- Laws of Form
**"Laws of Form**" (hereinafter "LoF") is a book byG. Spencer-Brown , published in 1969, that straddles the boundary betweenmathematics and ofphilosophy . "LoF" describes three distinctlogical system s:

* The "primary arithmetic" (described in Chapter 4), whose models include Boolean arithmetic;

* The "primaryalgebra " (Chapter 6), which interprets thetwo-element Boolean algebra (hereinafter abbreviated**2**),Boolean logic , and the classicalpropositional calculus ;

* "Equations of the second degree" (Chapter 11), whose interpretations includefinite automata andAlonzo Church 's Restricted Recursive Arithmetic (RRA).Spencer-Brown referred to the mathematical system of "Laws of Form" as the "primary algebra" and the "calculus of indications"; others have termed it boundary algebra. "Laws of Form" may refer to "LoF" or to the primary algebra (hereinafter abbreviated "pa").

**The book**"LoF" emerged out of work in electronic engineering its author did around 1960, and from 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 most recent a 1997 German translation, and has never gone out of print.The mathematics fills only about 55pp and is rather elementary. But "LoF"'s mystical and declamatory prose, and its love of

paradox , 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 Peirce ,Bertrand Russell , andAlfred North Whitehead .**Reception**Ostensibly a work of formal mathematics and philosophy, "LoF" became something of a

cult classic , praised in the "Whole Earth Catalog ". Those who agree point to "LoF" as embodying an enigmatic "mathematics ofconsciousness ," its algebraic symbolism capturing an (perhaps even "the") implicit root ofcognition : the ability to "distinguish". "LoF" argues that the "pa" reveals striking connections amonglogic , Boolean algebra, and arithmetic, and thephilosophy of language andmind .Some, e.g. [

*http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.ndjfl/1093888028?abstract= Banaschewski (1977),*] argue that the "pa" is nothing but new notation for Boolean algebra. It is true that**2**can be seen as the intended interpretation of the "pa". Nevertheless, [*http://www.lawsofform.org/docs/Meguire_LoF.pdf Meguire (2005)*] counters that "pa" notation:

* Fully exploits theduality characterizing not just Boolean algebras but all lattices;

*Highlights how syntactically distinct statements in logic and**2**can have identicalsemantics ;

* Dramatically simplifies Boolean algebra calculations, and proofs in sentential and syllogisticlogic .Moreover, the syntax of the "pa" can be extended to formal systems other than**2**and sentential logic, resulting in "boundary mathematics" (see Related Work below)."LoF" has influenced, among others,

Heinz von Foerster ,Louis Kauffman ,Niklas Luhmann ,Humberto Maturana ,Francisco Varela andWilliam Bricken . Some of these authors 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 "pa". Spencer-Brown eventually circulated a purported proof of the Four Color Theorem [*For a sympathetic evaluation, see [*] . The proof met with skepticism and Spencer-Brown's mathematical reputation, as well as that of "LoF", went into decline. (The Four Color Theorem and Fermat's Last Theorem were proved in 1976 and 1995, respectively, using methods owing nothing to "LoF".)*http://www.arxiv.org/math.CO/0112266 Kauffman (2001)*] .**The form (Chapter 1)**The symbol:

:

also called the

**"mark**" or**"cross**", is the essential feature of the Laws ofForm . In Spencer-Brown's inimitable and enigmatic fashion, the Mark symbolizes the root ofcognition , i.e., the dualistic Mark indicates thecapability of differentiating a "this" from "everything else "but" this."In "LoF", a Cross denotes the drawing of a "distinction", and can be thought of as signifying the following, all at once:

* The act of drawing a boundary around something, thus separating it from everything else;

* That which becomes distinct from everything by drawing the boundary;

* Crossing from one side of the boundary to the other.All three ways imply an "action" on the part of the cognitive entity (e.g., person) making the distinction. As "LoF" puts it:

"The first command:

* Draw a distinctioncan 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, 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 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."

