Laws of Form (hereinafterLoF) is a book byG. Spencer-Brown, published in 1969, that straddles the boundary betweenmathematics andphilosophy.LoF describes three distinctlogical systems:

• Theprimary arithmetic (described in Chapter 4 ofLoF), whose models includeBoolean arithmetic;
• Theprimary algebra (Chapter 6 ofLoF), whosemodels include thetwo-element Boolean algebra (hereinafter abbreviated2),Boolean logic, and the classicalpropositional calculus;
• Equations of the second degree (Chapter 11), whoseinterpretations includefinite automata andAlonzo Church's Restricted Recursive Arithmetic (RRA).

"Boundary algebra" is aMeguire (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 toLoF.

The preface states that the work was first explored in 1959, and Spencer Brown citesBertrand Russell as being supportive of his endeavour.^[a] He also thanksJ. C. P. Miller ofUniversity College London for helping with the proofreading 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 theUniversity of London, to deliver a course on the mathematics of logic.^{[citation needed]}

LoF emerged from work in electronic engineering its author did around 1960. Key ideas of theLOF were first outlined in his 1961 manuscriptDesign with the Nor, which remained unpublished until 2021,^[1] and further refined during subsequent lectures onmathematical logic he gave under the auspices of theUniversity of London's extension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,^[2] and the most recent being a 1997 German translation.LoF has never gone out of print.

LoF'smystical and declamatory prose and its love ofparadox make it a challenging read for all. Spencer-Brown was influenced byWittgenstein andR. D. Laing.LoF also echoes a number of themes from the writings ofCharles Sanders Peirce,Bertrand Russell, andAlfred North Whitehead.

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 somethingis, is the statement which says no such statements are used in this book.^[3] Furthermore, the claim asserts that except for this one sentence the book can be seen as an example ofE-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 verbto be throughout, and in all its grammatical forms, as may be seen both in the original and in quotes shown below.^[4]

Ostensibly a work of formal mathematics and philosophy,LoF became something of acult classic: it was praised byHeinz von Foerster when he reviewed it for theWhole Earth Catalog.^[5] Those who agree point toLoF as embodying an enigmatic "mathematics ofconsciousness", its algebraic symbolism capturing an (perhaps even "the") implicit root ofcognition: the ability to "distinguish".LoF argues that primary algebra reveals striking connections amonglogic,Boolean algebra, and arithmetic, and thephilosophy of language andmind.

Stafford Beer wrote in a review forNature, "When one thinks of all that Russell went through sixty years ago, to write thePrincipia, and all we his readers underwent in wrestling with those three vast volumes, it is almost sad".^[6]

Banaschewski (1977)^[7] argues that the primary algebra is nothing but new notation for Boolean algebra. Indeed, thetwo-element Boolean algebra2 can be seen as the intended interpretation of the primary algebra. Yet the notation of the primary algebra:

• Fully exploits theduality characterizing not justBoolean algebras but alllattices;
• Highlights how syntactically distinct statements in logic and2 can have identicalsemantics;
• Dramatically simplifies Boolean algebra calculations, and proofs insentential andsyllogisticlogic.

Moreover, the syntax of the primary algebra can be extended to formal systems other than2 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 andWilliam 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 thefour color theorem,Fermat's Last Theorem, and theGoldbach conjecture, are provable using extensions of the primary algebra. Spencer-Brown eventually circulated a purported proof of the four color theorem, but it was met with skepticism.^[8]

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 ofcognition, i.e., thedualistic Mark indicates the capability of differentiating a "this" from "everything elsebut this".

InLoF, 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. AsLoF 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 ofconsciousness andlanguage. 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.

Thesyntax of the primary arithmetic goes as follows. There are just twoatomic 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 beconcatenated.

Thesemantics of the primary arithmetic are perhaps nothing more than the sole explicitdefinition inLoF: "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:

${\displaystyle =}$

"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:

${\displaystyle =}$

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, besimplified 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 metatheorems of the primary arithmetic state that:

• Every finite expression has a unique simplification. (T3 inLoF);
• 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 inLoF).

Thus therelation oflogical equivalencepartitions all primary arithmetic expressions into twoequivalence 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 frommathematics andcomputer science:

• ADyck language with a null alphabet;
• The simplestcontext-free language in theChomsky hierarchy;
• Arewrite system that isstrongly normalizing andconfluent.

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

WhileLoF 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 calledcanons. 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 inmetalogic between the object language, the formal language of the logical system under discussion, and themetalanguage, 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 thecanons 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.

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 algebraformula. Letters so employed inmathematics andlogic are calledvariables. 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.

