Laws of Form(logic, wiki, imported, Proteus)
The book
Laws of Form (hereinafter abbreviated
LoF), by
G. Spencer-Brown, describes three distinct logical systems:
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:
Moreover, the syntax of the
pa can be extended to represent mathematical formalisms other than
2, resulting in
boundary mathematics (see Related Work below).
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 -
also called the
mark or
cross, is the essence of the Laws of
Form. In Spencer-Brown'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 a "that."
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:
- 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 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.
Thus the
syntax of the
primary arithmetic. The
semantics of the primary arithmetic are established by the only explicit
definition in
LoF:
Distinction is perfect continence.
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 -
missing image!
- Laws of Form - cross.gif -
=
missing image!
- Laws of Form - cross.gif -
Crossing from the unmarked state takes you to the marked state; crossing again from that marked state takes you back to the unmarked state. To recross is not to cross. Symbolically:
missing image!
- Laws of Form - double cross.gif -
=
Applying
A1 and
A2 repeatedly can reduce any expression consisting solely of Crosses to the expression's
simplification, either the marked or the unmarked state. The fundamental metatheorem of the primary arithmetic (T3-4 in
LoF) states that:
- 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.
Hence the
relation of
logical equivalence partitions all primary arithmetic formulas 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.
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)
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.
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.
R2 is employed very frequently in
pa demonstrations (see below), almost always silently. These rules are routinely yet unwittingly invoked in
logic and most of mathematics.
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 -
".
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)).
J2 is the familiar
distributive law of
sentential logic and
Boolean algebra. A more economical set of initials, also one friendlier to calculations, is;
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:
Once
A=
B has been demonstrated,
A=
B is a consequence and can be invoked to justify steps in subsequent demonstrations.
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).
The last step of a calculation always invokes
J0.
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).
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 (and hence of all
Boolean algebras with finite
carriers).
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.
Let Boolean:
If meet interprets
AB, then join interprets ~(~
A+~
B), or vice versa. Hence the
pa and
2 are isomorphic, and
2 emerges as a
model of the primary algebra. The primary arithmetic suggests that
2 can be axiomatized arithmetically by 1+1=1+0=0+1=1=~0, and 0+0=0=~1.
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?.
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:
missing image!
- Laws of Form - cross.gif -
=
True =
not False
missing image!
- Laws of Form - double cross.gif -
=
Not True =
False
The
pa interprets sentential logic as follows. A letter represents any given sentential expression. Thus:
missing image!
- Laws of Form - not a.gif -
interprets
Not A
missing image!
- Laws of Form - a or b.gif -
interprets
A Or B
missing image!
- Laws of Form - if a then b.gif -
interprets
Not A Or B
missing image!
- Laws of Form - a and b.gif -
interprets
Not (Not A or Not B)
or Not (If A Then Not B)
or A And B.
Thus any expression in
sentential logic has a
pa translation. 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. There exists a less tedious decision procedure for the
pa, more in the spirit of
Quine's "truth value analysis".
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.
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).
The
pa, like
groups, features a
unary operation,
complementation. By virtue of
J0 the
inverse element is (). Hence the primary algebra is a
boundary algebra, namely a ?--,(-),()? algebra of type ?2,1,0?. A boundary algebras would also be an
abelian group if the inverse and identity elements were identical instead of mutual inverses. Only
C2 clearly demarcates boundary algebras from other
magmas, as
C2 enables demonstrating two laws alien to magmas but central to
lattice theory: the
absorption law and the
distributive law.
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 19761, but they were not published in full until 19932,3
- 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.
Peirce's
semiotics may yet shed light on the philosophical aspects of
LoF.
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:
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
External links
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.
Laws Of Form
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(last updated by Proteus, 8:26pm EDT - Tue, Apr 03 2007)
