Editing Laws of Form

{{Short description|1969 non-fiction book by G. Spencer-Brown}}
{{Use dmy dates|date=January 2020}}
{{italic title}}
'''''Laws of Form''''' (hereinafter '''''LoF''''') is a book by [[G. Spencer-Brown]], published in 1969, that straddles the boundary between [[mathematics]] and [[philosophy]]. ''LoF'' describes three distinct [[logical system]]s:
* The '''primary arithmetic''' (described in Chapter 4 of ''LoF''), whose models include [[Two-element Boolean algebra#Some basic identities|Boolean arithmetic]];
* The '''primary algebra''' (Chapter 6 of ''LoF''), whose [[interpretation (logic)|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 [[interpretation (logic)|interpretations]] include [[finite automata]] and [[Alonzo Church]]'s Restricted Recursive Arithmetic (RRA).

"Boundary algebra" is a {{harvp|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''.

==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.{{efn|This is also mentioned by Russell in his autobiography, {{harvp|Russell|2014|p=664}}.}} He also thanks [[J. C. P. Miller]] of [[University 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 the [[University of London]], to deliver a course on the mathematics of logic.{{cn|date=December 2024}}

''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 Key ideas of the ''LOF'' were first outlined in his 1961 manuscript ''Design with the Nor'', which remained unpublished until 2021,{{sfnp|Spencer-Brown|2021}} and further refined during subsequent lectures on [[mathematical logic]] he gave under the auspices of the [[University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,<ref name="Eine Einführung">{{cite book |last1=Schönwälder-Kuntze |first1=Tatjana |last2=Wille |first2=Katrin |last3=Hölscher |first3=Thomas |last4=Spencer Brown |first4=George |title="George Spencer Brown: Eine Einführung in die ''Laws of Form'', 2. Auflage" |date=2009 |publisher=VS Verlag für Sozialwissenschaften |location=Wiesbaden |isbn=978-3-531-16105-1}}</ref> and the most recent being a 1997 German translation.  ''LoF'' has never gone out of print.

''LoF'''s [[mysticism|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]].

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.<ref>Felix Lau: ''Die Form der Paradoxie'', 2005 Carl-Auer Verlag, {{ISBN|9783896703521}}</ref> 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.{{sfnp|Spencer-Brown|1969}}

==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]]''.<ref name="Computing a Realit">{{cite journal |last1=Müller |first1=Albert |title=Computing a Reality Heinz von Foerster's Lecture at the A.U.M Conference in 1973 |journal=Constructivist Foundations |date=2008 |volume=4 |issue=1 |pages=62–69 |url=https://constructivist.info/articles/4/1/062.foerster.pdf}}</ref> 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 (logic)|Boolean algebra]], and arithmetic, and the [[philosophy of language]] and [[Philosophy of mind|mind]].

[[Stafford Beer]] wrote in a review for ''[[Nature (journal)|Nature]]'', "When one thinks of all that Russell went through sixty years ago, to write the ''[[Principia Mathematica|Principia]]'', and all we his readers underwent in wrestling with those three vast volumes, it is almost sad".<ref>{{Cite journal |last=Beer |first=Stafford |date=1969 |title=Maths Created |journal=Nature |volume=223 |issue=5213 |pages=1392–1393 |doi=10.1038/2231392b0|bibcode=1969Natur.223.1392B |s2cid=5223774 }}</ref>

Banaschewski (1977)<ref>{{cite journal | url=https://projecteuclid.org/download/pdf_1/euclid.ndjfl/1093888028 | author=B. Banaschewski | title=On G. Spencer Brown's Laws of Form | journal=Notre Dame Journal of Formal Logic | volume=18 | number=3 | pages=507–509 | date=Jul 1977 | doi=10.1305/ndjfl/1093888028 | doi-access=free }}</ref> 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 (mathematics)|duality]] characterizing not just [[Boolean algebra (structure)|Boolean algebra]]s but all [[Lattice (order)|lattice]]s;
*Highlights how syntactically distinct statements in logic and '''2''' can have identical [[Semantics of logic|semantics]];
* Dramatically simplifies Boolean algebra calculations, and proofs in [[sentential logic|sentential]] and [[syllogism|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 {{section link||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 was met with skepticism.<ref>For a sympathetic evaluation, see {{harvp|Kauffman|2001}}.</ref>

==The form (Chapter 1)==
The symbol:

:[[Image:Laws of Form - cross.gif]]

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 [[duality (mathematics)|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:
<blockquote>
"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)
</blockquote>

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:

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

[[Charles Sanders Peirce|C. S. Peirce]] came to a related insight in the 1890s; see {{slink||Related work}}.

