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Laws of Form (hereinafter LoF) is a book by G. Spencer-Brown, published in 1969, that straddles the boundary between mathematics and of philosophy. George Spencer-Brown (born April 2, 1923, Grimsby, Lincolnshire, England) is a Polymath best known as the author of Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language LoF describes three distinct logical systems:

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. In Universal algebra, a branch of pure Mathematics, an Algebraic structure is a variety or Quasivariety. "Laws of Form" may refer to LoF or to the primary algebra (hereinafter abbreviated pa).

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. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. 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. A paradox is a true statement or group of statements that leads to a Contradiction or a situation which defies intuition; or inversely Spencer-Brown was influenced by Wittgenstein and R. D. Laing. Ronald David Laing ( 7 October 1927 – 23 August 1989 was a Scottish Psychiatrist who wrote extensively on Mental illness LoF also echoes a number of themes from the writings of Charles Peirce, Bertrand Russell, and Alfred North Whitehead. Charles Sanders Peirce (pronounced purse) (September 10 1839 &ndash April 19 1914 was an American Logician mathematician, philosopher Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian Alfred North Whitehead, OM ( February 15 1861, Ramsgate, Kent, England &ndash December 30 1947,

Reception

Ostensibly a work of formal mathematics and philosophy, LoF became something of a cult classic, praised in the Whole Earth Catalog. A cult film is a Film that has acquired a highly devoted but relatively small group of fans. The Whole Earth Catalog was an American Counterculture catalog that granted "Access to Tools" published by Stewart Brand between 1968 and 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. Consciousness has been defined loosely as a constellation of attributes of Mind such as Subjectivity, Self-awareness, Sentience, and the Cognition is a concept used in different ways by different disciplines but is generally accepted to mean the process of awareness or thought LoF argues that the pa reveals striking connections among logic, Boolean algebra, and arithmetic, and the philosophy of language and mind. Logic is the study of the principles of valid demonstration and Inference. Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the late 1830s Philosophy of language is the reasoned inquiry into the nature origins and usage of Language. MIND ( Moving In New Directions) (est 1975 is an alternative education high school in Montreal, Quebec, Canada.

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:

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. Heinz von Foerster (Nov 13 1911 Vienna – Oct 2 2002 Pescadero California) was an Austrian American scientist combining Physics and Philosophy Louis H Kauffman ( 3 February, 1945) is an American Mathematician, topologist, and professor of Mathematics in the Department of Mathematics Niklas Luhmann ( December 8, 1927 - November 6, 1998) was a German Sociologist, administration expert and a prominent Humberto Maturana (born September 14, 1928, in Santiago Chile) is a Chilean Biologist. Francisco Javier Varela García ( Sept 7, 1946 &ndash May 28, 2001) was a Chilean biologist, philosopher and neuroscientist 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. 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 Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like Goldbach's conjecture is one of the oldest unsolved problems in Number theory and in all of Mathematics. 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. )

The form (Chapter 1)

The symbol:

Image:Laws of Form - cross.gif

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. Cognition is a concept used in different ways by different disciplines but is generally accepted to mean the process of awareness or thought e. , the dualistic Mark indicates the capability of differentiating a "this" from "everything else but this. Dualism denotes a state of two parts The word's origin is the Latin duo, "two". "

In LoF, a Cross denotes the drawing of a "distinction", and can be thought of as signifying the following, all at once:

All three ways imply an action on the part of the cognitive entity (e. g. , person) making the distinction. As LoF puts it:

"The first command:

can well be expressed in such ways as: Or:

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. Consciousness has been defined loosely as a constellation of attributes of Mind such as Subjectivity, Self-awareness, Sentience, and the A language is a dynamic set of visual auditory or tactile Symbols of Communication and the elements used to manipulate them 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. Charles Sanders Peirce (pronounced purse) (September 10 1839 &ndash April 19 1914 was an American Logician mathematician, philosopher

The primary arithmetic (Chapter 4)

The syntax of the primary arithmetic (PA) goes as follows. In Linguistics, syntax (from Ancient Greek grc συν- syn-, "together" and grc τάξις táxis, "arrangement" is the There are just two atomic expressions:

There are two inductive rules:

The semantics of the primary arithmetic are perhaps nothing more than the sole explicit definition in LoF: Distinction is perfect continence. Semantics is the study of meaning in communication The word derives from Greek σημαντικός ( semantikos) "significant" from A definition is a statement of the meaning of a Word or Phrase.

