory have already been included in AST. Using fuzzy type theory, we generalize basic concepts of rough set theory and the topological concepts of AST to become the concepts of the fuzzy set theory. We will give mostly syntactic proofs of the main properties and relations among all the considered concepts, thus showing that they are universally valid.


2015 ◽  
Vol 25 (5) ◽  
pp. 1040-1070 ◽  
Author(s):  
JEREMY AVIGAD ◽  
KRZYSZTOF KAPULKIN ◽  
PETER LEFANU LUMSDAINE

Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed by this approach to the formalizing homotopy-theoretic material. We also compare our constructions with the more classical approach to homotopy limits via fibration categories.


Author(s):  
Mai Gehrke ◽  
Carol Walker ◽  
Elbert Walker

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.


Author(s):  
Rajab Ali Borzooei ◽  
Narges Akhlaghinia ◽  
Mona Aaly Kologani ◽  
Xiao Long Xin

EQ-algebras were introduced by Nova ́k in [15] as an algebraic structure of truth values for fuzzy type theory (FFT). In this paper, we studied the category of EQ-algebras and showed that it is complete, but it is not cocomplete, in general. We proved that multiplicatively relative EQ-algebras have coequlizers and we calculate coprodut and pushout in a special case. Also, we construct a free EQ-algebra on a singleton.


2000 ◽  
Vol 113 (2) ◽  
pp. 161-183 ◽  
Author(s):  
Carl W. Entemann
Keyword(s):  

2015 ◽  
Vol 25 (5) ◽  
pp. 1278-1294 ◽  
Author(s):  
VLADIMIR VOEVODSKY

This is a short overview of an experimental library of Mathematics formalized in the Coq proof assistant using the univalent interpretation of the underlying type theory of Coq. I started to work on this library in February 2010 in order to gain experience with formalization of Mathematics in a constructive type theory based on the intuition gained from the univalent models (see Kapulkin et al. 2012).


2020 ◽  
Vol 5 (1) ◽  
Author(s):  
Adwitya Rai Paramaartha ◽  
Rikip Ginanjar
Keyword(s):  

Smartphone is one of the most important things for people nowadays. The increasing number of products as well as the amount information carried by each brand of smartphone can overload, there are many things to consider before buying a smartphone. Fuzzy logic is a form of many- valued logic in which the truth values of variables may be any real number between 0 and 1. Fuzzy logic is one method to analyze system containing uncertainty. This study aims to use fuzzy logic in helping people make decisions on buying the most suitable smartphone for them by producing an output value that can help the user to determine which smartphone will be purchased based on user’s ideal criteria of a smartphone.


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