Charles Peirce came to a related insight in the 1890s; see Related Work.**The primary arithmetic (Chapter 4)**The

syntax of the**primary arithmetic**(**PA**) goes as follows. There are just two atomic expressions:

* The empty Cross ;

*All or part of the blank page (the "void").There are two inductive rules:

* A Cross may be written over any expression;

* Any two expressions may be concatenated.Thesemantics of the primary arithmetic are perhaps nothing more than the sole explicitdefinition 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 of the unmarked or marked states 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**. Crossing twice from the unmarked to the marked state is indistinguishable from crossing once. To make a distinction twice has the same effect as making it once. For example, saying "Let there be light." and then saying "Let there be light." again, is the same as saying it once. Formally::: $=$

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

In both

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

* Every expression has a unique simplification. (**T3**in "LoF");

* Starting from an initial marked or unmarked state, "complicating" an expression by repeated application of**A1**and**A2**cannot yield an expression whose simplification differs from the initial state. (**T4**in "LoF").Thus the relation oflogical equivalence partitions all primary arithmetic expressions into twoequivalence 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 andcomputer science :

* ADyck language of order 1 with a null alphabet;

* The simplestcontext-free language in theChomsky hierarchy ;

* Arewrite system that isstrongly normalizing andconfluent .The phrase "calculus of indications" in "LoF" is a synonym for "primary arithmetic".

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

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

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

These excerpts relate to the distinction in

metalogic between theobject 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 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)****yntax**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 "pa"

formula . Letters so employed inmathematics andlogic are calledvariable s. A "pa" variable indicates a location where one can write the primitive value or its complement . Multiple instances of the same variable denote multiple locations of the same primitive value.**Rules governing logical equivalence**The sign '=' may link two logically equivalent expressions; the result is an

equation . By "logically equivalent" is meant that the two expressions have the same simplification.Logical equivalence is anequivalence relation over the set of "pa" 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 "pa" demonstrations (see below), almost always silently. These rules are routinely invoked inlogic and most of mathematics, nearly always unconsciously.The "pa" consists of

equations , i.e., pairs of formulae linked by an infix '='.**R1**and**R2**enable transforming one equation into another. Hence the "pa" is an "equational" formal system, like the manyalgebraic structures , includingBoolean algebra , that are varieties. Equational logic was common before "Principia Mathematica" (e.g., Peirce,^{1,2,3}Johnson 1892), and has present-day advocates (Gries and Schneider 1993).Conventional

mathematical logic consists of tautological formulae, signalled by a prefixedturnstile . To denote that the "pa" formula "A" is a tautology, simply write "A" = ". If one replaces '=' in**R1**and**R2**with thebiconditional , the resulting rules hold in conventional logic. However, conventional logic relies mainly on the rulemodus 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 "pa" equation verifiable by a

decision procedure and as such is "not" anaxiom . "LoF" lays down the initials:*$J1:\; ((A)A)\; =\; .$The absence of anything to the right of the "=" above, is deliberate.

*$J2:\; ((A)(B))C\; =\; ((AC)(BC)).$

**J2**is the familiardistributive law ofsentential logic and Boolean algebra.Another set of initials, friendlier to calculations, is:

*$J0:\; (())A\; =\; A.$

*$J1a:\; (A)A\; =\; ()$

*$C2:\; A(AB)=A(B).$

It is thanks to

**C2**that the "pa" is a lattice. By virtue of**J1a**, it is acomplemented lattice whose upper bound andinverse element is (). By**J0**, (()) is the corresponding lower bound andidentity element .**J0**is also an algebraic version of**A2**and makes clear the sense in which (()) aliases with the blank page.T13 in "LoF" generalizes

**C2**as follows. Any "pa" (or sentential logic) formula "B" can be viewed as an ordered tree with "branches". Then:**T13**: A subformula "A" can be copied at will into any depth of "B" greater than that of "A", as long as "A" and its copy are in the same branch of "B". Also, given multiple instances of "A" in the same branch of "B", all instances but the shallowest are redundant.While a proof of T13 would require

induction , the intuition underlying it should be clear.**C2**or its equivalent is named:

*"Generation" in "LoF";

*"Exclusion" in Johnson (1892);

*"Pervasion" in the work of William Bricken;

*"Mimesis" in the entrylogical nand .Charles Peirce 'sexistential graph s was perhaps the first formal system to appreciate the power of**C2**. His Rule of "(De)Iteration" combined T13 and "AA=A"."LoF" asserts that concatenation can be read as commuting and associating by default and hence need not be explicitly assumed or demonstrated. (Peirce made a similar assumption in his graphical logic.) Let a period denote grouping. That concatenation commutes and associates may then be demonstrated from the:

* Initial "AB.C"="BC.A" and the consequence "AA"="A" (Byrne 1946). This result holds for all lattices, because "AA"="A" is an easy consequence of theabsorption law , which holds for all lattices;

* Initials "AB.C"="AC.B" and**J0**. Since**J0**holds only for lattices with a lower bound, this method holds only forbounded lattice s (which include the "pa" and**2**). Commutativity is trivial; just set "A"=(()). Associativity: "AB.C" = "BA.C" = "BC.A" = "A.BC".Now that associativity is demonstrated, the period is no longer required.**Proof theory**The "pa" contains three kinds of proved assertions:

* "Consequence" is a "pa" equation verified by a "demonstration". A demonstration consists of a sequence of "steps", each step justified by an initial or a previously demonstrated consequence.

