Monday, July 9, 2018

Laws of Triple Form; 1 of 12

Laws of Triple Form
© Nathaniel Hellerstein, 2018

 Laws of Triple Form

                Synopsis                                                                  3

        Introduction                                                          5

          Chapter 1:  Brownian Forms                                 6
          Chapter 2:  The Third Form                                14
          Chapter 3: Kleenean Logic                                   18
          Chapter 4: Inner Order                                        21
          Chapter 5: Completeness Theorems                    25
          Chapter 6: Three Marks                                       29
          Chapter 7: Pivot                                                    33
          Chapter 8: Orderings                                            35
          Chapter 9: Trinary Logics                                    37
          Chapter 10: Self-Reference                                  42
Chapter 11: Voter’s Paradox                               45


          This book is a reply to George Spencer-Brown’s “Laws of Form”. It extends his form calculus to 3 values; and it covers the triple-form calculus’s laws, tables, axioms, theorems, normal forms, deductive completeness and self-reference. Then it introduces two more forms of distinction; these generate three logics on the three logics, with new laws, and with paradoxes that summarize deduction.

          Chapter 1:  Brownian Forms
          Form arithmetic
          Form algebra
          Isomorphic twice to Boolean logic
          Normal forms
          Complete deduction
          Incomplete re-entrance

          Chapter 2:  The Third Form
The Liar
          Ternary forms and algebra

          Chapter 3: Kleenean Logic
          Isomorphic to Kleenean logic
Bochvarian operators

          Chapter 4: Inner Order
          Inner order

          Chapter 5: Completeness Theorems
          Anchored normal forms
          Deductive completeness

          Chapter 6: Three Marks
          Three marks
          Trinary algebra

          Chapter 7: Pivot
          Pivot definition and laws

          Chapter 8: Orderings
          6 orders on the three forms

          Chapter 9: Trinary Logics
          Octohedral distribution
          The Hexagram

          Chapter 10: Self-Reference
          Kleenean and Bochvarian logics in the Hexagram
          Complete Self-Reference in the Hexagram
          Incomplete Self-Reference in the Hexagram

          Chapter 11: Voter’s Paradox
          Voter’s Paradox
Trilemma deduction
Syllogisms by Trilemma



          This book is a reply to George Spencer-Brown’s “Laws of Form”. His book describes a system with two values and two permutations. It is a bracket-based version of two-valued Boolean logic. This book describes a bracket-based system with three values and six permutations; a version of three-valued Kleenean logic.

This book has many “exercises for the student”, so as not to deny the ambitious student the pleasure of re-discovery of these calculations. It’s also because this book is a work in progress. I might include answers to some of these exercises in later editions of this book.

          Like “Laws of Form”, this book ends with a break in its order. “Laws of Form” ends by evoking the Liar Paradox, which subverts the Boolean deductive scheme previously described. But it also presents the Brownian modulator, whose workings are explained by paradox logic.
          This book ends by evoking the Voter’s Paradox, which destabilizes the fixedpoints previously constructed. But it also presents the Some-All-None Trilemma, which generates all of syllogistic logic.

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