## Monday, July 9, 2018

### Laws of Triple Form; 1 of 12

Laws of Triple Form

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

Synopsis

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
Differentials
Bochvarian operators

Chapter 4: Inner Order
Majority
Min
Inner order

Chapter 5: Completeness Theorems
Anchored normal forms
Deductive completeness
Self-reference

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
Majorities
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

Trilemma deduction
Syllogisms by Trilemma

Introduction

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.