* "Theorem " is a statement in themetalanguage verified by a "proof", i.e., an argument, formulated in the metalanguage, that is accepted by trained mathematicians and logicians.

* "Initial", defined above. Demonstrations and proofs invoke an initial as if it were an axiom.The distinction between consequence and

theorem holds for all formal systems, including mathematics and logic, but is usually not made explicit. A demonstration ordecision procedure can be carried out and verified by computer. The proof of atheorem cannot be.Let "A" and "B" be "pa"

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 ("A")"B" and ("B")"A" to . This is known as a "calculation".Once "A"="B" has been demonstrated, "A"="B" can be invoked to justify steps in subsequent demonstrations. "pa" demonstrations and calculations often require no more than**J1a**,**J2**,**C2**, and the consequences ()"A"=() (**C3**in "LoF"), (("A"))="A" (**C1**), and "AA"="A" (**C5**).The consequence ((("A")"B")"C") = ("AC")(("B")"C"),

**C7**in "LoF", enables analgorithm , sketched in "LoF"s proof of T14, that transforms an arbitrary "pa" formula to an equivalent formula whose depth does not exceed two. The result is a "normal form", the "pa" analog of theconjunctive normal form . "LoF" (T14-15) proves the "pa" analog of the well-known Boolean algebra theorem that every formula has a normal form.Let "A" be a subformula of some

formula "B". When paired with**C3**,**J1a**can be viewed as the closure condition for calculations: "B" is a tautologyif and only if "A" and ("A") both appear in depth 0 of "B". 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 (("A")"A").The last step of a calculation always invokes**J1a**."LoF" includes elegant new proofs of the following standard

metatheory :

* "Completeness ": all "pa" consequences are demonstrable from the initials (T17).

* "Independence":**J1**cannot be demonstrated from**J2**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 of**2**.

= Interpretations =If the Marked and Unmarked states are read as the Boolean values 1 and 0 (or

**True**and**False**), the "pa" interprets**2**(orsentential logic ). "LoF" shows how the "pa" can interpret thesyllogism . Each of these interpretations is discussed in a subsection below. Extending the "pa" so that it could interpret standardfirst-order logic has yet to be done, but Peirce's "beta"existential graph s suggest that this extension is feasible.**Two-element Boolean algebra 2**The "pa" is an elegant minimalist notation for the

two-element Boolean algebra **2**. Let:

* One of Booleanmeet (×) orjoin (+) interpretconcatenation ;

* The complement of "A" interpret

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

Boolean domain or "carrier". In the language ofuniversal algebra , the "pa" is thealgebraic structure $lang\; B,--,(-),()\; ang$ of type $lang\; 2,1,0\; ang$. The expressive adequacy of theSheffer stroke points to the "pa" also being a $lang\; B,(--),()\; ang$ algebra of type $lang\; 2,0\; ang$. In both cases, the identities are J1a, J0, C2, and "ACD=CDA". Since the "pa" and**2**areisomorphic ,**2**can be seen as a $lang\; B,+,lnot,1\; ang$ algebra of type $lang\; 2,1,0\; ang$. This description of**2**is simpler than the conventional one, namely an $lang\; B,+,\; imes,lnot,1,0\; ang$ algebra of type $lang\; 2,2,1,0,0\; ang$.**Sentential logic**Let the blank page denote

**True**or**False**, and let a Cross be read as**Not**. Then the primary arithmetic has the following sentential reading:::: =

**False**:: =

**True**=**not False**:: =

**Not True**=**False**The "pa" interprets sentential logic as follows. A letter represents any given sentential expression. Thus:

:: interprets

**Not A**:: interprets

**A Or B**:: interprets

**Not A Or B**or**If A Then B**.:: interprets

**Not (Not A Or Not B)**:::::or**Not (If A Then Not B)**:::::or**A And B**.$(((A)B)(A(B))),\; ((A)(B))(AB)$ both interpret

