Well-Structured Mathematical Logic

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1 Well-Structured Mathematical Logic Damon Scott Department of Mathematics Francis Marion University CAROLINA ACADEMIC PRESS Durham, North Carolina

2 Copyright 2013 Damon Scott All Rights Reserved Library of Congress Cataloging-in-Publication Data Scott, Damon, Well-structured mathematical logic / Damon Scott. pages cm Includes bibliographical references and index. ISBN Logic, Symbolic and mathematical. I. Title. QA9.S dc Carolina Academic Press 700 Kent Street Durham, North Carolina Telephone (919) Fax (919) Printed in the United States of America

3 Table of Contents Table of Contents v Detailed Outline vii Acknowledgements xix PART ONE: The New Formal Linguistics Chapter 01: Rectification of Phrase Structure Chapter 02: Escape from the Old Formal Linguistics Chapter 03: The Central Result of the New Formal Linguistics Chapter 04: The Relationship between Phrase Structure and User-Friendliness Chapter 05: Elementary Applications of the New Formal Linguistics Chapter 06: Newly Structured Programming PART TWO: Establishment of the Context-Oriented Language for Mathematics Chapter 07: Preliminaries Chapter 08: Definition of COLM, the Context-Oriented Language for Mathematics Chapter 09: Early Tutorial on Context-Oriented Symbolic Manipulation Chapter 10: The Substitution Operators Chapter 11: Small-Scale Applications: The Qualified Quantifier PART THREE: Fundamentals of Context-Oriented Mathematical Logic Chapter 12: The Syntactics of Well-Formed Mathematical Contexts Chapter 13: The Semantics of Well-Formed Mathematical Contexts Chapter 14: Ramification by Scale of Expression Chapter 15: The Central Correspondence between Contexts and Statements Chapter 16: The Relations of Strength and Weakness Chapter 17: Middle-Scale Applications: The Quote-and-Prove Rules PART FOUR: Development of Rules of Inference for the New System Chapter 18: Negations, Complements, and the Dual Symmetry Chapter 19: The Theorem on Monotonicity Chapter 20: The Local Calculus Rules Chapter 21: The Copy-Migration Rules: Logical Deduction at a Distance Chapter 22: The Look-and-Feel of Formal Deduction in the New System Chapter 23: Large-Scale Applications: Well-Structured Relativity v

4 vi Table of Contents PART FIVE: The Structure of Deduction and Proof in the New System Chapter 24: Overview of Context-Oriented Proof Theory Chapter 25: The First Formalization of Context-Oriented Proof Chapter 26: Interpolation Operators, Their Warrants and Phrase Structure Chapter 27: The Second Formalization of Context-Oriented Proof Chapter 28: Meta-Proof of Theoretical Soundness and Sufficiency Chapter 29: A Full Example of Context-Oriented Formal Proof in Action PART SIX: Epilogue Chapter 30: Epilogue Glossary Bibliography Index

5 Detailed Outline Front Matter Table of Contents v Detailed Outline vii Acknowledgements xix PART ONE: The New Formal Linguistics Chapter 1: Rectification of Phrase Structure Beyond Words 3 02 On the Title, Well-Structured Mathematical Logic 4 03 On the Names Old and New Formal Linguistics 5 04 On the Names Context-Oriented and Statement-Oriented Mathematical Logic 5 05 A Full Example of Rectified Phrase Structure 6 06 Second Example of Rectified Phrase Structure 8 07 Review of the Literature The Method of Gradual Presentation What Lies Ahead Departure Point for the Expedition 13 Chapter 2: Escape from the Old Formal Linguistics The Importance of Structure Incompatibility of Nested Structures with User-Friendliness Description (not Definition) of the term Phrase Different Phrase Structures on the Same Expression Various Scales of Phrase Structure Yet More Systems of Phrase Structure 20 vii

6 viii Front Matter Chapter 3: The Central Result of the New Formal Linguistics The Nature of the Central Result The Central Result Stated Demonstration of the Central Result Crossing into the New Formal Linguistics Consequences of the Central Result Elementary Analytic Applications of the Central Result Synthetic Applications of the Central Result 26 Chapter 4: The Relationship between Phrase Structure and User-Friendliness Hilbert s Meta-Standard Another Example of the Problem Implicit Phrase Structures Relationship of Phrase Structure to User-Friendliness, Shown by Example Completely-Nested Phrasing and Its User-Hostility Mostly-Flowing Phrasing and Its User-Friendliness Conscious and Subconscious Changes of Phrase Structure Random-Accessed Phrasing and Its User-Friendliness Comments on the Division between the Ground- and Meta-Languages 35 Chapter 5: Elementary Applications of the New Formal Linguistics Syntactic Craft and Craftsmanship A New Outlook Finding the Function Space A Simple Example A Pair of Pre-Compilers for Changing Phrase Structure Nest-and-Flow Syntax from Function Spaces with Multiple Inputs An Example of Creating a Nest-and-Flow Language 44 Chapter 6: Newly Structured Programming Rectification to the Rescue Setting Up the Problem to Be Solved Creation of Newly Structured Programming Example of Newly Structured Programming in Action Comments on the Example Expert Employment of the One Remaining Nested Form Fluvious Programming Languages A Night s Rest in the Highlands of Phrase Structure 52