The sign '=' may link two logically equivalent expressions; the result is anequation. By "logically equivalent" is meant that the two expressions have the same simplification.Logical equivalence is anequivalence 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 subformulaA:

• R1,Substitution of equals. Replaceone or more instances ofA inC byB, resulting inE. IfA=B, thenC=E.
• R2,Uniform replacement. Replaceall instances ofA inC andD withB.C becomesE andD becomesF. IfC=D, thenE=F. Note thatA=B is not required.

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

Theprimary algebra consists ofequations, i.e., pairs of formulae linked by aninfix operator '='.R1 andR2 enable transforming one equation into another. Hence theprimary algebra is anequational formal system, like the manyalgebraic structures, includingBoolean algebra, that arevarieties. Equational logic was common beforePrincipia Mathematica (e.g.Johnson (1892)), and has present-day advocates (Gries & Schneider (1993)).

Conventionalmathematical logic consists oftautological formulae, signalled by a prefixedturnstile. To denote that theprimary algebra formulaA is atautology, simply write "A = ". If one replaces '=' inR1 andR2 with thebiconditional, the resulting rules hold in conventional logic. However, conventional logic relies mainly on the rulemodus ponens; thus conventional logic isponential. The equational-ponential dichotomy distills much of what distinguishes mathematical logic from the rest of mathematics.

Aninitial is aprimary algebra equation verifiable by adecision procedure and as such isnot anaxiom.LoF lays down the initials:

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

J2 is the familiardistributive law ofsentential logic andBoolean algebra.

Another set of initials, friendlier to calculations, is:

It is thanks toC2 that theprimary algebra is alattice. By virtue ofJ1a, it is acomplemented lattice whose upper bound is. ByJ0, is the corresponding lower bound andidentity element.J0 is also an algebraic version ofA2 and makes clear the sense in which aliases with the blank page.

T13 inLoF generalizesC2 as follows. Anyprimary algebra (or sentential logic) formulaB can be viewed as anordered tree withbranches. Then:

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

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

C2 or its equivalent is named:

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

Perhaps the first instance of an axiom or rule with the power ofC2 was the "Rule of (De)Iteration", combining T13 andAA=A, ofC. S. Peirce'sexistential graphs.

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

• InitialAC.D=CD.A and the consequenceAA=A.^[9] This result holds for alllattices, becauseAA=A is an easy consequence of theabsorption law, which holds for all lattices;
• InitialsAC.D=AD.C andJ0. SinceJ0 holds only for lattices with a lower bound, this method holds only forbounded lattices (which include theprimary algebra and2). Commutativity is trivial; just setA=. Associativity:AC.D =CA.D =CD.A =A.CD.

Having demonstrated associativity, the period can be discarded.

The initials inMeguire (2011) areAC.D=CD.A, calledB1;B2, J0 above;B3, J1a above; andB4, C2. By design, these initials are very similar to the axioms for anabelian group,G1-G3 below.

Theprimary algebra contains three kinds of proved assertions:

• Consequence is aprimary algebra equation verified by ademonstration. A demonstration consists of a sequence ofsteps, each step justified by an initial or a previously demonstrated consequence.
• Theorem is a statement in themetalanguage verified by aproof, 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 andtheorem holds for all formal systems, including mathematics and logic, but is usually not made explicit. A demonstration ordecision procedure can be carried out and verified by computer. Theproof of atheorem cannot be.

LetA andB beprimary algebraformulas. A demonstration ofA=B may proceed in either of two ways:

• ModifyA in steps untilB is obtained, or vice versa;
• Simplify both and to. This is known as a "calculation".

OnceA=B has been demonstrated,A=B can be invoked to justify steps in subsequent demonstrations.primary algebra demonstrations and calculations often require no more thanJ1a,J2,C2, and the consequences (C3 inLoF), (C1), andAA=A (C5).

The consequence,C7' inLoF, enables analgorithm, sketched inLoFs proof of T14, that transforms an arbitraryprimary algebra formula to an equivalent formula whose depth does not exceed two. The result is anormal form, theprimary algebra analog of theconjunctive normal form.LoF (T14–15) proves theprimary algebra analog of the well-knownBoolean algebra theorem that every formula has a normal form.

LetA be asubformula of someformulaB. When paired withC3,J1a can be viewed as the closure condition for calculations:B is atautologyif and only ifA and (A) both appear in depth 0 ofB. A related condition appears in some versions ofnatural 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 invokesJ1a.

LoF includes elegant new proofs of the following standardmetatheory:

• Completeness: allprimary algebra consequences are demonstrable from the initials (T17).
• Independence:J1 cannot be demonstrated fromJ2 and vice versa (T18).

