Laws of Form (nonfiction)

Front cover of Laws for Form.

Laws of Form (hereinafter LoF) is a book by George Spencer-Brown, published in 1969, which defines a logical calculus which combines mathematics and philosophy.

Synopsis

LoF describes three distinct logical systems:

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

"Boundary algebra" is Meguire's (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

LoF emerged from 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 being a 1997 German translation, and has never gone out of print.[when?]

The mathematics fills only about 55pp and is rather elementary.[original research?] But LoF's mystical and declamatory prose, and its love of paradox, make it a challenging read for all. Spencer-Brown was influenced by Wittgenstein and R. D. Laing. LoF also echoes a number of themes from the writings of Charles Sanders Peirce, Bertrand Russell, and Alfred North Whitehead.

The entire book is written in an operational way, giving instructions to the reader instead of telling them what "is". 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. Except for this one sentence the book can be seen as an example of E-Prime.

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 of consciousness”, its algebraic symbolism capturing an (perhaps even "the") implicit root of cognition: the ability to "distinguish". LoF argues that primary algebra reveals striking connections among logic, Boolean algebra, and arithmetic, and the philosophy of language and mind.

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

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

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

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

LoF claimed that certain well-known mathematical conjectures of very long standing, such as the Four Color Theorem, Fermat's Last Theorem, and the Goldbach conjecture, are provable using extensions of the primary algebra. Spencer-Brown eventually circulated a purported proof of the Four Color Theorem, but it met with skepticism.

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

The primary algebra (Chapter 6)

Syntax

Given any valid primary arithmetic expression, insert into one or more locations any number of Latin letters bearing optional numerical subscripts; the result is a primary algebra formula. Letters so employed in mathematics and logic are called variables. A primary algebra variable indicates a location where one can write the primitive value Laws of Form - cross.gif or its complement 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 ‘=’. R1 and R2 enable transforming one equation into another. Hence the primary algebra is an equational formal system, like the many algebraic structures, including Boolean algebra, that are varieties. Equational logic was common before Principia Mathematica (e.g., 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 prefixed turnstile. To denote that the primary algebra formula A is a tautology, simply write “A =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.

Proof theory

The primary algebra contains three kinds of proved assertions:

• Consequence is a primary algebra equation verified by a demonstration. A demonstration consists of a sequence of steps, each step justified by an initial or a previously demonstrated consequence.
• Theorem is a statement in the metalanguage verified by a proof, i.e., an argument, formulated in the metalanguage, that is accepted by trained mathematicians and logicians.
• Initial, defined above. Demonstrations and proofs invoke an initial as if it were an axiom.

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

Interpretations

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

Two-element Boolean algebra 2

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

Syllogisms

Appendix 2 of LoF shows how to translate traditional syllogisms and sorites into the primary algebra. A valid syllogism is simply one whose primary algebra translation simplifies to an empty Cross.

Relation to magmas

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

Related work

Gottfried Leibniz, in memoranda not published before the late 19th and early 20th centuries, invented Boolean logic. His notation was isomorphic to that of LoF: concatenation read as conjunction, and "non-(X)" read as the complement of X. 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 (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, but they were not published in full until 1993. In a 1902 encyclopedia article, Peirce notated Boolean algebra and sentential logic in the manner of this entry, except that he employed two styles of brackets, toggling between ‘(’, ‘)’ and ‘[’, ‘]’ with each increment in formula depth.

The syntax of his alpha existential graphs is merely concatenation, read as conjunction, and enclosure by ovals, read as negation.[8] If primary algebra concatenation is read as conjunction, then these graphs are isomorphic to the primary algebra (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's semiotics, about which LoF is silent, may yet shed light on the philosophical aspects of LoF.

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

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

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

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

Second-generation cognitive science emerged in the 1970s, after LoF was written. On cognitive science and its relevance to Boolean algebra, logic, and set theory, see Lakoff (1987) (see index entries under "Image schema examples: container") and Lakoff 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.