==The primary arithmetic (Chapter 4)==
The [[syntax]] of the primary arithmetic goes as follows. There are just two [[atomic formula|atomic expressions]]:
* The empty Cross [[Image:Laws of Form - cross.gif]] ;
* All or part of the blank page (the "void").
There are two inductive rules:
* A Cross [[Image:Laws of Form - cross.gif]] may be written over any expression;
* Any two expressions may be [[concatenation|concatenated]].
The [[Semantics of logic|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" [[axiom]]s 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:

::[[Image:Laws of Form - cross.gif]] [[Image:Laws of Form - cross.gif]] <math>\ =</math>[[Image:Laws of Form - cross.gif]]

"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:

::[[Image:Laws of Form - double cross.gif]] <math>\ =</math>

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 metatheorems 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 (mathematics)|relation]] of [[logical equivalence]] [[partition of a set|partitions]] all primary arithmetic expressions into two [[equivalence class]]es: 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 [[confluence (abstract rewriting)|confluent]].

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

===The notion of canon===
While ''LoF'' does not formally define canon, the following two excerpts from the Notes to chpt. 2 are apt:

<blockquote>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.</blockquote>

<blockquote>...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.</blockquote>

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 [[Variable (mathematics)|variables]]. A primary algebra variable indicates a location where one can write the primitive value [[Image:Laws of Form - cross.gif]] or its complement [[Image:Laws of Form - double cross.gif]]. 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 (structure)|Boolean algebra]], that are [[variety (universal algebra)|varieties]]. Equational logic was common before ''Principia Mathematica'' (e.g. {{harvp|Johnson|1892}}), and has present-day advocates ({{harvp|Gries|Schneider|1993}}).

Conventional [[mathematical logic]] consists of [[Tautology (logic)|tautological]] formulae, signalled by a prefixed [[Turnstile (symbol)|turnstile]]. To denote that the ''primary algebra'' formula ''A'' is a [[Tautology (logic)|tautology]], simply write "''A'' =[[Image:Laws of Form - cross.gif]] ". 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:
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
|
{|
|-
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
| A
|}
| A
|}
|}
| = .
|}
The absence of anything to the right of the "=" above, is deliberate.

{|
|-
| 
*J2:
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
|
{|
|-
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
| A
|}
|
{|style="border-top: 2px solid black; border-right: 2px solid black;"
|-
| B
|}
|}
|}
| C
| =
|
{|style="border-top: 2px solid black; border-right: 2px solid black;"
|-
|
{|
|-
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
| A C
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
| B C
|}
|}
|}
|.
|}

'''J2''' is the familiar [[distributive law]] of [[sentential logic]] and [[Boolean algebra (structure)|Boolean algebra]].

Another set of initials, friendlier to calculations, is:

{|
|-
| 
*J0:
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
|
{|
|-
|
|}
|}
|}
| A
| =
| A.
|}

{|
|-
|
*J1a:
|
{|
|-
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
| A
|}
| A
|}
| =
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
|
{|
|-
|
|}
|}
|.
|}

{|
|-
| 
*C2:
| A
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
| A B
|}
| = 
| A
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
| B
|}
|.
|}

It is thanks to '''C2''' that the ''primary algebra'' is a [[lattice (order)|lattice]]. By virtue of '''J1a''', it is a [[complemented lattice]] whose upper bound is [[Image:Laws_of_Form_-_cross.gif]]. By '''J0''', [[Image:Laws_of_Form_-_double_cross.gif]] is the corresponding lower bound and [[identity element]]. '''J0''' is also an algebraic version of '''A2''' and makes clear the sense in which [[Image:Laws_of_Form_-_double_cross.gif]] 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 [[formula|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 [[mathematical induction|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 [[Charles Sanders Peirce|C. S. Peirce]]'s [[existential graph]]s.