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. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject The law of Calling. Crossing twice from the unmarked to the marked state is indistinguishable from crossing 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:

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


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:

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


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:

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. This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations In Logic, statements p and q are logically equivalent if they have the same logical content In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks " In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X

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:

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

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. Metalogic is the study of the Metatheory of Logic. While logic is the study of the manner in which logical systems can be used to decide the correctness Object language has meaning in contexts of computer programming and operation and in linguistics and logic In Logic and Linguistics, a metalanguage is a Language used to make statements about statements in another language which is called the Object The first quote seems to assert that the canons are part of the metalanguage. The second quote seems to assert that statements in the object language are essentially commands addressed to the reader by the author. Neither assertion holds in standard metalogic.

The primary algebra (Chapter 6)

Syntax

Given any valid primary arithmetic expression, insert into one or more locations any number of Latin letters bearing optional numerical subscripts; the result is a pa formula. In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information Letters so employed in mathematics and logic are called variables. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Logic is the study of the principles of valid demonstration and Inference. A variable (ˈvɛərɪəbl is an Attribute of a physical or an abstract System which may change its Value while it is under Observation. A pa variable indicates a location where one can write the primitive value Image:Laws of Form - cross.gif or its complement Image:Laws of Form - double cross.gif. Multiple instances of the same variable denote multiple locations of the same primitive value.

Rules governing logical equivalence

The sign '=' may link two logically equivalent expressions; the result is an equation. An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent 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. 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" Let C and D be formulae each containing at least one instance of the subformula A:

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. Logic is the study of the principles of valid demonstration and Inference.

The pa consists of equations, i. An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent 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. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Universal algebra, a variety of algebras is the class of all Algebraic structures of a given signature satisfying a given set of identities 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. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. In Propositional logic, a tautology (from the Greek word ταυτολογία is a Propositional formula that is true under any possible valuation A turnstile, also called a baffle gate, is a form of Gate which allows one person to pass at a time To denote that the pa formula A is a tautology, simply write "A =Image:Laws of Form - cross.gif ". In Propositional logic, a tautology (from the Greek word ταυτολογία is a Propositional formula that is true under any possible valuation If one replaces '=' in R1 and R2 with the biconditional, the resulting rules hold in conventional logic. In Logic and Mathematics, logical biconditional (sometimes also known as the material biconditional) is a Logical operator connecting two statements However, conventional logic relies mainly on the rule modus ponens; thus conventional logic is ponential. In Classical logic, modus ponendo ponens ( Latin: mode that affirms by affirming; often abbreviated to MP or modus ponens) is a The equational-ponential dichotomy distills much of what distinguishes mathematical logic from the rest of mathematics.

Initials

An initial is a pa equation verifiable by a decision procedure and as such is not an axiom. In Computability theory and Computational complexity theory, a decision problem is a question in some Formal system with a yes-or-no answer depending on In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject 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. In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.

Another set of initials, friendlier to calculations, is:

It is thanks to C2 that the pa is a lattice. In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' By virtue of J1a, it is a complemented lattice whose upper bound and inverse element is (). In the mathematical discipline of Order theory, and in particular in lattice theory, a complemented lattice is a bounded lattice (with In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to By J0, (()) is the corresponding lower bound and identity element. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that 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. In Computer science, a tree is a widely-used Data structure that emulates a Tree structure with a set of linked nodes 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. In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information 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:

Charles Peirce's existential graphs was perhaps the first formal system to appreciate the power of C2. Charles Sanders Peirce (pronounced purse) (September 10 1839 &ndash April 19 1914 was an American Logician mathematician, philosopher 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 His Rule of (De)Iteration combined T13 and AA=A.

LoF asserts that concatenation can be read as commuting and associating by default and hence need not be explicitly assumed or demonstrated. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, associativity is a property that a Binary operation can have (Peirce made a similar assumption in his graphical logic. 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 ) Let a period denote grouping. That concatenation commutes and associates may then be demonstrated from the:

Now that associativity is demonstrated, the period is no longer required.

