scholarly journals A logic for approximate reasoning

1994 ◽  
Vol 59 (3) ◽  
pp. 830-837 ◽  
Author(s):  
Mingsheng Ying
Keyword(s):  
Artificial Intelligence ◽  
Set Theory ◽  
Theorem Proving ◽  
Intuitionistic Fuzzy Set ◽  
Propositional Calculus ◽  
Forthcoming Paper ◽  
Approximate Reasoning ◽  
Approximate Methods ◽  
Truth Values ◽  
Antecedent Clause

Classical logic is not adequate to face the essential vagueness of human reasoning, which is approximate rather than precise in nature. The logical treatment of the concepts of vagueness and approximation is of increasing importance in artificial intelligence and related research. Consequently, many logicians have proposed different systems of many-valued logic as a formalization of approximate reasoning (see, for example, Goguen [G], Gerla and Tortora [GT], Novak [No], Pavelka [P], and Takeuti and Titani [TT]). As far as we know, all the proposals are obtained by extending the range of truth values of propositions. In these logical systems reasoning is still exact and to make a conclusion the antecedent clause of its rule must match its premise exactly. In addition. Wang [W] pointed out: “If we compare calculation with proving,... Procedures of calculation... can be made so by fairly well-developed methods of approximation; whereas... we do not have a clear conception of approximate methods in theorem proving.... The concept of approximate proofs, though undeniably of another kind than approximations in numerical calculations, is not incapable of more exact formulation in terms of, say, sketches of and gradual improvements toward a correct proof” (see pp, 224–225). As far as the author is aware, however, no attempts have been made to give a conception of approximate methods in theorem proving.The purpose of this paper is. unlike all the previous proposals, to develop a propositional calculus, a predicate calculus in which the truth values of propositions are still true or false exactly and in which the reasoning may be approximate and allow the antecedent clause of a rule to match its premise only approximately. In a forthcoming paper we shall establish set theory, based on the logic introduced here, in which there are ∣L∣ binary predicates ∈λ, λ ∈ L such that R(∈, ∈λ) = λ where ∈ stands for ∈1 and 1 is the greatest element in L, and x ∈λy is interpreted as that x belongs to y in the degree of λ, and relate it to intuitionistic fuzzy set theory of Takeuti and Titani [TT] and intuitionistic modal set theory of Lano [L]. In another forthcoming paper we shall introduce the resolution principle under approximate match and illustrate its applications in production systems of artificial intelligence.

scholarly journals A Theory of Approximate Reasoning with Type-2 Fuzzy Set

2021 ◽  
Vol 27 (1) ◽  
pp. 9-28
Author(s):  
Sudin Mandal ◽  
Injamam Ul Karim ◽  
Swapan Raha
Keyword(s):  
Fuzzy Logic ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Residuated Lattice ◽  
Approximate Reasoning ◽  
Truth Values ◽  
Logic Structure

In this paper, an attempt is made to study approximate reasoning based on a Type-2 fuzzy set theory. In the process, we have examined the underlying fuzzy logic structure on which the reasoning is formulated. We have seen that the partial/incomplete/imprecise truth-values of elements of a type-2 fuzzy set under consideration forms a lattice. We propose two new lattice operations which ultimately help us to define a residual and thereby making the structure of truth- values a residuated lattice. We have focused upon two typical rules of inference used mostly in ordinary approximate reasoning methodology based on Type-1 fuzzy set theory. Our proposal is illustrated with typical artificial examples.

Equalization of finite flowers

1988 ◽  
Vol 53 (1) ◽  
pp. 105-123
Author(s):  
Stefano Berardi
Keyword(s):  
Set Theory ◽  
Forthcoming Paper ◽  
Direct Limits

A dilator D is a functor from ON to itself commuting with direct limits and pull-backs. A dilator D is a flower iff D(x) is continuous. A flower F is regular iff F(x) is strictly increasing and F(f)(F(z)) = F(f(z)) (for f ϵ ON(x,y), z ϵ X).Equalization is the following axiom: if F, G ϵ Flr (class of regular flowers), then there is an H ϵ Flr such that F ° H = G ° H. From this we can deduce that if ℱ is a set ⊆ Flr, then there is an H ϵ Flr which is the smallest equalizer of ℱ (it can be said that H equalizes ℱ iff for every F, G ϵ ℱ we have F ° H = G ° H). Equalization is not provable in set theory because equalization for denumerable flowers is equivalent to -determinacy (see a forthcoming paper by Girard and Kechris).Therefore it is interesting to effectively find, by elementary means, equalizers even in the simplest cases. The aim of this paper is to prove Girard and Kechris's conjecture: “ is the (smallest) equalizer for Flr < ω” (where Flr < ω denotes the set of finite regular flowers). We will verify that is an equalizer of Flr < ω; we will sketch the proof that it is the smallest one at the end of the paper. We will denote by H.

scholarly journals Preface

2011 ◽  
Vol 21 (4) ◽  
pp. 671-677 ◽  
Author(s):  
GÉRARD HUET
Keyword(s):  
Mathematical Logic ◽  
Computer Science ◽  
20Th Century ◽  
Theorem Proving ◽  
Propositional Calculus ◽  
Interactive Theorem Proving ◽  
Predicate Calculus ◽  
Special Issue ◽  
Mathematical Structures ◽  
Infinite Domains

This special issue of Mathematical Structures in Computer Science is devoted to the theme of ‘Interactive theorem proving and the formalisation of mathematics’.The formalisation of mathematics started at the turn of the 20th century when mathematical logic emerged from the work of Frege and his contemporaries with the invention of the formal notation for mathematical statements called predicate calculus. This notation allowed the formulation of abstract general statements over possibly infinite domains in a uniform way, and thus went well beyond propositional calculus, which goes back to Aristotle and only allowed tautologies over unquantified statements.

