mar appendix laws of form existential graphs

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Appendix to The Laws of Form Controversy: Laws of Form via Existential Graphs, and Beyond

Randolph Dible

Professor Mar, PHI 490

“Here you are at a certain crossing and a certain imaginary intersection of the void.” -

Edmund Jabes, Book of the Absent; part 3, section 6.

On its own and through its influence on Schroder and Peano, the work of the great

logician Charles Sander Peirce (1839-1914) helped shape modern first-order logic. But in his

later years his focus was on his “Existential Graphs”, which he dubbed his chef d’oeuvre and

considered superior to his symbolic system. The Existential Graphs were not symbolic like the

standard logical notation he had so influenced, but rather diagrammatic, graphic and iconic,

meaning it bears a likeness to what it signifies, rather like the familiar Venn diagram, the reader

may find illustrative, and those of Euler.

“…in Euler diagrams, we can represent relations between sets iconically: the syntactic

(topological) properties of enclosure, exclusion and intersection of circles resemble the

set-theoretic properties of subset, disjointness, and intersection.” (John Howse, “Book

Review, The Iconic Logic of Peirce’s Graphs”)

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Peirce’s Existential Graphs, like the rest of his architectonic, is tripartite. The three

parts are called the Alpha Graphs, Beta Graphs, and Gamma Graphs, and these respectively

represent propositional, predicate, and modal logic (Sun-Jo Shin). Traditional symbolic logic

could be understood as expressing speech (it describes), in comparison to iconic logic which

could be said to express vision (it shows). And it is Peirce’s Alpha Graphs we will focus on, in

its likeness to the “Primary Algebra” of George Spencer-Brown’s Laws of Form, in its

interpretation of propositional logic. The Primary Algebra is a development of the Primary

Arithmetic, by the addition of variables to represent propositions. The Primary Arithmetic has

two initials:

Initial 1. Number – () () = ()

Initial 2. Order – (()) =

These initial equations begin Chapter 4 “The Primary Arithmetic”, but are the axioms

taken from Chapter 1 “The Form.” These axioms are, respectively, the “law of calling” and the

“law of crossing.” The former means that for any name, to recall is to call (“if a name is called

and then is called again, the value indicated by the two calls taken together is the value indicated

by one of them” and “the crossing of the boundary can be identified with the value of the

content” – Spencer-Brown, Chapter 1, Laws of Form), or as Spencer-Brown puts it, “the value of

a call made again is the value of the call” (ibid.) The latter axiom means that for any boundary,

to recross is not to cross (“if it is intended to cross a boundary and then it is intended to cross it

again, the value indicated by the two intentions taken together is the value indicated by none of

them” – Ibid.). In Chapter 2 “Forms Taken Out of the Form”, we learn that the former equation

is called “the form of condensation” and that latter is called “the form of cancellation.” Chapter

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3 “The Concept of Calculation” ends in a section called “The calculus of indications” where

these two axioms of the Laws of Form are first called the initials of a calculus, and the forms

generated from direct consequences of these initials he calls the Primary Arithmetic, also known

as the Calculus of Indications. And going back to Chapter 4, “The Primary Arithmetic,” we find

that these two initials of the Primary Arithmetic may be read either way: forwards, the former

reads as condensation of the form, and the latter as cancellation of the form; backwards, the

forms reads as confirmation, and the latter as compensation. Take a moment to consider how

these readings apply to the form of the initials by referring to the parenthetical forms of

enclosure I have given to represent the two axioms of the Laws of Form. Chapter 4 “The

Primary Arithmetic” explores primitive general statements called theorems that are formal

consequences of these initials. Eight theorems are developed therein; the first three are called

theorems of representation, the next three are called theorems of procedure, and the last two are a

gateway into a new calculus and are called theorems of connection. This new calculus is the

Primary Algebra, so similar to Peirce’s Existential Graphs.

An Alpha Graph consists of two kinds of primitive vocabulary; sentential or

propositional symbols and the cut , a simple closed plane curve. In Spencer-Brown’s Laws of

Form, the Primary Algebra consists of the Primary Arithmetic (which, itself, consists merely of

cuts) and variables, symbols of sentential or propositional symbols. In Laws of Form, a cut or

cross, a mark indicating the crossing of distinction represents negation of its contents, and

juxtaposition represents disjunction or alternation. In the Alpha Graphs, juxtaposition represents

conjunction, and the cut represents negation.

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The cut in both systems represents negation of elements and can be construed as any

closed curve, a circle is used in both systems, but in Spencer-Brown’s system, the right-angle

bracket (inverted capital L) is the standard symbol, called the Mark, which indicates the crossing

of the first distinction. The Primary Algebra consists of the introduction of variables

(representing variable expressions, which are interpreted for propositional or sentential logic)

into the Primary Arithmetic which itself consists of constants, and is a non-numerical arithmetic. These constants are prototypes or archetypal forms of indication of the primary constant, the first

distinction. The first distinction is drawn in an otherwise unmarked state or unmarked space, or

Void, and is the basis of Laws of Form. In the Existential Graphs, the primary state is the Page

of Assertion, like the unmarked state, which is the blank page upon which are drawn the cuts and

variables. In Existential Graphs, the Page of Assertion has the truth value of True. In Laws of

Form, the two equivalence classes are Mark and Void, and either may be interpreted as true or

false, so long as the interpretation is consistent throughout the form. An equivalence class

consists of all the forms that reduce to the same value (Mark or Void).

