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| | Workplace Relationships Adviser Sandy from Haliburton, likes to spend some time weather forecasting, diet and badge collecting. Discovers the beauty in traveling to destinations around the world, of late just returning from Historic Centre of Zacatecas. |
| {{italic title}}
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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 [[philosophy]]. ''LoF'' describes three distinct [[logical system]]s:
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| * The ''primary arithmetic'' (described in Chapter 4 of ''LoF''), whose models include [[Two-element Boolean algebra#Some basic identities|Boolean arithmetic]];
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| * <!-- Please see the discussion page. --> The ''primary [[algebra]]'' (Chapter 6 of ''LoF''), whose [[interpretation (logic)|models]] include the [[two-element Boolean algebra]] (hereinafter abbreviated '''2'''), [[Boolean logic]], and the classical [[propositional calculus]];
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| * ''Equations of the second degree'' (Chapter 11), whose [[interpretation (logic)|interpretations]] include [[finite automata]] and [[Alonzo Church]]'s Restricted Recursive Arithmetic (RRA).
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| '''Boundary algebra''' is [http://www.canterbury.ac.nz/spark/Researcher.aspx?researcherid=84894 Dr Philip Meguire]'s (2011) term<ref>Meguire, P. (2011) Boundary Algebra: A Simpler Approach to Basic Logic and Boolean Algebra. Saarbrücken: VDM Publishing Ltd. 168pp</ref> for the union of the primary algebra (hereinafter abbreviated ''pa'') and the primary arithmetic. "Laws of Form" sometimes loosely refers to the ''pa'' as well as to ''LoF''.
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| ==The book==
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| ''LoF'' emerged from 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 being a 1997 German translation, and has never gone out of print.
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| The mathematics fills only about 55pp and is rather elementary. But ''LoF'''s mystical and declamatory prose, and its love of [[paradox]], make it a challenging read for all. Spencer-Brown was influenced by [[Wittgenstein]] and [[R. D. Laing]]. ''LoF'' also echoes a number of themes from the writings of [[Charles Sanders Peirce]], [[Bertrand Russell]], and [[Alfred North Whitehead]].
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| The entire book is written in an operational way, giving instructions to the reader instead of telling him what ''is''. In accordance with G. Spencer-Brown's interest in paradoxes, the only sentence that makes a statement that something is, is the statement, which says no such statements are used in this book.<ref>Felix Lau: "Die Form der Paradoxie", 2005 Carl-Auer Verlag, ISBN 9783896703521 {{Please check ISBN|reason=Invalid length.}}</ref> Except for this one sentence the book can be seen as an example of [[E-Prime]].
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| ==Reception==
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| Ostensibly a work of formal mathematics and philosophy, ''LoF'' became something of a [[cult classic]], praised in the ''[[Whole Earth Catalog]]''. Those who agree point to ''LoF'' as embodying an enigmatic "mathematics of [[consciousness]]," its algebraic symbolism capturing an (perhaps even ''the'') implicit root of [[cognition]]: the ability to ''distinguish''. ''LoF'' argues that the ''pa'' (primary algebra) reveals striking connections among [[logic]], [[Boolean algebra (logic)|Boolean algebra]], and arithmetic, and the [[philosophy of language]] and [[Philosophy of mind|mind]].
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| [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.ndjfl/1093888028?abstract= Banaschewski (1977)] argues that the ''pa'' is nothing but new notation for Boolean algebra. Indeed, the [[two-element Boolean algebra]] '''2''' can be seen as the intended interpretation of the ''pa''. Yet the notation of the ''pa'':
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| * Fully exploits the [[Duality (mathematics)|duality]] characterizing not just [[Boolean algebra (structure)|Boolean algebra]]s but all [[Lattice (order)|lattice]]s;
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| *Highlights how syntactically distinct statements in logic and '''2''' can have identical [[semantics]];
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| * Dramatically simplifies Boolean algebra calculations, and proofs in [[sentential logic|sentential]] and [[syllogism|syllogistic]] [[logic]].
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| 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).
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| ''LoF'' has influenced, among others, [[Heinz von Foerster]], [[Louis Kauffman]], [[Niklas Luhmann]], [[Humberto Maturana]], [[Francisco Varela]] and [[William Bricken]]. Some of these authors have modified the primary algebra in a variety of interesting ways.
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| ''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, but it met with skepticism.<ref>For a sympathetic evaluation, see [http://www.arxiv.org/math.CO/0112266 Kauffman (2001)].</ref> (The Four Color Theorem and Fermat's Last Theorem were proved in 1976 and 1995, respectively, using methods owing nothing to ''LoF''.)
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| ==The form (Chapter 1)==
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| The symbol:
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| :[[Image:Laws of Form - cross.gif]]
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| also called the '''''mark''''' or '''''cross''''', is the essential feature of the Laws of Form. In Spencer-Brown's inimitable and enigmatic fashion, the Mark symbolizes the root of [[cognition]], i.e., the [[dualism|dualistic]] Mark indicates the capability of differentiating a "this" from "everything else ''but'' this."
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| In ''LoF'', a Cross denotes the drawing of a "distinction", and can be thought of as signifying the following, all at once:
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| * The act of drawing a boundary around something, thus separating it from everything else;
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| * That which becomes distinct from everything by drawing the boundary;
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| * Crossing from one side of the boundary to the other.
