## Wednesday, May 4, 2016

### 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

## Saturday, January 9, 2016

### 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.

## Saturday, May 24, 2014

### LoF: Recalling and Crossing from the Unmarked Binaric

LoF symbolizes one binaric (one of the two binaric entities) by no-mark.

Such symbolization is seemingly not only for elegance in calculation but expresses an asserted degenerate identity in LoF, if not binarics itself, of calculation and calculated, operation and operand, instruction and name.

But one consequence of such symbolization is that the law of calling as symbolized in LoF (here in parenthetic form)

( ) ( ) = ( )

appears to imply that that law applies only to the marked binaric, whereas it must apply to both binarics, as can be seen by symbolizing the marked and unmarked binarics as _m_ and _n_ respectively (as is done in LoF occasionally)

m = ( )

n =

and the law of calling with regard to each binaric as

m m = m

n n = n

The law of calling is more correctly and commodiously expressed in LoF's own algebraic Consequence 5 as

a a = a

Similarly, LoF's symbolization of the law of crossing

( ( ) ) =

again appears to imply that that law applies only to the marked binaric, whereas it must also apply to the unmarked binaric, symbolizable as

( ) = ( )

These last two symbolizations and points are perhaps better understood by again using the labels _m_ and _n_ (in part)

( m ) = n

and

( n ) = m

Note that last parenthetic

( ) = ( )

appears to correspond to the usual law of identity, more correctly and commodiously expressed in binarics as

m = m

n = n

and generally as

a = a

and which does indeed suggest an identity of crossing and identity.

## Tuesday, February 25, 2014

### Progress (Sort Of) Report

Progress has been slow in my study of ternarics and higher enarics (how complex simplicity can be!), but I hope to finish a small (about 20 pp.) paper I've been working sporadically on over the last couple of years on this subject called "Toward Ternarics" and put it in this year here and in the Yahoo group lawsofform. This paper as its title indicates will be very far from complete even so (more on this below). The present post is more a recitation of mea-culpas, if that's the phrase, if anything.

One recent realization, and this by way of at best qualified mea culpa, has to do with my invocation on various occasions in the Yahoo group of "groundedness" in enarics, which in fact arose from a misunderstanding on my part of the direction of indication in the calculus of form as presented in LoF: Indication is outward, in the opposite direction from depth. Although (the qualification referred to) grounded form may well prove to be in the end another variety of form.

Another and much more important realization has been that The Enary Plenary Parenthetics (TEPP) that I've put forth here and in the Yahoo group represent only a small subset of the varieties of form at each level above the binaric, and a logarithmically smaller subset at that with each increase in order, from ternaric to quaternaric, quaternaric to quinaric, and so on. There are explosions combinatorial and otherwise of properties absent or degenerate in Lof and the binaric as form and analysis increase in order above the binaric; I can barely do more at present than point to the fact (this is another excuse for my slow progress). Essentially, the enary plenaries treated in TEPP are "binaric-like" (I cannot describe it any better here). I strongly doubt any of us will live to see the complete explication of the properties of even the orders treated in TEPP, up to the undecenaric: The ternaric volume corresponding to LoF must necessarily be a larger and likely a much larger volume, and that corresponding to the quaternaric a multi-volume set. (I console myself with the reflection that this is not complexity but richness.)

"Toward Ternarics" will need to be made mutatis mutandis into a new but necessarily still provisional introduction to TEPP. I also need to fix the missing-forms problem above the nonaric in those tables; to generate and release the corresponding tables of what I've been calling "eversions" and now call "perspects" (the internal views in each form, taken as a series of interconnecting cells with the appropriate visibility) as well; to simplify and release the software tools involved (see the next paragraph); and to generate the numeric sequences corresponding to the enary plenaries and perspects in various ways, if only to help people realize when they're dealing with phenomena best modeled by enaric analysis. ("This is not complexity but richness" indeed.)

With the regard to the software tools involved, which are unavoidably going to eventually become very numerous and complex indeed, I tend to write portmanteau warez, which is a fundamental programming error, other than in software intended for nonprogrammer use: The correct "software tools philosophy" is that warez should consist of tightly-defined utilities each doing one thing well, which can then be cobbled together in various pipelines as needed, as in shell languages such as bash and the multimedia language Gstreamer. Jim Snyder-Grant of the Yahoo group wrote a program generating the initial forms at each order (before checking for duplicates and equiforms, eg "(())()" and "()(())") that is one such, as is my "eversions.c" (soon to become "perspects.c")  generating perspects from a supplied form (see the "Software" directory in the Yahoo group), although both should be rewritten to take their inputs from standard input (keyboard or redirected output), for the sake of pipelining. Such arrangement allows software to be used as flexibly as possible, and in ways unimagined by the original programmers. (I might as well add here that Jim's program generates parenthetics deepest on the left, and mine on the right, and while I've come around to believing that left-weighted is best, for ease of taxonomic and dictionary use—another revision of TEPP!—both and related warez should probably be written to work either way.)

