Laws of Form

The phrase Laws of Form refers to either of two things:

• A book, hereinafter abbreviated LoF, by the eclectic free-lance British intellectual G. Spencer-Brown.
• The mathematico-philosophical system set out in LoF, especially the "primary algebra" ('see below; hereinafter abbreviated pa), a provocative and economical notation for the two-element Boolean algebra (hereinafter abbreviated as 2).

The book

The first edition of LoF was published in 1969; the most recent edition in 1994. Its mathematics fills only about 55pp and is not difficult. But LoF's mystical and declamatory prose style, and its love of paradox, make it a challenging read for mathematicians and non-mathematicians alike. (In this and other respects, Spencer-Brown was much influenced by Wittgenstein and R. D. Laing. At the same time, Spencer-Brown also intended LoF to reprise a number of themes from Bertrand Russell's work.

Ostensibly a work of formal mathematics and philosophy, LoF became something of a holistic classic, praised in the Whole Earth Catalog. Those who agree point to the Laws of Form as embodying an enigmatic "mathematics of consciousness," its algebraic symbolism capturing an (perhaps even the) implicit root of cognition: the ability to distinguish. LoF argues that the pa reveals striking connections among logic, Boolean algebra and arithmetic, and the philosophy of language and mind.

Some (e.g., Banaschewski 1977 in the Notre Dame Journal of Formal Logic) argue that the pa is 'merely' Boolean algebra, but the implication that Boolean algebra is mathematically trivial is unwarranted. Proponents counter that the beauty of the pa stems from its having the expressive power of 2 despite its minimalist syntax. The pa simplifies truth functional and syllogistic logic as well as 2, and highlights how syntactically distinct statements in logic and 2 can have identical semantics. [1] argues that the real mathematical value of the pa is its dramatic simplification of Boolean proofs and calculations. Moreover, the syntax of the pa' can be extended, giving rise to boundary mathematics (cf. Related Work below).

LoF claims that certain well-known mathematical conjectures of very long standing, such as the Four Color Theorem, Fermat's Last Theorem, and the Goldbach conjecture, are provable using extensions of the pa. Spencer-Brown eventually published a claimed proof of Four Color; for a sympathetic evaluation, see [2]. Nevertheless, that claimed proof met with skepticism and Spencer-Brown's mathematical reputation, as well as that of LoF, went into decline. N.B. The Four Color Theorem and Fermat's Last Theorem were proved in 1976 and 1995, respectively, using methods owing nothing to LoF.

The Form

The symbol:

,

also called the Mark or Cross, is the essence of the Laws of Form.

In Spencer-Brown's initimable and enigmatic fashion, the Mark symbolizes the root of cognition, i.e., the dualistic Mark indicates the capability of differentiating a "this" from a "that." The brilliance of LoF largely stems from its having created an algebraic symbolism for pure, unjudged content-in-context which can logically be interpreted as a "truth," perhaps (for Spencer-Brown at least) the most fundamental one.

In LoF, a Cross denotes the drawing of a "distinction", and can be thought of as signifying the following, all at once:

1. The act of drawing a boundary around something, thus separating it from everything else;
2. That which becomes distinct from everything by drawing the boundary;
3. Crossing from one side of the boundary to the other.

All three ways imply an action on the part of the cognitive entity (e.g., person) making the distinction. The final sentence of LoF, excluding back matter, is: "...the first distinction, the Mark and the observer are not only interchangeable, but, in the form, identical."

The counterpoint to the Marked state is the Unmarked state, which is simply nothing, the void, represented by a blank space. It is simply the absence of a Cross. No distinction has been made and nothing has been crossed. The unmarked state nevertheless has a denotatum: nothing, undistinguished...implying neither content nor context.

“The first command,

• Draw a distinction

... can well be expressed in such ways as

• Let there be a distinction,
• Find a distinction,
• See a distinction,
• Describe a distinction,
• Define a distinction,

Or

• Let a distinction be drawn.” (LoF, Notes to chapter 2).