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| All three ways imply an ''action'' on the part of the cognitive entity (e.g., person) making the distinction. As ''LoF'' puts it:
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| <blockquote>
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| "The first command:
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| * Draw a distinction
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| can well be expressed in such ways as:
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| * Let there be a distinction,
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| * Find a distinction,
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| * See a distinction,
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| * Describe a distinction,
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| * Define a distinction,
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| Or:
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| * Let a distinction be drawn." (''LoF'', Notes to chapter 2)
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| </blockquote>
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| 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.
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| 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:
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| "...the first distinction, the Mark and the observer are not only interchangeable, but, in the form, identical."
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| [[Charles Sanders Peirce|C. S. Peirce]] came to a related insight in the 1890s; see [[#Related work|Related Work]].
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| ==The primary arithmetic (Chapter 4)==
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| The [[syntax]] of the '''primary arithmetic''' ('''PA''') goes as follows. There are just two [[atomic formula|atomic expressions]]:
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| * The empty Cross [[Image:Laws of Form - cross.gif]] ;
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| *All or part of the blank page (the "void").
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| There are two inductive rules:
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| * A Cross [[Image:Laws of Form - cross.gif]] may be written over any expression;
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| * Any two expressions may be [[concatenation|concatenated]].
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| The [[semantics]] of the primary arithmetic are perhaps nothing more than the sole explicit [[definition]] in ''LoF'': ''Distinction is perfect continence''.
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| 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" [[axiom]]s '''A1''' and '''A2''', which ground the primary arithmetic (and hence all of the Laws of Form):
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| '''A1. The law of Calling'''. Calling twice from a state is indistinguishable from calling once. To make a distinction twice has the same effect as making it once. For example, saying "Let there be light" and then saying "Let there be light" again, is the same as saying it once. Formally:
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| ::[[Image:Laws of Form - cross.gif]] [[Image:Laws of Form - cross.gif]] <math>\ =</math>[[Image:Laws of Form - cross.gif]]
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| '''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:
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| ::[[Image:Laws of Form - double cross.gif]] <math>\ =</math>
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| 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:
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| * Every finite expression has a unique simplification. ('''T3''' in ''LoF'');
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| * Starting from an initial marked or unmarked state, "complicating" an expression by a finite number of repeated application of '''A1''' and '''A2''' cannot yield an expression whose simplification differs from the initial state. ('''T4''' in ''LoF'').
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| Thus the [[relation (mathematics)|relation]] of ''[[logical equivalence]]'' [[partition of a set|partitions]] all primary arithmetic expressions into two [[equivalence class]]es: those that simplify to the Cross, and those that simplify to the void.
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| '''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.
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| The primary arithmetic is analogous to the following formal languages from [[mathematics]] and [[computer science]]:
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| * A [[Dyck language]] of order 1 with a null alphabet;
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| * The simplest [[context-free language]] in the [[Chomsky hierarchy]];
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| * A [[rewrite system]] that is [[strongly normalizing]] and [[confluence (abstract rewriting)|confluent]].
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| The phrase ''calculus of indications'' in ''LoF'' is a synonym for "primary arithmetic".
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| ===The notion of canon===
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| 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:
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| <blockquote>"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."</blockquote>
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| <blockquote>"...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."</blockquote>
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| These excerpts relate to the distinction in [[metalogic]] between the [[object language]], the formal language of the logical system under discussion, and the [[metalanguage]], a language (often a natural language) distinct from the object language, employed to exposit and discuss the object language. The first quote seems to assert that the ''canons'' are part of the metalanguage. The second quote seems to assert that statements in the object language are essentially commands addressed to the reader by the author. Neither assertion holds in standard metalogic.
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| ==The primary algebra (Chapter 6)==
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| ===Syntax===
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| Given any valid primary arithmetic expression, insert into one or more locations any number of Latin letters bearing optional numerical subscripts; the result is a ''pa'' [[formula]]. Letters so employed in [[mathematics]] and [[logic]] are called [[Variable (mathematics)|variables]]. 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.
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| ===Rules governing logical equivalence===
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| The sign '=' may link two logically equivalent expressions; the result is an [[equation]]. By "logically equivalent" is meant that the two expressions have the same simplification. [[Logical equivalence]] is an [[equivalence relation]] over the set of ''pa'' formulas, governed by the rules '''R1''' and '''R2'''. Let ''C'' and ''D'' be formulae each containing at least one instance of the subformula ''A'':
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| *'''R1''', ''Substitution of equals''. Replace ''one or more'' instances of ''A'' in ''C'' by ''B'', resulting in ''E''. If ''A''=''B'', then ''C''=''E''.
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| *'''R2''', ''Uniform replacement''. Replace ''all'' instances of ''A'' in ''C'' and ''D'' with ''B''. ''C'' becomes ''E'' and ''D'' becomes ''F''. If ''C''=''D'', then ''E''=''F''. Note that ''A''=''B'' is not required.
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| '''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.
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| The ''pa'' consists of [[equations]], i.e., pairs of formulae linked by an infix '='. '''R1''' and '''R2''' enable transforming one equation into another. Hence the ''pa'' is an ''equational'' formal system, like the many [[algebraic structures]], including [[Boolean algebra (structure)|Boolean algebra]], that are [[variety (universal algebra)|varieties]]. Equational logic was common before ''Principia Mathematica'' (e.g., Peirce,<sup>1,2,3</sup> Johnson 1892), and has present-day advocates (Gries and Schneider 1993).