''LoF'' asserts that concatenation can be read as [[commutativity|commuting]] and [[associativity|associating]] by default and hence need not be explicitly assumed or demonstrated. (Peirce made a similar assertion about his [[existential graph]]s.) 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''.{{sfnp|Byrne|1946}} This result holds for all [[lattice (order)|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 lattice]]s (which include the ''primary algebra'' and '''2'''). Commutativity is trivial; just set ''A''=[[Image:Laws_of_Form_-_double_cross.gif]]. Associativity: ''AC.D'' = ''CA.D'' = ''CD.A'' = ''A.CD''.
Having demonstrated associativity, the period can be discarded.

The initials in {{harvp|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 ''[[Mathematical proof|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 [[Mathematical proof|proof]] of a [[theorem]] cannot be.

Let ''A'' and ''B'' be ''primary algebra'' [[formula]]s. 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 [[Image:Laws of Form - (A)B.png|50px]] and [[Image:Laws of Form - (B)A.png|50px]] to [[Image:Laws of Form - cross.gif]]. 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 [[Image:Laws of Form - ()A=().png|80px]] ('''C3''' in ''LoF''), [[Image:Laws of Form - ((A))=A.png|80px]] ('''C1'''), and ''AA''=''A'' ('''C5''').

The consequence [[Image:Laws of Form - (((A)B)C)=(AC)((B)C).png|170px]], '''C7''' in ''LoF'', enables an [[algorithm]], sketched in ''LoF'''s 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 (logic)|Boolean algebra]] theorem that every formula has a normal form.

Let ''A'' be a [[formula|subformula]] of some [[formula]] ''B''. When paired with '''C3''', '''J1a''' can be viewed as the closure condition for calculations: ''B'' is a [[Tautology (logic)|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 [[Image:Laws of Form - ((A)A).png|50px]].
The last step of a calculation always invokes '''J1a'''.

''LoF'' includes elegant new proofs of the following standard [[metatheory]]:
* ''[[Completeness (logic)|Completeness]]'': all ''primary algebra'' consequences are demonstrable from the initials (T17).
* ''[[axiom|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 [[two-element Boolean algebra|Boolean]] values 1 and 0 (or '''True''' and '''False'''), the ''primary algebra'' [[interpretation (logic)|interprets]] '''[[two-element Boolean algebra|2]]''' (or [[sentential logic]]). ''LoF'' shows how the ''primary algebra'' can interpret the [[syllogism]]. Each of these [[interpretation (logic)|interpretations]] is discussed in a subsection below. Extending the ''primary algebra'' so that it could [[interpretation (logic)|interpret]] standard [[first-order logic]] has yet to be done, but [[Charles Sanders Peirce|Peirce]]'s ''beta'' [[existential graph]]s 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 (mathematics)|join]] (+) or [[meet (mathematics)|meet]] (×) interpret [[concatenation]];
* The [[Complement (order theory)|complement]] of ''A'' interpret [[Image:Laws of Form - not a.gif]]
* 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 <math>\overline{\overline{A |} \ \ \overline{C |}  \Big|}</math>. 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 theory|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 (mathematics)|set]] <math>\ B=\{</math>[[Image:Laws of Form - cross.gif]] <math>,</math> [[Image:Laws of Form - double cross.gif]]<math>\ \}</math> is the [[Boolean domain]] or ''carrier''. In the language of [[universal algebra]], the ''primary algebra'' is the [[algebraic structure]] <math>\lang B,-\ -,\overline{- \ |},\overline{\ \ |} \rang</math> of type <math>\lang 2,1,0 \rang</math>. The [[functional completeness|expressive adequacy]] of the [[Sheffer stroke]] points to the ''primary algebra'' also being a <math>\lang B,\overline{-\ - \ |},\overline{\ \ |}\rang</math> algebra of type <math>\lang 2,0 \rang</math>. 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 <math>\lang B,+,\lnot,1 \rang</math> algebra of type <math>\lang 2,1,0 \rang</math>. This description of '''2''' is simpler than the conventional one, namely an <math>\lang B,+,\times,\lnot,1,0 \rang</math> algebra of type <math>\lang 2,2,1,0,0 \rang</math>.

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 [[Image:Laws of Form - (A)A=().png|80px]]. Interpreting juxtaposition as OR and [[Image:Laws of Form - cross.gif|30px]] as 1, this translates to <math>\neg A \lor A = 1</math> which is true. Interpreting juxtaposition as AND and [[Image:Laws of Form - cross.gif|30px]] as 0, this translates to <math>\neg A \land A = 0</math> which is true as well (and the dual of <math>\neg A \lor A = 1</math>).