The Cross can be seen as denoting the distinction between two states, one "considered as a symbol" and another not so considered. From this fact arises a curious resonance with some theories of consciousness and language. Paradoxically, the form is at once Observer and Observed, and is also the creative act of making an observation. Charles Peirce came to a related insight in the 1890s; see Related Work below. Francisco Varela, a co-founder of the Integral Institute, and Humberto Maturana, citing LoF, also identify "distinction" as the elementary cognitive act.

Resonances in religion, philosophy, and science

The mathematical and logical content of LoF is wholly consistent with a secular point of view. Nevertheless, LoF's "first distinction", and the Notes to its chapter 12, bring to mind the following landmarks in religious, philosophical, and scientific beliefs (in rough order of appearance):

• Vedic, Hindu and Buddhist: Related ideas can be noted in the ancient Vedic Upanishads, which form the monastic foundations of Hinduism and later Buddhism. As stated in the Aitareya Upanishad ("The Microcosm of Man"), the Supreme Atman manifests itself as the objective Universe from one side, and as the subjective individual from the other side. In this process, things which are effects of God's creation become causes of our perceptions, by a reversal of the process. In the Svetasvatara Upanishad, the core concept of Vedicism and Monism is "Thou art That."
• Taoism, (Chinese Traditional Religion): "...The Tao that can be told is not the eternal Tao; The name that can be named is not the eternal name. The nameless is the beginning of heaven and earth..." (Tao Te Ching).
• Zoroastrianism: "This I ask Thee, tell me truly, Ahura. What artist made light and darkness?" (Gathas 44.5)
• Judaism (from the Tanakh, called Old Testament by Christians): "In the beginning when God created the heavens and the earth, the earth was a formless void... Then God said, 'Let there be light'; and there was light. ...God separated the light from the darkness. God called the light Day, and the darkness he called Night.

"...And God said, 'Let there be a dome in the midst of the waters, and let it separate the waters from the waters.' So God made the dome and separated the waters that were under the dome from the waters that were above the dome.

"...And God said, 'Let the waters under the sky be gathered together into one place, and let the dry land appear.' ...God called the dry land Earth, and the waters that were gathered together he called Seas.

"...And God said, 'Let there be lights in the dome of the sky to separate the day from the night...' God made the two great lights... to separate the light from the darkness." (Genesis 1:1-18; Revised Standard Version, emphasis added).

"I am that I am." (Exodus 3:14)

• Confucianism: Confucius claimed that he sought "a unity all pervading" (Analects XV.3) and that there was "one single thread binding my way together." (Ana. IV.15). The Analects also contain the following remarkable passage on how the social, moral, and aesthetic orders are grounded in right language, grounded in turn in the ability to "rectify names," i.e., to make correct distinctions: "Zilu said, ‘What would be master's priority?’ The master replied, ‘Rectifying names!’ ...If names are not rectified then language will not flow. If language does not flow, then affairs cannot be completed. If affairs are not completed, ritual and music will not flourish. If ritual and music do not flourish, punishments and penalties will miss their mark. When punishments and penalties miss their mark, people lack the wherewithal to control hand and foot." (Ana. XIII.3)
• Plato: Logos (see below) is also a fundamental technical term in the Platonic worldview.
• Christianity: "In the Beginning was the Word, and the Word was with God, and the Word was God." (John 1:1). "Word" translates logos in the koine original. Also the words of Jesus: "...that I am..." (John 8:24).
• Islamic philosophy distinguishes essence (Dhat) from attribute (Sifat), which are neither identical nor separate.
• Josiah Royce: "When logically analyzed, order turns out to be… inconceivable and incomprehensible to us unless we had the idea… expressed by the term ‘negation’. Thus it is that negation, which is always also something intensely positive, not only aids us in giving order to life, and in finding order in the world, but logically determines the very essence of order." ("Order" in Hasting, J., ed. (1917) Encyclopedia of Religion and Ethics. Scribner’s: 540).
• John Archibald Wheeler: "The boundary of a boundary is zero. This central principle of algebraic topology, identity, triviality, tautology though it is, is also the unifying theme of Maxwell’s electrodynamics, general relativity, and almost every version of modern field theory. That one can get so much out of so little, almost everything from almost nothing, inspires hope that we will someday complete the mathematization of physics and derive everything from nothing, all law from no law." ("It from Bit" in Wheeler, J. A. (1996) At Home in the Universe. American Institute of Physics Press: 302).