Thatsentential logic is complete is taught in every first university course inmathematical logic. But university courses in Boolean algebra seldom mention the completeness of2.

If the Marked and Unmarked states are read as theBoolean values 1 and 0 (orTrue andFalse), theprimary algebrainterprets2 (orsentential logic).LoF shows how theprimary algebra can interpret thesyllogism. Each of theseinterpretations is discussed in a subsection below. Extending theprimary algebra so that it couldinterpret standardfirst-order logic has yet to be done, butPeirce'sbetaexistential graphs suggest that this extension is feasible.

Theprimary algebra is an elegant minimalist notation for thetwo-element Boolean algebra2. Let:

• One of Booleanjoin (+) ormeet (×) interpretconcatenation;
• Thecomplement ofA interpret
• 0 (1) interpret the empty Mark if join (meet) interpretsconcatenation (because a binary operation applied to zero operands may be regarded as being equal to theidentity 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) interpretsAC, then meet (join) interprets${\displaystyle \overline{\overline{A|}\overline{C|}|}}$. Hence theprimary algebra and2 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 amodel of the primary algebra. The primary arithmetic suggests the following arithmetic axiomatization of2: 1+1=1+0=0+1=1=~0, and 0+0=0=~1.

Theset${\displaystyle B=\{}$${\displaystyle ,}$${\displaystyle \}}$ is theBoolean domain orcarrier. In the language ofuniversal algebra, theprimary algebra is thealgebraic structure${\displaystyle ⟨B,--,\overline{-|},\overline{|}⟩}$ of type${\displaystyle ⟨2,1,0⟩}$. Theexpressive adequacy of theSheffer stroke points to theprimary algebra also being a${\displaystyle ⟨B,\overline{--|},\overline{|}⟩}$ algebra of type${\displaystyle ⟨2,0⟩}$. In both cases, the identities are J1a, J0, C2, andACD=CDA. Since theprimary algebra and2 areisomorphic,2 can be seen as a${\displaystyle ⟨B,+,\mathrm{\neg },1⟩}$ algebra of type${\displaystyle ⟨2,1,0⟩}$. This description of2 is simpler than the conventional one, namely an${\displaystyle ⟨B,+,\times ,\mathrm{\neg },1,0⟩}$ algebra of type${\displaystyle ⟨2,2,1,0,0⟩}$.

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${\displaystyle \mathrm{\neg }A\vee A=1}$ which is true. Interpreting juxtaposition as AND and as 0, this translates to${\displaystyle \mathrm{\neg }A\wedge A=0}$ which is true as well (and the dual of${\displaystyle \mathrm{\neg }A\vee A=1}$).

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${\displaystyle m(x)}$ and${\displaystyle u(x)}$ for the marked and unmarked state, respectively: let${\displaystyle m(x)=1-max(\{0\}\cup x)}$ and${\displaystyle u(x)=max(\{0\}\cup x)}$, where${\displaystyle x}$ is a (possibly empty) set of boolean values.

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

Let the blank page denoteFalse, and let a Cross be read asNot. Then the primary arithmetic has the following sentential reading:

 =  False

 =  True  =  not False

 =  Not True  =  False

Theprimary algebra interprets sentential logic as follows. A letter represents any given sentential expression. Thus:

interpretsNot A

interpretsA Or B

interpretsNot A Or B orIf A Then B.

interpretsNot (Not A Or Not B)

orNot (If A Then Not B)

orA And B.

Thus any expression insentential logic has aprimary algebra translation. Equivalently, theprimary algebrainterprets sentential logic. Given an assignment of every variable to the Marked or Unmarked states, thisprimary 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 istautological orsatisfiable. This is an example of adecision procedure, one more or less in the spirit of conventional truth tables. Given someprimary algebra formula containingN variables, this decision procedure requires simplifying 2^N primary arithmetic formulae. For a less tedious decision procedure more in the spirit ofQuine's "truth value analysis", seeMeguire (2003).

Schwartz (1981) proved that theprimary algebra is equivalent —syntactically,semantically, andproof theoretically — with theclassical propositional calculus. Likewise, it can be shown that theprimary algebra is syntactically equivalent with expressions built up in the usual way from the classicaltruth valuestrue andfalse, thelogical connectives NOT, OR, and AND, and parentheses.

Interpreting the Unmarked State asFalse is wholly arbitrary; that state can equally well be read asTrue. All that is required is that the interpretation ofconcatenation change from OR to AND. IF A THEN B now translates as instead of. More generally, theprimary algebra is "self-dual", meaning that anyprimary algebra formula has twosentential orBoolean readings, each thedual of the other. Another consequence of self-duality is the irrelevance ofDe Morgan's laws; those laws are built into the syntax of theprimary algebra from the outset.

