Laws of Form (hereinafter LoF) is a book by G. SpencerBrown, published in 1969, that straddles the boundary between mathematics and of philosophy. LoF describes three distinct logical systems:
SpencerBrown 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).
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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. SpencerBrown 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.
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 of consciousness," its algebraic symbolism capturing an (perhaps even the) implicit root of cognition: the ability to distinguish. LoF argues that the pa reveals striking connections among logic, Boolean algebra, and arithmetic, and the philosophy of language and mind.
Some, e.g. Banaschewski (1977), argue that the pa is nothing but new notation for Boolean algebra. It is true that twoelement Boolean algebra (2) can be seen as the intended interpretation of the pa. Nevertheless, Meguire (2005) counters that pa notation:
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 and William Bricken. Some of these authors modified the primary algebra in a variety of interesting ways. LoF claimed that certain wellknown mathematical conjectures of very long standing, such as the Four Color Theorem, Fermat's Last Theorem, and the Goldbach conjecture, are provable using extensions of the pa. SpencerBrown eventually circulated a purported proof of the Four Color Theorem^{[1]}. The proof met with skepticism and SpencerBrown'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.)
The symbol:
also called the mark or cross, is the essential feature of the Laws of Form. In SpencerBrown's inimitable and enigmatic fashion, the Mark symbolizes the root of cognition, i.e., the dualistic Mark indicates the capability of differentiating a "this" from "everything else but this."
In LoF, a Cross denotes the drawing of a "distinction", and can be thought of as signifying the following, all at once:
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:
can well be expressed in such ways as:
 Draw a distinction
Or:
 Let there be a distinction,
 Find a distinction,
 See a distinction,
 Describe a distinction,
 Define a distinction,
 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 and language. Paradoxically, the Form is at once Observer and Observed, and is also the creative act of making an observation. LoF (excluding back matter) closes with the words:
"...the first distinction, the Mark and the observer are not only interchangeable, but, in the form, identical."
C. S. Peirce came to a related insight in the 1890s; see Related Work.
The syntax of the primary arithmetic (PA) goes as follows. There are just two atomic expressions:
There are two inductive rules:
The semantics of the primary arithmetic are perhaps nothing more than the sole explicit definition in LoF: Distinction is perfect continence.
Let the unmarked state be a synonym for the void. Let an empty Cross denote the marked state. To cross is to move from one of the unmarked or marked states to the other. We can now state the "arithmetical" axioms A1 and A2, which ground the primary arithmetic (and hence all of the Laws of Form):
A1. The law of Calling. Calling twice from a state is indistinguishable from calling once. To make a distinction twice has the same effect as making it once. For example, saying "Let there be light" and then saying "Let there be light" again, is the same as saying it once. Formally:
A2. The law of Crossing. After crossing from the unmarked to the marked state, crossing again ("recrossing") starting from the marked state returns one to the unmarked state. Hence recrossing annuls crossing. Formally:
In both A1 and A2, the expression to the right of '=' has fewer symbols than the expression to the left of '='. This suggests that every primary arithmetic expression can, by repeated application of A1 and A2, be simplified to one of two states: the marked or the unmarked state. This is indeed the case, and the result is the expression's simplification. The two fundamental metatheorems of the primary arithmetic state that:
Thus the relation of logical equivalence partitions all primary arithmetic expressions into two equivalence classes: those that simplify to the Cross, and those that simplify to the void.
A1 and A2 have loose analogs in the properties of series and parallel electrical circuits, and in other ways of diagramming processes, including flowcharting. A1 corresponds to a parallel connection and A2 to a series connection, with the understanding that making a distinction corresponds to changing how two points in a circuit are connected, and not simply to adding wiring.
The primary arithmetic is analogous to the following formal languages from mathematics and computer science:
The phrase calculus of indications in LoF is a synonym for "primary arithmetic".
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 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.
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 in mathematics and logic are called variables. 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.
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 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:
R2 is employed very frequently in pa demonstrations (see below), almost always silently. These rules are routinely invoked in logic 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 many algebraic structures, including Boolean algebra, that are varieties. Equational logic was common before Principia Mathematica (e.g., Peirce,^{1,2,3} Johnson 1892), and has presentday advocates (Gries and Schneider 1993).
Conventional mathematical logic consists of tautological formulae, signalled by a prefixed turnstile. To denote that the pa formula A is a tautology, simply write "A = ". If one replaces '=' in R1 and R2 with the biconditional, the resulting rules hold in conventional logic. However, conventional logic relies mainly on the rule modus ponens; thus conventional logic is ponential. The equationalponential dichotomy distills much of what distinguishes mathematical logic from the rest of mathematics.
An initial is a pa equation verifiable by a decision procedure and as such is not an axiom. LoF lays down the initials:
The absence of anything to the right of the "=" above, is deliberate.
J2 is the familiar distributive law of sentential logic and Boolean algebra.