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| Conventional [[mathematical logic]] consists of [[Tautology (logic)|tautological]] formulae, signalled by a prefixed [[Turnstile (symbol)|turnstile]]. To denote that the ''pa'' formula ''A'' is a [[Tautology (logic)|tautology]], simply write "''A'' =[[Image:Laws of Form - cross.gif]] ". If one replaces '=' in '''R1''' and '''R2''' with the [[biconditional]], the resulting rules hold in conventional logic. However, conventional logic relies mainly on the rule [[modus ponens]]; thus conventional logic is ''ponential''. The equational-ponential dichotomy distills much of what distinguishes mathematical logic from the rest of mathematics.
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| ===Initials===
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| An ''initial'' is a ''pa'' equation verifiable by a [[decision procedure]] and as such is ''not'' an [[axiom]]. ''LoF'' lays down the initials:
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| *<math> \ J1: ((A)A) = . </math>
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| The absence of anything to the right of the "=" above, is deliberate.
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| *<math> \ J2: ((A)(B))C = ((AC)(BC)).</math>
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| '''J2''' is the familiar [[distributive law]] of [[sentential logic]] and [[Boolean algebra (structure)|Boolean algebra]].
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| Another set of initials, friendlier to calculations, is:
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| *<math> \ J0: (())A = A.</math>
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| *<math> \ J1a: (A)A = ()</math>
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| *<math> \ C2: A(AB)=A(B).</math>
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| It is thanks to '''C2''' that the ''pa'' is a [[lattice (order)|lattice]]. By virtue of '''J1a''', it is a [[complemented lattice]] whose upper bound is (). By '''J0''', (()) is the corresponding lower bound and [[identity element]]. '''J0''' is also an algebraic version of '''A2''' and makes clear the sense in which (()) aliases with the blank page.
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| T13 in ''LoF'' generalizes '''C2''' as follows. Any ''pa'' (or sentential logic) formula ''B'' can be viewed as an [[tree data structure|ordered tree]] with ''branches''. Then:
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| '''T13''': A [[formula|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.
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| While a proof of T13 would require [[mathematical induction|induction]], the intuition underlying it should be clear.
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| '''C2''' or its equivalent is named:
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| *"Generation" in ''LoF'';
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| *"Exclusion" in Johnson (1892);
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| *"Pervasion" in the work of William Bricken;
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| *"Mimesis" in the entry [[logical nand]].
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| Perhaps the first instance of an axiom or rule with the power of '''C2''' was the "Rule of (De)Iteration," combining T13 and ''AA=A'', of [[Charles Sanders Peirce|C. S. Peirce]]'s [[existential graph]]s.
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| ''LoF'' asserts that concatenation can be read as [[commutativity|commuting]] and [[associativity|associating]] by default and hence need not be explicitly assumed or demonstrated. (Peirce made a similar assertion about his [[existential graph]]s.) Let a period be a temporary notation to establish grouping. That concatenation commutes and associates may then be demonstrated from the:
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| * Initial ''AC.D''=''CD.A'' and the consequence ''AA''=''A'' (Byrne 1946). This result holds for all [[lattice (order)|lattices]], because ''AA''=''A'' is an easy consequence of the [[absorption law]], which holds for all lattices;
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| * Initials ''AC.D''=''AD.C'' and '''J0'''. Since '''J0''' holds only for lattices with a lower bound, this method holds only for [[bounded lattice]]s (which include the ''pa'' and '''2'''). Commutativity is trivial; just set ''A''=(()). Associativity: ''AC.D'' = ''CA.D'' = ''CD.A'' = ''A.CD''.
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| Having demonstrated associativity, the period can be discarded.
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| The initials in Meguire (2011) are ''AC.D''=''CD.A'', called '''B1'''; '''B2''', J0 above; '''B3''', J1a above; and '''B4''', C2. By design, these initials are very similar to the axioms for an [[abelian group]], '''G1-G3''' below.
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| ===Proof theory===
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| The ''pa'' contains three kinds of proved assertions:
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| * ''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.
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| * ''[[Theorem]]'' is a statement in the [[metalanguage]] verified by a ''[[Mathematical proof|proof]]'', i.e., an argument, formulated in the metalanguage, that is accepted by trained mathematicians and logicians.
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| * ''Initial'', defined above. Demonstrations and proofs invoke an initial as if it were an axiom.
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| 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 [[Mathematical proof|proof]] of a [[theorem]] cannot be.
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| Let ''A'' and ''B'' be ''pa'' [[formula]]s. A demonstration of ''A''=''B'' may proceed in either of two ways:
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| * Modify ''A'' in steps until ''B'' is obtained, or vice versa;
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| * Simplify both (''A'')''B'' and (''B'')''A'' to [[Image:Laws of Form - cross.gif]]. This is known as a "calculation".
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| 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''').
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| The consequence (((''A'')''B'')''C'') = (''AC'')((''B'')''C''), '''C7''' in ''LoF'', enables an [[algorithm]], sketched in ''LoF''s 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 (logic)|Boolean algebra]] theorem that every formula has a normal form.
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| Let ''A'' be a [[formula|subformula]] of some [[formula]] ''B''. When paired with '''C3''', '''J1a''' can be viewed as the closure condition for calculations: ''B'' is a [[Tautology (logic)|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:
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| * Invoking T13 repeatedly to eliminate redundant subformulae;
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| * Erasing any subformulae having the form ((''A'')''A'').