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

This reveals that <math>u</math> is either the value 0 or the OR operator, while <math>m</math> is either the value 1 or the NOR operator, depending on whether <math>x</math> 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:

:::&nbsp;= &nbsp; '''False'''

::[[Image:Laws of Form - cross.gif]] &nbsp;= &nbsp;'''True''' &nbsp;= &nbsp;'''not False'''

::[[Image:Laws of Form - double cross.gif]] &nbsp;= &nbsp;'''Not True''' &nbsp;= &nbsp;'''False'''

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

::[[Image:Laws of Form - not a.gif]] interprets '''Not A'''

::[[Image:Laws of Form - a or b.gif]] interprets '''A Or B'''

::[[Image:Laws of Form - if a then b.gif]] interprets '''Not A Or B'''  or   '''If A Then B'''.

::[[Image:Laws of Form - a and b.gif]] interprets '''Not (Not A Or Not B)'''
:::::or '''Not (If A Then Not B)'''
:::::or '''A And B'''.

{|
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
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{| style="border-top: 2px solid black; border-right: 2px solid black;"
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{| style="border-top: 2px solid black; border-right: 2px solid black;"
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{| style="border-top: 2px solid black; border-right: 2px solid black;"
| <big><big><big>a</big></big></big>
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| <big><big><big>b</big></big></big>
|}
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{| style="border-top: 2px solid black; border-right: 2px solid black;"
| <big><big><big>a</big></big></big>
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{| style="border-top: 2px solid black; border-right: 2px solid black;"
| <big><big><big>b</big></big></big>
|}
|}
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|,
|
{|
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
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{| style="border-top: 2px solid black; border-right: 2px solid black;"
| <big><big><big>a</big></big></big>
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{| style="border-top: 2px solid black; border-right: 2px solid black;"
| <big><big><big>b</big></big></big>
|}
|}
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{| style="border-top: 2px solid black; border-right: 2px solid black;"
| <big><big><big>a&nbsp;b</big></big></big>
|}
|}
| both interpret '''A [[if and only if]] B''' or '''A is [[logical equivalence|equivalent]] to B'''.
|}

Thus any expression in [[sentential logic]] has a ''primary algebra'' translation. Equivalently, the ''primary algebra'' [[interpretation (logic)|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 [[Tautology (logic)|tautological]] or [[Satisfiability|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<sup>''N''</sup> primary arithmetic formulae. For a less tedious decision procedure more in the spirit of [[Willard Van Orman Quine|Quine]]'s "truth value analysis", see {{harvp|Meguire|2003}}.

{{harvp|Schwartz|1981}} proved that the ''primary algebra'' is equivalent — [[syntax|syntactically]], [[Semantics of logic|semantically]], and [[proof theory|proof theoretically]] — with the [[Propositional calculus|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 value]]s '''true''' and '''false''', the [[logical connective]]s 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 [[Image:Laws of Form - (A(B)).png|50px]] instead of [[Image:Laws of Form - (A)B.png|50px]]. More generally, the ''primary algebra'' is "self-[[Duality (mathematics)|dual]]", meaning that any ''primary algebra'' formula has two [[sentential logic|sentential]] or [[two-element Boolean algebra|Boolean]] readings, each the [[Duality (mathematics)|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, [[Logical complement|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 (mathematics)|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 [[syllogism]]s and [[polysyllogism|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 <math>\overline{A |}</math>, 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 [[syllogism|Barbara]] whose ''primary algebra'' equivalent is <math>\overline{A^* \ B |} \ \ \overline{\overline{B |} \ C^* \Big|} \ A^* \ C^* </math>. These 24 possible permutations include the 19 syllogistic forms deemed valid in [[Aristotelian logic|Aristotelian]] and [[medieval logic]]. This ''primary algebra'' translation of syllogistic logic also suggests that the ''primary algebra'' can [[interpretation (logic)|interpret]] [[monadic logic|monadic]] and [[term logic]], and that the ''primary algebra'' has affinities to the [[Boolean term schema]]ta of {{harvp|Quine|1982|loc=Part II}}.

