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The book Laws of Form (hereinafter abbreviated LoF), by G. Spencer-Brown, describes three distinct logical systems:
- The primary arithmetic (described in Chapter 4), which can be interpreted as Boolean arithmetic;
- The primary algebra (Chapter 6), which can be interpreted via two-element Boolean algebra (hereinafter abbreviated 2), Boolean logic, and propositional calculus;
- Equations of the second degree (Chapter 11), which can be interpreted through finite automata and Alonzo Church's Restricted Recursive Arithmetic (RRA).
The phrase Laws of Form may refer to LoF or to the primary algebra (hereinafter abbreviated pa).
The book
There are several editions of LoF, the first in 1969, the most recent (a German translation) in 1997. 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, regardless of mathematical training. In this and other respects, Spencer-Brown was much influenced by Wittgenstein and R. D. Laing. LoF also echoes a number of themes from the writings of Charles 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 a reformulation of Boolean algebra. Those who argue in this fashion seem to believe that Boolean algebra is mathematically uninteresting, or that new notations for existing formalisms are of little interest. Proponents, e.g. (Meguire 2005), of the pa counter that its notation:
- Dramatically simplifies proofs in Boolean algebra, and in sentential and syllogistic logic;
- 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.
LoF claims 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 pa. Spencer-Brown eventually circulated a purported proof of Four Color; for a sympathetic evaluation, see Kauffman (2001). Nevertheless, that claim of proof met with skepticism and Spencer-Brown's mathematical reputation, as well as that of LoF, went into decline. N.B. The Four Color Theorem and Fermat's Last Theorem were proved in 1976 and 1995, respectively, using methods owing nothing to LoF.
The form (Chapter 1)
The symbol:missing image!
- Laws of Form - cross.gif -
- Laws of Form - cross.gif -
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: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."
Charles Peirce came to a related insight in the 1890s; see Related Work below.
The primary arithmetic (Chapter 4)
LoF often uses the phrase calculus of indications in place of "primary arithmetic".Begin with the void. Then posit two inductive rules:
- Given any expression, a Cross can be written over it;
- Any two expressions can be concatenated.
The primary arithmetic (and all of the Laws of Form) are grounded in a mere two axioms, A1 and A2.
A1. The law of Calling. To make a distinction twice has the same effect as making it once. For instance, if you say "Let there be light." and then you say "Let there be light." again, it is the same as saying it once. Crossing twice from the unmarked state cannot be distinguished from crossing once. Symbolically:
missing image!
- Laws of Form - cross.gif -
- Laws of Form - cross.gif -
missing image!
- Laws of Form - cross.gif -
= - Laws of Form - cross.gif -
missing image!
- Laws of Form - cross.gif -
- Laws of Form - cross.gif -
missing image!
- Laws of Form - double cross.gif -
=- Laws of Form - double cross.gif -
- An expression has a unique simplification;
- The repeated application of A1 and A2 to either the marked or the unmarked state cannot yield an expression whose simplification differs from the initial state.
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.
More formally, the primary arithmetic is a Dyck language of order 1 with a null alphabet, and the simplest instance of a context-free language in the Chomsky hierarchy.
The notion of 'canon'
The notion of a canon is discussed in the following two excerpts from the Notes to Chapter 2 of LoF:"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 quotes 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 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, with or without 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 indicate multiple locations of the same primitive value. In Boolean algebras, the set of primitive values B={(),(())} is called the carrier.Rules governing logical equivalence
The sign '=' denotes that what appears to the left and right of = are logically equivalent, i.e., have the same simplification. An expression of the form "A=B" is an equation, meaning that A and B are logically equivalent. 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 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. A becomes E and B becomes F. If C=D, then E=F. Note that A=B is not required.
The pa consists of equations, i.e., pairs of formulae linked by an equivalence relation denoted by an infix '='. R1 and R2 enable transforming one equation into another. Hence the pa is an equational formal system, like Boolean and most other algebraic structures. Mathematical logic consists of tautological formulae, signalled by a prefixed turnstile. To variants of R1 and R2, conventional logic adds the rule modus ponens; thus conventional logic is ponential. The equational-ponential dichotomy summarizes much of what distinguishes mathematical logic from the rest of mathematics. To indicate that the pa formula A is a tautology, simply write "A =
missing image!
- Laws of Form - cross.gif -
".
- Laws of Form - cross.gif -
Initials
An initial is a pa equation verifiable by a decision procedure and as such is not an axiom. LoF lays down the initials:- J1: ((A)A) =
. - J2: ((A)(B))C = ((AC)(BC)).
- J0: (A)A=missing image!
- Laws of Form - cross.gif - - C2: A(AB)=A(B). C2 and generation are synonyms. Both terms are from LoF. William Bricken calls C2 pervasion.
