Laws of Form

[Laws of Form (hereinafter LoF) is a book by G. Spencer-Brown (GSB), published in 1969, that asserts to begin at the beginning, unlike the traditional expositions of mathematics and philosophy start in the middle. For this reason this article on LoF stated that it straddles the boundary between mathematics and philosophy. To say that "LoF describes three distinct logical systems:" is to minimize the import of actually starting at the beginning. However, since starting in the middle is the normal method of modern inquiry, It is a useful approach to leave most of the original article in place and point out what it missed by the previous authors of this article.[ These "uncited comments" are original to me and are based on actually studying the original work and not relying on the cited comments of others. LAA]

• The primary arithmetic (described in Chapter 4 of LoF), whose models include Boolean arithmetic;
• The primary algebra (Chapter 6 of LoF), whose models include the two-element Boolean algebra (hereinafter abbreviated 2), Boolean logic, and the classical propositional calculus;
• Equations of the second degree (Chapter 11), whose interpretations include finite automata and Alonzo Church's Restricted Recursive Arithmetic (RRA).

"Boundary algebra" is a Meguire (2011) term for the union of the primary algebra and the primary arithmetic. Laws of Form sometimes loosely refers to the "primary algebra" as well as to LoF.

[As GSB writes in A not on the mathematical approach: "The theme of this book is that a universe comes into being when a space is severed or taken apart. The skin of a living organism cuts off an outside from an inside. So does the circumference of a circle in a plane. By tracing the way we represent such a severance, we can begin to reconstruct, with an accuracy and coverage that appear almost uncanny, the basic forms underlying linguistic, mathematical, physical, and biological science, and can begin to see how the familiar laws of our own experience follow inexorably from the original act of severance." The philosophical implication of this is profound. If true, then this universe came into being when a space was severed or taken apart. The conclusion of the book is equally profound philosophically: "We see now that the first distinction, the mark, and the observer are not only interchangeable, but, in the form, identical." The concept of the Trinity, Three-in-One, emerges, as GSB, says, "of its own accord." ]

The book

The preface states that the work was first explored in 1959, and Spencer Brown cites Bertrand Russell as being supportive of his endeavour. He also thanks J. C. P. Miller of University College London for helping with the proof reading and offering other guidance. In 1963 Spencer Brown was invited by Harry Frost, staff lecturer in the physical sciences at the department of Extra-Mural Studies of the University of London, to deliver a course on the mathematics of logic.

[To say that Bertrand Russell was supportive of GSB's endeavor is to disingenuous in the extreme. In the Preface to the fifth English edition GSB states that LoF would not have been published at all without Russell's unqualified approval. LoF was the Book Russell wished he had written rather than Principia Mathematica which wrongly started with logic. Mathematics can't be based on logic because logic immediately starts using mathematical notations.]

LoF emerged from work in electronic engineering its author did around 1960. Key ideas of the LOF were first outlined in his 1961 manuscript Design with the Nor, which remained unpublished until 2021, and further refined during subsequent lectures on mathematical logic he gave under the auspices of the University of London's extension program. LoF has appeared in several editions. The second series of editions appeared in 1972 with the "Preface to the First American Edition", which emphasised the use of self-referential paradoxes, and the most recent being a 1997 German translation. LoF has never gone out of print.

[In the body of LoF, a paradoxical state is an oscillatory state: "We define the degree of an expression as the number of distinct stable states it can beshown to have. A paradoxical ( = 'oscillatory') state is defined to be 'stable' for this purpose." It arises from re-entrant algebraic equations: E2 f2 = ((f2)) and E3 f3= (f3), where parentheses represent GSB's Cross or Mark. Both of these resolve in the arithmetic to "identical" infinite expressions. While E2 has two stable solutions, marked and unmarked, E3, only has an "imaginary" solution of an oscillatory state which has been associated with i the square root of minus one, or the square root of negation. From this oscillatory state, time and frequency emerge, of their own accord.]

LoF's mystical and declamatory prose and its love of paradox make it a challenging read for all. Spencer-Brown was influenced by Wittgenstein and R. D. Laing. LoF also echoes a number of themes from the writings of Charles Sanders Peirce, Bertrand Russell, and Alfred North Whitehead.

[ While it is true that GSB was a poet, "Bertrand Russell loved my poems, and his favourite was my first, the one about the talking cows. He was terrified of my mathematics, but I was not to know this until he informed the world of it in his autobiography." His work does not "echo themes from Russell and Whitehead, it destroyed them. ]

The work has had curious effects on some classes of its readership; for example, on obscure grounds, it has been claimed that the entire book is written in an operational way, giving instructions to the reader instead of telling them what "is", and that in accordance with G. Spencer-Brown's interest in paradoxes, the only sentence that makes a statement that something is, is the statement which says no such statements are used in this book. Furthermore, the claim asserts that except for this one sentence the book can be seen as an example of E-Prime. What prompted such a claim, is obscure, either in terms of incentive, logical merit, or as a matter of fact, because the book routinely and naturally uses the verb to be throughout, and in all its grammatical forms, as may be seen both in the original and in quotes shown below.