The true nature of the distinction between theprimary algebra on the one hand, and2 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-formedprimary algebra expression, denoting the Marked state, a primitive value. Hence a nonempty Cross is anoperator, while an empty Cross is anoperand because it denotes a primitive value. Thus theprimary 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.

Appendix 2 ofLoF shows how to translate traditionalsyllogisms andsorites into theprimary algebra. A valid syllogism is simply one whoseprimary algebra translation simplifies to an empty Cross. LetA* denote aliteral, i.e., eitherA or${\displaystyle \overline{A|}}$, 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 ofBarbara whoseprimary algebra equivalent is${\displaystyle \overline{A^{\ast }B|}\overline{\overline{B|}{C}^{\ast }|}{A}^{\ast }{C}^{\ast }}$. These 24 possible permutations include the 19 syllogistic forms deemed valid inAristotelian andmedieval logic. Thisprimary algebra translation of syllogistic logic also suggests that theprimary algebra caninterpretmonadic andterm logic, and that theprimary algebra has affinities to theBoolean term schemata ofQuine (1982), Part II.

The following calculation ofLeibniz's nontrivialPraeclarum Theorema exemplifies the demonstrative power of theprimary algebra. Let C1 be${\displaystyle \overline{\overline{A|}|}}$ =A, C2 be${\displaystyle A\overline{AB|}=A\overline{B|}}$, C3 be${\displaystyle \overline{|}A=\overline{|}}$, J1a be${\displaystyle \overline{A|}A=\overline{|}}$, 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.${\displaystyle ◻}$

Theprimary algebra embodies a point noted byHuntington in 1933:Boolean algebra requires, in addition to oneunary operation, one, and not two,binary operations. Hence the seldom-noted fact that Boolean algebras aremagmas. (Magmas were calledgroupoids until the latter term was appropriated bycategory theory.) To see this, note that theprimary algebra is acommutative:

• Semigroup becauseprimary algebra juxtapositioncommutes andassociates;
• Monoid withidentity element, by virtue ofJ0.

Groups also require aunary operation, calledinverse, the group counterpart ofBoolean complementation. Let denote the inverse ofa. Let denote the groupidentity element. Then groups and theprimary algebra have the samesignatures, namely they are both${\displaystyle ⟨--,\overline{-|},\overline{|}⟩}$ algebras of type 〈2,1,0〉. Hence theprimary algebra is aboundary algebra. The axioms for anabelian group, in boundary notation, are:

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

FromG1 andG2, the commutativity and associativity of concatenation may be derived, as above. Note thatG3 andJ1a are identical.G2 andJ0 would be identical if    =    replacedA2. This is the defining arithmetical identity of group theory, in boundary notation.

Theprimary algebra differs from anabelian group in two ways:

• FromA2, it follows that ≠. If theprimary algebra were agroup, = would hold, and one of    a =    or   a  = a   would have to be aprimary algebra consequence. Note that and are mutualprimary algebra complements, as group theory requires, so that${\displaystyle \overline{\overline{\overline{|}|}|}=\overline{|}}$ is true of both group theory and theprimary algebra;
• C2 most clearly demarcates theprimary algebra from other magmas, becauseC2 enables demonstrating theabsorption law that defineslattices, and thedistributive law central toBoolean algebra.

BothA2 andC2 follow fromB's being anordered set.

Chapter 11 ofLoF introducesequations of the second degree, composed ofrecursive 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 betweentrue andfalse 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 theprimary algebra.

Turney (1986) shows how these recursive formulae can be interpreted viaAlonzo Church's Restricted Recursive Arithmetic (RRA). Church introduced RRA in 1955 as an axiomatic formalization offinite automata. Turney presents a general method for translating equations of the second degree into Church's RRA, illustrating his method using the formulaeE1,E2, andE4 in chapter 11 ofLoF. This translation into RRA sheds light on the names Spencer-Brown gave toE1 andE4, namely "memory" and "counter". RRA thus formalizes and clarifiesLoF's notion of an imaginary truth value.

This sectionpossibly containsoriginal research. Pleaseimprove it byverifying the claims made and addinginline citations. Statements consisting only of original research should be removed.(December 2024) (Learn how and when to remove this message)

Gottfried Leibniz, in memoranda not published before the late 19th and early 20th centuries, inventedBoolean logic. His notation was isomorphic to that ofLoF: concatenation read asconjunction, and "non-(X)" read as thecomplement ofX. Recognition of Leibniz's pioneering role inalgebraic logic was foreshadowed byLewis (1918) andRescher (1954). But a full appreciation of Leibniz's accomplishments had to await the work of Wolfgang Lenzen, published in the 1980s and reviewed inLenzen (2004).