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

• Mereology which typically has a lattice structure very similar to that of Boolean algebra. For a few authors, mereology is simply a model of Boolean algebra and hence of the primary algebra as well.
• Mereotopology, which is inherently richer than Boolean algebra;
• The system of Alfred North 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 ... TO_DO

See also

• Boolean algebra (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

Notes

• Meguire, P. (2011) Boundary Algebra: A Simpler Approach to Basic Logic and Boolean Algebra. Saarbrücken: VDM Publishing Ltd. 168pp
• Felix Lau: Die Form der Paradoxie, 2005 Carl-Auer Verlag, ISBN 9783896703521
• B. Banaschewski (July 1977). "On G. Spencer Brown's Laws of Form". Notre Dame Journal of Formal Logic. 18 (3): 507–509. doi:10.1305/ndjfl/1093888028.
• For a sympathetic evaluation, see Kauffman (2001).
• "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
• "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.
• Reprinted in Peirce, C.S. (1933) Collected Papers of Charles Sanders Peirce, Vol. 4, Charles Hartshorne and Paul Weiss, eds. Harvard University Press. Paragraphs 378–383
• The existential graphs are described at length in Peirce, C.S. (1933) Collected Papers, Vol. 4, Charles Hartshorne and Paul Weiss, eds. Harvard University Press. Paragraphs 347–529.

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 2008 Bohmeier Verlag, Leipzig, 5th international edition. ISBN 978-3-89094-580-4 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 (6): 751–78. doi:10.2307/3131383. JSTOR 3131383. 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. Louis H. Kauffman, 2001, “The Mathematics of C.S. Peirce”, Cybernetics and Human Knowing 8: 79–110.

---

, 2006, “Reformulating the Map Color Theorem.”

---

, 2006a. “Laws of Form – An Exploration in Mathematics and Foundations.” Book draft (hence big).

Lenzen, Wolfgang, 2004, “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.

---

and Rafael 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. CiteSeerX 10.1.1.106.634. doi:10.1080/0308107031000075690.

---

, 2011. Boundary Algebra: A Simpler Approach to Basic Logic and Boolean Algebra. VDM Publishing Ltd. ISBN 978-3639367492. The source for much of this entry, including the notation which encloses in parentheses what LoF places under a cross. Steers clear of the more speculative aspects of LoF.

Willard Quine, 1951. Mathematical Logic, 2nd ed. Harvard University Press.

---

, 1982. Methods of Logic, 4th ed. Harvard University Press.

Rescher, Nicholas (1954). "Leibniz's Interpretation of His Logical Calculi". Journal of Symbolic Logic. 18 (1): 1–13. doi:10.2307/2267644. JSTOR 2267644. Schwartz, Daniel G. (1981). "Isomorphisms of G. Spencer-Brown's Laws of Form and F. Varela's Calculus for Self-Reference". International Journal of General Systems. 6 (4): 239–55. doi:10.1080/03081078108934802. Turney, P. D. (1986). "Laws of Form and Finite Automata". International Journal of General Systems. 12 (4): 307–18. doi:10.1080/03081078608934939. 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. Dirk Baecker (ed.) (1993), Kalkül der Form. Suhrkamp; Dirk Baecker (ed.), Probleme der Form. Suhrkamp. Dirk Baecker (ed.) (1999), Problems of Form, Stanford University Press. Dirk Baecker (ed.) (2013), A Mathematics of Form, A Sociology of Observers, Cybernetics & Human Knowing, vol. 20, no. 3-4.

External links

• Laws of Form, archive of website by Richard Shoup.
• Spencer-Brown’s talks at Esalen, 1973. Self-referential forms are introduced in the section entitled "Degree of Equations and the Theory of Types".
• Louis H. Kauffman, “Box Algebra, Boundary Mathematics, Logic, and Laws of Form.”
• Kissel, Matthias, “A nonsystematic but easy to understand introduction to Laws of Form.”
• The Laws of Form Forum, where the primary algebra and related formalisms have been discussed since 2002.
• A meeting with G.S.B by Moshe Klein
• The Markable Mark, an introduction by easy stages to the ideas of Laws of Form
• The BF Calculus and the Square Root of Negation by Louis Kauffman and Arthur Collings; it extends the Laws of Form by adding an imaginary logical value. (Imaginary logical values are introduced in chapter 11 of the book Laws of Form.)