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Encyclopedia > Laws of form

Laws of Form (hereinafter LoF) is a book by G. Spencer Brown, published in 1969, that straddles the boundary between mathematics and of philosophy. LoF describes three distinct logical systems: G. Spencer-Brown (April 2, 1923) was born in Grimsby, Lincolnshire, England and is a British mathematician. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... For other uses, see Philosophy (disambiguation). ...

  • The primary arithmetic (described in Chapter 4), whose models include Boolean arithmetic;
  • The primary algebra (Chapter 6), whose models include 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).-1... This article is about the branch of mathematics. ... -1... Boolean logic is a complete system for logical operations. ... In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows to establish that certain formulas are theorems of the formal system. ... In the theory of computation, a finite state machine (FSM) or finite state automaton (FSA) is an abstract machine that has only a finite, constant amount of memory. ... ‹ The template below (Expand) is being considered for deletion. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of a set closed under one or more operations, satisfying a number of axioms, including none. ...

Contents

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. Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ...


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 Peirce, Bertrand Russell, and Alfred North Whitehead. Look up paradox in Wiktionary, the free dictionary. ... Ludwig Wittgenstein (1889-1951), pictured here in 1930, made influential contributions to Logic and the philosophy of language, critically examining the task of conventional philosophy and its relation to the nature of language. ... R.D.Laing; photo credit Robert E. Haraldsen Ronald David Laing (October 7, 1927–August 23, 1989), was a Scottish psychiatrist who wrote extensively on mental illness and particularly the experience of psychosis. ... Charles Sanders Peirce (IPA: /pɝs/), (September 10, 1839 – April 19, 1914) was an American polymath, physicist, and philosopher, born in Cambridge, Massachusetts. ... Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician, advocate for social reform, and pacifist. ... Alfred North Whitehead, OM (February 15, 1861 Ramsgate, Kent, England – December 30, 1947 Cambridge, Massachusetts, USA) was an English-born mathematician who became a philosopher. ...


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 reveals striking connections among logic, Boolean algebra, and arithmetic, and the philosophy of language and mind. This article is being considered for deletion in accordance with Wikipedias deletion policy. ... The Whole Earth Catalog was a sizeable catalog published twice a year from 1968 to 1972, and occasionally thereafter, until 1998. ... Consciousness is a quality of the mind generally regarded to comprise qualities such as subjectivity, self-awareness, sentience, sapience, and the ability to perceive the relationship between oneself and ones environment. ... Look up Cognition in Wiktionary, the free dictionary. ... Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... Boolean algebra is the finitary algebra of two values. ... Philosophy of language is the reasoned inquiry into the nature, origins, and usage of language. ... For other uses, see Mind (disambiguation). ...


Some, e.g. Banaschewski (1977), argue that the pa is nothing but new notation for Boolean algebra. It is true that 2 can be seen as the intended interpretation of the pa. Nevertheless, Meguire (2005) 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). The word duality has a variety of different meanings in different contexts: In several spiritual, religious, and philosophical doctrines, duality refers to a two-fold division also called dualism. ... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... A propositional calculus is a formal, deduction system, or proof theory for reasoning with propositional formulas as symbolic logic. ... A syllogism (Greek: — conclusion, inference), usually the categorical syllogism, is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises) of a certain form. ... Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ...


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. He is a twat He was born in Vienna and died in Pescadero, California. ... Louis Kauffman, topologist Louis Kauffman is a topologist, whose work is primarily in knot theory and connections with statistical mechanics, quantum theory, algebra, combinatorics and foundations. ... Niklas Luhmann (December 8, 1927 - November 6, 1998) was a German sociologist, administration expert, and social systems theorist, as well as one the most prominent modern day thinkers in the sociological systems theory. ... Humberto Maturana (born September 14, 1928 in Santiago) is a Chilean biologist whose work crosses over into philosophy and cognitive science. ... Francisco Varela (Santiago, September 7, 1946 – May 28, 2001, Paris) was a Chilean biologist and philosopher who, together with his teacher Humberto Maturana, is best known for introducing the concept of autopoiesis to biology. ...