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

1. Two papers he wrote in 1886 proposed a logical algebra employing but one symbol, thestreamer, nearly identical to the Cross ofLoF. 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,^[10] but they were not published in full until 1993.^[11]
2. In a 1902 encyclopedia article,^[12] 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. Thesyntax of his alphaexistential graphs is merelyconcatenation, read asconjunction, and enclosure by ovals, read asnegation.^[13] Ifprimary algebra concatenation is read asconjunction, then these graphs areisomorphic to theprimary algebra.^[14]

LoF cites vol. 4 of Peirce'sCollected 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'ssemiotics, about whichLoF is silent, may yet shed light on the philosophical aspects ofLoF.

Kauffman (2001) discusses another notation similar to that ofLoF, that of a 1917 article byJean Nicod, who was a disciple ofBertrand Russell's.

The above formalisms are, like theprimary algebra, all instances ofboundary 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, orpostfix 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 beforeEmil Post's landmark 1920 paper (whichLoF cites), proving thatsentential logic is complete, and beforeHilbert andŁukasiewicz showed how to proveaxiom independence usingmodels.

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 onalgebraic logic.

Second-generationcognitive science emerged in the 1970s, afterLoF was written. On cognitive science and its relevance to Boolean algebra, logic, andset theory, seeLakoff (1987) (see index entries under "Image schema examples: container") andLakoff & Núñez (2000). Neither book citesLoF.

The biologists and cognitive scientistsHumberto Maturana and his studentFrancisco Varela both discussLoF in their writings, which identify "distinction" as the fundamental cognitive act. The Berkeley psychologist and cognitive scientistEleanor 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 alattice structure very similar to that of Boolean algebra. For a few authors, mereology is simply amodel ofBoolean algebra and hence of the primary algebra as well.
• Mereotopology, which is inherently richer than Boolean algebra;
• The system ofWhitehead (1934), whose fundamental primitive is "indication".

The primary arithmetic and algebra are a minimalist formalism forsentential logic and Boolean algebra. Other minimalist formalisms having the power ofset theory include:

• Thelambda calculus;
• Combinatory logic with two (S andK) or even one (X) primitive combinators;
• Mathematical logic done with merely three primitive notions: one connective,NAND (whoseprimary algebra translation is${\displaystyle \overline{AB|}}$ or, dually,${\displaystyle \overline{A|}\overline{B|}}$), universalquantification, and onebinaryatomic formula, denotingset membership. This is the system ofQuine (1951).
• Thebetaexistential graphs, with a singlebinary predicate denoting set membership. This has yet to be explored. Thealpha graphs mentioned above are a special case of thebeta graphs.

• 1969. London: Allen & Unwin, hardcover.ISBN 0-04-510028-4
• 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, Oregon: Cognizer Company, paperback.ISBN 0-9639899-0-1
• 1997 German translation, titledGesetze der Form. Lübeck: Bohmeier Verlag.ISBN 3-89094-321-7
• 2008 Bohmeier Verlag, Leipzig, 5th international edition.ISBN 978-3-89094-580-4

• Boolean algebra – Algebraic manipulation of "true" and "false"
• Boolean algebras canonically defined – 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 logicPages displaying short descriptions of redirect targets
• Two-element Boolean algebra – Boolean algebra
• List of Boolean algebra topics

This article'suse ofexternal links may not follow Wikipedia's policies or guidelines. Pleaseimprove this article by removingexcessive orinappropriate external links, and converting useful links where appropriate intofootnote references.(November 2024) (Learn how and when to remove this message)

• 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 toLaws of Form."
• A meetingwith G.S.BArchived 17 February 2012 at theWayback Machine by Moshe Klein
• The Markable Mark, an introduction by easy stages to the ideas ofLaws 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 bookLaws 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.

• 1969 non-fiction books
• Algebra
• Books about consciousness
• Boolean algebra
• Finite-state machines
• Logic books
• Logical calculi
• Mathematical logic
• Philosophy of language literature
• Philosophy of mind literature

• Articles with short description
• Short description matches Wikidata
• Use dmy dates from January 2020
• All articles with unsourced statements
• Articles with unsourced statements from December 2024
• Articles that may contain original research from December 2024
• All articles that may contain original research
• Pages displaying short descriptions of redirect targets via Module:Annotated link
• CS1: long volume value
• Pages with missing ISBNs
• Wikipedia external links cleanup from November 2024
• Webarchive template wayback links