===An example of calculation===
The following calculation of [[Gottfried Wilhelm Leibniz|Leibniz]]'s nontrivial ''Praeclarum Theorema'' exemplifies the demonstrative power of the ''primary algebra''. Let C1 be <math>\overline{\overline{A |} \Big|}</math> =''A'', C2 be <math>A \ \overline{A \ B |} = A \ \overline{B |}</math>, C3 be <math>\overline{\ \ |} \ A = \overline{\ \ |}</math>, J1a be <math>\overline{A |} \ A = \overline{\ \ |}</math>, 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''.
|-
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| .
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| ''primary algebra'' translation
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| .
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| C1.
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| R
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| .
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| C1.
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| OI.
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| C2.
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| OI.
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{| style="border-top: 2px solid black; border-right: 2px solid black;"
| <span style="color:white;">B</span>
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{| style="border-top: 2px solid black; border-right: 2px solid black;"
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| C3. <math>\square</math>
|}

===Relation to magmas===
The ''primary algebra'' embodies a point noted by [[Edward Vermilye Huntington|Huntington]] in 1933: [[Boolean algebra (logic)|Boolean algebra]] requires, in addition to one [[unary operation]], one, and not two, [[binary operation]]s. Hence the seldom-noted fact that Boolean algebras are [[magma (algebra)|magmas]]. (Magmas were called [[groupoid]]s 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 [[Commutative property|commute]]s and [[associative property|associates]];
*[[Monoid]] with [[identity element]] [[Image:Laws of Form - double cross.gif]], by virtue of '''J0'''.

[[group (mathematics)|Groups]] also require a [[unary operation]], called [[Group (mathematics)#Definition|inverse]], the group counterpart of [[Boolean algebra (logic)|Boolean complementation]]. Let [[Image:Laws of Form - (a).png|20px]] denote the inverse of ''a''. Let [[Image:Laws of Form - cross.gif]] denote the group [[identity element]]. Then groups and the ''primary algebra'' have the same [[signature (logic)|signatures]], namely they are both <math>\lang - \ -, \overline{- \ |}, \overline{\ \ |} \rang</math> algebras of type 〈2,1,0〉. Hence the ''primary algebra'' is a [[list of algebraic structures|boundary algebra]]. The axioms for an [[abelian group]], in boundary notation, are:
* '''G1'''. ''abc'' = ''acb'' (assuming association from the left);
* '''G2'''. [[Image:Laws of Form - ()a=a.png|80px]]
* '''G3'''. [[Image:Laws of Form - (a)a=().png|80px]].
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 &nbsp;&nbsp;[[Image:Laws of Form - double cross.gif|25px]]&nbsp;=&nbsp;[[Image:Laws of Form - cross.gif|20px]]&nbsp;&nbsp; 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 [[Image:Laws of Form - double cross.gif]] ≠ [[Image:Laws of Form - cross.gif]]. If the ''primary algebra'' were a [[group (mathematics)|group]], [[Image:Laws of Form - double cross.gif]] = [[Image:Laws of Form - cross.gif]] would hold, and one of &nbsp;&nbsp;[[Image:Laws of Form - (a).png|20px]]&nbsp;''a''&nbsp;=&nbsp;[[Image:Laws of Form - double cross.gif|30px]]&nbsp;&nbsp; or &nbsp;&nbsp;''a''&nbsp;[[Image:Laws of Form - cross.gif|30px]]&nbsp;=&nbsp;''a''&nbsp;&nbsp; would have to be a ''primary algebra'' consequence. Note that [[Image:Laws of Form - cross.gif|20px]] and [[Image:Laws of Form - double cross.gif|25px]] are mutual ''primary algebra'' complements, as group theory requires, so that <math>\overline{\ \overline{\ \overline{\ \ |} \ \Big|} \ \Bigg|} = \overline{\ \ |}</math> 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 [[lattice (order)|lattices]], and the [[distributive law]] central to [[Boolean algebra (structure)|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 [[recursion|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''.

{{harvp|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==
{{original research|section|date=December 2024}}
[[Gottfried Leibniz]], in memoranda not published before the late 19th and early 20th centuries, invented [[Boolean algebra (logic)|Boolean logic]]. His notation was isomorphic to that of ''LoF'': concatenation read as [[Logical conjunction|conjunction]], and "non-(''X'')" read as the [[Logical complement|complement]] of ''X''. Recognition of Leibniz's pioneering role in [[algebraic logic]] was foreshadowed by {{harvp|Lewis|1918}} and {{harvp|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 {{harvp|Lenzen|2004}}.