7 Detailed Outline ix PART TWO: Establishment of the Context-Oriented Language for Mathematics Chapter 7: Preliminaries Before the Threshold Some Overarching Concepts Several Specific Conventions and Notations The Presumed Equivalence Relation on Expressions Rules of Precedence Thinking at the Proper Granularity 61 Chapter 8: Definition of COLM, the Context-Oriented Language for Mathematics User-Friendliness from the Inside Out Overview of Basic COLM Definition of the Operators Given and Required, G and R Definition of the Quantifiers, A and E The Qualifier Such-That, st, and the Qualified Operators Digression on the Paradigm of Hypothesis and Conclusion Confirmation of Ideas of Phrase Structure The Russian Doll Maneuver The Negation Operator, n The Comment Operator, Called Mark, M The Empty Operator, Called Ita, i The Wipe-Out Operators, W, Y, w0, and w Operators of Convenience: and, or, if, and iff in a New Phrase Structure The Substitution Operators, U and V Comments on the Notation of Punctuation Translation Guide for the New System Sufficiency of Basic COLM for First-Order Expression 75 Chapter 9: Early Tutorial Context-Oriented Manipulation The Need for a Tutorial A Starter Set of Context-Oriented Rules of Inference Throwing the Main Clause of a Statement into a Context Atomic Translations to Context-Oriented Form Solving our Tutorial Problem Comparison and Contrast with Traditional Logic Preview of Coming Attractions Upward and Outward 82

8 x Front Matter Chapter 10: The Substitution Operators The Place of the Substitution Operators in the Grand Scheme of Things Overview of Ideas The Definition of the Substitution Operators, Part One: Substantial Meanings Simple Substitutions The Definition of the Substitution Operators, Part Two: Vacuous Meanings Actually Performing the Substitution Advantages of the New Substitution Operators Technical Discussion on Issues of Substitutability Technical Section: How to say Uniqueness in the New Language Technical Comments about the Substitution Operators 92 Chapter 11: Small-Scale Applications: The Qualified Quantifier Improvement as Harder Than It Looks The Qualified Quantifier The Qualified Quantifier as an Aid to Repairing Fractured Standard Syntax A Multi-Purpose Qualification COLM as an Assistance to Parsing Dense Technical Prose Comments on the Substitution Operators How Well-Structured Mathematical Logic Was Built from the Qualified Quantifier 98

9 Detailed Outline xi PART THREE: Fundamentals of Context-Oriented Mathematical Logic Chapter 12: The Syntactics of Well-Formed Mathematical Contexts Definitions of Clauses. Species of Clauses. Interiors of Clauses Definition of Well-Formed Mathematical Contexts Composition of Contexts with Contexts and with Statements Phrase Structure of Contexts 105 Chapter 13: The Semantics of Well-Formed Mathematical Contexts The Existence of Mathematical Situations The Idea of Scale of Expression The Semantics of G- and R-Clauses The Semantics of A- and E-Clauses The Semantics of U- and V-Clauses The Semantics of Negation Aspects of Contexts Overloaded Use of Some Words in Informal Mathematics The Semantics of Contexts Formed by Composition The Necessity of Context-Oriented Thinking for Actually Doing Mathematics Further Comments on the Paradigm of Hypothesis and Conclusion Limitations to What is Formalized as a Well-Formed Mathematical Context Issues of Semantic Soundness and Adequacy Technical Section: Contexts of Higher Complexity 118 Chapter 14: Ramification by Scale of Expression Peano s Dream Ramification by Scale of Expression Scales of Expression Again Examples of Each of the Three Main Scales of Expression Self-Similarity of Both Formally and Informally Written Mathematics Seeing the Self-Similarity by Zooming In on a Piece of Syntax The One-to-One Correspondence between Nestedness and Scale of Expression Ramification of Rules of Inference COLM as a Useful Shorthand for Writing Mathematics On the Intermingling of Hypotheses and Conclusions On the Division between Small- and Middle-Scale Expression Ramified Thinking 130