Revisiting the Biblical analogy, "Let there be light" is the same as

• "...and there was light" - the light itself;
• "...morning and evening" - the boundaries of the light;
• "...called the light Day" - the manifestation of the light.

A Cross denotes a distinction made, and the absence of a Cross means that no distinction has been made. In the Biblical example, light is distinct from the void – the absence of light. The Cross and the Void are the two primitive values of the Laws of Form.

The primary arithmetic and its axioms

Begin with the void, the only "atomic" expression. Then posit two inductive rules:

• Given any expression, a Cross can be written over it;
• Any two expressions can be concatenated.

Thus the syntax of the primary arithmetic, a Dyck language of order 1 with a null alphabet, and the simplest instance of a context-free language in the Chomsky hierarchy. LoF often uses the phrase calculus of indications in place of "primary arithmetic".

The primary arithmetic and algebra begin with a definition, Distinction is perfect continence, and the axioms A1 and A2.

A1. The law of Calling. To make a distinction twice has the same effect as making it once. For instance, if you say “Let there be light” and then you say “Let there be light” again, it is the same as saying it once. Crossing twice from the unmarked state cannot be distinguished from crossing once. Symbolically:

  = 

A2. The law of Crossing. Crossing from the unmarked state takes you to the marked state; crossing again from that marked state takes you back to the unmarked state. To recross is not to cross. Symbolically:

 =

A2 captures the essence of Wheeler's sentence "The boundary of a boundary is zero." quoted above.

The repeated application of A1 and A2 can reduce any expression consisting solely of Crosses to the expression's simplification, either the marked or the unmarked state. The fundamental metatheorem of the primary arithmetic (T3-4 in LoF) states that:

• An expression has a unique simplification;
• The repeated application of A1 and A2 to either the marked or the unmarked state cannot yield an expression whose simplification differs from the initial state.

A1 and A2 have loose analogs in the properties of series and parallel electrical circuits, and in other ways of diagramming processes, including flowcharting. A1 corresponds to a parallel connection, and A2 to a series connection, with the understanding that making a distinction corresponds to changing the state of connectedness between two points in a circuit, and not simply to adding wiring.

The notion of 'canon'

An unconventional aspect of LoF is its enigmatic notion of a canon, exemplified by the following two excerpts from the Notes to chapter 2 of LoF:

"The more important structures of command are sometimes called canons. They are the ways in which the guiding injunctions appear to group themselves in constellations, and are thus by no means independent of each other. A canon bears the distinction of being outside (i.e., describing) the system under construction, but a command to construct (e.g., 'draw a distinction'), even though it may be of central importance, is not a canon. A canon is an order, or set of orders, to permit or allow, but not to construct or create."

"...the primary form of mathematical communication is not description but injunction… Music is a similar art form, the composer does not even attempt to describe the set of sounds he has in mind, much less the set of feelings occasioned through them, but writes down a set of commands which, if they are obeyed by the performer, can result in a reproduction, to the listener, of the composer’s original experience."

Prevailing practice in logic and mathematics is innocent of the notion of canon.

The primary algebra

Given any valid primary arithmetic expression, insert into one or more locations any number of Latin letters, with or without numerical subscripts; the result is a pa formula. Variable is the conventional name for a letter of this sort. A pa variable signifies that location where one can write either primitive value. Multiple instances of the same variable stand for multiple locations where the same primitive value must be written.

The sign '=' denotes that what appears to the left and right of = are logically equivalent, i.e., have the same simplification. An expression of the form "A=B" is an equation, meaning that A and B are logically equivalent. Logical equivalence is an equivalence relation governed by the rules R1 and R2. Let C and D be formulae containing at least one instance of the subformula A:

• R1, Substitution of equals. Replace one or more instances of A in C by B, resulting in E. If A=B, then C=E.
• R2, Uniform replacement. Replace all instances of A in C and D with B. A becomes E and B becomes F. If C=D, then E=F. Note that A=B is not required.

Uniform replacement is invoked very frequently in pa demonstrations (see below), almost always silently. These rules should also be familiar to any logician, and are routinely but unwittingly invoked in much of mathematics.