Another set of initials, friendlier to calculations, is:
It is thanks to C2 that the pa is a lattice. By virtue of J1a, it is a complemented lattice whose upper bound is (). By J0, (()) is the corresponding lower bound and identity element. J0 is also an algebraic version of A2 and makes clear the sense in which (()) aliases with the blank page.
T13 in LoF generalizes C2 as follows. Any 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:
Perhaps the first instance of an axiom or rule with the power of C2 was the "Rule of (De)Iteration," combining T13 and AA=A, of C. S. Peirce's existential graphs.
LoF asserts that concatenation can be read as commuting and associating by default and hence need not be explicitly assumed or demonstrated. (Peirce made a similar assertin about his existential graphs.) Let a period be a temporary notation to establish grouping. That concatenation commutes and associates may then be demonstrated from the:
Having demonstrated associativity, the period can be discarded.
The pa contains three kinds of proved assertions:
The distinction between consequence and theorem holds for all formal systems, including mathematics and logic, but is usually not made explicit. A demonstration or decision procedure can be carried out and verified by computer. The proof of a theorem cannot be.
Let A and B be pa formulas. A demonstration of A=B may proceed in either of two ways:
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 an algorithm, sketched in LoFs 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 the conjunctive normal form. LoF (T1415) proves the pa analog of the wellknown Boolean algebra theorem that every formula has a normal form.
Let A be a subformula of some formula B. When paired with C3, J1a can be viewed as the closure condition for calculations: B is a tautology if and only if A and (A) both appear in depth 0 of B. A related condition appears in some versions of natural deduction. A demonstration by calculation is often little more than:
The last step of a calculation always invokes J1a.
LoF includes elegant new proofs of the following standard metatheory:
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.
If the Marked and Unmarked states are read as the Boolean values 1 and 0 (or True and False), the pa interprets 2 (or sentential logic). LoF shows how the pa can interpret the syllogism. Each of these interpretations is discussed in a subsection below. Extending the pa so that it could interpret standard firstorder logic has yet to be done, but Peirce's beta existential graphs suggest that this extension is feasible.
The pa is an elegant minimalist notation for the twoelement Boolean algebra 2. Let:
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 , is the Boolean domain or carrier. In the language of universal algebra, the pa is the algebraic structure of type . The expressive adequacy of the Sheffer stroke points to the pa also being a algebra of type . In both cases, the identities are J1a, J0, C2, and ACD=CDA. Since the pa and 2 are isomorphic, 2 can be seen as a algebra of type . This description of 2 is simpler than the conventional one, namely an algebra of type .
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:
The pa interprets sentential logic as follows. A letter represents any given sentential expression. Thus:
both interpret A if and only if B or 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 PA 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).
Schwartz (1981) proved that the pa is equivalent  syntactically, semantically, and proof theoretically  with the classical propositional calculus. Likewise, it can be shown that the pa is syntactically equivalent with expressions built up in the usual way from the classical truth values true and false, the logical connectives NOT, OR, and AND, and parentheses.
Interpreting the Unmarked State as False is wholly arbitrary; that state can equally well be read as True. All that is required is that the interpretation of concatenation change from OR to AND. IF A THEN B now translates as (A(B)) instead of (A)B. More generally, the pa is "selfdual," meaning that any pa formula has two sentential or Boolean readings, each the dual of the other. Another consequence of selfduality is the irrelevance of De Morgan's laws; those laws are built into the syntax of the pa 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 wellformed. But an empty Cross is a wellformed pa expression, denoting the Marked state, a primitive value. Hence a nonempty Cross is an operator, while an empty Cross is an operand because it denotes a primitive value. Thus the pa reveals that the heretofore distinct mathematical concepts of operator and operand are in fact merely different facets of a single fundamental action, the making of a distinction.
Appendix 2 of LoF shows how to translate traditional syllogisms 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 and medieval logic. This pa translation of syllogistic logic also suggests that the pa can interpret monadic and term logic, and that the pa has affinities to the Boolean term schemata of Quine (1982: Part II).
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 is conjunction, it is simpler to translate the Theorema into the pa using the dual interpretation. The objective then becomes one of simplifying that translation to (()).
Remarks:
Given some standard notions from mathematical logic and some suggestions in Bostock (1997: 83, fn 11, 12), {} and may be interpreted as the classical bivalent truth values. Let the extension of an nplace atomic formula be the set of ordered ntuples of individuals that satisfy it (i.e., for which it comes out true). Let a sentential variable be a 0place atomic formula, whose extension is a classical truth value, by definition. An ordered 2tuple is an ordered pair, whose standard set theoretic definition is <a,b> = {{a},{a,b}}, where a,b are individuals. Ordered ntuples for any n>2 may be obtained from ordered pairs by a wellknown recursive construction. Dana Scott has remarked that the extension of a sentential variable can also be seen as the empty ordered pair (ordered 0tuple), {{},{}} = because {a,a}={a} for all a. Hence has the interpretation True. Reading {} as False follows naturally.
