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Multi-Valued Logic, Neutrosophy, and Schrödinger Equation

By Smarandache, Florentin

Book Id:WPLBN0002828444 Format Type:PDF (eBook) File Size:1.60 mb Reproduction Date:7/31/2013

Smarandache, F., & Christianto, V. (2013). Multi-Valued Logic, Neutrosophy, and Schrödinger Equation. Retrieved from http://gutenberg.us/

Description
This book was intended to discuss some paradoxes in Quantum Mechanics from the viewpoint of Multi-Valued-logic pioneered by Lukasiewicz, and a recent concept Neutrosophic Logic. Essentially, this new concept offers new insights on the idea of ‘identity’, which too often it has been accepted as given.

Summary
The book is motivated by observation that despite almost eight decades, there is indication that some of those paradoxes known in Quantum Physics are not yet solved. In our knowledge, this is because the solution of those paradoxes requires re-examination of the foundations of logic itself, in particular on the notion of identity and multi-valuedness of entity.

Excerpt
2 Lukasiewicz Multi-Valued-logic: History and Introduction to Multi- Valued Algebra
2.1 Introduction to trivalent logic and plurivalent logic
We all have heard of typical binary logic, Yes or No. Or in a famous phrase by Shakespeare: “To be or not to be.” In the same way all computer hardwares from early sixties up to this year are built upon the same binary logic.
It is known that the Classical Logic, also called Bivalent Logic for taking only two values {0, 1}, or Boolean Logic from British mathematician George Boole (1815-64), was named by the philosopher Quine (1981) “sweet simplicity.” [57] But this typical binary logic is not without problems. In the light of aforementioned ‘garment analogue’, we can compare this binary logic with a classic black-and-white tuxedo. It is timeless design, but of course you will not wear it for all occasions. Aristotle himself apparently knew this problem; therefore he introduced new terms ‘contingency’ and ‘possibility’ into his modal logic [5]. And then American logician Lewis first formulated these concepts of logical modality.

Table of Contents
Contents
Foreword 6
1 Introduction: Paradoxes, Lukasiewicz, Multi-Valued logic 7
2 Lukasiewicz Multi-Valued Logic: History and Introduction to Multi-Valued
Algebra 10
2.1. Introduction to trivalent logic and plurivalent logic 10
2.2. History of Lukasiewicz and Multi-Valued Logic 12
2.3. Introduction to Multi-Valued Algebra, Chang’s Notation 15
2.4. Linkage between Multi-Valued Logic and Quantum Mechanics 15
2.5. Exercise 17
3 Neutrosophy 25
3.1. Introduction to Neutrosophy 25
3.2. Introduction to Non-Standard Analysis 26
3.3. Definition of Neutrosophic Components 27
3.4. Formalization 28
3.5. Evolution of an Idea 30
3.6. Definition of Neutrosophic Logic 31
3.7. Differences between Neutrosophic Logic and IFL 32
3.8. Operations with Sets 33
3.9. Generalizations 34
4 Schrödinger Equation 39
4.1. Introduction 39
4.2. Quantum wave dynamics and classical dynamical system 43
4.3. A new derivation of Schrödinger-type Equation 45
5 Solution to Schrödinger’s Cat Paradox 47
5.1. Standard interpretation 47
5.2. Schrödinger’s Cat Paradox 48
5.3. Hidden-variable hypothesis 50
5.4. Hydrodynamic viewpoint and diffusion interpretation 50
5.5. Some less known interpretations 51
5.6. How Neutrosophy could offer solution to Schrödinger’s Cat Paradox 52
6 Quantum Sorites Paradox and Quantum Quasi-Paradoxes 54
6.1. Introduction 55
6.2. Sorites Paradox and Quantum Sorites Paradox 56
7 Epistemological Aspects of Multi-Valued-Logic 57
7.1. Non-standard real numbers and real sets 62
7.2. Epimenidean Paradox and Grelling Paradox 62
7.3. Neutrosophic Statistics and Neutrosophic Probability Space 64
7.4. Generalization of Other Probability 65
8 Postscript: Schrödinger Equation and Quantization of Celestial Systems 67
Epilogue 76
Acknowledgment 78
References 79
Appendix I: An example of self-referential code 88
Appendix 2: Summary of bibliography of Non-Standard Logic 93
Appendix 3: List of some known paradoxes 95
Appendix 4: A few basic notations 103
Appendix 5: A few historical achievements in foundation of set theory 105