**A**orif and only if B**A is equivalent to B**.Thus any expression in sentential logic has a "pa" translation. Equivalently, the "pa" interprets sentential logic. Given an assignment of every variable to the Marked or Unmarked states, this "pa" translation reduces to a primary arithmetic expression, which can be simplified. Repeating this exercise for all possible assignments of the two primitive values to each variable, reveals whether the original expression is tautological or satisfiable. This is an example of a

decision procedure , one more or less in the spirit of conventional truth tables. Given some "pa" formula containing "N" variables, this decision procedure requires simplifying 2^{"N"}PA formulae. For a less tedious decision procedure more in the spirit of Quine's "truth value analysis," see Meguire (2003).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 ofconcatenation change from OR to AND. IF A THEN B now translates as ("A"("B")) instead of ("A")"B". More generally, the "pa" is "self-dual," meaning that any "pa" formula has two sentential or Boolean readings, each the dual of the other. Another consequence of self-duality is the irrelevance ofDeMorgan's laws ; those laws are built into the "pa" syntax from the outset.The true nature of the distinction between the "pa" on the one hand, and

**2**and sentential logic on the other, now emerges. In the latter formalisms,complementation /negation operating on "nothing" is not well-formed. But an empty Cross is a well-formed "pa" expression, denoting the Marked state, a primitive value. Hence a nonempty Cross is anoperator , while an empty Cross is anoperand by virtue of denoted a primitive value. Thus the "pa" reveals that the heretofore distinct mathematical concepts of operator and operand are in fact merely different facets of a single fundamental action, the making of a distinction.**yllogisms**Appendix 2 of "LoF" shows how to translate traditional

syllogism s and sorites into the "pa". A valid syllogism is simply one whose "pa" translation simplifies to an empty Cross. Let "A"* denote a "literal", i.e., either "A" or ("A"), indifferently. Then all syllogisms that do not require that one or more terms be assumed nonempty are one of 24 possible permutations of a generalization of Barbara whose "pa" equivalent is ("A"*"B")(("B")"C"*)"A"*"C"*. These 24 possible permutations include the 19 syllogistic forms deemed valid in Aristotelian andmedieval logic . This "pa" translation of syllogistic logic also suggests that the "pa" can interpret monadic andterm logic , and that the "pa" has affinities to theBoolean term schema ta of Quine (1982: Part II).**An example of calculation**The following calculation of Leibniz's nontrivial "Praeclarum Theorema" exemplifies the demonstrative power of the "pa". Let C1 be (("A"))="A", and let OI mean that variables and subformulae have been reordered in a way that commutativity and associativity permit. Because the only

commutative connective appearing in the "Theorema" isconjunction , it is simpler to translate the "Theorema" into the "pa" using the dual interpretation. The objective then becomes one of simplifying that translation to (()).* [("P"→"R")∧("Q"→"S")] → [("P"∧"Q")→("R"∧"S")] . "Praeclarum Theorema".

* (("P"("R"))("Q"("S"))(("PQ"("RS")))). "pa" translation.

*= ((**P**("R"))"P"(**Q**("S"))"Q"("RS")). OI; C1.

*= ((("R"))(("S"))"PQ"("RS"). Invoke C2 2x to eliminate the bold letters in the previous expression; OI.

*= ("RSPQ"("RS")). C1,2x.

*= (("RSPQ")"RSPQ"). C2; OI.

*= (()). J1.$square$"Remarks":

* C1 (C2) is repeatedly invoked in a fairly mechanical way to eliminate nested parentheses (variable instances). This is the essence of the calculation method;

* A single invocation of J1 (or, in other contexts, J1a) terminates the calculation. This too is typical;

* Experienced users of the "pa" are free to invoke OI silently. OI aside, the demonstration requires a mere 7 steps.**A technical aside**Given some standard notions from

mathematical logic and some suggestions in Bostock (1997: 83, fn 11, 12), {} and may be interpreted as the classical bivalenttruth value s. Let theextension of an "n"-placeatomic formula be the set of ordered "n"-tuples ofindividual s that satisfy it (i.e., for which it comes out true). Let asentential variable be a 0-place atomic formula, whose extension is a classical truth value, by definition. An ordered 2-tuple is anordered pair , whose standard set theoretic definition is <"a","b"> = "a"},{"a","b", where "a","b" are individuals. Ordered "n"-tuples for any "n">2 may be obtained from ordered pairs by a well-known recursive construction.Dana Scott has remarked that the extension of a sentential variable can also be seen as the empty ordered pair (ordered 0-tuple), },{ = because {"a","a"}={"a"} for all "a". Hence has the interpretation**True**. Reading {} as**False**follows naturally.**Relation to groupoids**The "pa" can be seen as the logical endpoint of a point noted by Huntington in 1933: Boolean algebra requires two, not three, operations, one binary and one unary. Hence the seldom-noted fact that Boolean algebras are magmas (a.k.a. groupoids). To see this, note that the "pa" is a

commutative :