[[Charles Sanders Peirce]] (1839–1914) anticipated the ''primary algebra'' in three veins of work:
#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,{{sfnp|Peirce|1976|loc=101-15.1}} but they were not published in full until 1993.<ref>"Qualitative Logic", MS 582 (1886) in Kloesel, Christian et al., eds., 1993. ''[[Charles Sanders Peirce bibliography#W|Writings of Charles S. Peirce: A Chronological Edition]], Vol. 5, 1884–1886''. [[Indiana University Press]]: 323-71. "The Logic of Relatives: Qualitative and Quantitative", MS 584 (1886) in Kloesel, Christian et al., eds., 1993. ''Writings of Charles S. Peirce: A Chronological Edition, Vol. 5, 1884–1886''. Indiana University Press: 372-78.</ref>
#In a 1902 encyclopedia article,<ref>Reprinted in Peirce, C. S. (1933) ''[[Charles Sanders Peirce bibliography#CP|Collected Papers of Charles Sanders Peirce]], Vol. 4'', [[Charles Hartshorne]] and [[Paul Weiss (philosopher)|Paul Weiss]], eds. [[Harvard University Press]]. Paragraphs 378–383</ref> 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.
#The [[syntax]] of his alpha [[existential graph]]s is merely [[concatenation]], read as [[Logical conjunction|conjunction]], and enclosure by ovals, read as [[negation]].<ref>The existential graphs are described at length in Peirce, C. S. (1933) ''Collected Papers, Vol. 4'', [[Charles Hartshorne]] and [[Paul Weiss (philosopher)|Paul Weiss]], eds. Harvard University Press. Paragraphs 347–529.</ref> If ''primary algebra'' concatenation is read as [[Logical conjunction|conjunction]], then these graphs are [[isomorphic]] to the ''primary algebra''.{{sfnp|Kauffman|2001}}

''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''.

{{harvp|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, [[Polish notation|prefix]], or [[Reverse Polish notation|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 [[David Hilbert|Hilbert]] and [[Jan Łukasiewicz|Łukasiewicz]] showed how to prove [[axiom independence]] using [[model theory|model]]s.

{{harvp|Craig|1979}} argued that the world, and how humans perceive and interact with that world, has a rich Boolean structure. [[William Craig (logician)|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 {{harvp|Lakoff|1987}} (see index entries under "Image schema examples: container") and {{harvp|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 (order)|lattice]] structure very similar to that of Boolean algebra. For a few authors, mereology is simply a [[model theory|model]] of [[Boolean algebra (structure)|Boolean algebra]] and hence of the primary algebra as well.
*[[Mereotopology]], which is inherently richer than Boolean algebra;
*The system of {{harvp|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, [[Sheffer stroke|NAND]] (whose ''primary algebra'' translation is <math>\overline{A \ \ B \ |}</math> or, dually, <math>\overline{A |} \ \ \overline{B |}</math>), universal [[Quantification (logic)|quantification]], and one [[binary relation|binary]] [[atomic formula]], denoting [[Set (mathematics)|set]] membership. This is the system of {{harvp|Quine|1951}}.
* The ''beta'' [[existential graph]]s, 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. {{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, 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==
*{{annotated link|Boolean algebra}}
*{{annotated link|Boolean algebras canonically defined}}
*{{annotated link|Entitative graph}}
*{{annotated link|Existential graph}}
*{{annotated link|Mark and space}}
*{{annotated link|Programming and Metaprogramming in the Human Biocomputer|''Programming and Metaprogramming''}}
*{{annotated link|Propositional calculus}}
*{{annotated link|Two-element Boolean algebra}}
*[[List of Boolean algebra topics]]

