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Encyclopedia > Logical graph

A logical graph is a special type of graph-theoretic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic. A diagram of a graph with 6 vertices and 7 edges. ... Syntax, originating from the Greek words συν (syn, meaning co- or together) and τάξις (táxis, meaning sequence, order, arrangement), can be described as the study of the rules, or patterned relations that govern the way the words in a sentence come together. ... Charles Sanders Peirce Charles Sanders Peirce (September 10, 1839 – April 19, 1914) was an American logician, philosopher, scientist, and mathematician. ... Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. ...


In his papers on qualitative logic, entitative graphs, and existential graphs, Peirce developed several versions of a graphical formalism, or a graph-theoretic formal language, designed to be interpreted for logic. An existential graph is a type of diagrammatic or visual notation for logical expressions, invented by Charles Peirce. ... In logic, mathematics, and computer science, a formal system is a formal grammar used for modelling purposes. ... In mathematics, logic and computer science, a formal language is a set of finite-length words (i. ...


In the century since Peirce initiated this line of development, a variety of formal systems have branched out from what is abstractly the same formal base of graph-theoretic structures. This article examines the common basis of these formal systems from a bird's eye view, focusing on those aspects of form that are shared by the entire family of algebras, calculi, or languages, however they happen to be viewed in a given application.

Contents


Abstract point of view

Wollust ward dem Wurm gegeben ...
(Friedrich Schiller, An die Freude)

The bird's eye view in question is more formally known as the perspective of formal equivalence, from which remove one cannot see many distinctions that appear momentous from lower levels of abstraction. In particular, expressions of different formalisms whose syntactic structures are isomorphic from the standpoint of algebra or topology are not recognized as being different from each other in any significant sense. Though we may note in passing such historical details as the circumstance that Charles Sanders Peirce used a streamer-cross symbol where George Spencer Brown used a carpenter's square marker, the theme of principal interest at the abstract level of form is neutral with regard to variations of that order. Friedrich Schiller Johann Christoph Friedrich von Schiller (November 10, 1759 – May 9, 1805), usually known as Friedrich Schiller, was a German poet, philosopher, historian, and dramatist. ... To Joy (An die Freude in German, in English often familiarly called the Ode to Joy rather than To Joy) is an ode written in 1785 by the German poet and historian Friedrich Schiller, and known especially for its musical setting by Beethoven in the fourth and final movement of... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... Algebra is a branch of mathematics which studies structure and quantity. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... G. Spencer-Brown (April 2, 1923) was born in Grimsby, Lincolnshire, England and is a British mathematician. ...


Duality, logical and topological

In mathematics, duality has numerous meanings. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...

Axioms

 o-----------------------------------------------------------o | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` /` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` O ` ` ` ` = ` ` ` ` O ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | o-----------------------------------------------------------o | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` `( ) ( )` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | o-----------------------------------------------------------o | Axiom I_1.` ` Distract <--- | ---> Condense ` ` ` ` ` ` ` | o-----------------------------------------------------------o 
 o-----------------------------------------------------------o | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` O ` ` ` ` = ` ` ` ` O ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | o-----------------------------------------------------------o | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` (( )) ` ` ` = ` ` ` ` ` ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | o-----------------------------------------------------------o | Axiom I_2.` ` ` Unfold <--- | ---> Refold ` ` ` ` ` ` ` ` | o-----------------------------------------------------------o 
 o-----------------------------------------------------------o | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` a O ` ` ` ` = ` ` ` ` O ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | o-----------------------------------------------------------o | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` a(a)` ` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | o-----------------------------------------------------------o | Axiom J_1.` ` ` Insert <--- | ---> Delete ` ` ` ` ` ` ` ` | o-----------------------------------------------------------o 
 o-----------------------------------------------------------o | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` `ab ` ac` ` ` ` ` ` ` b ` c ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` /` ` ` ` ` ` ` ` ` /` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` O ` ` ` ` = ` ` ` a O ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | o-----------------------------------------------------------o | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` `((ab)(ac)) ` ` = ` ` a((b)(c)) ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | o-----------------------------------------------------------o | Axiom J_2.` Distribute <--- | ---> Collect` ` ` ` ` ` ` ` | o-----------------------------------------------------------o 

Here is one way of reading these formal axioms under the existential interpretation:

  • I1. false and false <=> false.
  • I2. not false <=> true.
  • J1. a and not a <=> false.
  • J2. [a and b] or [a and c] <=> a and [b or c]

(Text in preparation, 07 January 2006)


References

See also

The phrase Laws of Form refers to either of two things: The book, hereinafter abbreviated LoF: G. Spencer-Brown, 1979. ... The phrase Laws of Form refers to either of two things: The book, hereinafter abbreviated LoF: G. Spencer-Brown, 1979. ... G. Spencer-Brown (April 2, 1923) was born in Grimsby, Lincolnshire, England and is a British mathematician. ...

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