The pa embodies a point noted by Huntington in 1933: Boolean algebra requires, in addition to one unary operation, one, and not two, binary operations. Hence the seldomnoted fact that Boolean algebras are magmas. (Magmas were called groupoids until the latter term was appropriated by category theory.) To see this, note that the pa is a commutative:
Groups also require a unary operation, called inverse, the group counterpart of Boolean complementation. Let (a) denote the inverse of a. Let () denote the group identity element. Then groups and the pa have the same signatures, namely they are both 〈,(),()〉 algebras of type 〈2,1,0〉. Hence the pa is a boundary algebra. The axioms for an abelian group, in boundary notation, are:
From G1 and G2, the commutativity and associativity of concatenation may be derived, as above. Note that G3 and J1a are identical. G2 and J0 would be identical if (())=() replaced A2. This is the defining arithmetical identity of group theory, in boundary notation.
The pa differs from an abelian group in two ways:
Both A2 and C2 follow from B 's being an ordered set.
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 of finite 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 SpencerBrown gave to E1 and E4, namely "memory" and "counter". RRA thus formalizes and clarifies LoF 's notion of an imaginary truth value.
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:
Returning to the Bible, the injunction "Let there be light" conveys:
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.
Gottfried Leibniz, in memoranda not published before the late 19th and early 20th centuries, invented Boolean logic. His notation was isomorphic to that of LoF: concatenation read as conjunction, and "non(X)" read as the complement of X. Leibniz's pioneering role in algebraic 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 Lenzen (2004).
Charles Sanders Peirce (18391914) anticipated the pa in three veins of work:
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's semiotics, about which LoF is silent, may yet shed light on the philosophical aspects of LoF.
Kauffman (2001) discusses another notation similar to that of LoF, that of a 1917 article by Jean Nicod, who was a disciple of Bertrand Russell's.
The above formalisms are, like the 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 wellknown curly braces ('{', '}') of set theory can be seen as a boundary notation.
The work of Leibniz, Peirce, and Nicod is innocent of metatheory, as they wrote before Emil Post's landmark 1920 paper (which LoF cites), proving that sentential logic is complete, and before Hilbert and Lukasiewicz showed how to prove axiom independence using models.
Craig (1979) argued that the world, and how humans perceive and interact with that world, has a rich Boolean structure. Craig was an orthodox logician and an authority on algebraic logic.
Secondgeneration cognitive science emerged in the 1970s, after LoF was written. On cognitive science and its relevance to Boolean algebra, logic, and set theory, see Lakoff (1987) (see index entries under "Image schema examples: container") and Lakoff and Núñez (2001). 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.
The 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 primitive binary operations: concatenation, read as Boolean OR, and infix "#", read as XOR. 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:
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:
Laws of Form is a book by George SpencerBrown published in 1969. It is about logic, mathematics, and philosophy. The mathematical systems that SpencerBrown presented in the book are known by the names "calculus of indications", "distinction calculus", and often just "LOF".
Laws of Form grew out of the author's work in electronic engineering. The book has been published in several editions and translations and has never gone out of print. A short book, its mathematical part is only 55 pages long.
SpencerBrown's philosophy was influenced by Ludwig Wittgenstein, R.D. Laing, Charles Sanders Peirce, Bertrand Russell, and Alfred North Whitehead.
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Laws of Form was listed in the Whole Earth Catalog in 1969 and quickly became a cult classic. The calculus of indications and the primary algebra may be regarded as a way to think about a fundamental activity of the mind, namely, the ability to distinguish or to draw distinctions. The book argues that this ability is the foundation of human cognition and consciousness. According to SpencerBrown, the primary arithemetic and primary algebra reveal new connections among logic, mathematics, the philosophy of language, and the philosophy of mind.
Let 0 and 1 be the two basic primitive values of Boolean algebra. Let AB denote a binary operation of Boolean algebra. Let (X) stand for the Boolean complement of X. Then the calculus of indications is simply Boolean arithmetic reduced to the two equations 11=1 and (1)=0. These are the only "axioms" in LoF.
The primary algebra is mainly a simpler notation for Boolean algebra, except for one thing. In Boolean algebra, () is not defined. () is "empty" complementation (the complementation of "nothing"). On the other hand, in the primary algebra () is defined, and stands for one of 0 or 1. (()) stands for the other primitive value, and is the same thing as the blank page.
Let A and B be any two expressions of the primary algebra. The primary algebra is made up of equations of the form A=B, and these equations are treated in the same way as the equations of the number algebra taught in all schools. Standard methods of logic seldom use equations. LoF argues that doing elementary logic with the primary algebra is easier. In particular, if A is a tautology in logic, then one of A=() or A=(()) holds in the primary algebra.
Laws of Form proves the following fact about the primary algebra:
Hence the primary algebra is a wellbehaved piece of mathematics. It can be useful even if the philosophy and cognitive science of LoF are wrong or uninteresting.
