Florentin Smarandache Neutrosophic Perspectives: Triplets, Duplets, Multisets, Hybrid Operators, Modal Logic, Hedge Algebras. And Applications Peer-Reviewers: Dr. Azeddine Elhassouny Fulbright postdoctoral fellow, Department of Mathematics and Science, University of New Mexico, Gallup Campus, USA. Prof. S. Jafari College of Vestsjaelland South, Herrestraede 11, 4200 Slagelse, Denmark. Dr. Mohamed Abdel-Basset Faculty of Computers and Informatics, Zagazig University, Egypt. Florentin Smarandache Neutrosophic Perspectives: Triplets, Duplets, Multisets, Hybrid Operators, Modal Logic, Hedge Algebras. And Applications Pons Publishing House Brussels, 2017 Pons Publishing House / Pons asbl Quai du Batelage, 5 1000 - Bruxelles Belgium DTP: George Lukacs ISBN 978-1-59973-527-6 Florentin Smarandache Neutrosophic Perspectives TABLE OF CONTENTS PREFACE Neutrosophic Perspectives 9 I.1. Positively or Negatively Qualitative Neutrosophic Components 12 I.2. The Average Positive Qualitative Neutrosophic Function and The Average Negative Qualitative Neutrosophic Function 13 II.1. Neutrosophic Overnumbers / Undernumbers / Offnumbers 18 II.2. Spherical Neutrosophic Numbers 21 III.1. Neutrosophic Indeterminacy of Second Type 23 IV.1. n-Refined Neutrosophic Set and Logic and Its Applications to Physics 26 V.1. Operations with Neutrosophic Numbers 45 V.2. Addition of Multiple Single-Valued Neutrosophic Numbers 48 V.3. Interval-Valued Neutrosophic Number Operations 50 V.4. Operations with (t, i, f)-Neutrosophic Matrices 53 VI.1. Neutrosophic Hybrid Operators 62 VII.1. Neutrophic Perfect/Imperfect Triplets 67 VII.2. Neutrosophic Triplet Function 76 5 Florentin Smarandache Neutrosophic Perspectives VII.3. Theorems on Neutrosophic Triplets 77 VII.4. Neutrosophic Triplet Group vs. Generalized Group. 91 VII.5. Neutrosophic Triplet Multiple Order 94 VIII.1. Neutrosophic Triplet Ring 95 VIII.2. Hybrid Neutrosophic Triplet Ring 97 VIII.3. Neutrosophic Triplet Field 99 VIII.4. Hybrid Neutrosophic Triplet Field 103 VIII.5. Neutrosophic Triplet Loop 104 VIII.6. Neutrosophic Triplet Structures 106 IX.1. Neutrosophic Duplets 111 IX.2. Neutrosophic Duplet Set and Neutrosophic Duplet Structures 113 X.1. Neutrosophic Multiset 117 X.2. Neutrosophic Multiset Applied in Physical Processes 122 X.3. Neutrosophic Complex Multiset 124 XI.1. Classical Logic and Neutrosophic Logic. Answers to K. Georgiev. 126 XI.2. Neutrosophic Tautological, Contradictional, and Provability Tresholds, and Neutrosophic Completeness. Answers to U. Rivieccio 147 XII.1. Neutrosophic Predicate Logic 178 XII.2. Neutrosophic Decidability System 182 XII.3. Neutrosophic Modal Logic 184 6 Florentin Smarandache Neutrosophic Perspectives XII.4. Neutrosophic Hedge Algebras 212 XIII.1. Neutrosophic MCDM 228 XIII.2 Neutrosophic Psychology 229 XIII.3. Neutrosophic Function as The Equatorial Virtual Line 230 XIII.4. PCR5 and Neutrosophic Probability in Target Identification 231 XIII.5. Easier to Break a Dynamic System from Inside than from Outside 245 XIII.6. Neutrosophic Quantum Computer 281 XIII.7. Theory of Neutrosophic Evolution: Degrees of Evolution, Indeterminacy, and Involution 297 XIII.8. Neutrosophic Triplet Structures in Practice 325 7 Florentin Smarandache Neutrosophic Perspectives PREFACE Neutrosophic Perspectives This book is part of the book-series dedicated to the advances of neutrosophic theories and their applications, started by the author in 1998. Its aim is to present the last developments in the field. For the first time, we now introduce: — Neutrosophic Duplets and the Neutrosophic Duplet Structures; — Neutrosophic Multisets (as an extension of the classical multisets); — Neutrosophic Spherical Numbers; — Neutrosophic Overnumbers / Undernumbers / Offnumbers; — Neutrosophic Indeterminacy of Second Type; — Neutrosophic Hybrid Operators (where the heterogeneous t-norms and t-conorms may be used in designing neutrosophic aggregations); — Neutrosophic Triplet Loop; — Neutrosophic Triplet Function; — Neutrosophic Modal Logic; — and Neutrosophic Hedge Algebras. 9