Laws of Form

From Wikipedia, the free encyclopedia

This article uses abbreviations that may be confusing or ambiguous. Specific

concerns may be found on the Talk page. Please improve this article if you can. (August 2009)

Laws of Form (hereinafter LoF) is a book by G. Spencer-Brown, published in 1969,

that straddles the boundary between mathematics and philosophy. LoF describes three

distinct logical systems:

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

arithmetic;

The primary algebra (Chapter 6), which interprets 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).

Spencer-Brown referred to the mathematical system of Laws of Form as the "primary

algebra" and the "calculus of indications"; others have termed it boundary algebra.

"Laws of Form" may refer to LoF or to the primary algebra (hereinafter abbreviated pa).

Contents

[hide]

1 The book

2 Reception

3 The form (Chapter 1)

4 The primary arithmetic (Chapter 4)

o 4.1 The notion of canon

5 The primary algebra (Chapter 6)

o 5.1 Syntax

o 5.2 Rules governing logical equivalence

o 5.3 Initials

o 5.4 Proof theory

o 5.5 Interpretations

5.5.1 Two-element Boolean algebra 2

5.5.2 Sentential logic

5.5.3 Syllogisms

o 5.6 An example of calculation

o 5.7 A technical aside

o 5.8 Relation to magmas

6 Equations of the second degree (Chapter 11)

7 Resonances in religion, philosophy, and science

8 Related work

9 See also

10 Notes

11 References

12 External links

[edit] The book

LoF emerged out of work in electronic engineering its author did around 1960, and from

subsequent lectures on mathematical logic he gave under the auspices of the University

of London's extension program. LoF has appeared in several editions, the most recent a

1997 German translation, and has never gone out of print.

The mathematics fills only about 55pp and is rather elementary. But LoF's mystical and

declamatory prose, and its love of paradox, make it a challenging read for all. Spencer-

Brown was influenced 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 him what is. In accordance with G. S. Browns 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.[1]

Except for this one sentence the book can be

seen as an example for the use of E-Prime.

[edit] 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 the pa (primary algebra) 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 two-element Boolean algebra (2) can be seen as the

intended interpretation of the pa. Nevertheless, Meguire (2005)[unreliable source?]

counters

that pa notation:

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 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 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

the Four Color Theorem.[2]

The proof met with skepticism and Spencer-Brown's

mathematical reputation, as well as that of LoF, went into decline. (The Four Color

Theorem and Fermat's Last Theorem were proved in 1976 and 1995, respectively, using

methods owing nothing to LoF.)

[edit] The form (Chapter 1)

The symbol:

also called the mark or cross, is the essential feature 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

"everything else but this."

In LoF, a Cross denotes the drawing of a "distinction", and can be thought of as

signifying the following, all at once:

The act of drawing a boundary around something, thus separating it from

everything else;

That which becomes distinct from everything by drawing the boundary;

Crossing from one side of the boundary to the other.

All three ways imply an action on the part of the cognitive entity (e.g., person) making

the distinction. As LoF puts it:

"The first command:

Draw a distinction

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."

C. S. Peirce came to a related insight in the 1890s; see Related Work.

[edit] The primary arithmetic (Chapter 4)

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

expressions:

The empty Cross ;

All or part of the blank page (the "void").

There are two inductive rules:

A Cross may be written over any expression;

Any two expressions may be concatenated.

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:

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

Starting from an initial marked or unmarked state, "complicating" an expression

by a finite number of repeated application of A1 and A2 cannot yield an

expression whose simplification differs from the initial state. (T4 in LoF).

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:

A Dyck language of order 1 with a null alphabet;

The simplest context-free language in the Chomsky hierarchy;

A rewrite system that is strongly normalizing and confluent.

The phrase calculus of indications in LoF is a synonym for "primary arithmetic".

[edit] 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.

[edit] The primary algebra (Chapter 6)

[edit] 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 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.