J0 is simply the complement of J1. The first formal system to incorporate anything like C2 was Peirce's existential graphs, who gave the name (De)Iteration to a combination of T13 and AA=A. C2 is called mimesis in logical nand.
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.
LoF asserts that the commutativity and associativity of concatenation can be taken as true by default and hence need not be explicitly assumed. (Peirce made a similar assumption in his graphical logic.) Commutativity and associativity also follow from ABC = BCA, taken as an initial.
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 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.
Let A and B be pa formulas. A demonstration of A=B may proceed in either of two ways:
- Modify A in steps until B is obtained, or vice versa;
- Simplify ((A)(B))(AB) to missing image!. This is known as a "calculation".
- Laws of Form - cross.gif -
pa demonstrations are usually quite easy, usually requiring no more than the initials, A2, and the consequences ((A))=A, AA=A, and (((A)B)C) = (AC)((B)C). This last consequence 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 (T14-15) proves the pa analog of a theorem, well-known in Boolean algebra, that every formula has a normal form.
Let A be a subformula of some formula B. When paired with the consequence ()A=() (C3 in LoF), J0 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:
- Invoking T13 repeatedly to eliminate redundant subformulae;
- Erasing any subformulae having the form ((A)A).
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).
Interpretations
The Marked and Unmarked states can be read as the Boolean values 1 and 0, or as True and False. The first reading transforms the pa into a notation for 2; the second into a notation for sentential logic. Extending the pa so that it would have standard first-order logic as a model has yet to be done, but Peirce's beta existential graphs suggest that the extension should be straightforward.Two-element Boolean algebra 2
Let Boolean:- Meet or join interpret AB;
- The complement of A interpret missing image!;
- Laws of Form - not a.gif - - 0 or 1 interpret the empty Mark.
In the language of universal algebra, the pa is the algebraic structure ?B,--,(-),()? of type ?2,1,0?, whose equational identities are J0, C2, and ABC=BCA. Since the pa and 2 are isomorphic, 2 can be seen as a ?B,?,~,1? algebra of type ?2,1,0?. This description of 2 is simpler than the conventional one, ?{0,1},?,?.~,1,0? of type ?2,2,1,0,0?. The expressive adequacy of the Sheffer stroke points to the pa also being a ?B,(--),()? algebra of type ?2,0?.
Propositional calculus
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
missing image!
- Laws of Form - cross.gif -
= True = not False- Laws of Form - cross.gif -
missing image!
- Laws of Form - double cross.gif -
= Not True = False- Laws of Form - double cross.gif -
missing image!
- Laws of Form - not a.gif -
interprets Not A- Laws of Form - not a.gif -
missing image!
- Laws of Form - a or b.gif -
interprets A Or B - Laws of Form - a or b.gif -
missing image!
- Laws of Form - if a then b.gif -
interprets Not A Or B- Laws of Form - if a then b.gif -
or If A Then B.
missing image!
- Laws of Form - a and b.gif -
interprets Not (Not A or Not B) - Laws of Form - a and b.gif -
or Not (If A Then Not B)
or A And B.
or A And B.
The interpretations above assume that the Unmarked State is read as False. This reading is wholly arbitrary; the Unmarked state can equally well denote 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, duality refers to there being two pa translations of any given statement of sentential logic or equation of 2. Likewise, the pa is "self-dual," meaning that any pa formula has two sentential or Boolean readings.
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 with an empty scope is not defined. Meanwhile a Cross, interpretable as complementation/negation, with nothing under itself denotes the Marked state, a primitive value. This is the sense in which 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.
Syllogism
Appendix 2 of LoF shows how to translate traditional syllogisms and sorites (and hence term and monadic logic, although this is not made explicit) 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. It can then be shown that all syllogisms not requiring that some terms be assumed nonempty are one of 24 permutations of a generalization of Barbara, the form (A*B)((B)C*)A*C*. This suggests that monadic logic is also a model of the pa, and that the pa has affinities to the Boolean term schemata of Quine's Methods of Logic.Boundary algebra is a magma
The pa can be seen as the culmination of a point Huntington highlighted in 1933: Boolean algebra merely requires one binary operation and one unary operation. Hence the seldom-noted fact that Boolean algebras are magmas (groupoids). To see this, recall that:- pa concatenation commutes and associates (the pa is a commutative semigroup);
- (()) is the pa identity element by virtue of the (elementary) pa consequence A(()) = A (the pa is a commutative monoid).