Proof theory

The pa contains three kinds of proved assertions:

The distinction between consequence and theorem holds for all formal systems, including mathematics and logic, but is usually not made explicit. In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements A demonstration or decision procedure can be carried out and verified by computer. In Computability theory and Computational complexity theory, a decision problem is a question in some Formal system with a yes-or-no answer depending on The proof of a theorem cannot be. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements

Let A and B be pa formulas. In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information A demonstration of A=B may proceed in either of two ways:

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. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation The result is a normal form, the pa analog of the conjunctive normal form. In Boolean logic, a Formula is in conjunctive normal form (CNF or cnf if it is a conjunction of clauses, where a clause is a disjunction LoF (T14-15) proves the pa analog of the well-known Boolean algebra theorem that every formula has a normal form. Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the late 1830s

Let A be a subformula of some formula B. In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information 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. In Propositional logic, a tautology (from the Greek word ταυτολογία is a Propositional formula that is true under any possible valuation ↔ A related condition appears in some versions of natural deduction. In Philosophical logic, natural deduction is an approach to Proof theory that attempts to provide a Deductive system which is a formal model of logical A demonstration by calculation is often little more than:

The last step of a calculation always invokes J1a.

LoF includes elegant new proofs of the following standard metatheory:

That sentential logic is complete is taught in every first university course in mathematical logic. This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. But university courses in Boolean algebra seldom mention the completeness of 2.

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). In Logic an interpretation gives meaning to an artificial or Formal language or to a sentence of such a language by assigning a denotation (extension In Mathematics and Abstract algebra, the two-element Boolean algebra is the Boolean algebra whose underlying set (or Universe or In Logic an interpretation gives meaning to an artificial or Formal language or to a sentence of such a language by assigning a denotation (extension In Mathematics and Abstract algebra, the two-element Boolean algebra is the Boolean algebra whose underlying set (or Universe or This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" LoF shows how the pa can interpret the syllogism. A syllogism, or logical appeal, (συλλογισμός &mdash "conclusion" "inference" (usually the categorical syllogism) is a kind of Each of these interpretations is discussed in a subsection below. In Logic an interpretation gives meaning to an artificial or Formal language or to a sentence of such a language by assigning a denotation (extension 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. In Logic an interpretation gives meaning to an artificial or Formal language or to a sentence of such a language by assigning a denotation (extension First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science Charles Sanders Peirce (pronounced purse) (September 10 1839 &ndash April 19 1914 was an American Logician mathematician, philosopher 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

Two-element Boolean algebra 2

The pa is an elegant minimalist notation for the two-element Boolean algebra 2. In Mathematics and Abstract algebra, the two-element Boolean algebra is the Boolean algebra whose underlying set (or Universe or Let:

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. In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models 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 \ B=\{Image:Laws of Form - cross.gif , Image:Laws of Form - double cross.gif\ \} is the Boolean domain or carrier. In Mathematics and Abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and In the language of universal algebra, the pa is the algebraic structure \lang B,--,(-),() \rang of type \lang 2,1,0 \rang. Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, The expressive adequacy of the Sheffer stroke points to the pa also being a \lang B,(--),()\rang algebra of type \lang 2,0 \rang. In Logic, a set S of Logical connectives is functionally complete (also expressively adequate or simply adequate) if every possible Definition The NAND operation is a Logical operation on two Logical values typically the values of two Propositions that produces a value 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 \lang B,+,\lnot,1 \rang algebra of type \lang 2,1,0 \rang. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective This description of 2 is simpler than the conventional one, namely an \lang B,+,\times,\lnot,1,0 \rang algebra of type \lang 2,2,1,0,0 \rang.

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
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:

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))), ((A)(B))(AB) \ both interpret A if and only if B or A is equivalent to B. ↔ In Logic, statements p and q are logically equivalent if they have the same logical content


Thus any expression in sentential logic has a pa translation. This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" Equivalently, the pa interprets sentential logic. In Logic an interpretation gives meaning to an artificial or Formal language or to a sentence of such a language by assigning a denotation (extension 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. In Propositional logic, a tautology (from the Greek word ταυτολογία is a Propositional formula that is true under any possible valuation This is an example of a decision procedure, one more or less in the spirit of conventional truth tables. In Computability theory and Computational complexity theory, a decision problem is a question in some Formal system with a yes-or-no answer depending on 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). Willard Van Orman Quine (June 25 1908 Akron, Ohio &ndash December 25 2000 (known to intimates as "Van"

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. For concatenation of general lists see Append. In Computer programming, string concatenation is the operation of joining two character 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. This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" In Mathematics and Abstract algebra, the two-element Boolean algebra is the Boolean algebra whose underlying set (or Universe or Another consequence of self-duality is the irrelevance of DeMorgan's laws; those laws are built into the pa syntax from the outset. In Logic, De Morgan's laws or De Morgan's theorem are rules in Formal logic relating pairs of dual Logical operators in a systematic manner expressed

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. In Logic and Mathematics, negation or not is an operation on Logical values for example the logical value of a Proposition 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 by virtue of denoted a primitive value. In Mathematics, an operator is a function which operates on (or modifies another function In Mathematics, an operand is one of the inputs (arguments of an Operator. 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.