[edit] 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 pa formulas, governed by

the rules R1 and R2. Let C and D be formulae each containing at least one instance of

the subformula A:

R1, Substitution of equals. Replace one or more instances of A in C by B,

resulting in E. If A=B, then C=E.

R2, Uniform replacement. Replace all instances of A in C and D with B. C

becomes E and D becomes F. If C=D, then E=F. Note that A=B is not required.

R2 is employed very frequently in pa demonstrations (see below), almost always

silently. These rules are routinely invoked 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 present-day 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 equational-ponential dichotomy

distills much of what distinguishes mathematical logic from the rest of mathematics.

[edit] Initials

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:

"Generation" in LoF;

"Exclusion" in Johnson (1892);

"Pervasion" in the work of William Bricken;

"Mimesis" in the entry logical nand.

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:

Initial AC.D=CD.A and the consequence AA=A (Byrne 1946). This result holds

for all lattices, because AA=A is an easy consequence of the absorption law,

which holds for all lattices;

Initials AC.D=AD.C and J0. Since J0 holds only for lattices with a lower bound,

this method holds only for bounded lattices (which include the pa and 2).

Commutativity is trivial; just set A=(()). Associativity: AC.D = CA.D = CD.A =

A.CD.

Having demonstrated associativity, the period can be discarded.

[edit] 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 both (A)B and (B)A to . This is known as a "calculation".

Once A=B has been demonstrated, A=B can be invoked to justify steps in subsequent

demonstrations. pa demonstrations and calculations often require no more than J1a, J2,

C2, and the consequences ()A=() (C3 in LoF), ((A))=A (C1), and AA=A (C5).

The consequence (((A)B)C) = (AC)((B)C), C7 in LoF, enables 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 the well-known

Boolean algebra theorem that every formula has a normal form.

Let A be a subformula of some formula B. When paired with C3, J1a can be viewed as

the closure condition for calculations: B is a 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 J1a.

LoF includes elegant new proofs of the following standard metatheory:

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

Independence: J1 cannot be demonstrated from J2 and vice versa (T18).

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.

[edit] Interpretations

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 first-order logic has yet to be done, but Peirce's

beta existential graphs suggest that this extension is feasible.

[edit] Two-element Boolean algebra 2

The pa is an elegant minimalist notation for the two-element Boolean algebra 2. Let:

One of Boolean meet (×) or join (+) interpret concatenation;

The complement of A interpret

0 (1) interpret the empty Mark if meet (join) interprets concatenation.

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 .

[edit] Sentential logic

Let the blank page denote True or False, and let a Cross be read as Not. Then the

primary arithmetic has the following sentential reading:

= False

= True = not False

= Not True = False

The pa interprets sentential logic as follows. A letter represents any given sentential

expression. Thus:

interprets Not A

interprets A Or B

interprets Not A Or B or If A Then B.

interprets Not (Not A Or Not B)

or Not (If A Then Not B)

or A And B.

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 2N 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 "self-dual," meaning that any pa formula has two sentential or

Boolean readings, each the dual of the other. Another consequence of self-duality is the

irrelevance 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 well-formed. But an empty Cross is a well-formed 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.

[edit] Syllogisms

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).

[edit] An example of calculation

The following calculation of Leibniz's nontrivial Praeclarum Theorema exemplifies the

demonstrative power of the pa. Let C1 be ((A))=A, and let OI mean that variables and

subformulae have been reordered in a way that commutativity and associativity permit.

Because the only commutative connective appearing in the Theorema 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)]. Praeclarum Theorema.

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

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

= (((R))((S))PQ(RS). Invoke C2 2x to eliminate the bold letters in the previous

expression; OI.

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

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

= (()). J1.

Remarks:

C1 (C2) is repeatedly invoked in a fairly mechanical way to eliminate nested

parentheses (variable instances). This is the essence of the calculation method;

A single invocation of J1 (or, in other contexts, J1a) terminates the calculation.