Reception

Ostensibly a work of formal mathematics and philosophy, LoF became something of a cult classic: it was praised by Heinz von Foerster when he reviewed it for the Whole Earth Catalog. Those who agree point to LoF 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 primary algebra reveals striking connections among logic, Boolean algebra, and arithmetic, and the philosophy of language and mind.

Stafford Beer wrote in a review for Nature, "When one thinks of all that Russell went through sixty years ago, to write the Principia, and all we his readers underwent in wrestling with those three vast volumes, it is almost sad".

Banaschewski (1977) argues that the primary algebra is nothing but new notation for Boolean algebra. Indeed, the two-element Boolean algebra 2 can be seen as the intended interpretation of the primary algebra. Yet the notation of the primary algebra:

• Fully exploits the duality characterizing not just Boolean algebras but all lattices;
• Highlights how syntactically distinct statements in logic and 2 can have identical semantics;
• Dramatically simplifies Boolean algebra calculations, and proofs in sentential and syllogistic logic.

Moreover, the syntax of the primary algebra can be extended to formal systems other than 2 and sentential logic, resulting in boundary mathematics (see § Related work below).

LoF has influenced, among others, Heinz von Foerster, Louis Kauffman, Niklas Luhmann, Humberto Maturana, Francisco Varela and William Bricken. Some of these authors have modified the primary algebra in a variety of interesting ways.

LoF claimed 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 primary algebra. Spencer-Brown eventually circulated a purported proof of the four color theorem, but it met with skepticism.

The form (Chapter 1)

[Chapter 1 does not begin with "The symbol", but with the words: "We take as given the idea of distinction and the idea of indication, and that we cannot make an indication without drawing a distinction. We take, therefore, the form of distinction for the form." The concept of the observer "we" is implicit in these statements as is the concept that there can be at least two observers: one making the distinction and one observing the indication of the distinction although the maker of the distinction could be doodling for his own amusement with no intention to communicate the distinction to another observer. Intention is another key concept: "There can be no distinction without motive, and there can be no motive unless contents are seen to differ in value." Intention is indicated by indication: "Once a distinction is drawn, the spaces, states, or contents on each side of the boundary, being distinct, can be indicated." GSB indicate the concept of a name for a value, and that calling of the name can be identified with the value of the content.

Chapter 1 then introduces two Axioms, or fundamental rules for the act of calling "The value of a call made again is the value of the call." No matter how many times a name is called, it still represents the same value. GSB then connects the concept of crossing the boundary to the value of the call. This leads to the concept of the crossing and its axiom: "The value of a crossing made again is not the value of the crossing." This introduces the concept of direction in of the crossing. If the initial crossing is from outside to inside, then the recross is from inside to outside.]

The symbol:

Also called the "mark" or "cross", is the essential feature of the Laws of Form. In Spencer-Brown's inimitable and enigmatic fashion, the Mark symbolizes the root of cognition, i.e., the dualistic Mark indicates the capability of differentiating a "this" from "everything else but this".

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

• The act of drawing a boundary around something, thus separating it from everything else;
• That which becomes distinct from everything by drawing the boundary;
• 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. As LoF puts it:

"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 counterpoint to the Marked state is the Unmarked state, which is simply nothing, the void, or the un-expressable infinite represented by a blank space. It is simply the absence of a Cross. No distinction has been made and nothing has been crossed. The Marked state and the void are the two primitive values of the Laws of Form.

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. LoF (excluding back matter) closes with the words:

...the first distinction, the Mark and the observer are not only interchangeable, but, in the form, identical.

C. S. Peirce came to a related insight in the 1890s; see § Related work.

The primary arithmetic (Chapter 4)

[Likewise, while the Primary Arithmetic is described in Chapter 4, there are two important Chapters before it that introduce the terms and concepts that will be used in Chapter 4. Chapter 2 introduces the basic concepts and terms that will be used to perform calculations. Skipping over these misses the point of the work entirely. GWB define "construction" as "Draw a distinction." He defines "content" as: "Call it the first distinction Call the space in which it is drawn the space severed or cloven by the distinction. Call the parts of the space shaped by the severance or cleft the sides of the distinction or, alternatively, the spaces, states, or contents distinguished by the distinction. {Laws, 2011, page 3}"

The next concept "intent" is the critical difference between a mathematics based on "logic" and a mathematics based on Laws of Form. "Let any mark, token, or sign be taken in any way with or with regard to the distinction as a signal. Call the use of any signal its intent." Let the intent of a signal be limited to the use allowed to it. Call this the convention of intention. In general, what is not allowed is forbidden.{Laws, 2011, page 3}" This concept, the convention of intention, is what will allow for calculation in the form of discreet steps of substituting one arrangement of marks for another according to allowed rules. Calculation in Laws of Form is based on allowed transitions from one form of arrangement to another. There will be two kinds of steps: allowed, and forbidden.

This is our first glimmer into the fact there are distinctions that are not merely marks on a page, but changes in some quality that is not physical. It is also where we first encounter the concept of Late binding where a term is used without a complete understanding of its meaning. GSB comments on several overt instances of this in the notes to Chapter 3 {Laws, 2011, page 69} here he expands on the definition of the form "Call the space cloven by any distinction, together with the entire content of the space, the form of the distinction. Call the form of the first distinction the form.{Laws, 2011, page 3}" The form in not just a mark but the entire space within which the mark is made.