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[1]. 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.) Example of a four-colored map The four color theorem (also known as the four color map theorem) states that given any plane separated into regions, such as a political map of the states of a country, the regions may be colored using no more than four colors in such... Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ... In mathematics, Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. ...


The form (Chapter 1)

The symbol:

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 "everything else but this." Laws of Form - Cross File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... It has been suggested that this article be split into multiple articles accessible from a disambiguation page. ... Look up Cognition in Wiktionary, the free dictionary. ... This article does not cite any references or sources. ... This article or section should be merged with Capability (computers) and Capability. ...


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." Charles Peirce came to a related insight in the 1890s; see Related Work below. Charles Sanders Peirce (IPA: /pɝs/), (September 10, 1839 – April 19, 1914) was an American polymath, physicist, and philosopher, born in Cambridge, Massachusetts. ...

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. For other uses, see Syntax (disambiguation). ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... Look up definition in Wiktionary, the free dictionary. ...


The primary arithmetic (and all of the Laws of Form) are grounded in a mere two axioms, A1 and A2. This article is about a logical statement. ...



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:

Image:Laws of Form - cross.gif Image:Laws of Form - cross.gif  =  Image:Laws of Form - cross.gif


A2. The law of Crossing. 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: Laws of Form - Cross File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Laws of Form - Cross File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Laws of Form - Cross File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

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: Laws of Form - Double Cross File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

  • 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. In mathematics, the concept of a relation is a generalization of 2-place relations, such as the relation of equality, denoted by the sign = in a statement like 5 + 7 = 12, or the relation of order, denoted by the sign < in a statement like 5 < 12. Relations that involve two... In logic, statements p and q are logically equivalent if they have the same logical content. ... A partition of U into 6 blocks: an Euler diagram representation. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...


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. In the theory of formal languages of computer science, mathematics, and linguistics, the Dyck language (Dyck being pronounced deek) is the language consisting of those balanced strings of parentheses [ and ]. It is important in the parsing of expressions that must have a correctly nested sequence of parentheses, such as arithmetic... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... The Chomsky hierarchy is a containment hierarchy of classes of formal grammars that generate formal languages. ...


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 metalogic of a system of logic is the formal proof supporting its soundness. ... For an account of the concept of object language in mathematical logic, see formal system. ... In logic and linguistics, a metalanguage is a language used to make statements about other languages (object languages). ...


The primary algebra (Chapter 6)

The primary algebra is an elegant minimalist notation for the two-element Boolean algebra, very similar to the entitative and existential graphs, and other formal systems Charles Peirce devised in work written 1885-1905. However, Peirce's work that most resembles the pa was not published until after the first edition of Laws of Form.-1... An entitative graph is an element of the graphical syntax for logic that Charles Sanders Peirce developed under the name of qualitative logic in the 1880s, taking the coverage of the formalism only as far as the propositional or sentential aspects of logic are concerned. ... An existential graph is a type of diagrammatic or visual notation for logical expressions, proposed by Charles Sanders Peirce, who wrote his first paper on graphical logic in 1882 and continued to develop the method until his death in 1914. ... Charles Sanders Peirce (IPA: /pɝs/), (September 10, 1839 – April 19, 1914) was an American polymath, physicist, and philosopher, born in Cambridge, Massachusetts. ...


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. In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... In computer science and mathematics, a variable (IPA pronunciation: ) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression. ... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ... In general, a carrier is a system or process with a specific property or is attributed of something (in physical or in abstract sense). ...


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: An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... In logic, statements p and q are logically equivalent if they have the same logical content. ... In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being equivalent in some way. ...

  • 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. Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ...