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| The last step of a calculation always invokes '''J1a'''.
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| ''LoF'' includes elegant new proofs of the following standard [[metatheory]]:
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| * ''[[Completeness]]'': all ''pa'' consequences are demonstrable from the initials (T17).
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| * ''[[axiom|Independence]]'': '''J1''' cannot be demonstrated from '''J2''' and vice versa (T18).
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| 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'''.
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| ===Interpretations===
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| If the Marked and Unmarked states are read as the [[two-element Boolean algebra|Boolean]] values 1 and 0 (or '''True''' and '''False'''), the ''pa'' [[interpretation (logic)|interprets]] '''[[two-element Boolean algebra|2]]''' (or [[sentential logic]]). ''LoF'' shows how the ''pa'' can interpret the [[syllogism]]. Each of these [[interpretation (logic)|interpretations]] is discussed in a subsection below. Extending the ''pa'' so that it could [[interpretation (logic)|interpret]] standard [[first-order logic]] has yet to be done, but [[Charles Sanders Peirce|Peirce]]'s ''beta'' [[existential graph]]s suggest that this extension is feasible.
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| ====Two-element Boolean algebra 2====
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| The ''pa'' is an elegant minimalist notation for the [[two-element Boolean algebra]] '''2'''. Let:
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| * One of Boolean [[meet (mathematics)|meet]] (×) or [[join (mathematics)|join]] (+) interpret [[concatenation]];
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| * The [[Complement (order theory)|complement]] of ''A'' interpret [[Image:Laws of Form - not a.gif]]
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| * 0 (1) interpret the empty Mark if meet (join) interprets [[concatenation]].
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| 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 theory|model]] of the primary algebra. The primary arithmetic suggests the following arithmetic axiomatization of '''2''': 1+1=1+0=0+1=1=~0, and 0+0=0=~1.
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| The [[Set (mathematics)|set]] <math>\ B=\{</math>[[Image:Laws of Form - cross.gif]] <math>,</math> [[Image:Laws of Form - double cross.gif]]<math>\ \}</math> is the [[Boolean domain]] or ''carrier''. In the language of [[universal algebra]], the ''pa'' is the [[algebraic structure]] <math>\lang B,--,(-),() \rang</math> of type <math>\lang 2,1,0 \rang</math>. The [[functional completeness|expressive adequacy]] of the [[Sheffer stroke]] points to the ''pa'' also being a <math>\lang B,(--),()\rang</math> algebra of type <math>\lang 2,0 \rang</math>. 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 <math>\lang B,+,\lnot,1 \rang</math> algebra of type <math>\lang 2,1,0 \rang</math>. This description of '''2''' is simpler than the conventional one, namely an <math>\lang B,+,\times,\lnot,1,0 \rang</math> algebra of type <math>\lang 2,2,1,0,0 \rang</math>.
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| ====Sentential logic====
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| 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:
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| ::: = '''False'''
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| ::[[Image:Laws of Form - cross.gif]] = '''True''' = '''not False'''
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| ::[[Image:Laws of Form - double cross.gif]] = '''Not True''' = '''False'''
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| The ''pa'' interprets sentential logic as follows. A letter represents any given sentential expression. Thus:
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| ::[[Image:Laws of Form - not a.gif]] interprets '''Not A'''
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| ::[[Image:Laws of Form - a or b.gif]] interprets '''A Or B'''
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| ::[[Image:Laws of Form - if a then b.gif]] interprets '''Not A Or B''' or '''If A Then B'''.
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| ::[[Image:Laws of Form - a and b.gif]] interprets '''Not (Not A Or Not B)'''
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| :::::or '''Not (If A Then Not B)'''
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| :::::or '''A And B'''.
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| <math> \ (((A)B)(A(B))), ((A)(B))(AB) \ </math> both interpret '''A [[if and only if]] B''' or '''A is [[logical equivalence|equivalent]] to B'''.
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| Thus any expression in [[sentential logic]] has a ''pa'' translation. Equivalently, the ''pa'' [[interpretation (logic)|interprets]] sentential logic. Given an assignment of every variable to the Marked or Unmarked states, this ''pa'' translation reduces to a PA expression, which can be simplified. Repeating this exercise for all possible assignments of the two primitive values to each variable, reveals whether the original expression is [[Tautology (logic)|tautological]] or [[Satisfiability|satisfiable]]. This is an example of a [[decision procedure]], one more or less in the spirit of conventional truth tables. Given some ''pa'' formula containing ''N'' variables, this decision procedure requires simplifying 2<sup>''N''</sup> PA formulae. For a less tedious decision procedure more in the spirit of [[Willard Van Orman Quine|Quine]]'s "truth value analysis," see Meguire (2003).
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| Schwartz (1981) proved that the ''pa'' is equivalent -- [[syntax|syntactically]], [[semantics|semantically]], and [[proof theory|proof theoretically]]—with the [[Propositional calculus|classical propositional calculus]]. Likewise, it can be shown that the ''pa'' is syntactically equivalent with expressions built up in the usual way from the classical [[truth value]]s '''true''' and '''false''', the [[logical connective]]s NOT, OR, and AND, and parentheses.