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 symmetric 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 (()).- [(P?R)?(Q?S)]?[(P?Q)?(R?S)] Theorema
- ((P(R))(Q(S))((PQ(RS)))) pa translation
- = ((P(R))P(Q(S))Q(RS)) OI; C1
- = (((R))((S))PQ(RS) C2,2x (C2 eliminates the bold letters in the previous expression); OI
- = (RSPQ(RS)) C1,2x
- = ((RSPQ)RSPQ) C2; OI
- = (()) J1.square
Remarks
- C1 and C2 are repeatedly invoked in a fairly mechanical way to eliminate boundaries and variables, respectively. This is typical of calculations;
- A single invocation of J1 (or, in other contexts, J0) 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 bivalent truth values. Let the extension of an n-place atomic formula be the set of ordered n-tuples of individuals that satisfy it (i.e., for which it comes out true). Let a sentential variable be a 0-place atomic formula, whose extension is a classical truth value, by definition. An ordered 2-tuple is an ordered 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.Equations of the second degree (Chapter 11)
Chapter 11 of LoF introduces equations of the second degree, which include 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 as the depth is even or odd. Such formulas implicitly add the notion of time to the pa. Specifically, certain recursive formulae can be interpreted as oscillating between true and false over successive intervals of time, in which case LoF deems a formula to have an "imaginary" truth value.Interpretation
Turney (1986) shows how the recursive formulae in LoF can be interpreted through 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 formulas E1, E2, and E4 in Chapter 11 of LoF. This translation into RRA explains Spencer-Brown's names for E1 and E4, namely "memory" and "counter". RRA provides a formal interpretation of Spencer Brown's notion of an imaginary truth value.Related work
Gottfried Leibniz, in memoranda not published until the late 19th and early 20th centuries, invented Boolean algebra. His notation was isomorphic to that of LoF: concatenation interpreted as conjunction and "non-(X)" intepreted as the complement of X. Clarence Irving Lewis (1918) came to a partial understanding of this fact, as did Rescher (1954), but a full appreciation of Leibniz's accomplishment had to await the work of Wolfgang Lenzen in the 1980s; click here for Lenzen's summary in English of his work.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^{1}, but they were not published in full until 1993^{2,3}
- A closely related notation appears in an encyclopedia article he published in 1902, reprinted in vol. 4 of his Collected Papers, paragraphs 378-383.
- His alpha existential graphs are isomorphic to the pa (Kauffman 2001). This work was virtually unknown at the time when (1960s) and in the place where (UK) LoF was written. Ironically, LoF cites vol. 4 of Peirce's Collected Papers, where (paragraphs 347-529) the existential graphs are described in detail.
Kauffman discusses another notation similar to that of LoF, that of a 1917 article by Jean Nicod, a disciple of Bertrand Russell's. Other formal systems with possible affinities to the Laws of Form are mereology and mereotopology.
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.
The pa and Peirce's graphical logic are instances of boundary mathematics, i.e., mathematics done with boundary notation, one restricted to variables and brackets (enclosing devices). In particular, 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.
That the world, and how humans perceive and interact with that world, has a rich Boolean structure has been noted by at least one orthodox logician, Craig (1979).
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 primary arithmetic and algebra is but one of several minimalist approaches to logic and the foundations of mathematics, or parts thereof. Other, and more powerful, minimalist approaches include:
- The lambda calculus;
- Combinatory logic with S and K as the sole primitive combinators;
- Mathematical logic done with merely three primitive notions: one connective, logical nand (whose pa translation is either (AB) or (A)(B) ), universal quantification, and primitive atomic formulae of one kind, set membership. Willard Quine employed this system in his 1951 Mathematical Logic.
References
- "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.
- "Qualitative Logic", MS 582 (1886) in Kloesel, Christian et al, eds., 1993. Writings of Charles S. Peirce: A Chronological Edition, Volume 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, Volume 5, 1884-1886. Indiana University Press: 372-78.
Bibliography
- 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.
- Craig, William, 1979, "Boolean Logic and the Everyday Physical World," Proceedings and Addresses of the American Philosophical Association 52: 751-78.
- 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).
- 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 revision. The notation of this paper differs from that of LoF in that it encloses in parentheses what LoF places under a cross. Steers clear of the more speculative aspects of LoF.
- Nicholas Rescher, 1954, "Leibniz's Interpretation of His Logical Calculi," Journal of Symbolic Logic 18: 1-13.
- Turney, P. D., 1986, "Laws of Form and Finite Automata," International Journal of General Systems 12: 307-18.
See also
- Boolean algebra
- Boolean logic
- List of Boolean algebra topics
- Two-element Boolean algebra
- existential graph
- entitative graph
External links
- Laws of Form web site.
- 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."
External links
- "Philosopher," the German "Lovecraftian" Death Metal Band, pay a musical tribute to G. Spencer-Brown's work on their EP Laws of Form.
Some content adapted from the Wikinfo article "Laws of Form" under the GNU Free Documentation License.
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