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 = Image:Laws of Form - cross.gif". This article is about equations in mathematics. ... In abstract algebra, an algebraic structure consists of a set together with a collection of operations or relations defined on it which satisfy certain axioms. ... Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ... Within the study of logic, a tautology is a statement containing more than one sub-statement, that is true regardless of the truth values of its parts. ... This article is about the pedestrian gate. ... In logic, modus ponens (Latin: mode that affirms; often abbreviated MP) is a valid, simple argument form. ... Within the study of logic, a tautology is a statement containing more than one sub-statement, that is true regardless of the truth values of its parts. ... Laws of Form - Cross File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...


Initials

An initial is a pa equation verifiable by a decision procedure and as such is not an axiom. LoF lays down the initials: In logic, a decision problem is determining whether or not there exists a decision procedure or algorithm for a class S of questions requiring a Boolean value (i. ... This article is about a logical statement. ...

  • J1: ((A)A) = <void>.
  • J2: ((A)(B))C = ((AC)(BC)).

J2 is the familiar distributive law of sentential logic and Boolean algebra. In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... A propositional calculus is a formal, deduction system, or proof theory for reasoning with propositional formulas as symbolic logic. ... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ...


Another set of initials, friendlier to calculations, is:

  • J0: (A)A= Image:Laws of Form - cross.gif
  • J1a: (())A = A
  • C2: A(AB)=A(B).

J0 establishes that the pa is a complemented lattice whose upper bound is (). J1a is an algebraic version of A2, and makes clear the sense in which (()) aliases with the blank page. J0 and J1a also establish the inverse element and identity element of the pa. J1 can be derived from J1a and J0. Laws of Form - Cross File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In the mathematical discipline of order theory, and in particular, in lattice theory, a complemented lattice is a bounded lattice in which each element x has a complement, defined as a unique element ~ x such that and A Boolean algebra may be defined as a complemented distributive lattice. ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ... For other uses, see identity (disambiguation). ...


C2 and generation are synonyms. Both terms are from LoF. William Bricken calls C2 pervasion. C2 is called mimesis in logical nand. 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. NAND Logic Gate The Sheffer stroke, |, is the negation of the conjunction operator. ... An existential graph is a type of diagrammatic or visual notation for logical expressions, proposed by Charles Sanders Peirce, who wrote his first paper on graphical logic in 1882 and continued to develop the method until his death in 1914. ...


T13 in LoF generalizes C2 as follows. Any pa (or sentential logic) formula B can be viewed as an ordered tree with branches. Then: In computer science, a tree is a widely-used computer data structure that emulates a tree structure with a set of linked nodes. ...


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. In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ...


While a proof of T13 would require induction, the intuition underlying it should be clear. Look up induction in Wiktionary, the free dictionary. ...


LoF asserts that concatenation can be read as commuting and associating by default and hence need not be explicitly assumed. (Peirce made a similar assumption in his graphical logic.) That concatenation commutes and associates can also be derived as consequences from such compact initials as ABC = BCA or ABC = ACB. See Meguire (2003). Example showing the commutativity of addition (3 + 2 = 2 + 3) For other uses, see Commute (disambiguation). ... In mathematics, associativity is a property that a binary operation can have. ... An existential graph is a type of diagrammatic or visual notation for logical expressions, proposed by Charles Sanders Peirce, who wrote his first paper on graphical logic in 1882 and continued to develop the method until his death in 1914. ...


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. Look up theorem in Wiktionary, the free dictionary. ... In logic and linguistics, a metalanguage is a language used to make statements about other languages (object languages). ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... Look up theorem in Wiktionary, the free dictionary. ... In logic, a decision problem is determining whether or not there exists a decision procedure or algorithm for a class S of questions requiring a Boolean value (i. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... Look up theorem in Wiktionary, the free dictionary. ...


Let A and B be pa formulas. A demonstration of A=B may proceed in either of two ways: In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ...

  • Modify A in steps until B is obtained, or vice versa;
  • Simplify both (A)B and (B)A to Image:Laws of Form - cross.gif. 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 J0, J1, J2, C2, and the consequences ()A=(), ((A))=A, and AA=A (in LoF, C3, C1, and C5, respectively). Laws of Form - Cross File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...