Syllogisms

Appendix 2 of LoF shows how to translate traditional syllogisms and sorites into the pa. A syllogism, or logical appeal, (συλλογισμός &mdash "conclusion" "inference" (usually the categorical syllogism) is a kind of A polysyllogism (also called multi-premise syllogism, climax, or gradatio) is a string of any number of Syllogisms such that the conclusion of 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*. A syllogism, or logical appeal, (συλλογισμός &mdash "conclusion" "inference" (usually the categorical syllogism) is a kind of These 24 possible permutations include the 19 syllogistic forms deemed valid in Aristotelian and medieval logic. The Organon is the name given by Aristotle 's followers the Peripatetics to the standard collection of his six works on Logic. Medieval philosophy is the Philosophy of Europe and the Middle East in the era now known as Medieval or the Middle Ages, the period roughly extending from 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). In Logic an interpretation gives meaning to an artificial or Formal language or to a sentence of such a language by assigning a denotation (extension In Logic, the monadic predicate calculus is the fragment of Predicate calculus in which all predicate letters are monadic (that is they take In Philosophy, term logic, also known as traditional logic, is a loose name for the way of doing logic that began with Aristotle, and that was dominant

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. In Mathematics, commutativity is the ability to change the order of something without changing the end result The objective then becomes one of simplifying that translation to (()).

Remarks:

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. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. In Logic and Mathematics, a logical value, also called a truth value, is a value indicating the extent to which a Proposition is true Let the extension of an n-place atomic formula be the set of ordered n-tuples of individuals that satisfy it (i. In Mathematical logic, an atomic formula (also known simply as an atom) is a formula with no deeper Propositional structure that is a formula As commonly used, individual refers to a Person or to any specific object in a collection 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. In Mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is a Variable which can either be An ordered 2-tuple is an ordered pair, whose standard set theoretic definition is <a,b> = {{a},{a,b}}, where a,b are individuals. In Mathematics, an ordered pair is a collection of two distinguishable objects one of which is identified as the first coordinate (or the first entry Ordered n-tuples for any n>2 may be obtained from ordered pairs by a well-known recursive construction. Recursion, in Mathematics and Computer science, is a method of defining functions in which the function being defined is applied within its own definition 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. Dana Stewart Scott (born 1932 is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Hence {{}} has the interpretation True. Reading {} as False follows naturally.

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. Edward Vermilye Huntington ( April 26 1874 Clinton New York, USA -- November 25 1952, Cambridge Massachusetts, USA was an American Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the late 1830s In its simplest meaning in Mathematics and Logic, an operation is an action or procedure which produces a new value from one or more input values Hence the seldom-noted fact that Boolean algebras are magmas (a. In Abstract algebra, a magma (or groupoid) is a basic kind of Algebraic structure. k. a. groupoids). To see this, note that the pa is a commutative:

Groups also require a unary operation, called inverse, whose inverse element is at once the inverse of, and equal to, the identity element. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a unary operation is an operation with only one Operand, i In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to Complementation is the pa unary operation corresponding to group inverse. Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the late 1830s By J1a, 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〉. In Logic, especially Mathematical logic, a signature lists and describes the Non-logical symbols of a Formal language. Hence the pa is a boundary algebra. In Universal algebra, a branch of pure Mathematics, an Algebraic structure is a variety or Quasivariety.

The axioms and initials of the pa distinguish it from an abelian group in two ways:

In boundary terms, the defining arithmetical fact of group theory is (())=(). The PA counterpart to that equation is ((()))=().

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. Recursion, in Mathematics and Computer science, is a method of defining functions in which the function being defined is applied within its own definition 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). Alonzo Church ( June 14, 1903 – August 11, 1995) was an American Mathematician and logician 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.

Resonances in religion, philosophy, and science

The mathematical and logical content of LoF is wholly consistent with a secular point of view. Secular humanism is a humanist philosophy that upholds Reason, Ethics and Justice, and specifically rejects the Supernatural 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:

". . . 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)
"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. " [3]