10 xii Front Matter Chapter 15: The Central Correspondence between Contexts and Statements A Connection between Two Ways of Thinking The Central Correspondence Stated Details of the Central Correspondence and Why It Is True Consequences of the Central Correspondence Technical Section: Proof of the Central Correspondence 134 Chapter 16: The Relations of Strength and Weakness The Prevalence of the Relations Hence, Because, and Hence-and-Conversely The Relations of Para-Strength and Para-Weakness for Statements The Relations of Para-Strength and Para-Weakness for Contexts The Algebraic Nature of Contexts Turning Contexts On and Off Para-Positive, Para-Negative, and Zeroish Contexts Non-Strict, or Para-, Strength and Weakness as Universal Defaults Fundamental Results Concerning the Composition of Contexts Relation between Para-Strengthening and the Processes of Proof 139 Chapter 17: Middle-Scale Applications: The Quote-and-Prove Rules Attaining Ramifiable Closure for How-to-Prove-It Rules The Quote-and-Prove Words The Quote-and-Prove Rules Stated Initial Comments on the Quote-and-Prove Rules The Quote-and-Prove Rules Proved Examples of the Quote-and-Prove Rules in Action The Quote-and-Prove Rules for Negation Miscellaneous Comments on the Quote-and-Prove Rules Technical Section: On the Closing Sections of Informal Proofs A Time for the Expedition Party to Reflect, Relax, and Rest 151

11 Detailed Outline xiii PART FOUR: Development of Rules of Inference for the New System Chapter 18: Negations, Complements, and the Dual Symmetry Arrival at the Construction Site General Comments on Negation and the Dual Symmetry The Calculus of Negation in COLM Tutorial Examples Negation in the New and in Traditional Logic Compared and Contrasted Complementary Operators Technical Section: Tutorial on Forming Duals 160 Chapter 19: The Theorem on Monotonicity A Show Horse and a Work Horse The Theorem on Monotonicity, Lofty Form The Theorem on Monotonicity, Nuts-and-Bolts Statement-Oriented Version The Theorem on Monotonicity, Nuts-and-Bolts Context-Oriented Version Miscellaneous Results A Tutorial Example Digression on Logical Variational Analysis The Theorem on Monotonicity and the Application of Calculus Rules Calculability of Monotonicity Comments on the Weakness of Absolute Truth Comments on the Strength of Absolute Falsehood 168

12 xiv Front Matter Chapter 20: The Local Calculus Rules The Bricks and Mortar of Logical Deduction On the Non-Minimalism of the System Collapsing Rules Rules for Contraposition Contraction Rules Rules for Bubble-Busting Rules Governing the Insertion of G-, R-, A-, and E-Clauses Commutativity and Semi-Commutativity of Clauses Further Commutativities When Statements Are Independent of Variables Pervasions of Quantifiers, Qualified Quantifiers, and Other Operators Rules Governing the Substitution Operator Seeing the Substitution Operator in Action Stray Rules of Inference One Last Rule Application of the Local Calculus Technical Section: Proof of the Relationship between Independence and Commutativity of Clauses 185 Chapter 21: The Copy-Migration Rules: Logical Deduction at a Distance The Nature of the Long-Distance Calculus Rules The Copy Rules The Copy-Migration Rules Stated for Hypotheses and Conclusions Copy-Migrations of Quantifiers The Copy-Erasure Rule The Accumulated Context at a Point Technical Section: Proof of the Copy-Migration Rules 191

13 Detailed Outline xv Chapter 22: The Look-and-Feel of Formal Deduction in the New System A New Method Is Born Tutorial on a Collapsing Rule The Definition of Collapsible Clauses and Contexts Tutorial on Another Collapsing Rule An Ad-Hoc Expository Method Proving Our First Substantial Tautology Tutorial on Getting Results Out of Syntax Bubbles Tutorial on Performing Copy-Migrations with Wild Abandon Tutorial on Contraposition Example: Showing the Associativity of Disjunction The Reduction of Provable Implication Seeing the Forest after Seeing the Trees 201 Chapter 23: Large-Scale Applications: Well-Structured Relativity The Paradigmatic Status of Formal Logic The Truth of Statements Relative to Contexts Relative Deductive Validity Validity Relative to Contradictory Contexts Meta-Contexts Layered Relativities On the Scopes of Variables On the Scopes of Hypotheses and Conclusions The Difference between the Tiers in the Big Chart Reformalization of Logical Axioms as Rules of Inference Third Resting Place for the Expedition Party 212

