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Neutrality and Many-Valued Logics

By Smarandache, Florentin

Book Id: WPLBN0002828460
Format Type: PDF (eBook)
File Size: 657.15 kb
Reproduction Date: 7/31/2013

Title:	Neutrality and Many-Valued Logics
Author:	Smarandache, Florentin
Volume:
Language:	English
Subject:	Non Fiction, Education, Neutrosophic Systems
Collections:	Mathematics, Authors Community, Literature, Most Popular Books in China
Historic Publication Date:	2013
Publisher:	World Public Library

Member Page:	Florentin Smarandache
Citation APA MLA Chicago Smarandache, F., & Schumann, A. (2013). Neutrality and Many-Valued Logics. Retrieved from http://gutenberg.us/

Description
In this book, we consider various many-valued logics: standard, linear, hyperbolic, parabolic, non-Archimedean, p -adic, interval, neutrosophic, etc. We survey also results which show the tree different proof-theoretic frameworks for many-valued logics, e.g. frameworks of the following deductive calculi: Hilbert ’s style, sequent, and hypersequent. Recall that hypersequents are a natural generalization of Gentzen ’s style sequents that was introduced independently by Avron and Pottinger . In particular, we examine Hilbert ’s style, sequent, and hypersequent calculi for infinite-valued logics based on the three fundamental continuous t-norms: Lukasiewicz ’s, Godel ’s, and Product logics.

Summary
This book written by A. Schumann & F. Smarandache is devoted to advances of non-Archimedean multiple-validity idea and its applications to logical reasoning. In this book, Schumann & Smarandache define such multiple-validities and describe the basic properties of non-Archimedean and p-adic valued logical systems proposed by them. In this book non-Archimedean and p-adic multiple-validities idea is regarded as one of possible approaches to explicate the neutrality concept.

Excerpt
1.1 Neutrality concept in logic Every point of view A tends to be neutralized, diminished, balanced by Non-A. At the same time, in between A and Non-A there are infinitely many points of view Neut-A. Let’s note by A an idea, or proposition, theory, event, concept, entity, by Non-A what is not A, and by Anti-A the opposite of A. Neut-A means what is neither A nor Anti-A, i.e. neutrality is also in between the two extremes.

Table of Contents
1 Introduction. . . . . . . . . . . . . . . 9 1.1 Neutrality concept in logic . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Neutrality and non-Archimedean logical multiple-validity . . . . 10 1.3 Neutrality and neutrosophic logic . . . . . . . . . . . . . . . . . . 13 2 First-order logical language. . . . . . . . . . . . . . . 17 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Hilbert’s type calculus for classical logic . . . . . . . . . . . . . . 20 2.3 Sequent calculus for classical logic . . . . . . . . . . . . . . . . . 22 3 n -valued Lukasiewicz’s logics. . . . . . . . . . . . . . . 27 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Originality of (p + 1)-valued Lukasiewicz’s logics . . . . . . . . . 30 3.3 n-valued Lukasiewicz’s calculi of Hilbert’s type . . . . . . . . . . 32 3.4 Sequent calculi for n-valued Lukasiewicz’s logics . . . . . . . . . . 33 3.5 Hypersequent calculus for 3-valued Lukasiewicz’s propositional logic . . . . . . . . . . . .37 4 Infinite valued Lukasiewicz’s logics. . . . . . . . . . . . . . . 39 4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2 Hilbert’s type calculus for infinite valued Lukasiewicz’s logic . . . 40 4.3 Sequent calculus for infinite valued Lukasiewicz’s propositional logic . . . . . . . . . . . . 41 4.4 Hypersequent calculus for infinite valued Lukasiewicz’s propositional logic. . . . . . . . . . 43 5 G¨odel’s logic. . . . . . . . . . . . . . . 45 5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2 Hilbert’s type calculus for G¨odel’s logic . . . . . . . . . . . . . . 45 5.3 Sequent calculus for G¨odel’s propositional logic . . . . . . . . . . 47 5.4 Hypersequent calculus for G¨odel’s propositional logic . . . . . . . 48 6 Product logic. . . . . . . . . . . . . . . 51 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.2 Hilbert’s type calculus for Product logic . . . . . . . . . . . . . . 51 6.3 Sequent calculus for Product propositional logic . . . . . . . . . . 53 6.4 Hypersequent calculus for Product propositional logic . . . . . . 54 7 Nonlinear many valued logics. . . . . . . . . . . . . . . 57 7.1 Hyperbolic logics . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.2 Parabolic logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 8 Non-Archimedean valued logics. . . . . . . . . . . . . . . 61 8.1 Standard many-valued logics . . . . . . . . . . . . . . . . . . . . 61 8.2 Many-valued logics on DSm models . . . . . . . . . . . . . . . . . 61 8.3 Hyper-valued partial order structure . . . . . . . . . . . . . . . . 65 8.4 Hyper-valued matrix logics . . . . . . . . . . . . . . . . . . . . . 68 8.5 Hyper-valued probability theory and hyper-valued fuzzy logic . . 70 9 p -Adic valued logics. . . . . . . . . . . . . . . 73 9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 9.2 p-Adic valued partial order structure . . . . . . . . . . . . . . . . 74 9.3 p-Adic valued matrix logics . . . . . . . . . . . . . . . . . . . . . 75 9.4 p-Adic probability theory and p-adic fuzzy logic . . . . . . . . . . 76 10 Fuzzy logics. . . . . . . . . . . . . . . 81 10.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 10.2 Basic fuzzy logic BL8 . . . . . . . . . . . . . . . . . . . . . . . . 84 10.3 Non-Archimedean valued BL-algebras . . . . . . . . . . . . . . . 85 10.4 Non-Archimedean valued predicate logical language . . . . . . . . 87 10.5 Non-Archimedean valued basic fuzzy propositional logic BL1 . . 89 11 Neutrosophic sets. . . . . . . . . . . . . . . 93 11.1 Vague sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 11.2 Neutrosophic set operations . . . . . . . . . . . . . . . . . . . . . 94 12 Interval neutrosophic logic. . . . . . . . . . . . . . . 99 12.1 Interval neutrosophic matrix logic . . . . . . . . . . . . . . . . . . 99 12.2 Hilbert’s type calculus for interval neutrosophic propositional logic. . . . . . . . . . . . . 100 13 Conclusion . . . . . . . . . . . . 103