One result of reflection upon the ternaric and higher enaric properties and explosions referred to above has been a correction of another foolish error of mine with regard to LoF and the binaric.

GSB says in LoF, "Notes", "Chapter 2", that "In conceiving the calculus of indications, we begin at a point of such degeneracy as to find that the ideas of description, indication, name, and instruction can amount to the same thing."

I misremembered this as embodying a claim that form and symbolization are identical or degenerate in LoF and the binaric, with which I casually if not sloppily (even mystically) did not disagree.

Call an expression combining several subexpressions such as "()()" a "symplex", or "symplexed" (and an involved expression such as "(())" an "implex" or "implexed"—this terminology is from "Toward Ternarics").

The symbolization of symplexes in LoF is (borrowing a term from arithmetic theory) "commutative", eg

(())() = ()(())

but the form of course is not, the form represented on both sides of the equation being exactly the same (which is, after all, exactly what that equation symbolizes):

Form and symbol are therefore not identical or degenerate even in LoF and the binaric.

Which is one explosion one can point to resulting from the increase in enaric and analytical order from the unaric to the binaric.

Finally, I am troubled by what I call the "low-hanging fruit" problem with regard to LoF and binarics themselves: Generally, whenever a new form of analysis is introduced, there will be a quick harvesting of problems the solutions of which are then rendered easier or even possible by the new analysis. We have not seen this with regard to LoF and binarics in the half-century since LoF was published. It is true that many varieties of binaric analysis were already in use long before LoF, and indeed LoF represents a pushing downward hierarchy-of-analysis-wise from one such, the algebraic treatment of propositional-logical truth-values, and the application of the algebra of form to the solution of propositional logical problems in Appendix 2 of LoF represents only a beautiful simplification of what was already used and known as "binary resolution". But what about the use of binaric imaginaries, in logic and elsewhere? GSB used one such in a switching-problem. Where are the others? Even GSB has not produced such (and the last I read was totting up seemingly metaphysical ternaries like a Welsh bard, although it's true that taxonomy comes first, hence TEPP). One can point to Maxwell's originally-controversial use of arithmetic imaginaries in his electromagnetic field equations which relatively quickly led to acceptance and widespread use thereof.

Where are the applications of binarics and (more importantly) binaric and logical imaginaries likewise?

## Monday, February 4, 2013

### Between Unity and LoF

The simplest of all cases would be undifferentiable or at least undifferentiated unity.

Unity was primitively recognized and denominated as "abyss", "void", "nameless" and "chaos".

And the simplest of all analyses would be such what might be called "unarics", limited to such recognition and denomination of unity, with no calculation possible.

The simplest of all cases which permit calculation are those analyzable in terms of two indexes, which analyses, calculuses, arithmetics and algebras are called binarics here, and of which George Spencer Brown's Laws of Form (LoF) of course presents a most elegant and thorough description and development.

Note, however that LoF's law of calling

[ ] = [ ] [ ]

can be, but law of crossing

[ [ ] ] =

cannot be, represented on a line, by segments of different colors.

This proves that there exist still simpler cases and corresponding binarics than that presented in LoF, binarics in which the law of calling but not the law of crossing is valid, but which cases and binarics like that of unity and "unarics" are incalculable.

And it also suggests some correspondence of form and representation on the one hand and dimensionality on the other.

But note too that calling involving more than two indexes and therefore similarly simplified what are called here enarics of number of indexes N > 2 can be represented on a line, by use of line-segments of more than two colors, which proves that there exist cases and such enarics in which the law of calling but not that of crossing is valid.

And could there not equally be cases and enarics where the law of crossing but not calling is valid?

Keywords: abyss, analysis, binarics, calculation, calling, chaos, crossing, dimension, enarics, George Spencer Brown, Laws of Form, LoF, nameless, unity, void

## Saturday, June 30, 2012

### A Possibly New Primary Arithmetic Theorem:"Volution"

The primary arithmetic theorem "Variance" (T9) of Laws of Form (hereafter "LoF") can be written in square-bracket notation as

[[pr][qr]] = [[p][q]]r

(square-bracket is preferable to paren notation in text, since it doesn't conflict with the literary use of parentheses).

(The theorem interpreted as logic—see Appendix 2 of LoF—might read " '( p OR r ) AND ( q OR r )' is equivalent to '( p AND q ) OR r'.)

The theorem itself is proven in LoF by application of Theorem 16, "The bridge" (see the final remark below), showing it holds in either case or value of r, reducing to

[] = []

when r is dominant ( r = [] ) and

[[p][q]] = [[p][q]]

when r is recessive ( r = ).