“Despite the great interest shown in diagrams, nevertheless a negative

attitude toward diagrams has been prevalent among logicians and mathematicians. They consider any nonlinguistic form of representation to be a heuristic tool only. No

diagram or collection of diagrams is considered a valid proof at all. It is more

interesting to note that nobody has shown any legitimate justification for this attitude

toward diagrams. Let me call this traditional attitude, that is, that diagrams can be only

heuristic tools but not valid proofs, the general prejudice against diagrams.” - Sun-Joo

Shin, Introduction, The Logical Status of Diagrams

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This prejudice is behind the prejudice against Laws of Form, besides the general

discomfort of investigating unfamiliar territory. In Shin’s book, by demonstrating the soundness

and completeness of his modified Venn diagrams, he shows that the prejudice against diagrams

and visualization is unjustified. In my paper, “The Laws of Form Controversy,” I come to the

conclusion that most problems the authors of the key critical paper, Flaws of Form (Frank and

Cull) come to are consequences of inappropriately imposing the criteria of symbolic logic on the

iconic logic of the Primary Algebra.

Spencer-Brown’s “exploration became a quest that arrived at the vision that the

various forms encountered in our experience and existence arise in stages out of formlessness by

drawing a distinction and then arranging tokens of that distinction.” (Jack Engstrom, “C. S.

Peirce’s Precursors to Laws of Form,” pp. 1) In his article, Engstrom shows that Laws of Form

is significant to a metaphysical and mystical understanding of knowledge because “the primary

ground of Laws of Form is unmarked space [or undivided wholeness], and that forms can not

only be created (constructed) out of this unmarked space [or undivided wholeness], but may also

be voided (deconstructed) back into this unmarked space [or undivided wholeness].” (Ibid., pp.

3) This is called the “voidability of relations.” Even the basis of forms, the drawing of a

distinction, is merely the crossing of a supposed distinction, and the first crossing is an event

rather than an entity, making this metaphysical interpretation a process-metaphysics with an

ontology of becoming; it is the primary act .

Spencer-Brown’s system is also tripartite. The mark of distinction (the inverted capital

‘L’, or right-angle bracket) consists of three points which remind us that the mark itself is the

third element between the two states distinguished in the primitive act of drawing a distinction or

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crossing a boundary. The marked state is the content or inside of the mark, and unmarked state is

the outside of the mark. By a series of four experiments (thought experiments) of alternatively

marking the inside, outside, both and neither sides of the two primitive equations in the last

chapter of the book, certain fascinating conclusions are reached, but for reasons of brevity, the

triplicity is shown here in the last conclusion of the main corpus of the book: “We see now that

the first distinction, the mark, and the observer are not only interchangeable, but, in the form,

identical” (Ibid.).

The Mark, in its thirdness, is the operative constant (in the calculus there are two

constants, but only one operative constant; the Mark) and what it indicates is the first distinction. The ontological status of the first distinction not explicitly mentioned, that is, there is nowhere in

Laws of Form a discussion of ontology by that label. But ontology is always present implicitly,

and in Laws of Form the first distinction is the only constant, from which are built the existential

precursors or prototypes of the forms of articulation of the universe. These archetypes are

mentioned in the author’s explanation of Laws of Form in another of his books, “Only Two Can

Play This Game” (1972). There, he describes Laws of Form as an account of the emergence of

physical archetypes by starting with nothing and making one mark and tracing all the eternal

forms. “From these we obtain two axioms, and proceed from here to develop theorems.” The

consequences of just having drawn one mark in an otherwise unmarked space, he there claims,

are the principles underlying Boolean algebra. He states that he take “Only Two” to be a

complimentary kind of book, a companion to the rigorous mathematical treatise Laws of Form,

and indeed it does enlighten one as to the further interpretations of the Laws of Form in the

mystical form. In particular, the triple-identity (akin to Peirce’s metaphysical doctrine of

Thirdness) is likened to the Trinity of not only Christian religion, but also of Vedanta, and

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explains it also in terms of Space, Time and the Void. This occurs in the Notes, in the midst of

his discussion of the five orders or levels of eternity. The literature of this genre is fecund with

metaphysical significance, applicable to mathematics and sciences that have yet to be developed,

and one thing is certain, that time will tell.

Works Cited

Engstrom, Jack, “Precursors to Laws of Form in C. S. Peirce’s Collected Papers”

Shin, Sun-Joo, The Iconic Logic of Peirce’s Graphs

Spencer-Brown, George, Laws of Form

Spencer-Brown, George, Only Two Can Play This Game

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