14 xvi Front Matter PART FIVE: The Structure of Deduction and Proof in the New System Chapter 24: Overview of Context-Oriented Proof Theory Overview of Proof Theory in General Statement- and Context-Oriented Formal Proof Compared and Contrasted Very Broad Overview of Different Context-Oriented Proof Apparatus Discussion of the First Formalization of Formal Proof Discussion of the Second Formalization of Formal Proof On the Meta-Proofs of Theoretical Soundness and Sufficiency On the Status of Positive Normal Form and of Negative Contexts 220 Chapter 25: The First Formalization of Context-Oriented Proof Rules of Inference Reviewed The First Formalization of Context-Oriented Proof 222 Chapter 26: Interpolation Operators, Their Warrants and Phrase Structure The Long, Long Roads of Proof The Interpolation Operator eq The Interpolation Operators lms, gms, lmw, and gmw Warrants of Interpolations Personal Phrase Structures of Interpolations Example: An Interpolation to Do a Proof by Contraposition Example: An Interpolation to Do a Proof by Cases Example: Interpolations to Insert Interior Deductions Example: Interpolations to Specify Variables Example: Interpolations to Add Auxiliary Constructions Species of Interpolations 232 Chapter 27: The Second Formalization of Context-Oriented Proof The Second Formalization Shown by Example Positive Normal Form Formalization of Sequitur The Second Formalization of Context-Oriented Proof Technical Section: Informal Sequiturs and Their Taxonomy Formal Deductions The Relation between Formal and Informal Deduction 242

15 Detailed Outline xvii Chapter 28: Meta-Proof of Theoretical Soundness and Sufficiency The Inevitable, at Last Meta-Statement of the Soundness and Sufficiency of the Second Formalization of Formal Proof Technical Section: Review of the Meta-Proof of Soundness Technical Section: Meta-Proof of Sufficiency, Part 1: Setting Up the Works Technical Section: Meta-Proof of Sufficiency, Part 2: Creating the Context-Oriented Proof from the Statement-Oriented Proof Technical Section: Meta-Proof of Sufficiency, Part 3: Marking All the Clauses Sequitur Technical Section: Meta-Proof of Sufficiency, Part 4: Meta-Proof Closure Grand Conclusion of the Meta-Proof of Theoretical Soundness and Sufficiency 247 Chapter 29: A Full Example of Context-Oriented Formal Proof in Action Arrival at Our Final Destination Euclid s Proof of the Pythagorean Theorem Translated into Context-Oriented Form Comments on the Full Example Concluding Remarks 253 PART SIX: Epilogue Chapter 30: Epilogue Dismissal of the Expedition Party 257 End Material Glossary The Greek Alphabet 259 Arithmetic Operators 261 Number Systems 259 Other Symbols 261 Symbols of Traditional Logic 260 Terms 261 Symbols of Set Theory 260 Bibliography Authorities on Traditional, Statement-Oriented Logic Prior Work in Context-Oriented Logic Sources Cited in the Text Other Books of Interest 265 Index

16

17 Acknowledgements A grateful mind By owing owes not, but still pays, at once Indebted and discharg d. Milton. 1 The author would like to thank first, foremost, and especially Richard E. Hodel of Duke University for his support of this endeavor. Dr. Hodel has been a friend to these researches from their earliest infancy through to this their published form. Even the reader has cause to thank Dr. Hodel. Many of the chapters of the current book had their content presented to Dr. Hodel at weekly meetings. With Dr. Hodel acting as first student of the new system, the author was able to tell how to present and how not to present the material in a manner that would be clear and understandable to someone first learning to think in the new way. Dr. Hodel also made numerous suggestions for improvement of the text, and he lent his expertise on occasion to some of the finer points of traditional logic. Barry Lobb, currently at Lynchburg College, should be mentioned because he presented certain calculations in flowing forms and encouraged the author to adopt flowing ways of thinking when the author was an undergraduate student at Butler University. Where Dr. Lobb got these ideas we do not know, whether from somewhere else or from some internal genius, but they form a precursor to the methodical treatment of the flowing form to be found in the following pages. Thanks also are due to Francis Marion University, which has supported this work by granting the author a summer research stipend and a sabbatical leave for the purpose of bringing this work to its present state. Keith Devlin offered the author some crucial advice when the research project was stuck and not advancing properly, and Edward Arroyo and Matthew Turner provided useful comments on drafts of the present work. This work would not have occurred had not Gregory F. Lawler taken the author in as a student and supervised his doctoral dissertation in probability. Credit must also be given to the memory of the late Joseph R. Shoenfield, who taught the author traditional logic during his graduate studies at Duke University. 1. Paradise Lost, Book IV, lines xix

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