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| Interpreting the Unmarked State as '''False''' is wholly arbitrary; that state can equally well be read as '''True'''. All that is required is that the interpretation of [[concatenation]] change from OR to AND. IF A THEN B now translates as (''A''(''B'')) instead of (''A'')''B''. More generally, the ''pa'' is "self-[[Duality (mathematics)|dual]]," meaning that any ''pa'' formula has two [[sentential logic|sentential]] or [[two-element Boolean algebra|Boolean]] readings, each the [[Duality (mathematics)|dual]] of the other. Another consequence of self-duality is the irrelevance of [[De Morgan's laws]]; those laws are built into the syntax of the ''pa'' from the outset.
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| 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, [[Logical complement|complementation]]/[[negation]] operating on "nothing" is not well-formed. But an empty Cross is a well-formed ''pa'' expression, denoting the Marked state, a primitive value. Hence a nonempty Cross is an [[Operator (mathematics)|operator]], while an empty Cross is an [[operand]] because it denotes a primitive value. Thus the ''pa'' reveals that the heretofore distinct mathematical concepts of operator and operand are in fact merely different facets of a single fundamental action, the making of a distinction.
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| ====Syllogisms====
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| Appendix 2 of ''LoF'' shows how to translate traditional [[syllogism]]s and [[polysyllogism|sorites]] into the ''pa''. A valid syllogism is simply one whose ''pa'' translation simplifies to an empty Cross. Let ''A''* denote a ''literal'', i.e., either ''A'' or (''A''), indifferently. Then all syllogisms that do not require that one or more terms be assumed nonempty are one of 24 possible permutations of a generalization of [[syllogism|Barbara]] whose ''pa'' equivalent is (''A''*''B'')((''B'')''C''*)''A''*''C''*. These 24 possible permutations include the 19 syllogistic forms deemed valid in [[Aristotelian logic|Aristotelian]] and [[medieval logic]]. This ''pa'' translation of syllogistic logic also suggests that the ''pa'' can [[interpretation (logic)|interpret]] [[monadic logic|monadic]] and [[term logic]], and that the ''pa'' has affinities to the [[Boolean term schema]]ta of Quine (1982: Part II).
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| ===An example of calculation===
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| The following calculation of [[Gottfried Wilhelm Leibniz|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]] [[Logical connective|connective]] appearing in the ''Theorema'' is [[Logical conjunction|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 (()).
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| * [(''P''→''R'')∧(''Q''→''S'')]→[(''P''∧''Q'')→(''R''∧''S'')]. ''Praeclarum Theorema''.
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| * ((''P''(''R''))(''Q''(''S''))((''PQ''(''RS'')))). ''pa'' translation.
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| *= (('''P'''(''R''))''P''('''Q'''(''S''))''Q''(''RS'')). OI; C1.
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| *= ((''R''))((''S''))''PQ''(''RS''). Invoke C2 2x to eliminate the bold letters in the previous expression; OI.
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| *= (''RSPQ''(''RS'')). C1,2x.
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| *= ((''RSPQ'')''RSPQ''). C2; OI.
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| *= (()). J1.<math>\square</math>
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| ''Remarks'':
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| * C1 (C2) is repeatedly invoked in a fairly mechanical way to eliminate nested parentheses (variable instances). This is the essence of the calculation method;
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| * A single invocation of J1 (or, in other contexts, J1a) terminates the calculation. This too is typical;
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| * Experienced users of the ''pa'' are free to invoke OI silently. OI aside, the demonstration requires a mere 7 steps.
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| ===A technical aside===
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| 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 value]]s. Let the [[Extension (predicate logic)|extension]] of an ''n''-place [[atomic formula]] be the set of ordered ''n''-tuples of [[individual]]s 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 ([[Kazimierz Kuratowski|Kuratowski]]'s definition) [[set theory|set theoretic]] definition is <a,b> = <nowiki>{{a},{{a,b}}</nowiki>, where ''a'',''b'' are individuals. Ordered ''n''-tuples for any ''n''>2 may be obtained from ordered pairs by a well-known [[recursion|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.
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| ===Relation to magmas===
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| The ''pa'' embodies a point noted by [[Edward Vermilye Huntington|Huntington]] in 1933: [[Boolean algebra (logic)|Boolean algebra]] requires, in addition to one [[unary operation]], one, and not two, [[binary operation]]s. Hence the seldom-noted fact that Boolean algebras are [[magma (algebra)|magmas]]. (Magmas were called [[groupoid]]s until the latter term was appropriated by [[category theory]].) To see this, note that the ''pa'' is a [[commutative]]:
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| *[[Semigroup]] because ''pa'' juxtaposition [[Commutative property|commute]]s and [[associative property|associates]];
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| *[[Monoid]] with [[identity element]] (()), by virtue of '''J0'''.
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| [[group (mathematics)|Groups]] also require a [[unary operation]], called [[Inverse (mathematics)|inverse]], the group counterpart of [[Boolean algebra (logic)|Boolean complementation]]. Let (''a'') denote the inverse of ''a''. Let () denote the group [[identity element]]. Then groups and the ''pa'' have the same [[signature (logic)|signatures]], namely they are both 〈--,(-),()〉 algebras of type 〈2,1,0〉. Hence the ''pa'' is a [[list of algebraic structures|boundary algebra]]. The axioms for an [[abelian group]], in boundary notation, are:
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| * '''G1'''. ''abc'' = ''acb'' (assuming association from the left);
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| * '''G2'''. ()''a'' = ''a'';
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| * '''G3'''. (''a'')''a'' = ().