==Notes==
{{notelist}}

==References==
{{Reflist}}

===Works cited===
{{refbegin}}
*{{cite journal |last=Byrne |first=Lee |date=1946 |title=Two Formulations of Boolean Algebra |journal=Bulletin of the American Mathematical Society |pages=268–71}}
*{{Cite journal | doi = 10.2307/3131383 | last1 = Craig | first1 = William |author1-link=William Craig (logician) |year = 1979 | title = Boolean Logic and the Everyday Physical World | jstor = 3131383| journal = Proceedings and Addresses of the American Philosophical Association | volume = 52 | issue = 6| pages = 751–78 }}
*{{cite book |last1=Gries |first1=David |author1-link=David Gries |last2=Schneider |first2=F. B. |year=1993 |title=A Logical Approach to Discrete Math |publisher=Springer-Verlag |isbn=978-0-387-94115-8}}
*{{cite journal |last=Johnson |first=William Ernest |author-link=William Ernest Johnson |date=1892 |title=The Logical Calculus |journal=Mind |volume=1 (n.s.) |pages=3–30}}
*{{cite journal |first=Louis H. |last=Kauffman |date=2001 |url=http://www2.math.uic.edu/~kauffman/CHK.pdf |title=The Mathematics of C. S. Peirce |journal=Cybernetics and Human Knowing |volume=8 |pages=79–110}}
*{{cite book |author-link=George Lakoff |last=Lakoff |first=George |year=1987 |title=Women, Fire, and Dangerous Things: What Categories Reveal about the Mind |publisher=University of Chicago Press}}{{ISBN?}}
*{{cite book |last1=Lakoff |first1=George |last2=Núñez |first2=Rafael E. |author2-link=Rafael E. Núñez |year=2000 |title=Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being |title-link=Where Mathematics Comes From |publisher=Basic Books |isbn=978-0-465-03770-4}}
*{{cite book |last=Lenzen |first=Wolfgang |year=2004 |chapter-url=http://www.philosophie.uni-osnabrueck.de/Publikationen%20Lenzen/Lenzen%20Leibniz%20Logic.pdf |chapter=Leibniz's Logic |editor1-last=Gabbay |editor1-first=D. |editor2-last=Woods |editor2-first=J. |title=The Rise of Modern Logic: From Leibniz to Frege |series=Handbook of the History of Logic |volume=3 |place=Amsterdam |publisher=Elsevier |isbn=978-0-08-053287-5 |pages=1–83}}
*{{cite book |last=Lewis |first=C. I. |author-link=C. I. Lewis |year=1918 |url=https://archive.org/details/asurveyofsymboli00lewiuoft |url-access=registration |title=A Survey of Symbolic Logic |place=Berkeley |publisher=University of California Press}} Republished in part by Dover in 1960.
*{{cite journal | last1 = Meguire | first1 = P. G. | year = 2003 | title = Discovering Boundary Algebra: A Simplified Notation for Boolean Algebra and the Truth Functors | journal = International Journal of General Systems | volume = 32 | pages = 25–87 | doi=10.1080/0308107031000075690| citeseerx = 10.1.1.106.634 | s2cid = 9460101 }}
*{{cite book |last1=Meguire |first1=P. G. |year=2011 |title=Boundary Algebra: A Simpler Approach to Basic Logic and Boolean Algebra |publisher=VDM Publishing |isbn=978-3-639-36749-2}}
*{{cite book |last=Peirce |first=Charles S. |author-link=Charles Sanders Peirce |chapter=Qualitative Logic, MS 736 (c. 1886) |editor-last=Eisele |editor-first=Carolyn |year=1976 |title=[[Charles Sanders Peirce bibliography#NEM|The New Elements of Mathematics by Charles S. Peirce]] |volume=4, Mathematical Philosophy |place=The Hague |publisher=Mouton}}
*{{cite book |last1=Quine |first1=Willard |author1-link=Willard Quine |year=1951 |title=Mathematical Logic |edition=2nd |publisher=Harvard University Press}}
*{{cite book |last1=Quine |first1=Willard |year=1982 |title=Methods of Logic |edition=4th |publisher=Harvard University Press |isbn=978-0-674-57176-1}}
*{{Cite journal | last1 = Rescher | first1 = Nicholas | authorlink = Nicholas Rescher | date = 1954 | title = Leibniz's Interpretation of His Logical Calculi | journal = Journal of Symbolic Logic | volume = 18 | issue = 1| pages = 1–13 | doi=10.2307/2267644| jstor = 2267644 | s2cid = 689315 }}
*{{cite book |last=Russell |first=Bertrand |author-link=Bertrand Russell |year=2014 |orig-year=1967 |title=The Autobiography of Bertrand Russell |publisher=Taylor & Francis |isbn=978-1-317-83504-2}}
*{{Cite journal | doi = 10.1080/03081078108934802 | last1 = Schwartz | first1 = Daniel G. | date = 1981 | title = Isomorphisms of G. Spencer-Brown's ''Laws of Form'' and F. Varela's Calculus for Self-Reference | journal = International Journal of General Systems | volume = 6 | issue = 4| pages = 239–55 }}
*{{cite book |first=George |last=Spencer-Brown |author-link=George Spencer-Brown |title=Laws of Form |year=1969 |place=London |publisher=George Allen and Unwin Ltd. |isbn=978-0-04-510028-6 |url=https://archive.org/details/lawsofform0000spen |url-access=registration}}
*{{Cite book |last=Spencer-Brown |first=G. |chapter=Design with the NOR |title=George Spencer Brown's "Design with the NOR": With Related Essays |publisher=Emerald Publishing |year=2021 |doi=10.1108/9781839826108 |isbn=978-1-83982-611-5 |editor-last1=Roth |editor-last2=Heidingsfelder |editor-last3=Clausen |editor-last4=Laursen |editor-first1=Steffen |editor-first2=Markus |editor-first3=Lars |editor-first4=Klaus Brønd |pages=7–22}}
*{{Cite journal | doi = 10.1080/03081078608934939 | last1 = Turney | first1 = P. D. | date = 1986 | title = ''Laws of Form'' and Finite Automata | journal = International Journal of General Systems | volume = 12 | issue = 4| pages = 307–18 }}
*{{cite journal |last1=Whitehead |first1=A. N. |author1-link=A. N. Whitehead |date=1934 |title=Indication, classes, number, validation |journal=Mind |volume=43 (n.s.) |issue=171 |pages=281–97, 543|doi=10.1093/mind/XLIII.171.281 }} The corrigenda on p.&nbsp;543 are numerous and important, and later reprints of this article do not incorporate them.
{{refend}}