This too is typical;

Experienced users of the pa are free to invoke OI silently. OI aside, the

demonstration requires a mere 7 steps.

[edit] 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 (Kuratowski's definition) 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.

[edit] Relation to magmas

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 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 pa is a

commutative:

Semigroup because pa juxtaposition commutes and associates;

Monoid with identity element (()), by virtue of J0.

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:

G1. abc = acb (assuming association from the left);

G2. ()a = a;

G3. (a)a = ().

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:

From A2, it follows that (()) ≠ (). If the pa were a group, (())=() would hold, and

one of (a)a=(()) or a()=a would have to be a pa consequence. Note that () and

(()) are mutual pa complements, as group theory requires, so that ((())) = () is

true of both group theory and the pa;

C2 most clearly demarcates the pa from other magmas, because C2 enables

demonstrating the absorption law that defines lattices, and the distributive law

central to Boolean algebra.

Both A2 and C2 follow from B 's being an ordered set.

[edit] Equations of the second degree (Chapter 11)

Chapter 11 of LoF introduces equations of the second degree, composed of recursive

formulae that can be seen as having "infinite" depth. Some recursive formulae simplify

to the marked or unmarked state. Others "oscillate" indefinitely between the two states

depending on whether a given depth is even or odd. Specifically, certain recursive

formulae can be interpreted as oscillating between true and false over successive

intervals of time, in which case a formula is deemed to have an "imaginary" truth value.

Thus the flow of time may be introduced into the pa.

Turney (1986) shows how these recursive formulae can be interpreted via Alonzo

Church's Restricted Recursive Arithmetic (RRA). Church introduced RRA in 1955 as

an axiomatic formalization 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 Spencer-Brown gave to E1 and E4, namely "memory"

and "counter". RRA thus formalizes and clarifies LoF 's notion of an imaginary truth

value.

[edit] Resonances in religion, philosophy, and science

This article may contain original research. Please improve it by verifying the

claims made and adding references. Statements consisting only of original

research may be removed. More details may be available on the talk page. (March

2011)

The mathematical and logical content of LoF is wholly consistent with a secular point

of view. Nevertheless, LoF's "first distinction", and the Notes to its chapter 12, bring to

mind the following landmarks in religious belief, and in philosophical and scientific

reasoning, presented in rough historical order:

Vedic, Hindu and Buddhist: Related ideas can be noted in the ancient Vedic

Upanishads, which form the monastic foundations of Hinduism and later

Buddhism. As stated in the Aitareya Upanishad ("The Microcosm of Man"), the

Supreme Atman manifests itself as the objective Universe from one side, and as

the subjective individual from the other side. In this process, things which are

effects of God's creation become causes of our perceptions, by a reversal of the

process. In the Svetasvatara Upanishad, the core concept of Vedicism and

Monism is "Thou art That."

Taoism, (Chinese Traditional Religion): "...The Tao that can be told is not the

eternal Tao; The name that can be named is not the eternal name. The nameless

is the beginning of heaven and earth..." (Tao Te Ching).

Zoroastrianism: "This I ask Thee, tell me truly, Ahura. What artist made light

and darkness?" (Gathas 44.5)

Judaism (from the Tanakh, called Old Testament by Christians): "In the

beginning when God created the heavens and the earth, the earth was a formless

void... Then God said, 'Let there be light'; and there was light. ...God separated

the light from the darkness. God called the light Day, and the darkness he called

Night.

"...And God said, 'Let there be a dome in the midst of the waters, and let it

separate the waters from the waters.' So God made the dome and separated the

waters that were under the dome from the waters that were above the dome.

"...And God said, 'Let the waters under the sky be gathered together into one

place, and let the dry land appear.' ...God called the dry land Earth, and the

waters that were gathered together he called Seas.

"...And God said, 'Let there be lights in the dome of the sky to separate the day

from the night...' God made the two great lights... to separate the light from the

darkness." (Genesis 1:1-18; Revised Standard Version, emphasis added).