Name is another term that has been used and is now more fully defined. "Let there be a form distinct from the form. Let the mark of distinction be copied out of the form into such another form. Call any such copy of the mark a token of the mark. Let any token of the mark be called as a name of the marked state. Let the name indicate the state."

How can there be a form distinct from the form (the first distinction)? The space to be cloven by the distinctions of the "distinct" form must be cloven from some space by a distinction. GSB calls this the "unwritten cross". On page 35 he comments: "Nonmathematicians have attributed some deep philosophical meaning to the unwritten cross, but it is merely a device to collect the whole expression into one bracket. - Author." The unwritten cross becomes then the edges of the sheet of paper on which calculations will be made. The important concept is that we are permitted to select such a space and copy the mark of distinction into it. Draw a mark has become copy a mark. With the copy of the mark comes the signal of that mark, and its intention. The act of copying a mark is itself a distinction from the act of making a mark. The concept of procedural distinctions has been introduced very quietly of its own accord. This will be formalized into the concept of a step, in Chapter 3.]

The syntax of the primary arithmetic goes as follows. There are just two atomic expressions:

• The empty Cross  ;
• All or part of the blank page (the "void").

There are two inductive rules:

• A Cross may be written over any expression;
• Any two expressions may be concatenated.

The semantics of the primary arithmetic are perhaps nothing more than the sole explicit definition in LoF: "Distinction is perfect continence".

Let the "unmarked state" be a synonym for the void. Let an empty Cross denote the "marked state". To cross is to move from one value, the unmarked or marked state, to the other. We can now state the "arithmetical" axioms A1 and A2, which ground the primary arithmetic (and hence all of the Laws of Form):

"A1. The law of Calling". Calling twice from a state is indistinguishable from calling once. To make a distinction twice has the same effect as making it once. For example, saying "Let there be light" and then saying "Let there be light" again, is the same as saying it once. Formally:

$\ =$

"A2. The law of Crossing". After crossing from the unmarked to the marked state, crossing again ("recrossing") starting from the marked state returns one to the unmarked state. Hence recrossing annuls crossing. Formally:

$\ =$

In both A1 and A2, the expression to the right of '=' has fewer symbols than the expression to the left of '='. This suggests that every primary arithmetic expression can, by repeated application of A1 and A2, be simplified to one of two states: the marked or the unmarked state. This is indeed the case, and the result is the expression's "simplification". The two fundamental meta-theorems of the primary arithmetic state that:

• Every finite expression has a unique simplification. (T3 in LoF);
• Starting from an initial marked or unmarked state, "complicating" an expression by a finite number of repeated application of A1 and A2 cannot yield an expression whose simplification differs from the initial state. (T4 in LoF).

Thus the relation of logical equivalence partitions all primary arithmetic expressions into two equivalence classes: those that simplify to the Cross, and those that simplify to the void.

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 how two points in a circuit are connected, and not simply to adding wiring.

The primary arithmetic is analogous to the following formal languages from mathematics and computer science:

• A Dyck language with a null alphabet;
• The simplest context-free language in the Chomsky hierarchy;
• A rewrite system that is strongly normalizing and confluent.

The phrase "calculus of indications" in LoF is a synonym for "primary arithmetic".

The notion of canon

A concept peculiar to LoF is that of "canon". While LoF does not formally define canon, the following two excerpts from the Notes to chpt. 2 are apt:

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.

These excerpts relate to the distinction in metalogic between the object language, the formal language of the logical system under discussion, and the metalanguage, a language (often a natural language) distinct from the object language, employed to exposit and discuss the object language. The first quote seems to assert that the canons are part of the metalanguage. The second quote seems to assert that statements in the object language are essentially commands addressed to the reader by the author. Neither assertion holds in standard metalogic.

The primary algebra (Chapter 6)

Syntax

Given any valid primary arithmetic expression, insert into one or more locations any number of Latin letters bearing optional numerical subscripts; the result is a primary algebra formula. Letters so employed in mathematics and logic are called variables. A primary algebra variable indicates a location where one can write the primitive value or its complement . Multiple instances of the same variable denote multiple locations of the same primitive value.

Rules governing logical equivalence

The sign '=' may link two logically equivalent expressions; the result is an equation. By "logically equivalent" is meant that the two expressions have the same simplification. Logical equivalence is an equivalence relation over the set of primary algebra formulas, governed by the rules R1 and R2. Let "C" and "D" be formulae each 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. C becomes E and D becomes F. If C=D, then E=F. Note that A=B is not required.

R2 is employed very frequently in primary algebra demonstrations (see below), almost always silently. These rules are routinely invoked in logic and most of mathematics, nearly always unconsciously.

The primary algebra consists of equations, i.e., pairs of formulae linked by an infix operator '='. R1 and R2 enable transforming one equation into another. Hence the primary algebra is an equational formal system, like the many algebraic structures, including Boolean algebra, that are varieties. Equational logic was common before Principia Mathematica (e.g. Johnson (1892)), and has present-day advocates (Gries & Schneider (1993)).