*Semigroup because "pa" juxtapositioncommute s andassociate s;

*Monoid withidentity element (()), by virtue of**J0**.Groups also require a

unary operation , calledinverse , whoseinverse element is at once the inverse of, and equal to, the identity element. Complementation is the "pa" unary operation corresponding to group inverse. By**J1a**, the "pa" inverse element is (). Groups and the "pa" have signatures of the same form, namely they both are ⟨--,(-),()⟩ algebras of type ⟨2,1,0⟩. Hence the "pa" is a boundary algebra.The axioms and initials of the "pa" distinguish it from an

abelian group in two ways:

*While the "pa" inverse element () and identity element (()) are mutual complements, as group theory requires,**A2**rules out their being identical. This follows from "B" being anordered set . If the "pa" were a group, one of ("a")"a"=(()) or "a"()="a" would have to be a "pa" consequence;

***C2**demarcates the "pa" from other magmas, because**C2**enables demonstrating the defining lattice property, theabsorption law , and thedistributive law central toBoolean algebra .In boundary terms, the defining arithmetical fact of group theory is (())=(). The "PA" counterpart to that equation is ((()))=().**Equations of the second degree (Chapter 11)**Chapter 11 of "LoF" introduces "equations of the second degree", composed of recursive formulae that can be seen as having "infinite" depth. Some recursive formulae simplify to the marked or unmarked state. Others "oscillate" indefinitely between the two states depending on whether a given depth is even or odd. Specifically, certain recursive formulae can be interpreted as oscillating between

**true**and**false**over successive intervals of time, in which case a formula is deemed to have an "imaginary" truth value. Thus the flow of time may be introduced into the "pa".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 offinite automata . Turney (1986) 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.**Resonances in religion, philosophy, and science**The mathematical and logical content of "LoF" is wholly consistent with a secular point of view. Nevertheless, "LoF"'s "first distinction", and the Notes to its chapter 12, bring to mind the following landmarks in religious belief, and in philosophical and scientific reasoning, presented in rough historical order:

* Vedic,

Hindu andBuddhist : Related ideas can be noted in the ancient VedicUpanishads , which form the monastic foundations ofHinduism and laterBuddhism . As stated in the "Aitareya Upanishad" ("The Microcosm of Man"), the Supreme Atman manifests itself as the objectiveUniverse from one side, and as the subjective individual from the other side. In this process, things which are "effects" of God's creation become "causes" of our perceptions, by a reversal of the process. In the "Svetasvatara Upanishad", the core concept of Vedicism and Monism is "Thou art That."

*Taoism , (Chinese Traditional Religion): "...The Tao that can be told is not the eternal Tao; The name that can be named is not the eternal name. The nameless is the beginning of heaven and earth..." (Tao Te Ching ).

*Zoroastrianism : "This I ask Thee, tell me truly, Ahura. What artist made light and darkness?" (Gathas 44.5)

*Judaism (from theTanakh , calledOld Testament by Christians): "In the beginning when God created the heavens and the earth, the earth was a formless "void"... Then God said, 'Let there be light'; and there was light. ...God "separated" the light from the darkness. God called the light Day, and the darkness he called Night.::"...And God said, 'Let there be a dome in the midst of the waters, and let it "separate" the waters from the waters.' So God made the dome and "separated" the waters that were under the dome from the waters that were above the dome.::"...And God said, 'Let the waters under the sky be gathered together into one place, and let the dry land appear.' ...God called the dry land Earth, and the waters that were gathered together he called Seas.::"...And God said, 'Let there be lights in the dome of the sky to "separate" the day from the night...' God made the two great lights... to "separate" the light from the darkness." (Genesis 1:1-18; Revised Standard Version, emphasis added).::"And the whole earth was of "one" language, and of "one" speech." (Genesis 11:1; emphasis added).::"I am; that is who I am." (Exodus 3:14)

*Confucianism :Confucius claimed that he sought "a unity all pervading" ("Analects" XV.3) and that there was "one single thread binding my way together." ("Ana". IV.15). The "Analects" also contain the following remarkable passage on how the social, moral, and aesthetic orders are grounded in right language, grounded in turn in the ability to "rectify names," i.e., to make correct distinctions: "Zilu said, 'What would be master's priority?" The master replied, "Rectifying names! ...If names are not rectified then language will not flow. If language does not flow, then affairs cannot be completed. If affairs are not completed, ritual and music will not flourish. If ritual and music do not flourish, punishments and penalties will miss their mark. When punishments and penalties miss their mark, people lack the wherewithal to control hand and foot." ("Ana". XIII.3)

*Heraclitus : Pre-socratic philosopher, credited with forming the idea oflogos . "He who hears not me but the "logos" will say: "All is one"." Further: "I am as I am not."