The pa consists of equations, i.e., pairs of formulae linked by an equivalence relation denoted by an infix '='. R1 and R2 enable transforming one equation into another. Hence the pa is an equational formal system, like Boolean and most other algebras. Mathematical logic consists of tautological formulae, signalled by a prefixed turnstile. To variants of R1 and R2, conventional logic adds the rule modus ponens; thus conventional logic is ponential. The equational-ponential dichotomy summarizes much of what sets mathematical logic off from the rest of mathematics. Incidentally, to indicate that the pa formula A is a tautology, write "A =".

The pa contains three kinds of proved assertions:

• Initial is a pa equation invoked in demonstrations and proofs like a mathematical axiom. An initial is verifiable by a decision procedure and hence is not an axiom.
• Consequence is a pa equation verified by a demonstration. A demonstration consists of a sequence of steps, each step justified by an initial or a previously demonstrated consequence.
• Theorem is a statement in the metalanguage verified by a prooof, i.e., an argument accepted by trained mathematicians and logicians.

The distinction between consequence and theorem holds for all mathematics and logic, but is usually not as explicit as it should be. A demonstration or decision procedure can be carried out and verified on a computer. The proof of a theorem cannot be.

Let A and B be pa formulae. A demonstration of A=B may proceed in either of two ways:

• Modify A in steps until B is obtained, or vice versa;
• Simplify ((A)(B))(AB) to . This is known as a "calculation".

Once A=B has been demonstrated, A=B is a consequence and can be invoked to justify steps in subsequent demonstrations.

LoF lays down the initials:

• I1: ((A)A)=<void>
• I2: ((A)(B))C = ((AC)(BC)).

I2 is the familiar distributive law of logic and and Boolean algebra. A more economical set of initials, also one friendlier to calculations, is;

• (A)A=, the "complement" of I1, and
• A(AB)=A(B), i.e., mimesis (see Sheffer stroke).

LoF asserts that the commutativity and associativity of concatenation can be taken as true by default and hence need not be explicitly assumed. (Peirce made a similar assumption in his graphical logic.) Commutativity and associativity are also both implied by ABC = BCA, taken as an initial.

pa demonstrations are usually quite easy, usually requiring no more than the initials, A2, and the consequences ((A))=A, AA=A, and (((A)B)C) = (AC)((B)C). This last enables an algorithm, sketched in LoFs proof of T14, that transforms an arbitrary pa formula to an equivalent one whose depth does not exceed two. The result is a normal form, the pa analog of the conjunctive normal form. LoF (T14-15) proves that every pa formula has a normal form.

LoF includes elegant new proofs of the following standard metatheory:

• Completeness: all pa consequences are demonstrable from the initials (T17).
• Independence: I1 cannot be demonstrated from I2 and vice versa (T18).

That sentential logic is complete is part of every first university course in mathematical logic. But the completeness of 2 (and hence of all Boolean algebras with finite carriers) is seldom mentioned.

Applying the form to Boolean algebra and logic

The marked and unmarked state can be read as the Boolean values 1 and 0, or as True and False. The first reading transforms the pa into a notation for 2; the second into a notation for sentential logic. Since the Cross also denotes crossing the boundary of a distinction, it may be read as Not. This may seem odd at first blush; however True is equivalent to Not False and both True and Not False are represented the same way — with a Cross.

 =   False

 =  True  =  not False

 =  Not True  =  False

Variables may represent any expression whose value is undetermined. Thus:

interprets Not A

interprets A Or B

interprets Not A Or B

or If A Then B.

interprets Not (Not A or Not B)

or Not (If A Then Not B)

or A And B.

Let AB in the pa translate either A+B, the meet of A and B, or the join thereof. Let denote the Boolean complement of A. 2 now emerges as a model of the primary algebra. The primary arithmetic suggests that 2 can be axiomatized arithmetically by 1+1=1+0=0+1=1=~0, and 0+0=0=~1. Then define AB as ~(~A+~B).