Conventional mathematical logic consists of tautological formulae, signalled by a prefixed turnstile. To denote that the primary algebra formula A is a tautology, simply write "A = ". If one replaces '=' in R1 and R2 with the biconditional, the resulting rules hold in conventional logic. However, conventional logic relies mainly on the rule modus ponens; thus conventional logic is ponential. The equational-ponential dichotomy distills much of what distinguishes mathematical logic from the rest of mathematics.

Initials

An initial is a primary algebra equation verifiable by a decision procedure and as such is not an axiom. LoF lays down the initials:

J1:

A A

= .

The absence of anything to the right of the "=" above, is deliberate.

J2:

A B

C

=

A C B C

.

J2 is the familiar distributive law of sentential logic and Boolean algebra.

Another set of initials, friendlier to calculations, is:

J0:

A

=

A.

J1a:

A A

=

.

C2:

A

A B

=

A

B

.

It is thanks to C2 that the primary algebra is a lattice. By virtue of J1a, it is a complemented lattice whose upper bound is . By J0, is the corresponding lower bound and identity element. J0 is also an algebraic version of A2 and makes clear the sense in which aliases with the blank page.

T13 in LoF generalizes C2 as follows. Any primary algebra (or sentential logic) formula B can be viewed as an ordered tree with branches. Then:

T13: A subformula A can be copied at will into any depth of B greater than that of A, as long as A and its copy are in the same branch of B. Also, given multiple instances of A in the same branch of B, all instances but the shallowest are redundant.

While a proof of T13 would require induction, the intuition underlying it should be clear.

C2 or its equivalent is named:

• "Generation" in LoF;
• "Exclusion" in Johnson (1892);
• "Pervasion" in the work of William Bricken.

Perhaps the first instance of an axiom or rule with the power of C2 was the "Rule of (De)Iteration", combining T13 and AA=A, of C. S. Peirce's existential graphs.

LoF asserts that concatenation can be read as commuting and associating by default and hence need not be explicitly assumed or demonstrated. (Peirce made a similar assertion about his existential graphs.) Let a period be a temporary notation to establish grouping. That concatenation commutes and associates may then be demonstrated from the:

• Initial AC.D=CD.A and the consequence AA=A. This result holds for all lattices, because AA=A is an easy consequence of the absorption law, which holds for all lattices;
• Initials AC.D=AD.C and J0. Since J0 holds only for lattices with a lower bound, this method holds only for bounded lattices (which include the primary algebra and 2). Commutativity is trivial; just set A=. Associativity: AC.D = CA.D = CD.A = A.CD.

Having demonstrated associativity, the period can be discarded.

The initials in Meguire (2011) are AC.D=CD.A, called B1; B2, J0 above; B3, J1a above; and B4, C2. By design, these initials are very similar to the axioms for an abelian group, G1-G3 below.

Proof theory

The primary algebra contains three kinds of proved assertions:

• Consequence is a primary algebra 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 proof, i.e., an argument, formulated in the metalanguage, that is accepted by trained mathematicians and logicians.
• Initial, defined above. Demonstrations and proofs invoke an initial as if it were an axiom.

The distinction between consequence and theorem holds for all formal systems, including mathematics and logic, but is usually not made explicit. A demonstration or decision procedure can be carried out and verified by computer. The proof of a theorem cannot be.

Let A and B be primary algebra formulas. A demonstration of A=B may proceed in either of two ways:

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

Once A=B has been demonstrated, A=B can be invoked to justify steps in subsequent demonstrations. primary algebra demonstrations and calculations often require no more than J1a, J2, C2, and the consequences (C3 in LoF), (C1), and AA=A (C5).

The consequence , C7' in LoF, enables an algorithm, sketched in LoFs proof of T14, that transforms an arbitrary primary algebra formula to an equivalent formula whose depth does not exceed two. The result is a normal form, the primary algebra analog of the conjunctive normal form. LoF (T14–15) proves the primary algebra analog of the well-known Boolean algebra theorem that every formula has a normal form.

Let A be a subformula of some formula B. When paired with C3, J1a can be viewed as the closure condition for calculations: B is a tautology if and only if A and (A) both appear in depth 0 of B. A related condition appears in some versions of natural deduction. A demonstration by calculation is often little more than:

• Invoking T13 repeatedly to eliminate redundant subformulae;
• Erasing any subformulae having the form .

The last step of a calculation always invokes J1a.

LoF includes elegant new proofs of the following standard metatheory:

• Completeness: all primary algebra consequences are demonstrable from the initials (T17).
• Independence: J1 cannot be demonstrated from J2 and vice versa (T18).

That sentential logic is complete is taught in every first university course in mathematical logic. But university courses in Boolean algebra seldom mention the completeness of 2.

Interpretations

If the Marked and Unmarked states are read as the Boolean values 1 and 0 (or True and False), the primary algebra interprets 2 (or sentential logic). LoF shows how the primary algebra can interpret the syllogism. Each of these interpretations is discussed in a subsection below. Extending the primary algebra so that it could interpret standard first-order logic has yet to be done, but Peirce's beta existential graphs suggest that this extension is feasible.