"And the whole earth was of one language, and of one speech." (Genesis 11:1;

emphasis added).

"I am; that is who I am." (Exodus 3:14)

Confucianism: Confucius claimed that he sought "a unity all pervading"

(Analects XV.3) and that there was "one single thread binding my way

together." (Ana. IV.15). The Analects also contain the following remarkable

passage on how the social, moral, and aesthetic orders are grounded in right

language, grounded in turn in the ability to "rectify names," i.e., to make correct

distinctions: "Zilu said, 'What would be master's priority?" The master replied,

"Rectifying names! ...If names are not rectified then language will not flow. If

language does not flow, then affairs cannot be completed. If affairs are not

completed, ritual and music will not flourish. If ritual and music do not flourish,

punishments and penalties will miss their mark. When punishments and

penalties miss their mark, people lack the wherewithal to control hand and foot."

(Ana. XIII.3)

Heraclitus: Pre-socratic philosopher, credited with forming the idea of logos.

"He who hears not me but the logos will say: All is one." Further: "I am as I am

not."

Parmenides: Argued that the every-day perception of reality of the physical

world is mistaken, and that the reality of the world is 'One Being': an

unchanging, ungenerated, indestructible whole.

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

Christianity: "In the Beginning was the Word, and the Word was with God, and

the Word was God." (John 1:1). "Word" translates logos in the koine original.

"If you do not believe that I am, you will die in your sins." (John 8:24). "The

Father and I are one." (John 10:30). "That they all may be one; as thou, Father,

art in me, and I in thee, that they may also be one in us: that the world may

believe that thou has sent me." (John 17:21). (emphases added)

Object relations theory, psychodynamics: The primary separation experienced

by infants between self and other objects, distinguishing of reality from

phantasy.

Islamic philosophy distinguishes essence (Dhat) from attribute (Sifat), which are

neither identical nor separate.

Leibniz: "All creatures derive from God and from nothingness. Their self-being

is of God, their nonbeing is of nothing. Numbers too show this in a wonderful

way, and the essences of things are like numbers. No creature can be without

nonbeing; otherwise it would be God... The only self-knowledge is to

distinguish well between our self-being and our nonbeing... Within our selfbeing

there lies an infinity, a footprint or reflection of the omniscience and

omnipresence of God."[3]

Josiah Royce: "Without negation, there is no inference. Without inference, there

is no order, in the strictly logical sense of the word. The fundamentally

significant position of the idea of negation in determining and controlling our

idea of the orderliness of both the natural and the spiritual order, becomes, in the

light of all these considerations, as momentous as it is, in our ordinary popular

views of this subject, neglected. ...From this point of view, negation appears as

one of the most significant. ideas that lie at the base of all the exact sciences. By

virtue of the idea of negation we are able to define processes of inference-

processes which, in their abstract form, the purely mathematical sciences

illustrate, and which, in their natural expression, the laws of the physical world,

as known to our inductive science, exemplify."

"When logically analyzed, order turns out to be something that would be

inconceivable and incomprehensible to us unless we had the idea which is

expressed by the term 'negation'. Thus it is that negation, which is always also

something intensely positive, not only aids us in giving order to life, and in

finding order in the world, but logically determines the very essence of order." [4]

Returning to the Bible, the injunction "Let there be light" conveys:

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

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

"… morning and evening" — the boundaries of the light.

A Cross denotes a distinction made, and the absence of a Cross means that no

distinction has been made. In the Biblical example, light is distinct from the void – the

absence of light. The Cross and the Void are, of course, the two primitive values of the

Laws of Form.

[edit] 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 pa in three veins of work:

1. 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,[5]

but they were not published in full until 1993.[6]

2. In a 1902 encyclopedia article,[7]

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.