A Hesitant Intuitionistic Fuzzy Set Approach to Study Ideals of Semirings

2021 ◽  
Vol 10 (3) ◽  
pp. 1-17
Author(s):  
Debabrata Mandal
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Cartesian Product ◽  
Intuitionistic Fuzzy Set ◽  
Ideal Theory ◽  
Hesitant Fuzzy Set ◽  
Neutrosophic Set ◽  
Intuitionistic Fuzzy ◽  
Interval Valued ◽  
The Ideal

The classical set theory was extended by the theory of fuzzy set and its several generalizations, for example, intuitionistic fuzzy set, interval valued fuzzy set, cubic set, hesitant fuzzy set, soft set, neutrosophic set, etc. In this paper, the author has combined the concepts of intuitionistic fuzzy set and hesitant fuzzy set to study the ideal theory of semirings. After the introduction and the priliminary of the paper, in Section 3, the author has defined hesitant intuitionistic fuzzy ideals and studied several properities of it using the basic operations intersection, homomorphism and cartesian product. In Section 4, the author has also defined hesitant intuitionistic fuzzy bi-ideals and hesitant intuitionistic fuzzy quasi-ideals of a semiring and used these to find some characterizations of regular semiring. In that section, the author also has discussed some inter-relations between hesitant intuitionistic fuzzy ideals, hesitant intuitionistic fuzzy bi-ideals and hesitant intuitionistic fuzzy quasi-ideals, and obtained some of their related properties.

Heterogeneous Approximate Reasoning with Graded Truth Values

2017 ◽  
pp. 61-82 ◽  
Author(s):  
Francesco Luca De Angelis ◽  
Giovanna Di Marzo Serugendo ◽  
Barbara Dunin-Kęplicz ◽  
Andrzej Szałas
Keyword(s):  
Approximate Reasoning ◽  
Truth Values

scholarly journals Applications of Non-Standard analysis in Topoi to Mathematical Neurosciences and Artificial Intelligence: Infons, Energons, Receptons (I)

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2048
Author(s):  
Ileana Ruxandra Badea ◽  
Carmen Elena Mocanu ◽  
Florin F. Nichita ◽  
Ovidiu Păsărescu
Keyword(s):  
Mathematical Modeling ◽  
Artificial Intelligence ◽  
Mathematical Model ◽  
Set Theory ◽  
Standard Analysis ◽  
New Methods ◽  
Internal Set ◽  
Human Thinking ◽  
Global Brain ◽  
Non Standard Analysis

The purpose of this paper is to promote new methods in mathematical modeling inspired by neuroscience—that is consciousness and subconsciousness—with an eye toward artificial intelligence as parts of the global brain. As a mathematical model, we propose topoi and their non-standard enlargements as models, due to the fact that their logic corresponds well to human thinking. For this reason, we built non-standard analysis in a special class of topoi; before now, this existed only in the topos of sets (A. Robinson). Then, we arrive at the pseudo-particles from the title and to a new axiomatics denoted by Intuitionistic Internal Set Theory (IIST); a class of models for it is provided, namely, non-standard enlargements of the previous topoi. We also consider the genetic–epigenetic interplay with a mathematical introduction consisting of a study of the Yang–Baxter equations with new mathematical results.

Decision Making Modeled as a Theorem Proving Process

2012 ◽  
Vol 4 (3) ◽  
pp. 1-11
Author(s):  
Jacques Calmet ◽  
Marvin Oliver Schneider
Keyword(s):  
Artificial Intelligence ◽  
Decision Making ◽  
Information Technology ◽  
Multiagent Systems ◽  
Theorem Proving ◽  
Mathematical Knowledge ◽  
Knowledge Bases ◽  
Advanced Mathematics ◽  
Knowledge Communities ◽  
Computational Systems

The authors introduce a theoretical framework enabling to process decisions making along some of the lines and methodologies used to mechanize mathematics and more specifically to mechanize the proofs of theorems. An underlying goal of Decision Support Systems is to trust the decision that is designed. This is also the main goal of their framework. Indeed, the proof of a theorem is always trustworthy. By analogy, this implies that a decision validated through theorem proving methodologies brings trust. To reach such a goal the authors have to rely on a series of abstractions enabling to process all of the knowledge involved in decision making. They deal with an Agent Oriented Abstraction for Multiagent Systems, Object Mechanized Computational Systems, Abstraction Based Information Technology, Virtual Knowledge Communities, topological specification of knowledge bases using Logical Fibering. This approach considers some underlying hypothesis such that knowledge is at the heart of any decision making and that trust transcends the concept of belief. This introduces methodologies from Artificial Intelligence. Another overall goal is to build tools using advanced mathematics for users without specific mathematical knowledge.

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