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. In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... In boolean logic, a formula is in conjunctive normal form (CNF) if it is a conjunction of clauses, where a clause is a disjunction of literals. ... Boolean algebra is the finitary algebra of two values. ...


Let A be a subformula of some formula B. When paired with C3, 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: In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... Within the study of logic, a tautology is a statement containing more than one sub-statement, that is true regardless of the truth values of its parts. ... In mathematical logic, natural deduction is an approach to proof theory that attempts to provide a formal model of logical reasoning as it naturally occurs. ...

  • 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: A metatheory is a theory which concerns itself with another theory, or theories. ...

  • 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. Look up completeness in Wiktionary, the free dictionary. ... A propositional calculus is a formal, deduction system, or proof theory for reasoning with propositional formulas as symbolic logic. ... Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ...


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.-1... -1... A propositional calculus is a formal, deduction system, or proof theory for reasoning with propositional formulas as symbolic logic. ... First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... Charles Sanders Peirce (IPA: /pɝs/), (September 10, 1839 – April 19, 1914) was an American polymath, physicist, and philosopher, born in Cambridge, Massachusetts. ... An existential graph is a type of diagrammatic or visual notation for logical expressions, proposed by Charles Sanders Peirce, who wrote his first paper on graphical logic in 1882 and continued to develop the method until his death in 1914. ...


Two-element Boolean algebra 2

Let Boolean:

  • Meet or join interpret AB;
  • The complement of A interpret Image:Laws of Form - not a.gif;
  • 0 or 1 interpret the empty Mark.

If meet interprets AB, then join interprets ~(~A+~B), or vice versa. Hence the pa and 2 are isomorphic, and2 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. See lattice for other mathematical as well as non-mathematical meanings of the term. ... Look up join in Wiktionary, the free dictionary. ... A complementation test is used in genetics to decide if two recessive mutant phenotypes are determined by mutations in the same gene or two different genes. ... Laws of Form - Not A File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ...


In the language of universal algebra, the pa is the algebraic structureB,--,(-),()〉 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〉. Universal algebra (sometimes called General algebra) is the field of mathematics that studies the ideas common to all algebraic structures. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... NAND Logic gate The Sheffer stroke, written | or ↑, denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as not both. It is also called the alternative denial, since it says in effect that at least one of its operands is false. ...


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
Image:Laws of Form - cross.gif  =  True  =  not False
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: Laws of Form - Cross File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Laws of Form - Double Cross File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

Image:Laws of Form - not a.gif interprets Not A
Image:Laws of Form - a or b.gif interprets A Or B
Image:Laws of Form - if a then b.gif interprets Not A Or B or If A Then B.
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.
  (((A)B)(A(B)))  and   ((A)(B))(AB)  both interpret   A  &lrarr; (if and only if)   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. Meguire (2003) sets out a less tedious decision procedure for the pa, more in the spirit of Quine's "truth value analysis". Laws of Form - Not A File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Laws of Form -- A or B File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Laws of Form -- If A then B File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Laws of Form - A and B File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... ↔ ⇔ ≡ logical symbols representing iff. ... A propositional calculus is a formal, deduction system, or proof theory for reasoning with propositional formulas as symbolic logic. ... Within the study of logic, a tautology is a statement containing more than one sub-statement, that is true regardless of the truth values of its parts. ... The Boolean satisfiability problem (SAT) is a decision problem considered in complexity theory. ... In logic, a decision problem is determining whether or not there exists a decision procedure or algorithm for a class S of questions requiring a Boolean value (i. ... For people named Quine, see Quine (surname). ...


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, the pa is "self-dual," meaning that any pa formula has two sentential or Boolean readings, each the dual of the other. Concatenation is a standard operation in computer programming languages (a subset of formal language theory). ... The word duality has a variety of different meanings in different contexts: In several spiritual, religious, and philosophical doctrines, duality refers to a two-fold division also called dualism. ... A propositional calculus is a formal, deduction system, or proof theory for reasoning with propositional formulas as symbolic logic. ... -1... The word duality has a variety of different meanings in different contexts: In several spiritual, religious, and philosophical doctrines, duality refers to a two-fold division also called dualism. ...