Two-element Boolean algebra 2

The primary algebra is an elegant minimalist notation for the two-element Boolean algebra 2. Let:

• One of Boolean join (+) or meet (×) interpret concatenation;
• The complement of A interpret
• 0 (1) interpret the empty Mark if join (meet) interprets concatenation (because a binary operation applied to zero operands may be regarded as being equal to the identity element of that operation; or to put it in another way, an operand that is missing could be regarded as acting by default like the identity element).

If join (meet) interprets AC, then meet (join) interprets ${\overline {{\overline {A|}}\ \ {\overline {C|}}{\Big |}}}$. Hence the primary algebra and 2 are isomorphic but for one detail: primary algebra complementation can be nullary, in which case it denotes a primitive value. Modulo this detail, 2 is a model of the primary algebra. The primary arithmetic suggests the following arithmetic axiomatization of 2: 1+1=1+0=0+1=1=~0, and 0+0=0=~1.

The set $\ B=\{$ $,$ $\ \}$ is the Boolean domain or carrier. In the language of universal algebra, the primary algebra is the algebraic structure $\langle B,-\ -,{\overline {-\ |}},{\overline {\ \ |}}\rangle$ of type $\langle 2,1,0\rangle$. The expressive adequacy of the Sheffer stroke points to the primary algebra also being a $\langle B,{\overline {-\ -\ |}},{\overline {\ \ |}}\rangle$ algebra of type $\langle 2,0\rangle$. In both cases, the identities are J1a, J0, C2, and ACD=CDA. Since the primary algebra and 2 are isomorphic, 2 can be seen as a $\langle B,+,\lnot ,1\rangle$ algebra of type $\langle 2,1,0\rangle$. This description of 2 is simpler than the conventional one, namely an $\langle B,+,\times ,\lnot ,1,0\rangle$ algebra of type $\langle 2,2,1,0,0\rangle$.

The two possible interpretations are dual to each other in the Boolean sense. (In Boolean algebra, exchanging AND ↔ OR and 1 ↔ 0 throughout an equation yields an equally valid equation.) The identities remain invariant regardless of which interpretation is chosen, so the transformations or modes of calculation remain the same; only the interpretation of each form would be different. Example: J1a is . Interpreting juxtaposition as OR and as 1, this translates to $\neg A\lor A=1$ which is true. Interpreting juxtaposition as AND and as 0, this translates to $\neg A\land A=0$ which is true as well (and the dual of $\neg A\lor A=1$).

operator-operand duality

The marked state, , is both an operator (e.g., the complement) and operand (e.g., the value 1). This can be summarized neatly by defining two functions $m(x)$ and $u(x)$ for the marked and unmarked state, respectively: let $m(x)=1-\max(\{0\}\cup x)$ and $u(x)=\max(\{0\}\cup x)$, where $x$ is a (possibly empty) set of boolean values.

This reveals that $u$ is either the value 0 or the OR operator, while $m$ is either the value 1 or the NOR operator, depending on whether $x$ is the empty set or not. As noted above, there is a dual form of these functions exchanging AND ↔ OR and 1 ↔ 0.

Sentential logic

Let the blank page denote False, and let a Cross be read as Not. Then the primary arithmetic has the following sentential reading:

 =   False

 =  True  =  not False

 =  Not True  =  False

The primary algebra interprets sentential logic as follows. A letter represents any given sentential expression. 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.

a b a b

,

a b a b

both interpret A if and only if B or A is equivalent to B.

Thus any expression in sentential logic has a primary algebra translation. Equivalently, the primary algebra interprets sentential logic. Given an assignment of every variable to the Marked or Unmarked states, this primary algebra translation reduces 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 whether the original expression is tautological or satisfiable. This is an example of a decision procedure, one more or less in the spirit of conventional truth tables. Given some primary algebra formula containing N variables, this decision procedure requires simplifying 2^N primary arithmetic formulae. For a less tedious decision procedure more in the spirit of Quine's "truth value analysis", see Meguire (2003).

Schwartz (1981) proved that the primary algebra is equivalent — syntactically, semantically, and proof theoretically — with the classical propositional calculus. Likewise, it can be shown that the primary algebra is syntactically equivalent with expressions built up in the usual way from the classical truth values true and false, the logical connectives NOT, OR, and AND, and parentheses.

Interpreting the Unmarked State as False is wholly arbitrary; that state can equally well be read as True. All that is required is that the interpretation of concatenation change from OR to AND. IF A THEN B now translates as instead of . More generally, the primary algebra is "self-dual", meaning that any primary algebra formula has two sentential or Boolean readings, each the dual of the other. Another consequence of self-duality is the irrelevance of De Morgan's laws; those laws are built into the syntax of the primary algebra from the outset.

The true nature of the distinction between the primary algebra on the one hand, and 2 and sentential logic on the other, now emerges. In the latter formalisms, complementation/negation operating on "nothing" is not well-formed. But an empty Cross is a well-formed primary algebra expression, denoting the Marked state, a primitive value. Hence a nonempty Cross is an operator, while an empty Cross is an operand because it denotes a primitive value. Thus the primary algebra 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.