Let p be dominant instead, and the theorem reduces to

qr = qr

while letting q be dominant yields

pr = pr

But let p be recessive, and the theorem reduces to

[[r][qr]] = r

while letting q be recessive yields

[[pr][r]] = r

The left-hand expressions of these equations are of course equiformal, since

[[r][qr]] = [[qr][r]]

and the equations therefore represent a single form of equation.

This equation can be rewritten as

[[a][ab]] = a

(since ab = ba).

It holds for all cases; dominant a yields

[] = []

while recessive a yields

=

and both cases of b yield

a = a

This equation is not presented in LoF as a primary arithmetic theorem, nor does it seem to be a consequence of the algebra.

This suggests that it is a new primary arithmetic theorem and might, therefore, be a new, third initial of the algebra as well, opening a whole new world of consequences.

(The theorem interpreted as logic might read " 'a AND ( a OR b )' is equivalent to 'a' ".)

Should the theorem prove indeed to be new, it is suggested that it be called "Volution", its forward step

[[a][ab]] -> a

"devolution", and its reverse

[[a][ab]] <- a

"evolution": This nomenclature is itself suggested by the reflection (LoF primary algebraic consequence C1) and internal iteration (C5) implicit as first steps in the latter, which recall the genetic duplication so critical to biological evolution, allowing mutation of a duplicate gene without losing the function of the protein encoded by the original.

The theorem might be written

If successive spaces sn, sn+1, sn+2 are arranged so that sn, sn+1 are distinguished by one cross, and sn+1, sn+2 are distinguished by two crosses, sn+2 being thus in two divisions, both divisions sharing an expression, one containing an unshared expression, then the value of the resultant expression in sn is the shared expression.

Note that

[[ab][ac]] = a

does not hold for recessive a or dominant b or c.

Finally, the above arose from a continuing skeptical consideration of LoF Theorem 16, "The bridge".

Keywords: binarics, binaries, enarics, enaries, George Spencer Brown, Laws of Form, LoF

## Tuesday, September 13, 2011

### First Forms: The Enary Plenary Parenthetics

In 1969, George Spencer Brown published Laws of Form (abbreviated among the cognoscenti as “LoF”), the classic and exhaustive study of the simplest possible analysis, involving two indexes or indices and transition between those indices, providing an elegant and powerful calculus for such analysis; extending it to the corresponding binary arithmetic and algebra; treating questions both fundamental and advanced about such analysis, calculus, arithmetic and algebra; and applying that algebra in Appendix 2 to the binary resolution or analysis of propositional logical arguments and to set analysis.

The book has gone through many editions since, and deservedly so.

Since the publication of LoF, there has been no comparable study of the next least complex or most simple analysis, three-index or ternary analysis, still less four-index or quaternary, five- or quinary, and least of all the generalized N-ary or enary (nor should one discount the validity and eventual elucidation of the transfinary).

The companion volume to this one, Enaries (now, July ’11, in progress), begins the study of enarics or enary analysis, by examining the enary plenaries, the fundamental forms that become available as order of analysis increases, from unary to binary, binary to ternary, ternary to quaternary, and so on, up to the septenary (N = 7), each new plenary form using each and every index available at its order once and only once.

The companion data CD to this volume and that, The Enaries CD (also available as a download) (also in progress), contains various digital forms of both volumes, the separate figures, graphics and tables, extended graphics and tables, and the software used to generate all of the above (generally more or less simple original programs written in the very simple, ancient, beautiful and powerful C programming language).

This volume simply presents the enary plenaries, as parenthetics, from the unary (N = 1) to the terdecenaries (N = 13), simply enumerated and tabulated, for their beauty alone, certainly, but also for examination of their fundamental characteristics and properties and consequent taxonomizings.

Note that each abstract plenary form here represents N! equiforms; for example, the ternary plenaries “(())” and “()()” each represents 3! = 3*2*1 = 6 possible forms, eg "A(B(C))", "A(C(B))", "B(A(C))", "B(C(A))", "C(A(B))" and "C(B(A))" in the first case, and "A(B)(C)", "A(C)(B)", "B(A)(C)", "B(C)(A)", "C(A)(B)" and "C(B)(A)" in the second.

The Enary Plenary Parenthetics

(As usual, this research could use some financial support.)

[20110927 Jim Snyder-Grant of the Yahoo group lawsofform has found several missing enary plenaries in the nonaries, eg "(()()())((()()))", so that means the posted list is incomplete.

[The list will be expanded as soon as possible.]

keywords: enarics, enaries, enary analysis, foundations of analysis, George Spencer Brown, laws of form, LoF, mathematics, philosophy, science