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| From '''G1''' and '''G2''', the commutativity and associativity of concatenation may be derived, as above. Note that '''G3''' and '''J1a''' are identical. '''G2''' and '''J0''' would be identical if (())=() replaced '''A2'''. This is the defining arithmetical identity of group theory, in boundary notation.
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| The ''pa'' differs from an [[abelian group]] in two ways:
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| *From '''A2''', it follows that (()) ≠ (). If the ''pa'' were a [[group (mathematics)|group]], (())=() would hold, and one of (''a'')''a''=(()) or ''a''()=''a'' would have to be a ''pa'' consequence. Note that () and (()) are mutual ''pa'' complements, as group theory requires, so that ((())) = () is true of both group theory and the ''pa'';
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| *'''C2''' most clearly demarcates the ''pa'' from other magmas, because '''C2''' enables demonstrating the [[absorption law]] that defines [[lattice (order)|lattices]], and the [[distributive law]] central to [[Boolean algebra (structure)|Boolean algebra]].
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| Both '''A2''' and '''C2''' follow from ''B'' 's being an [[ordered set]].
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| ==Equations of the second degree (Chapter 11)==
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| Chapter 11 of ''LoF'' introduces ''equations of the second degree'', composed of [[recursion|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''.
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| 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.
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| ==Resonances in religion, philosophy, and science==
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| {{Original research|date=March 2011}}
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| The mathematical and logical content of ''LoF'' is wholly consistent with a [[secular humanist|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:
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| * [[Historical Vedic religion|Vedic]], [[Hindu]] and [[Buddhist]]: Related ideas can be noted in the ancient [[Historical Vedic religion|Vedic]] [[Upanishads]], which form the [[Monism|monastic]] foundations of [[Hinduism]] and later [[Buddhism]]. As stated in the ''Aitareya Upanishad'' ("The Microcosm of Man"), the Supreme [[Atman (Buddhism)|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."
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| * [[Taoism]], ([[Chinese traditional religion|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]]).
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| * [[Zoroastrianism]]: "This I ask Thee, tell me truly, Ahura. What artist made light and darkness?" ([[Gathas]] 44.5)
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| * [[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.
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| ::"...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.
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| ::"...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.
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| ::"...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).
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| ::"And the whole earth was of ''one'' language, and of ''one'' speech." (Genesis 11:1; emphasis added).
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| ::"I am; that is who I am." (Exodus 3:14)
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| * [[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)
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| * [[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."
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| * [[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.
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| * [[Plato]]: ''Logos'' is also a fundamental technical term in the Platonic worldview.
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| * [[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)
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| * [[Object relations theory]], [[psychodynamics]]: The primary separation experienced by infants between self and other objects, distinguishing of reality from phantasy.
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| * [[Islamic philosophy]] distinguishes essence (''Dhat'') from attribute (''Sifat''), which are neither identical nor separate.
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| * [[Gottfried Wilhelm Leibniz|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."<ref>"On the True ''Theologia Mystica''" in Loemker, Leroy, ed. and trans., 1969. ''Leibniz: Philosophical Papers and Letters''. Reidel: 368.</ref>
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| * [[Josiah Royce]]: "Without negation, there is no inference. Without inference, there is no order, in the strictly logical sense of the word. The fundamentally significant position of the idea of negation in determining and controlling our idea of the orderliness of both the natural and the spiritual order, becomes, in the light of all these considerations, as momentous as it is, in our ordinary popular views of this subject, neglected. ...From this point of view, negation appears as one of the most significant. ideas that lie at the base of all the exact sciences. By virtue of the idea of negation we are able to define processes of inference-processes which, in their abstract form, the purely mathematical sciences illustrate, and which, in their natural expression, the laws of the physical world, as known to our inductive science, exemplify."
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| :"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." <ref>"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.</ref>
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| Returning to the Bible, the injunction "Let there be light" conveys:
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| * "... and there was light" — the light itself;
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| * "... called the light Day" — the manifestation of the light;
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| * "... morning and evening" — the boundaries of the light.
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| 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.
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| ==Related work==
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| [[Gottfried Leibniz]], in memoranda not published before the late 19th and early 20th centuries, invented [[Boolean algebra (logic)|Boolean logic]]. His notation was isomorphic to that of ''LoF'': concatenation read as [[Logical conjunction|conjunction]], and "non-(''X'')" read as the [[Logical complement|complement]] of ''X''. Leibniz's pioneering role in [[algebraic logic]] was foreshadowed by [[Clarence Irving Lewis|Lewis]] (1918) and [[Nicholas Rescher|Rescher]] (1954). But a full appreciation of Leibniz's accomplishments had to await the work of Wolfgang Lenzen, published in the 1980s and reviewed in [http://www.philosophie.uni-osnabrueck.de/Publikationen%20Lenzen/Lenzen%20Leibniz%20Logic.pdf Lenzen (2004).]