==Further reading==
{{refbegin}}
*{{cite book |editor-first=Dirk |editor-last=Baecker |year=1999 |title=Problems of Form |publisher=Stanford University Press}}
*{{cite journal |editor-first=Dirk |editor-last=Baecker |date=2013 |title=Foreword: A Mathematics of Form, A Sociology of Observers |url=http://www.ingentaconnect.com/content/imp/chk/2013/00000020/f0020003 |journal=Cybernetics & Human Knowing |volume=20 |number=3–4}}
*{{cite book |last=Bostock |first=David |year=1997 |title=Intermediate Logic |publisher=Oxford University Press}}
*{{cite arXiv |first=Louis H. |last=Kauffman |date=2006 |eprint=math.CO/0112266 |title=Reformulating the Map Color Theorem}}
*{{cite book |first=Louis H. |last=Kauffman |year=2006a |url=http://www.math.uic.edu/~kauffman/Laws.pdf |title=Laws of Form: An Exploration in Mathematics and Foundations}}
*{{cite journal |editor-first=Louis H. |editor-last=Kauffman |year=2019 |url=http://chkjournal.com/node/326 |journal=Cybernetics & Human Knowing |volume=26 |number=2–3 |title=Laws of Form: Spencer-Brown at Esalen, 1973}}
{{refend}}

==External links==
{{external links|date=November 2024}}
* [https://web.archive.org/web/20150315203300/http://lawsofform.org/lof.html Laws of Form], archive of website by Richard Shoup.
* [https://web.archive.org/web/20060821204917/http://lawsofform.org/aum/session1.html Spencer-Brown's talks at Esalen, 1973.] Self-referential forms are introduced in the section entitled "Degree of Equations and the Theory of Types".
* [https://www.kurtvonmeier.com/blog-1/2018/3/19/on-audio-the-aum-conference-opening-session-1973?rq=aum Audio recording of the opening session, 1973 AUM Conference at Esalen]. 
* [http://www.math.uic.edu/~kauffman/ Louis H. Kauffman], "[http://www.math.uic.edu/~kauffman/Arithmetic.htm Box Algebra, Boundary Mathematics, Logic, and Laws of Form.]"
* Kissel, Matthias, "[https://web.archive.org/web/20070310071916/http://de.geocities.com/matthias_kissel/gdf/LoF.html  A nonsystematic but easy to understand introduction to ''Laws of Form''.]"
* A meeting  [http://www.omath.org.il/112431/4CT with G.S.B] by Moshe Klein
* [http://markability.net The Markable Mark], an introduction by easy stages to the ideas of ''Laws of Form''
* [https://arxiv.org/abs/1905.12891 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 - [https://www.youtube.com/playlist?list=PLoK3NtWr5NbqEOdjQrWaq1sDweF7NJ5NB 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.

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