*Parmenides : Argued that the every-day perception of reality of the physical world is mistaken, and that the reality of the world is 'One Being': an unchanging, ungenerated, indestructible whole.

*Plato : "Logos" is also a fundamental technical term in the Platonic worldview.

*Christianity : "In the Beginning was the Word, and the Word was with God, and the Word was God." (John 1:1). "Word" translateslogos in thekoine original. "If you do not believe "that I am", you will die in your sins." (John 8:24). "The Father and I are "one"." (John 10:30). "That they all may be "one"; as thou, Father, art in me, and I in thee, that they may also be "one" in us: that the world may believe that thou has sent me." (John 17:21). (emphases added)

*Islamic philosophy distinguishes essence ("Dhat") from attribute ("Sifat"), which are neither identical nor separate.

*Leibniz : "All creatures derive from God and from nothingness. Their self-being is of God, their nonbeing is of nothing. Numbers too show this in a wonderful way, and the essences of things are like numbers. No creature can be without nonbeing; otherwise it would be God... The only self-knowledge is to distinguish well between our self-being and our nonbeing... Within our selfbeing there lies an infinity, a footprint or reflection of the omniscience and omnipresence of God." [*"On the True "Theologia Mystica" in Loemker, Leroy, ed. and trans., 1969. "Leibniz: Philosophical Papers and Letters". Reidel: 368.*]

*Josiah Royce : "Without negation, there is no inference. Without inference, there is no order, in the strictly logical sense of the word. The fundamentally significant position of the idea of negation in determining and controlling our idea of the orderliness of both the natural and the spiritual order, becomes, in the light of all these considerations, as momentous as it is, in our ordinary popular views of this subject, neglected. ...From this point of view, negation appears as one of the most significant. ideas that lie at the base of all the exact sciences. By virtue of the idea of negation we are able to define processes of inference-processes which, in their abstract form, the purely mathematical sciences illustrate, and which, in their natural expression, the laws of the physical world, as known to our inductive science, exemplify.":"When logically analyzed, order turns out to be something that would be inconceivable and incomprehensible to us unless we had the idea which is expressed by the term 'negation'. Thus it is that negation, which is always also something intensely positive, not only aids us in giving order to life, and in finding order in the world, but logically determines the very essence of order." [*"Order" in Hasting, J., ed., 1917. "*]Encyclopedia of Religion and Ethics ". Scribner's: 540. Reprinted in Robinson, D. S., ed., 1951, "Royce's Logical Essays". Dubuque IA: Wm. C. Brown: 230-31.Returning to the Bible, the injunction "Let there be light" conveys:

* "… and there was light" — the light itself;

* "… called the light Day" — the manifestation of the light;

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

**Related work**Gottfried Leibniz , in memoranda not published before the late 19th and early 20th centuries, invented Boolean logic. His notation was isomorphic to that of "LoF": concatenation read asconjunction , and "non-("X")" read as thecomplement of "X". Leibniz's pioneering role inalgebraic logic was foreshadowed by Lewis (1918) and Rescher (1954). But a full appreciation of Leibniz's accomplishments had to await the work of Wolfgang Lenzen, published in the 1980s and reviewed in [*http://www.philosophie.uni-osnabrueck.de/Publikationen%20Lenzen/Lenzen%20Leibniz%20Logic.pdf Lenzen (2004).*]Charles Peirce (1839-1914) anticipated the "pa" 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, [*"Qualitative Logic", MS 736 (c. 1886) in Eisele, Carolyn, ed. 1976. "The New Elements of Mathematics by Charles S. Peirce. Vol. 4, Mathematical Philosophy". (The Hague) Mouton: 101-15.1*] but they were not published in full until 1993. [*"Qualitative Logic", MS 582 (1886) in Kloesel, Christian et al, eds., 1993. "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.*]