3. The syntax of his alpha existential graphs is merely concatenation, read as

conjunction, and enclosure by ovals, read as negation.[8]

If pa concatenation is

read as conjunction, then these graphs are isomorphic to the pa (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 pa, all instances of boundary mathematics, i.e.,

mathematics whose syntax is limited to letters and brackets (enclosing devices). A

minimalist syntax of this nature is a "boundary notation." Boundary notation is free of

infix, prefix, or postfix operator symbols. The very well known curly braces ('{', '}') of

set theory can be seen as a boundary notation.

The work of Leibniz, Peirce, and Nicod is innocent of metatheory, as they wrote before

Emil Post's landmark 1920 paper (which LoF cites), proving 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.

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.

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:

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 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: one connective,

NAND (whose pa translation is (AB) or—dually -- (A)(B) ), universal

quantification, and one binary atomic formula, denoting set membership. This is

the system of Quine (1951).

The beta existential graphs, with a single binary predicate denoting set

membership. This has yet to be explored. The alpha graphs mentioned above are

a special case of the beta graphs.

[edit] 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

[edit] Notes

1. ^ Felix Lau: "Die Paradoxie der Form", 2005 Carl-Auer Verlag, ISBN 978-

89670-352-1

2. ^ For a sympathetic evaluation, see Kauffman (2001).

3. ^ "On the True Theologia Mystica" in Loemker, Leroy, ed. and trans., 1969.

Leibniz: Philosophical Papers and Letters. Reidel: 368.

4. ^ "Order" in Hasting, J., ed., 1917. Encyclopedia of Religion and Ethics.

Scribner's: 540. Reprinted in Robinson, D. S., ed., 1951, Royce's Logical Essays.

Dubuque IA: Wm. C. Brown: 230-31.

5. ^ "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

6. ^ "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.

7. ^ 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

8. ^ 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.

[edit] References

Editions of Laws of Form:

o 1969. London: Allen & Unwin, hardcover.

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

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

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

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

1

o 1997 German translation, titled Gesetze der Form. Lübeck: Bohmeier

Verlag. ISBN 3-89094-321-7

Bostock, David, 1997. Intermediate Logic. Oxford Univ. Press.

Byrne, Lee, 1946, "Two Formulations of Boolean Algebra," Bulletin of the

American Mathematical Society: 268-71.

Craig, William (1979). "Boolean Logic and the Everyday Physical World".

Proceedings and Addresses of the American Philosophical Association 52 (6):

751–78. doi:10.2307/3131383.

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. Revision. Steers clear of the more speculative aspects of

LoF. The source for the notation of much of this entry, which encloses in

parentheses what LoF places under a cross.

Willard Quine, 1951. Mathematical Logic, 2nd ed. Harvard 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: 1–13.

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: 239–55. doi:10.1080/03081078108934802.

Turney, P. D. (1986). "Laws of Form and Finite Automata". International

Journal of General Systems 12: 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.

[edit] External links

Laws of Form web site, 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."

Draw a distinction... The space of imagination based on LoF.

The Multiple Form Logic, by G.A. Stathis, owes much to the primary algebra.

The Laws of Form Forum, where the primary algebra and related formalisms

have been discussed since 2002.

View page ratings

Rate this page

What's this?

Trustworthy

Objective

Complete

Well-written

I am highly knowledgeable about this topic (optional)

Categories: Algebra | Boolean algebra | Logic | Logical calculi | Mathematical logic

Log in / create account

Article

Discussion

Read

Edit

View history

Main page

Contents

Featured content

Current events

Random article

Donate to Wikipedia

Interaction

Help

About Wikipedia

Community portal

Recent changes

Contact Wikipedia

Toolbox

Print/export

Languages

Deutsch

한국어

日本語

Simple English

This page was last modified on 22 June 2011 at 06:42.

Text is available under the Creative Commons Attribution-ShareAlike License;

additional terms may apply. See Terms of use for details.

Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-

profit organization.

Contact us

Privacy policy

About Wikipedia

Disclaimers