Any expression in sentential (truth functional) logic has a pa translation. Given some assignment of every variable to the Marked or Unmarked states, reduces this pa translation to a primary arithmetic expression, which can be simplified. Repeating this exercise for all possible assignments of the two primitive values to each variable, reveals the truth value of the original expression, in particular whether it is tautological or satisfiable. This is an example of a decision procedure. A decision procedure akin to Quine's "truth value analysis" can be extracted from the Laws of Form, one less combinatorially tedious than conventional truth tables.

The interpretations above assume that the unmmarked state is read as False. This assumption is wholly arbitrary; the unmarked state can equally well denote True. All that is required is that the interpretation of concatenation change from OR to AND. IF A THEN B now translates as (A(B)) instead of . More generally, any statement in sentential logic or 2 translates into the pa in one of two ways. Duality is the name of this fact. "Self-duality" refers to the fact that any pa formula can be read in either of two ways, each the dual of the other.

The true nature of the distinction between the pa on the one hand, and 2 and sentential logic on the other, now emerges. In the latter formalisms, complementation/negation with an empty scope is not defined. Meanwhile a Cross, interpretable as complementation/negation, with nothing under itself denotes the Marked state, a primitive value. This is the sense in which the pa reveals that the heretofore distinct mathematical concepts of operator and operand are in fact merely different facets of a single fundamental action, the making of a distinction.

Appendix 2 of LoF shows how to translate traditional syllogisms into the pa. A valid syllogism is simply one whose pa translation simplifies to an empty Cross. This suggests that the monadic predicate calculus is yet another model of the pa, and that the pa has affinities to the Boolean term schemata of Quine's Methods of Logic. Extending the pa so that one of its models would be standard first order logic has yet to be done, but Peirce's beta existential graphs suggest that the extension should be straightforward.

Related work

LoF would have benefited from knowing Charles Peirce's semiotics and existential graphs. Peirce (1839-1914) anticipated the pa in three veins of work:

• Two papers he wrote in 1886 employed a notation almost identical to that of LoF. These papers were published only in 1993, in vol. 5 of the Writings of C. S. Peirce (Indiana Uni. Press).
• A closely related notation appears in an encyclopedia article he published in 1902, reprinted in vol. 4 of his Collected Papers (Harvard Uni. Press).
• His alpha existential graphs are isomorphic to the pa. This work was virtually unknown at the time when (1960s) and in the place where (UK) LoF was written. (Ironically, LoF cites vol. 4 of Peirce's Collected Papers. in which the existential graphs are described in detail.)

Another notation similar to that of LoF is that of a 1917 article by Jean Nicod, a disciple of Bertrand Russell's. (This paragraph owes much to [3].)

The pa and Peirce's graphical logic are instances of boundary mathematics, i.e., mathematics done with boundary notation, one restricted to ("variables" excepted) enclosing devices, mainly brackets. In particular, boundary notation is free of infix, prefix, or postfix operator symbols. The curly braces of standard set theory can be seen as an instance of a boundary notation.

The work of Peirce and Nicod is innocent of metatheory, as they wrote before Emil Post proved in 1920 that sentential logic is complete (cited in LoF), and before Hilbert and Lukasiewicz showed how to prove axiom independence using models.

It is to be regretted that second generation cognitive science emerged only in the 1970s, after LoF was written. On cognitive science and its relevance to Boolean algebra, logic, and set theory, see:

• Lakoff, George (1987) Women, Fire, and Dangerous Things. University of Chicago Press.
• Lakoff, George, and Rafael E. Núñez (2001) Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Basic Books.

Lakoff does not cite LoF.

The primary arithmetic and algebra is but one of several minimalist approaches to logic and the foundations of mathematics, or parts thereof. Other, and more powerful, minimalist approaches include:

• The lambda calculus;
• Combinatory logic with S and K as the sole primitive combinators;
• Mathematical logic done with merely the Sheffer stroke (A|B translates as either (AB) or (A)(B) ), and one type of primitive formula--set membership--containing tacitly quantified variables.

External links

• lawsofform.org. The definitive Laws of Form web site.
• Spencer-Brown's talks at Esalen 1973 Note the introduction to self-referential forms in the section titled "Degree of Equations and the Theory of Types"
• [4]. This long and difficult academic paper, the source for much of this entry, is not a good introduction to the more speculative aspects of LoF. There is also notational difference; what LoF places under a Cross is here enclosed in parentheses.