#In a 1902 encyclopedia article, [*Reprinted in Peirce, C.S. (1933) "Collected Papers, Vol. 4",*] 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.Charles Hartshorne andPaul Weiss , eds. Harvard Univ. Press. Paragraphs 378-383

#Thesyntax of his alphaexistential graph s is merelyconcatenation , read asconjunction , and enclosure by ovals, read asnegation . [*The existential graphs are described at length in Peirce, C.S. (1933) "Collected Papers, Vol. 4",*] If "pa" concatenation is read asCharles Hartshorne andPaul Weiss , eds. Harvard Univ. Press. Paragraphs 347-529.conjunction , then these graphs areisomorphic to the "pa" [*http://www2.math.uic.edu/~kauffman/CHK.pdf (Kauffman 2001).*] Ironically, "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'ssemiotics , about which "LoF" is silent, may yet shed light on the philosophical aspects of "LoF".[

*http://www2.math.uic.edu/~kauffman/CHK.pdf Kauffman (2001)*] discusses another notation similar to that of "LoF", that of a 1917 article byJean Nicod , who was a disciple ofBertrand Russell 's.The above formalisms are, like the "pa", 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 , prefix, or postfix operator symbols. The very well-known curly braces ('{', '}') of set theory can be seen as a boundary notation.The work of Leibniz, Peirce, and Nicod is innocent of metatheory, as they wrote before

Emil Post 's landmark 1920 paper (which "LoF" cites), proving thatsentential logic is complete, and beforeHilbert andLukasiewicz showed how to proveaxiom independence using models.Craig (1979) argued that the world, and how humans perceive and interact with that world, has a rich Boolean structure. Craig was an orthodox logician and an authority on

algebraic logic .Second-generation

cognitive science emerged in the 1970s, after "LoF" was written. On cognitive science and its relevance to Boolean algebra, logic, andset theory , see Lakoff (1987) (see index entries under "Image schema examples: container") and Lakoff and Núñez (2001). Neither book cites "LoF".The biologists and cognitive scientists

Humberto Maturana and his studentFrancisco Varela both discuss "LoF" 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.The [

*http://multiforms.netfirms.com Multiple Form Logic,*] by G.A. Stathis, "generalises [the primary algebra] into Multiple Truth Values" so as to be "more consistent with Experience." Multiple Form Logic, which is "not" a boundary formalism, employs two primitivebinary operation s: concatenation, read as BooleanOR , andinfix "#", read asXOR . The primitive values are 0 and 1, and the corresponding arithmetic is 11=1 and 1#1=0. The axioms are 1A=1, A#X#X = A, and A(X#(AB)) = A(X#B).Other formal systems with possible affinities to the primary algebra include:

*Mereology which typically has a lattice structure very similar to that of Boolean algebra. For a few authors, mereology is simply a model ofBoolean algebra and hence of the primary algebra as well.

*Mereotopology , which is inherently richer than Boolean algebra;

*The system of Whitehead (1934), whose fundamental primitive is "indication."The primary arithmetic and algebra are a minimalist formalism for

sentential logic andBoolean algebra . Other minimalist formalisms having the power ofset theory include:

* Thelambda calculus ;

*Combinatory logic with two (**S**and**K**) or even one (**X**) primitive combinators;

*Mathematical logic done with merely three primitive notions: one connective,NAND (whose "pa" translation is ("AB") or--dually--("A")("B") ), universalquantification , and one binaryatomic formula , denoting set membership. This is the system of Quine (1951).

* The "beta"existential graph s, with a singlebinary predicate denoting set membership. This has yet to be explored. The "alpha" graphs mentioned above are a special case of the "beta" graphs.In 1981 D.G. Schwartz proved that the "primary algebra" is equivalent---syntactically, semantically, and proof theoretically---with Classical Propositional Calculus; similar technques can be used to show that the "primary arithemetic" is equivalent with the set of all expressions that can be built up from the truth symbols "true" and "false" in the usual way by means to the logical connectives NOT, OR, and AND and parentheses (cf. Schwartz reference cited below).