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| [[Charles Sanders Peirce]] (1839–1914) anticipated the ''pa'' in three veins of work:
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| #Two papers he wrote in 1886 proposed a logical algebra employing but one symbol, the ''streamer'', nearly identical to the Cross of ''LoF''. The semantics of the streamer are identical to those of the Cross, except that Peirce never wrote a streamer with nothing under it. An excerpt from one of these papers was published in 1976,<ref>"Qualitative Logic", MS 736 (c. 1886) in Eisele, Carolyn, ed. 1976. ''[[Charles Sanders Peirce bibliography#NEM|The New Elements of Mathematics by Charles S. Peirce]]. Vol. 4, Mathematical Philosophy''. (The Hague) Mouton: 101-15.1</ref> but they were not published in full until 1993.<ref>"Qualitative Logic", MS 582 (1886) in Kloesel, Christian et al., eds., 1993. ''[[Charles Sanders Peirce bibliography#W|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.</ref>
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| #In a 1902 encyclopedia article,<ref>Reprinted in Peirce, C.S. (1933) ''[[Charles Sanders Peirce bibliography#CP|Collected Papers of Charles Sanders Peirce]], Vol. 4'', [[Charles Hartshorne]] and [[Paul Weiss (philosopher)|Paul Weiss]], eds. [[Harvard University Press]]. Paragraphs 378-383</ref> 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.
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| #The [[syntax]] of his alpha [[existential graph]]s is merely [[concatenation]], read as [[Logical conjunction|conjunction]], and enclosure by ovals, read as [[negation]].<ref>The existential graphs are described at length in Peirce, C.S. (1933) ''Collected Papers, Vol. 4'', [[Charles Hartshorne]] and [[Paul Weiss (philosopher)|Paul Weiss]], eds. Harvard University Press. Paragraphs 347-529.</ref> If ''pa'' concatenation is read as [[Logical conjunction|conjunction]], then these graphs are [[isomorphic]] to the ''pa'' [http://www2.math.uic.edu/~kauffman/CHK.pdf (Kauffman 2001).]
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| Ironically, ''LoF'' cites vol. 4 of Peirce's ''Collected Papers,'' the source for the formalisms in (2) and (3) above.
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| (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''.
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| [http://www2.math.uic.edu/~kauffman/CHK.pdf 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.
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| 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]], [[Polish notation|prefix]], or [[Reverse Polish notation|postfix]] operator symbols. The very well known curly braces ('{', '}') of set theory can be seen as a boundary notation.
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| 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 [[David Hilbert|Hilbert]] and [[Jan Łukasiewicz|Lukasiewicz]] showed how to prove [[axiom independence]] using [[model theory|model]]s.
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| [[William Craig (logician)|Craig]] (1979) argued that the world, and how humans perceive and interact with that world, has a rich Boolean structure. Craig was an orthodox logician and an authority on [[algebraic logic]].
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| Second-generation [[cognitive science]] emerged in the 1970s, after ''LoF'' was written. On cognitive science and its relevance to Boolean algebra, logic, and [[set theory]], see Lakoff (1987) (see index entries under "Image schema examples: container") and Lakoff and Núñez (2001). Neither book cites ''LoF''.
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| 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.
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| The [http://multiforms.netfirms.com 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 operation]]s: [[concatenation (mathematics)|concatenation]], read as Boolean [[logical disjunction|OR]], and [[infix]] "#", read as [[XOR]]. The primitive values are 0 and 1, and the corresponding arithmetic is 11=1 and 1#1=0. The axioms are 1A=1, A#X#X = A, and A(X#(AB)) = A(X#B).
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| Other formal systems with possible affinities to the primary algebra include:
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| *[[Mereology]] which typically has a [[lattice (order)|lattice]] structure very similar to that of Boolean algebra. For a few authors, mereology is simply a [[model theory|model]] of [[Boolean algebra (structure)|Boolean algebra]] and hence of the primary algebra as well.
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| *[[Mereotopology]], which is inherently richer than Boolean algebra;
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| *The system of Whitehead (1934), whose fundamental primitive is "indication."
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| The primary arithmetic and algebra are a minimalist formalism for [[sentential logic]] and Boolean algebra. Other minimalist formalisms having the power of [[set theory]] include:
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| * The [[lambda calculus]];
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| * [[Combinatory logic]] with two ('''S''' and '''K''') or even one ('''X''') primitive combinators;
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| * [[Mathematical logic]] done with merely three primitive notions: one connective, [[Sheffer stroke|NAND]] (whose ''pa'' translation is (''AB'') or—dually -- (''A'')(''B'') ), universal [[quantification]], and one [[binary relation|binary]] [[atomic formula]], denoting [[Set (mathematics)|set]] membership. This is the system of [[Willard Quine|Quine]] (1951).
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| * The ''beta'' [[existential graph]]s, with a single [[binary predicate]] denoting set membership. This has yet to be explored. The ''alpha'' graphs mentioned above are a special case of the ''beta'' graphs.
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| ==See also==
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| * [[:simple:Boolean algebra|Boolean algebra]] (Simple English Wikipedia)
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| *[[Boolean algebra (introduction)]]
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| *[[Boolean algebra (logic)]]
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| *[[Boolean algebra (structure)]]
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| *[[Boolean algebras canonically defined]]
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| *[[Boolean logic]]
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| *[[Entitative graph]]
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| *[[Existential graph]]
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| *[[List of Boolean algebra topics]]
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| *[[Propositional calculus]]
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| *[[Two-element Boolean algebra]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *Editions of ''Laws of Form'':
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| **1969. London: Allen & Unwin, hardcover.