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. In the pa, a Cross, interpretable as complementation/negation, with nothing under itself denotes the Marked state, 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. A complementation test is used in genetics to decide if two recessive mutant phenotypes are determined by mutations in the same gene or two different genes. ... Negation (i. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... In mathematics, an operand is one of the inputs (arguments) of an operator. ...


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. A syllogism (Greek: — conclusion, inference), usually the categorical syllogism, is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises) of a certain form. ... A polysyllogism, sometimes called multi-premise syllogism, is a string of any number of syllogisms such that the conclusion of one is a premise for the next, and so on. ... Traditional logic, also known as term logic, is a loose term for the logical tradition that originated with Aristotle and survived broadly unchanged until the advent of modern predicate logic in the late nineteenth century. ... In mathematics, a monadic logic is one that employs only unary relations. ... In mathematics, a monadic logic is one that employs only unary relations. ...


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 (()). Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ...

  • [(PR)∧(QS)]→[(PQ)→(RS)] 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. Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ... In logic, a truth value, or truth-value, is a value indicating to what extent a statement is true. ... In metaphysics, extension is the property of taking up space; see Extension (metaphysics). ... In mathematical logic, an atomic formula or atom is a formula with no underlying propositional structure. ... As commonly used, individual refers to a person or to any specific object in a collection. ... In mathematics, an ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element (the first and second elements are also known as left and right projections). ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... This article is about the concept of recursion. ... Dana Stewart Scott (born 1932) is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California. ...


Relation to groupoids

The pa can be seen as the logical endpoint of a point noted by Huntington in 1933: Boolean algebra requires two, not three, operations, one binary and one unary. Hence the seldom-noted fact that Boolean algebras are magmas (a.k.a. groupoids). To see this, note that the pa is a commutative: Edward Vermilye Huntington (April 26 1874, Clinton, New York, USA -- November 25, 1952, Cambridge, Massachusetts, USA) was an American mathematician. ... Boolean algebra is the finitary algebra of two values. ... In logic and mathematics, an operation ω is a function of the form ω : X1 × … × Xk → Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ... In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...

Groups also require a unary operation, called inverse, whose inverse element is at once the inverse of, and equal to, the identity element. Complementation is the pa unary operation corresponding to group inverse. By J0, the pa inverse element is (). Groups and the pa have signatures of the same form, namely they both are 〈--,(-),()〉 algebras of type 〈2,1,0〉. Hence the pa is a boundary algebra. In mathematics, a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation. ... In quantum Mechanics, we define: [A,B]=AB-BA If [A,B]=0, then we say A, B is commute. ... To join as a partner, ally, or friend. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... For other uses, see identity (disambiguation). ... This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, a unary operation is an operation with only one operand. ... Look up inverse in Wiktionary, the free dictionary. ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ... Boolean algebra is the finitary algebra of two values. ... In mathematics, a signature for an algebraic structure A over a set S is a list of the operations that characterize A, along with their arities. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of a set closed under one or more operations, satisfying a number of axioms, including none. ...


The axioms and initials of the pa distinguish it from an abelian group in two ways: In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ...

  • While the pa inverse element () and identity element (()) are mutual complements, as group theory requires, A2 rules out their being identical. This follows from B being an ordered set. If the pa were a group, one of (a)a=(()) or a()=a would have to be a pa consequence;
  • C2 powerfully demarcates the pa from other magmas, because C2 enables demonstrating the absorption law central to lattice theory, and the distributive law central to Boolean algebra.

In boundary terms, the defining arithmetical fact of group theory is (())=(). The PA counterpart of that equation is ((()))=(). ... Look up group in Wiktionary, the free dictionary. ... In algebra, the absorption law is an identity linking a pair of binary operations. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ...


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. This article is about the concept of recursion. ...


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. ‹ The template below (Expand) is being considered for deletion. ... In the theory of computation, a finite state machine (FSM) or finite state automaton (FSA) is an abstract machine that has only a finite, constant amount of memory. ...