**Notes****ee also*** (Simple English Wikipedia)

*Boolean algebra (introduction)

*Boolean algebra (logic)

*Boolean algebra (structure)

*Boolean algebras canonically defined

*Boolean logic

*entitative graph

*existential graph

*List of Boolean algebra topics

*Propositional calculus

*Two-element Boolean algebra **References***Editions of "Laws of Form":

**1969. London: Allen & Unwin, hardcover.

**1972. Crown Publishers, hardcover: ISBN 0-517-52776-6

**1973. Bantam Books, paperback. ISBN 0-553-07782-1

**1979. E.P. Dutton, paperback. ISBN 0-525-47544-3

**1994. Portland OR: Cognizer Company, paperback. ISBN 0-9639899-0-1

**1997 German translation, titled "Gesetze der Form". Lübeck: Bohmeier Verlag. ISBN 3-89094-321-7*Bostock, David, 1997. "Intermediate Logic". Oxford Univ. Press.

*Byrne, Lee, 1946, "Two Formulations of Boolean Algebra," "Bulletin of the American Mathematical Society": 268-71.

*Craig, William, 1979, "Boolean Logic and the Everyday Physical World," "Proceedings and Addresses of the American Philosophical Association" 52: 751-78.

*David Gries , and Schneider, F B, 1993. "A Logical Approach to Discrete Math". Springer-Verlag.

*William Ernest Johnson , 1892, "The Logical Calculus," "Mind" 1 (n.s.): 3-30.

* [*http://www.math.uic.edu/~kauffman/ Louis H. Kauffman,*] 2001, " [*http://www2.math.uic.edu/~kauffman/CHK.pdf The Mathematics of C.S. Peirce*] ", "Cybernetics and Human Knowing" 8: 79-110.

* ------, 2006, " [*http://www.arxiv.org/math.CO/0112266 Reformulating the Map Color Theorem.*] "

* ------, 2006a. " [*http://www.math.uic.edu/~kauffman/Laws.pdf Laws of Form - An Exploration in Mathematics and Foundations.*] " Book draft (hence big).

* Lenzen, Wolfgang, 2004, " [*http://www.philosophie.uni-osnabrueck.de/Publikationen%20Lenzen/Lenzen%20Leibniz%20Logic.pdf Leibniz's Logic*] " in Gabbay, D., and Woods, J., eds., "The Rise of Modern Logic: From Leibniz to Frege (Handbook of the History of Logic - Vol. 3)". Amsterdam: Elsevier, 1-83.

*Lakoff, George, 1987 "Women, Fire, and Dangerous Things". University of Chicago Press.

*-------- andRafael E. Núñez , 2001. "Where Mathematics Comes From : How the Embodied Mind Brings Mathematics into Being". Basic Books.

* Meguire, P. G., 2003, "Discovering Boundary Algebra: A Simplified Notation for Boolean Algebra and the Truth Functors," "International Journal of General Systems" 32: 25-87. [*http://www.lawsofform.org/docs/Meguire_LoF.pdf Revision.*] Steers clear of the more speculative aspects of "LoF". The source for the notation of much of this entry, which encloses in parentheses what "LoF" places under a cross.

*Willard Quine , 1951. "Mathematical Logic", 2nd ed. Harvard Univ. Press.

*--------, 1982. "Methods of Logic", 4th ed. Harvard Univ. Press.

*Nicholas Rescher , 1954, "Leibniz's Interpretation of His Logical Calculi," "Journal of Symbolic Logic" 18: 1-13.

* Schwartz, Daniel G., "Isomorphisms of G. Spencer-Brown's Laws of Form and F. Varela's Calculus for Self-Reference", "International Journal of General Systems", 6 (1981) 239--255.

* Turney, P. D., 1986, "Laws of Form and Finite Automata," "International Journal of General Systems" 12: 307-18.

*A. N. Whitehead , 1934, "Indication, classes, number, validation," "Mind" 43 (n.s.): 281-97, 543. The corrigenda on p. 543 are numerous and important, and later reprints of this article do not incorporate them.**External links*** "Laws of Form" [

*http://www.lawsofform.org/ web site,*] by Richard Shoup.

* [*http://www.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."

* [*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, " [*http://de.geocities.com/matthias_kissel/gdf/LoF.html A nonsystematic but easy to understand introduction to "Laws of Form".*] "

* [*http://www.transordinator.de/edition/draw/0.html Draw a distinction...*] The space of imagination based on "LoF".

* The [*http://multiforms.netfirms.com Multiple Form Logic,*] by G.A. Stathis, owes much to the primary algebra.

* The [*http://groups.yahoo.com/group/lawsofform Laws of Form Forum*] , where the primary algebra and related formalisms have been discussed since 2002.**Trivia*** " [

*http://www.philosopher-music.de Philosopher,*] " the German "Lovecraftian" Death Metal Band, pay a musical tribute to G. Spencer-Brown's work on their EP "Laws of Form".

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2010.*