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| **1972. Crown Publishers, hardcover: ISBN 0-517-52776-6
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| **1973. Bantam Books, paperback. ISBN 0-553-07782-1
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| **1979. E.P. Dutton, paperback. ISBN 0-525-47544-3
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| **1994. Portland OR: Cognizer Company, paperback. ISBN 0-9639899-0-1
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| **1997 German translation, titled ''Gesetze der Form''. Lübeck: Bohmeier Verlag. ISBN 3-89094-321-7
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| *Bostock, David, 1997. ''Intermediate Logic''. Oxford Univ. Press.
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| *Byrne, Lee, 1946, "Two Formulations of Boolean Algebra," ''Bulletin of the American Mathematical Society'': 268-71.
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| *{{Cite journal | doi = 10.2307/3131383 | last1 = Craig | first1 = William | year = 1979 | title = Boolean Logic and the Everyday Physical World | jstor = 3131383| journal = Proceedings and Addresses of the American Philosophical Association | volume = 52 | issue = 6| pages = 751–78 }}
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| * [[David Gries]], and Schneider, F B, 1993. ''A Logical Approach to Discrete Math''. Springer-Verlag.
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| *[[William Ernest Johnson]], 1892, "The Logical Calculus," ''Mind'' 1 (n.s.): 3-30.
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| * [http://www.math.uic.edu/~kauffman/ Louis H. Kauffman], 2001, "[http://www2.math.uic.edu/~kauffman/CHK.pdf The Mathematics of C.S. Peirce]", ''Cybernetics and Human Knowing'' 8: 79-110.
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| * ------, 2006, "[http://www.arxiv.org/math.CO/0112266 Reformulating the Map Color Theorem.]"
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| * ------, 2006a. "[http://www.math.uic.edu/~kauffman/Laws.pdf Laws of Form - An Exploration in Mathematics and Foundations.]" Book draft (hence big).
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| * Lenzen, Wolfgang, 2004, "[http://www.philosophie.uni-osnabrueck.de/Publikationen%20Lenzen/Lenzen%20Leibniz%20Logic.pdf 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.
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| *[[George Lakoff|Lakoff, George]], 1987. ''Women, Fire, and Dangerous Things''. University of Chicago Press.
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| *-------- and [[Rafael E. Núñez]], 2001. ''[[Where Mathematics Comes From]]: How the Embodied Mind Brings Mathematics into Being''. Basic Books.
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| * 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.
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| *--------, 2011. ''Boundary Algebra: A Simpler Approach to Basic Logic and Boolean Algebra''. VDM Publishing Ltd. IBSN 978-3639367492. The source for much of this entry, including the notation which encloses in parentheses what ''LoF'' places under a cross. Steers clear of the more speculative aspects of ''LoF''.
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| *[[Willard Quine]], 1951. ''Mathematical Logic'', 2nd ed. Harvard University Press.
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| *--------, 1982. ''Methods of Logic'', 4th ed. Harvard University Press.
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| * {{Cite journal | last1 = Rescher | first1 = Nicholas | authorlink = Nicholas Rescher | year = 1954 | title = Leibniz's Interpretation of His Logical Calculi | url = | journal = Journal of Symbolic Logic | volume = 18 | issue = | pages = 1–13 }}
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| * {{Cite journal | doi = 10.1080/03081078108934802 | last1 = Schwartz | first1 = Daniel G. | year = 1981 | title = [[Isomorphism]]s of [[G. Spencer-Brown]]'s ''Laws of Form'' and [[Francisco Varela|F. Varela's]] Calculus for Self-Reference | url = | journal = International Journal of General Systems | volume = 6 | issue = 4| pages = 239–55 }}
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| * {{Cite journal | doi = 10.1080/03081078608934939 | last1 = Turney | first1 = P. D. | year = 1986 | title = ''Laws of Form'' and Finite Automata | url = | journal = International Journal of General Systems | volume = 12 | issue = 4| pages = 307–18 }}
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| *[[A. N. Whitehead]], 1934, "Indication, classes, number, validation," ''Mind'' 43 (n.s.): 281-97, 543. The corrigenda on p. 543 are numerous and important, and later reprints of this article do not incorporate them.
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| ==External links==
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| * ''Laws of Form'' [http://www.lawsofform.org/ web site], by Richard Shoup.
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| * [http://www.lawsofform.org/aum/session1.html Spencer-Brown's talks at Esalen, 1973.] Self-referential forms are introduced in the section entitled "Degree of Equations and the Theory of Types."
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| * [http://www.math.uic.edu/~kauffman/ Louis H. Kauffman,] "[http://www.math.uic.edu/~kauffman/Arithmetic.htm Box Algebra, Boundary Mathematics, Logic, and Laws of Form.]"
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| * Kissel, Matthias, "[http://web.archive.org/web/20070310071916/http://de.geocities.com/matthias_kissel/gdf/LoF.html A nonsystematic but easy to understand introduction to ''Laws of Form''.]"
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| * The [http://multiforms.netfirms.com Multiple Form Logic], by G.A. Stathis, owes much to the primary algebra.
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| * The [http://groups.yahoo.com/group/lawsofform Laws of Form Forum], where the primary algebra and related formalisms have been discussed since 2002.
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| * A meeting [http://www.omath.org.il/112431/4CT with G.S.B] by Moshe Klein
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| {{Use dmy dates|date=September 2010}}
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| {{DEFAULTSORT:Laws Of Form}}
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| [[Category:Algebra]]
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| [[Category:Boolean algebra]]
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| [[Category:Logic books]]
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| [[Category:Logical calculi]]
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| [[Category:Mathematical logic]]
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