Resonances in religion, philosophy, and science

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)
  • 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." ("On the True Theologia Mystica" in Loemker, Leroy, ed. and trans., 1969. Leibniz: Philosophical Papers and Letters. Reidel: 368.)
  • 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... as momentous as it is, in our ordinary popular views... neglected. ...negation appears as one of the most significant... ideas that lie at the base of all the exact sciences..."
"When logically analyzed, order turns out to 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." ("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.)
  • John Archibald Wheeler: "The boundary of a boundary is zero. This central principle of algebraic topology, identity, triviality, tautology though it is, is also the unifying theme of Maxwell's electrodynamics, general relativity, and almost every version of modern field theory. That one can get so much out of so little, almost everything from almost nothing, inspires hope that we will someday complete the mathematization of physics and derive everything from nothing, all law from no law." ("It from Bit" in Wheeler, J. A. (1996) At Home in the Universe. American Institute of Physics Press: 302).

Returning to the Bible, the injunction "Let there be light" conveys: This article discusses the historical religious practices in the Vedic time period; see Dharmic religions for details of contemporary religious practices. ... This article discusses the adherents of Hinduism. ... A replica of an ancient statue found among the ruins of a temple at Sarnath Buddhism is a philosophy based on the teachings of the Buddha, Siddhārtha Gautama, a prince of the Shakyas, whose lifetime is traditionally given as 566 to 486 BCE. It had subsequently been accepted by... This article discusses the historical religious practices in the Vedic time period; see Dharmic religions for details of contemporary religious practices. ... The Upanishads (&#2313;&#2346;&#2344;&#2367;&#2359;&#2342;&#2381;, Upani&#351;ad) are part of the Hindu Shruti scriptures which primarily discuss meditation and philosophy and are seen as religious instructions by most schools of Hinduism. ... For other uses, see Monist (disambiguation). ... Hinduism (known as in modern Indian languages[1]) is a religious tradition[2] that originated in the Indian subcontinent. ... A silhouette of a Buddha statue at Ayutthaya, Thailand. ... Atman is a Sanskrit word, normally translated as soul or self (also ego). ... For other uses, see Universe (disambiguation). ... Taoism (Daoism) is the English name referring to a variety of related Chinese philosophical and religious traditions and concepts. ... It has been suggested that Chinese folk religion be merged into this article or section. ... The Tao Te Ching (&#36947;&#24503;&#32147;, Pinyin: D J&#299;ng, thus sometimes rendered in recent works as Dao De Jing; archaic pre-Wade-Giles rendering: Tao Teh Ching; roughly translated as The Book of the Way and its Virtue (see dedicated chapter below on translating the title)) is... Zoroastrianism is the religion and philosophy based on the teachings ascribed to the prophet Zoroaster (Zarathustra, Zartosht). ... The Gathas (Gāθās) are the most sacred of the texts of the Zoroastrian faith, and are traditionally believed to have been composed by Zarathushtra (Zoroaster) himself. ... This article or section does not cite its references or sources. ... For the musical collective, see Tanakh (band). ... Note: Judaism commonly uses the term Tanakh to refer to its canon, which corresponds to the Protestant Old Testament. ... Wenmiao Temple, a Confucian Temple in Wuwei, Gansu, Peoples Republic of China. ... Confucius (Chinese: ; pinyin: ; Wade-Giles: Kung-fu-tzu), lit. ... Heraclitus of Ephesus (Ancient Greek - Herákleitos ho Ephésios (Herakleitos the Ephesian)) (about 535 - 475 BC), known as The Obscure (Ancient Greek - ho Skoteinós), was a pre-Socratic Greek philosopher, a native of Ephesus on the coast of Asia Minor. ... This article is about logos (logoi) in ancient Greek philosophy, mathematics, rhetoric, Theophilosophy, and Christianity. ... Parmenides of Elea (Greek: , early 5th century BC) was an ancient Greek philosopher born in Elea, a Hellenic city on the southern coast of Italy. ... PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ... Topics in Christianity Movements · Denominations · Other religions Ecumenism · Preaching · Prayer Music · Liturgy · Calendar Symbols · Art · Criticism Important figures Apostle Paul · Church Fathers Constantine · Athanasius · Augustine Anselm · Aquinas · Palamas · Luther Calvin · Wesley Arius · Marcion of Sinope Archbishop of Canterbury · Catholic Pope Coptic Pope · Ecumenical Patriarch Christianity Portal This box:      Christianity is... This article is about logos (logoi) in ancient Greek philosophy, mathematics, rhetoric, Theophilosophy, and Christianity. ... The literal meaning of the Greek word koine (κοινή) is common. It is used in several senses: Koiné Greek (Κοινή Ἑλληνική), a Greek dialect that developed from the Attic dialect (of Athens) and became the spoken language of Greece at the time of the Empire of Alexander the Great. ... Islamic philosophy (الفلسفة الإسلامية) is a branch of Islamic studies, and is a longstanding attempt to create harmony between philosophy (reason) and the religious teachings of Islam (faith). ... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... Josiah Royce (November 20, 1855, Grass Valley, California. ... Encyclopedia of Religion and Ethics is an extensive work by James Hastings, written between 1908 and 1927, covering religion, comprised of entries by many contributors. ... John Archibald Wheeler (born July 9, 1911) is an eminent American theoretical physicist. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... Within the study of logic, a tautology is a statement containing more than one sub-statement, that is true regardless of the truth values of its parts. ... James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and theoretical physicist from Edinburgh, Scotland, UK. His most significant achievement was aggregating a set of equations in electricity, magnetism and inductance — eponymously named Maxwells equations — including an important modification (extension) of the Ampères... Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... Field theory (mathematics), the theory of the algebraic concept of field. ...

