A Simple Ternaric Analysis

In binaric (two-value) systems and analyses, as classically defined in Spencer Brown's _ Laws of Form _ (LoF), direction is alternation, with only two crossings possible, from either value to the other.

But in ternaric (three-value) systems and analyses, there are two possible crossings to and from each of the three values involved, six in all.

That is, given N values, there are N * ( N - 1 ) possible crossings.

Take values A, B and C, and lexical order to be positive direction, and symbolize crossings by original value with "+" and "-" appended.

Then

A+ = B

A- = C

The ternaric law of calling (see LoF) is of course

A = AA

And the ternaric law of crossing is

A = A+- = A-+

But the ternaric law of crossing extends also to what might be called the law of cycling

A = A+++ = A---

Taken together, these laws allow such single-valued expressions and equations as

A = AB- = AC+ = AB-C+

(Geometricians may visualize a triangle, and algebraicists see a (proto-) ring, here.)

All this elides the general enaric issue of definition of value and direction of crossing in terms of (opposed or negative) reference to (all) others, and it leaves the status of forms and values such as AB in question, whether allowed or disallowed, or, if allowed, whether conditional or imaginary.

And it is not the grand unified enarics we seek, applied to the ternaric, since it does not extend to quaternaric and higher form, other than where crossing is appropriately limited, or direction appropriately defined (eg, in the quaternaric as ">", "<" and "^").

But as far as it goes, it is valid and simple.


Keywords: analysis,binarics,enarics,form,GSB,Laws of Form,LoF,mathematics,ternarics


[20160506--Revised already!]

The Law of Form

Let the mark, marked value and marked state be represented by a pair of square brackets

[ ]

and the unmarked by none.

Thus LoF's Chapter 1's Axiom 1, "The Law of Calling"; Chapter 2's "form of condensation"; and Chapter 3's primary calculus' primitive equation and Chapter 4's primary arithmetic's Initial 1, "Number", are all represented by

[ ] [ ] = [ ]

while Chapter 1's Axiom 2, "The Law of Crossing"; Chapter 2's "form of cancellation"; and Chapter 3's primary calculus' primitive equation and Chapter 4's prmary arithmetic's Initial 2, "Order", are all represented by

[ [ ] ] = 

Now let M stand for the mark, marked value and marked state

M = [ ]

and N for the unmarked

N = 

Then

MN = M

This is used as part of Chapter 4, Sixth Canon, "Rule of Dominance" (the canon itself used as lemma to Theorem 3, "Agreement"), and is one of the two primary arithmetic possibilities of Chapter 6's primary algebraic Consequence 3, "Integration".

But it seems a definite, profound and beautiful law by itself, one at least as primitive and fundamental as the laws of calling and crossing:

Call it the law of form.