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

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

Gottfried Leibniz, in memoranda not published before the late 19th and early 20th centuries, invented Boolean logic. Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the late 1830s 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). In Mathematical logic, algebraic logic formalizes logic using the methods of Abstract algebra. Clarence Irving Lewis ( April 12, 1883 Stoneham Massachusetts - February 3, 1964 Cambridge Massachusetts) usually Nicholas Rescher (born July 15, 1928 in Hagen, Germany) is an American philosopher, affiliated for many years with the 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 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. Charles Sanders Peirce (pronounced purse) (September 10 1839 &ndash April 19 1914 was an American Logician mathematician, philosopher 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,[4] but they were not published in full until 1993. [5]
  2. In a 1902 encyclopedia article,[6] 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. In Linguistics, syntax (from Ancient Greek grc συν- syn-, "together" and grc τάξις táxis, "arrangement" is the 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 For concatenation of general lists see Append. In Computer programming, string concatenation is the operation of joining two character In Logic and Mathematics, negation or not is an operation on Logical values for example the logical value of a Proposition [7] If pa concatenation is read as conjunction, then these graphs are isomorphic to the pa (Kauffman 2001).

Ironically, LoF cites vol. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective 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. Semiotics, semiotic studies, or semiology is the study of sign processes (semiosis or signification and communication signs and Symbols both

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. Jean George Pierre Nicod (c 1893 - 16 February 1924) was a French Philosopher and Logician. Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian

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. An infix is an Affix inserted inside a stem (an existing word Polish notation, also known as prefix notation, is a form of notation for Logic, Arithmetic, and Algebra. Reverse Polish notation (or just RPN) by analogy with the related Polish notation, a prefix notation introduced in 1920 by the Polish mathematician 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. Emil Leon Post, PhD, ( February 11 1897, Augustów – April 21 1954, New York City) was a Mathematician This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Basic Definition and Requirements An Axiom P is independent if there is no other axiom Q such that Q implies P In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models

Craig (1979) argued that the world, and how humans perceive and interact with that world, has a rich Boolean structure. William Craig (born 1918 is Emeritus professor of Philosophy at University of California Berkeley in Berkeley, California. Craig was an orthodox logician and an authority on algebraic logic. In Mathematical logic, algebraic logic formalizes logic using the methods of Abstract algebra.

Second-generation cognitive science emerged in the 1970s, after LoF was written. Cognitive science may be broadly defined as the multidisciplinary study of mind and behavior 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. Humberto Maturana (born September 14, 1928, in Santiago Chile) is a Chilean Biologist. Francisco Javier Varela García ( Sept 7, 1946 &ndash May 28, 2001) was a Chilean biologist, philosopher and neuroscientist The Berkeley psychologist and cognitive scientist Eleanor Rosch has written extensively on the closely related notion of categorization. Eleanor Rosch (once known as Eleanor Rosch Heider) is a professor of Psychology at the University of California Berkeley, specializing in Cognitive

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. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In Mathematics, concatenation is the joining of two vectors, that is when vectors a and b are concatenated they form the combined vector An infix is an Affix inserted inside a stem (an existing word 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:

The primary arithmetic and algebra are a minimalist formalism for sentential logic and Boolean algebra. This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" Other minimalist formalisms having the power of set theory include:

Notes

  1. ^ For a sympathetic evaluation, see Kauffman (2001).
  2. ^ "On the True Theologia Mystica" in Loemker, Leroy, ed. and trans. , 1969. Leibniz: Philosophical Papers and Letters. Reidel: 368.
  3. ^ "Order" in Hasting, J. , ed. , 1917. Encyclopedia of Religion and Ethics. Encyclopedia of Religion and Ethics is a 12-volume work (plus an index volume edited by James Hastings, written between 1908 and 1927 and composed of entries by many contributors Scribner's: 540. Reprinted in Robinson, D. S. , ed. , 1951, Royce's Logical Essays. Dubuque IA: Wm. C. Brown: 230-31.
  4. ^ "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
  5. ^ "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.
  6. ^ Reprinted in Peirce, C. S. (1933) Collected Papers, Vol. 4, Charles Hartshorne and Paul Weiss, eds. Charles Hartshorne ( June 5, 1897 &ndash October 9, 2000) was a prominent American philosopher who concentrated primarily on the Philosophy Paul S Weiss (born October 10, 1959) is a leading American Nanoscientist at the Pennsylvania State University. Harvard Univ. Press. Paragraphs 378-383
  7. ^ The existential graphs are described at length in Peirce, C. S. (1933) Collected Papers, Vol. 4, Charles Hartshorne and Paul Weiss, eds. Charles Hartshorne ( June 5, 1897 &ndash October 9, 2000) was a prominent American philosopher who concentrated primarily on the Philosophy Paul S Weiss (born October 10, 1959) is a leading American Nanoscientist at the Pennsylvania State University. Harvard Univ. Press. Paragraphs 347-529.

See also

References

External links

Trivia

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