  • "...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.


Related work

Lenzen (2004) reviews in English his extensive work in German showing that Gottfried Leibniz, in memoranda not published until the late 19th and early 20th centuries, invented Boolean logic. Leibniz's notation was isomorphic to that of LoF: concatenation interpreted as conjunction and "non-(X)" interpreted as the complement of X. Leibniz's pioneering role in algebraic logic was adumbrated by Clarence Irving Lewis (1918) and by Rescher (1954). But a full appreciation of Leibniz's accomplishment had to await Lenzen's work in the 1980s. “Leibniz” redirects here. ... Boolean algebra is the finitary algebra of two values. ... Look up Complement in Wiktionary, the free dictionary. ... Clarence Irving Lewis (April 12, 1883 Stoneham, Massachusetts - February 3, 1964 Cambridge, Massachusetts) was an American academic philosopher. ... Nicholas Rescher (born July 15, 1928 in Hagen, Germany) is an American philosopher, affiliated for many years with the University of Pittsburgh, where he is currently University Professor of Philosophy and Chairman of the Center for the Philosophy of Science. ...


Charles Peirce (1839-1914) anticipated the pa in three veins of work: Charles Sanders Peirce (IPA: /pɝs/), (September 10, 1839 – April 19, 1914) was an American polymath, physicist, and philosopher, born in Cambridge, Massachusetts. ...

  • 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 by Peirce 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. An existential graph is a type of diagrammatic or visual notation for logical expressions, proposed by Charles Sanders Peirce, who wrote his first paper on graphical logic in 1882 and continued to develop the method until his death in 1914. ... Semiotics, semiotic studies, or semiology is the study of signs and symbols, both individually and grouped into sign systems. ...


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. An infix is an affix inserted inside an existing word. ... It has been suggested that this article or section be merged with Reverse Polish notation. ... Postfix notation is a mathematical notation wherein every operator follows all of its operands. ...


Kauffman discusses another notation similar to that of LoF, that of a 1917 article by Jean Nicod, a disciple of Bertrand Russell's. Jean George Pierre Nicod (c. ... Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician, advocate for social reform, and pacifist. ...