Syllogisms

Appendix 2 of LoF shows how to translate traditional syllogisms and sorites into the primary algebra. A valid syllogism is simply one whose primary algebra translation simplifies to an empty Cross. Let A* denote a literal, i.e., either A or ${\overline {A|}}$, indifferently. Then every syllogism that does not require that one or more terms be assumed nonempty is one of 24 possible permutations of a generalization of Barbara whose primary algebra equivalent is ${\overline {A^{*}\ B|}}\ \ {\overline {{\overline {B|}}\ C^{*}{\Big |}}}\ A^{*}\ C^{*}$. These 24 possible permutations include the 19 syllogistic forms deemed valid in Aristotelian and medieval logic. This primary algebra translation of syllogistic logic also suggests that the primary algebra can interpret monadic and term logic, and that the primary algebra has affinities to the Boolean term schemata of Quine (1982), Part II.

An example of calculation

The following calculation of Leibniz's nontrivial Praeclarum Theorema exemplifies the demonstrative power of the primary algebra. Let C1 be ${\overline {{\overline {A|}}{\Big |}}}$ =A, C2 be $A\ {\overline {A\ B|}}=A\ {\overline {B|}}$, C3 be ${\overline {\ \ |}}\ A={\overline {\ \ |}}$, J1a be ${\overline {A|}}\ A={\overline {\ \ |}}$, and let OI mean that variables and subformulae have been reordered in a way that commutativity and associativity permit.

[(P→R)∧(Q→S)]→[(P∧Q)→(R∧S)].	Praeclarum Theorema.
P R Q S P Q R S .	primary algebra translation
P R Q S P Q R S .	C1.
P R Q S P Q R S .	C1.
P P R Q S Q R S .	OI.
P R Q S Q R S .	C2.
P R Q Q S R S .	OI.
P R Q S R S .	C2.
P Q S R R S .	OI.
P Q S R S .	C2.
P Q S R S .	C1.
P Q S S R .	OI.
P Q B R .	J1a.
B P Q R .	OI.
B	C3. $\square$

Relation to magmas

The primary algebra embodies a point noted by Huntington in 1933: Boolean algebra requires, in addition to one unary operation, one, and not two, binary operations. Hence the seldom-noted fact that Boolean algebras are magmas. (Magmas were called groupoids until the latter term was appropriated by category theory.) To see this, note that the primary algebra is a commutative:

• Semigroup because primary algebra juxtaposition commutes and associates;
• Monoid with identity element , by virtue of J0.

Groups also require a unary operation, called inverse, the group counterpart of Boolean complementation. Let denote the inverse of a. Let denote the group identity element. Then groups and the primary algebra have the same signatures, namely they are both $\langle -\ -,{\overline {-\ |}},{\overline {\ \ |}}\rangle$ algebras of type 〈2,1,0〉. Hence the primary algebra is a boundary algebra. The axioms for an abelian group, in boundary notation, are:

• G1. abc = acb (assuming association from the left);
• G2.
• G3. .

From G1 and G2, the commutativity and associativity of concatenation may be derived, as above. Note that G3 and J1a are identical. G2 and J0 would be identical if    =    replaced A2. This is the defining arithmetical identity of group theory, in boundary notation.

The primary algebra differs from an abelian group in two ways:

• From A2, it follows that ≠ . If the primary algebra were a group, = would hold, and one of    a =    or   a  = a   would have to be a primary algebra consequence. Note that and are mutual primary algebra complements, as group theory requires, so that ${\overline {\ {\overline {\ {\overline {\ \ |}}\ {\Big |}}}\ {\Bigg |}}}={\overline {\ \ |}}$ is true of both group theory and the primary algebra;
• C2 most clearly demarcates the primary algebra from other magmas, because C2 enables demonstrating the absorption law that defines lattices, and the distributive law central to Boolean algebra.

Both A2 and C2 follow from B's being an ordered set.

Equations of the second degree (Chapter 11)

Chapter 11 of LoF introduces equations of the second degree, composed of recursive formulae that can be seen as having "infinite" depth. Some recursive formulae simplify to the marked or unmarked state. Others "oscillate" indefinitely between the two states depending on whether a given depth is even or odd. Specifically, certain recursive formulae can be interpreted as oscillating between true and false over successive intervals of time, in which case a formula is deemed to have an "imaginary" truth value. Thus the flow of time may be introduced into the primary algebra.

Turney (1986) shows how these recursive formulae can be interpreted via Alonzo Church's Restricted Recursive Arithmetic (RRA). Church introduced RRA in 1955 as an axiomatic formalization of finite automata. Turney presents a general method for translating equations of the second degree into Church's RRA, illustrating his method using the formulae E1, E2, and E4 in chapter 11 of LoF. This translation into RRA sheds light on the names Spencer-Brown gave to E1 and E4, namely "memory" and "counter". RRA thus formalizes and clarifies LoF's notion of an imaginary truth value.