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. Emil Leon Post (February 11, 1897 - April 21, 1954) was a Polish-American mathematician and logician. ... A propositional calculus is a formal, deduction system, or proof theory for reasoning with propositional formulas as symbolic logic. ... David Hilbert David Hilbert (January 23, 1862 &#8211; February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... Notable people named Łukasiewicz include: Ignacy Łukasiewicz (1822-1882), a Polish pharmacist and inventor of kerosene Jan Łukasiewicz (1878-1956), a Polish mathematician This is a disambiguation page, a list of pages that otherwise might share the same title. ... An axiom P is independent if there is no other axiom Q such that Q implies P. In many cases independency is desired, either to reach the [[[conclusion]]] of a reduced set of axioms, or to be able to replace an independent axiom to create a more unique system (for... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ...


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, William Craig (1979). The Right Honourable William Craig (b. ...


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: Cognitive science is usually defined as the scientific study either of mind or of intelligence (e. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

  • Lakoff, George (1987) Women, Fire, and Dangerous Things. University of Chicago Press. See index entries under "Image schema examples: container."
  • Lakoff, George, and Rafael E. Núñez (2001) Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Basic Books.

Neither book cites LoF. This article or section does not cite any references or sources. ... This article or section does not cite any references or sources. ... Rafael E. Núñez is a professor of Cognitive science at the University of California, San Diego and is well known for promoting the idea of embodied cognition. ... Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (hereinafter WMCF) is a book by George Lakoff, a cognitive linguist, and Rafael E. Núñez, a psychologist. ...


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. Humberto Maturana (born September 14, 1928 in Santiago) is a Chilean biologist whose work crosses over into philosophy and cognitive science. ... Francisco Varela (Santiago, September 7, 1946 – May 28, 2001, Paris) was a Chilean biologist and philosopher who, together with his teacher Humberto Maturana, is best known for introducing the concept of autopoiesis to biology. ... Eleanor Rosch is a professor of psychology at The University of California, Berkeley. ...


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:

Other formal systems with possible affinities to the Laws of Form are mereology and mereotopology. The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ... Not to be confused with combinational logic, a topic in digital electronics. ... Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ... NAND Logic Gate The Sheffer stroke, |, is the negation of the conjunction operator. ... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... W. V. Quine Willard Van Orman Quine (June 25, 1908 - December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. ... Mereology is a collection of axiomatic formal systems dealing with parts and their respective wholes. ... Mereotopology is a formal theory, combining mereology and topology, of the topological relationships among wholes, parts, and the boundaries between parts. ...


References

  1. "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.
  2. "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.
  3. "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).
  • 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.
  • 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.

Nicholas Rescher (born July 15, 1928 in Hagen, Germany) is an American philosopher, affiliated for many years with the University of Pittsburgh, where he is currently University Professor of Philosophy and Chairman of the Center for the Philosophy of Science. ...

See also

Boolean logic is a complete system for logical operations. ... Algebra of sets George Boole Boolean algebra Boolean function Boolean logic Boolean homomorphism Boolean Implicant Boolean prime ideal theorem Boolean-valued model Boolean satisfiability problem Booles syllogistic canonical form (Boolean algebra) compactness theorem Complete Boolean algebra connective -- see logical operator de Morgans laws Augustus De Morgan duality (order... -1... An existential graph is a type of diagrammatic or visual notation for logical expressions, proposed by Charles Sanders Peirce, who wrote his first paper on graphical logic in 1882 and continued to develop the method until his death in 1914. ... An entitative graph is an element of the graphical syntax for logic that Charles Sanders Peirce developed under the name of qualitative logic in the 1880s, taking the coverage of the formalism only as far as the propositional or sentential aspects of logic are concerned. ...

External links

An existential graph is a type of diagrammatic or visual notation for logical expressions, proposed by Charles Sanders Peirce, who wrote his first paper on graphical logic in 1882 and continued to develop the method until his death in 1914. ...

Trivia

  • "Philosopher," the German "Lovecraftian" Death Metal Band, pay a musical tribute to G. Spencer-Brown's work on their EP Laws of Form.

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