Related work

Gottfried Leibniz, in memoranda not published before the late 19th and early 20th centuries, invented Boolean logic. His notation was isomorphic to that of LoF: concatenation read as conjunction, and "non-(X)" read as the complement of X. Recognition of Leibniz's pioneering role in algebraic logic was foreshadowed Lewis (1918) and Rescher (1954). But a full appreciation of Leibniz's accomplishments had to await the work of Wolfgang Lenzen, published in the 1980s and reviewed in Lenzen (2004).

Charles Sanders Peirce (1839–1914) anticipated the primary algebra in three veins of work:

1. Two papers he wrote in 1886 proposed a logical algebra employing but one symbol, the streamer, nearly identical to the Cross of LoF. The semantics of the streamer are identical to those of the Cross, except that Peirce never wrote a streamer with nothing under it. An excerpt from one of these papers was published in 1976, but they were not published in full until 1993.
2. In a 1902 encyclopedia article, Peirce notated Boolean algebra and sentential logic in the manner of this entry, except that he employed two styles of brackets, toggling between '(', ')' and '[', ']' with each increment in formula depth.
3. The syntax of his alpha existential graphs is merely concatenation, read as conjunction, and enclosure by ovals, read as negation. If primary algebra concatenation is read as conjunction, then these graphs are isomorphic to the primary algebra.

LoF cites vol. 4 of Peirce's Collected Papers, the source for the formalisms in (2) and (3) above. (1)-(3) were virtually unknown at the time when (1960s) and in the place where (UK) LoF was written. Peirce's semiotics, about which LoF is silent, may yet shed light on the philosophical aspects of LoF.

Kauffman (2001) discusses another notation similar to that of LoF, that of a 1917 article by Jean Nicod, who was a disciple of Bertrand Russell's.

The above formalisms are, like the primary algebra, all instances of boundary mathematics, i.e., mathematics whose syntax is limited to letters and brackets (enclosing devices). A minimalist syntax of this nature is a "boundary notation". Boundary notation is free of infix operators, prefix, or postfix operator symbols. The very well known curly braces ('{', '}') of set theory can be seen as a boundary notation.

The work of Leibniz, Peirce, and Nicod is innocent of metatheory, as they wrote before Emil Post's landmark 1920 paper (which LoF cites), proving that sentential logic is complete, and before Hilbert and Łukasiewicz showed how to prove axiom independence using models.

Craig (1979) argued that the world, and how humans perceive and interact with that world, has a rich Boolean structure. Craig was an orthodox logician and an authority on algebraic logic.

Second-generation cognitive science emerged in the 1970s, after LoF was written. On cognitive science and its relevance to Boolean algebra, logic, and set theory, see Lakoff (1987) (see index entries under "Image schema examples: container") and Lakoff & Núñez (2000). Neither book cites LoF.

The biologists and cognitive scientists Humberto Maturana and his student Francisco Varela both discuss LoF in their writings, which identify "distinction" as the fundamental cognitive act. The Berkeley psychologist and cognitive scientist Eleanor Rosch has written extensively on the closely related notion of categorization.

Other formal systems with possible affinities to the primary algebra include:

• Mereology which typically has a lattice structure very similar to that of Boolean algebra. For a few authors, mereology is simply a model of Boolean algebra and hence of the primary algebra as well.
• Mereotopology, which is inherently richer than Boolean algebra;
• The system of Whitehead (1934), whose fundamental primitive is "indication".

The primary arithmetic and algebra are a minimalist formalism for sentential logic and Boolean algebra. Other minimalist formalisms having the power of set theory include:

• The lambda calculus;
• Combinatory logic with two (S and K) or even one (X) primitive combinators;
• Mathematical logic done with merely three primitive notions: one connective, NAND (whose primary algebra translation is ${\overline {A\ \ B\ |}}$ or, dually, ${\overline {A|}}\ \ {\overline {B|}}$), universal quantification, and one binary atomic formula, denoting set membership. This is the system of Quine (1951).
• The beta existential graphs, with a single binary predicate denoting set membership. This has yet to be explored. The alpha graphs mentioned above are a special case of the beta graphs.

Editions

• 1969. London: Allen & Unwin, hardcover.
• 1972. Crown Publishers, hardcover: ISBN 0-517-52776-6
• 1973. Bantam Books, paperback. ISBN 0-553-07782-1
• 1979. E. P. Dutton, paperback. ISBN 0-525-47544-3
• 1994. Portland OR: Cognizer Company, paperback. ISBN 0-9639899-0-1
• 1997 German translation, titled Gesetze der Form. Lübeck: Bohmeier Verlag. ISBN 3-89094-321-7
• 2008 Bohmeier Verlag, Leipzig, 5th international edition. ISBN 978-3-89094-580-4

See also

• Boolean algebra – Algebraic manipulation of "true" and "false"
• Boolean algebras canonically defined – A technical treatment of Boolean algebras
• Entitative graph – Type of diagrammatic notation for propositional logicPages displaying short descriptions of redirect targets
• Existential graph – Type of diagrammatic notation for propositional logic
• Mark and space – States of a communications signal
• Programming and Metaprogramming – 1968 non-fiction book by John C. Lilly
• Propositional calculus – Branch of logic
• Two-element Boolean algebra – Boolean algebra
• List of Boolean algebra topics

Notes

Works cited

• Byrne, Lee (1946). "Two Formulations of Boolean Algebra". Bulletin of the American Mathematical Society: 268–71.
• Craig, William (1979). "Boolean Logic and the Everyday Physical World". Proceedings and Addresses of the American Philosophical Association. 52 (6): 751–78. doi:10.2307/3131383. JSTOR 3131383.
• Gries, David; Schneider, F. B. (1993). A Logical Approach to Discrete Math. Springer-Verlag. ISBN 978-0-387-94115-8.
• Johnson, William Ernest (1892). "The Logical Calculus". Mind. 1 (n.s.): 3–30.
• Kauffman, Louis H. (2001). "The Mathematics of C. S. Peirce" (PDF). Cybernetics and Human Knowing. 8: 79–110.
• Lakoff, George (1987). Women, Fire, and Dangerous Things: What Categories Reveal about the Mind. University of Chicago Press.
• Lakoff, George; Núñez, Rafael E. (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Basic Books. ISBN 978-0-465-03770-4.
• Lenzen, Wolfgang (2004). "Leibniz's Logic" (PDF). In Gabbay, D.; Woods, J. (eds.). The Rise of Modern Logic: From Leibniz to Frege. Handbook of the History of Logic. Vol. 3. Amsterdam: Elsevier. pp. 1–83. ISBN 978-0-08-053287-5.
• Lewis, C. I. (1918). A Survey of Symbolic Logic. Berkeley: University of California Press. Republished in part by Dover in 1960.
• Meguire, P. G. (2003). "Discovering Boundary Algebra: A Simplified Notation for Boolean Algebra and the Truth Functors". International Journal of General Systems. 32: 25–87. CiteSeerX 10.1.1.106.634. doi:10.1080/0308107031000075690. S2CID 9460101.
• Meguire, P. G. (2011). Boundary Algebra: A Simpler Approach to Basic Logic and Boolean Algebra. VDM Publishing. ISBN 978-3-639-36749-2.
• Peirce, Charles S. (1976). "Qualitative Logic, MS 736 (c. 1886)". In Eisele, Carolyn (ed.). The New Elements of Mathematics by Charles S. Peirce. Vol. 4, Mathematical Philosophy. The Hague: Mouton.
• Quine, Willard (1951). Mathematical Logic (2nd ed.). Harvard University Press.
• Quine, Willard (1982). Methods of Logic (4th ed.). Harvard University Press. ISBN 978-0-674-57176-1.
• Rescher, Nicholas (1954). "Leibniz's Interpretation of His Logical Calculi". Journal of Symbolic Logic. 18 (1): 1–13. doi:10.2307/2267644. JSTOR 2267644. S2CID 689315.
• Schwartz, Daniel G. (1981). "Isomorphisms of G. Spencer-Brown's Laws of Form and F. Varela's Calculus for Self-Reference". International Journal of General Systems. 6 (4): 239–55. doi:10.1080/03081078108934802.
• Turney, P. D. (1986). "Laws of Form and Finite Automata". International Journal of General Systems. 12 (4): 307–18. doi:10.1080/03081078608934939.
• Whitehead, A. N. (1934). "Indication, classes, number, validation". Mind. 43 (n.s.) (171): 281–97, 543. doi:10.1093/mind/XLIII.171.281. The corrigenda on p. 543 are numerous and important, and later reprints of this article do not incorporate them.

Further reading

• Baecker, Dirk, ed. (1999). Problems of Form. Stanford University Press.
• Baecker, Dirk, ed. (2013). "Foreword: A Mathematics of Form, A Sociology of Observers". Cybernetics & Human Knowing. 20 (3–4).
• Bostock, David (1997). Intermediate Logic. Oxford University Press.
• Kauffman, Louis H. (2006). "Reformulating the Map Color Theorem". arXiv:math.CO/0112266.
• Kauffman, Louis H. (2006a). Laws of Form: An Exploration in Mathematics and Foundations (PDF).
• Kauffman, Louis H., ed. (2019). "Laws of Form: Spencer-Brown at Esalen, 1973". Cybernetics & Human Knowing. 26 (2–3).

External links

• Laws of Form, archive of website by Richard Shoup.
• Spencer-Brown's talks at Esalen, 1973. Self-referential forms are introduced in the section entitled "Degree of Equations and the Theory of Types".
• Audio recording of the opening session, 1973 AUM Conference at Esalen.
• Louis H. Kauffman, "Box Algebra, Boundary Mathematics, Logic, and Laws of Form."
• Kissel, Matthias, "A nonsystematic but easy to understand introduction to Laws of Form."
• A meeting with G.S.B by Moshe Klein
• The Markable Mark, an introduction by easy stages to the ideas of Laws of Form
• The BF Calculus and the Square Root of Negation by Louis Kauffman and Arthur Collings; it extends the Laws of Form by adding an imaginary logical value. (Imaginary logical values are introduced in chapter 11 of the book Laws of Form.)
• Laws of Form Course - a free on-line course taking people through the main body of the text of Laws of Form by Leon Conrad, Spencer-Brown's last student, who studied the work with the author.