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MyWikiBiz, Author Your Legacy — Friday November 01, 2024
Jump to navigationJump to searchspacing in TeX <math>a \texttt{( )} = \texttt{( )},~\!</math>
<font size="3">☞</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
In his papers on ''[[qualitative logic]]'', ''[[entitative graph]]s'', and ''[[existential graph]]s'', Peirce developed several versions of a graphical formalism, or a graph-theoretic formal language, designed to be interpreted for logic.
A '''logical graph''' is a graph-theoretic structure in one of the systems of graphical syntax that Charles Sanders Peirce developed for logic.
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.
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.
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.
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| width="40%" | ''Wollust ward dem Wurm gegeben …''
| width="40%" | ''Wollust ward dem Wurm gegeben …''
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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.
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.
==In lieu of a beginning==
==In lieu of a beginning==
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For the time being these two forms of transformation may be referred to as ''[[axioms]]'' or ''initial equations''.
For the time being these two forms of transformation may be referred to as ''axioms'' or ''initial equations''.
==Duality : logical and topological==
==Duality : logical and topological==
There are two types of duality that have to be kept separately mind in the use of logical graphs — logical duality and topological duality.
There are two types of duality that have to be kept separately mind in the use of logical graphs — logical duality and topological duality.
There is a standard way that graphs of the order that Peirce considered, those embedded in a continuous [[manifold]] like that commonly represented by a plane sheet of paper — with or without the paper bridges that Peirce used to augment its topological genus — can be represented in linear text as what are called ''parse strings'' or ''traversal strings'' and parsed into ''pointer structures'' in computer memory.
There is a standard way that graphs of the order that Peirce considered, those embedded in a continuous manifold like that commonly represented by a plane sheet of paper — with or without the paper bridges that Peirce used to augment its topological genus — can be represented in linear text as what are called ''parse strings'' or ''traversal strings'' and parsed into ''pointer structures'' in computer memory.
A blank sheet of paper can be represented in linear text as a blank space, but that way of doing it tends to be confusing unless the logical expression under consideration is set off in a separate display.
A blank sheet of paper can be represented in linear text as a blank space, but that way of doing it tends to be confusing unless the logical expression under consideration is set off in a separate display.
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This can be written inline as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}</math> or set off in a text display as follows:
This can be written inline as <math>{}^{\backprime\backprime} \texttt{( ( ) )} = \quad {}^{\prime\prime}\!</math> or set off in a text display as follows:
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
| width="33%" | <math>\texttt{(~(~)~)}</math>
| width="33%" | <math>\texttt{( ( ) )}\!</math>
| width="34%" | <math>=\!</math>
| width="34%" | <math>=\!</math>
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When we turn to representing the corresponding expressions in computer memory, where they can be manipulated with utmost facility, we begin by transforming the planar graphs into their topological duals. The planar regions of the original graph correspond to nodes (or points) of the [[dual graph]], and the boundaries between planar regions in the original graph correspond to edges (or lines) between the nodes of the dual graph.
When we turn to representing the corresponding expressions in computer memory, where they can be manipulated with utmost facility, we begin by transforming the planar graphs into their topological duals. The planar regions of the original graph correspond to nodes (or points) of the dual graph, and the boundaries between planar regions in the original graph correspond to edges (or lines) between the nodes of the dual graph.
For example, overlaying the corresponding dual graphs on the plane-embedded graphs shown above, we get the following composite picture:
For example, overlaying the corresponding dual graphs on the plane-embedded graphs shown above, we get the following composite picture:
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Though it's not really there in the most abstract topology of the matter, for all sorts of pragmatic reasons we find ourselves compelled to single out the outermost region of the plane in a distinctive way and to mark it as the ''[[root node]]'' of the corresponding dual graph. In the present style of Figure the root nodes are marked by horizontal strike-throughs.
Though it's not really there in the most abstract topology of the matter, for all sorts of pragmatic reasons we find ourselves compelled to single out the outermost region of the plane in a distinctive way and to mark it as the ''root node'' of the corresponding dual graph. In the present style of Figure the root nodes are marked by horizontal strike-throughs.
Extracting the dual graphs from their composite matrices, we get this picture:
Extracting the dual graphs from their composite matrices, we get this picture:
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It is easy to see the relationship between the parenthetical expressions of Peirce's logical graphs, that somewhat clippedly picture the ordered containments of their formal contents, and the associated dual graphs, that constitute the species of [[rooted tree]]s here to be described.
It is easy to see the relationship between the parenthetical expressions of Peirce's logical graphs, that somewhat clippedly picture the ordered containments of their formal contents, and the associated dual graphs, that constitute the species of rooted trees here to be described.
In the case of our last example, a moment's contemplation of the following picture will lead us to see that we can get the corresponding parenthesis string by starting at the root of the tree, climbing up the left side of the tree until we reach the top, then climbing back down the right side of the tree until we return to the root, all the while reading off the symbols, in this case either <math>{}^{\backprime\backprime} \texttt{(} {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \texttt{)} {}^{\prime\prime},</math> that we happen to encounter in our travels.
In the case of our last example, a moment's contemplation of the following picture will lead us to see that we can get the corresponding parenthesis string by starting at the root of the tree, climbing up the left side of the tree until we reach the top, then climbing back down the right side of the tree until we return to the root, all the while reading off the symbols, in this case either <math>{}^{\backprime\backprime} \texttt{(} {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} \texttt{)} {}^{\prime\prime},\!</math> that we happen to encounter in our travels.
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This ritual is called ''[[traversing]]'' the tree, and the string read off is called the ''[[traversal string]]'' of the tree. The reverse ritual, that passes from the string to the tree, is called ''[[parsing]]'' the string, and the tree constructed is called the ''[[parse graph]]'' of the string. The speakers thereof tend to be a bit loose in this language, often using ''[[parse string]]'' to mean the string that gets parsed into the associated graph.
This ritual is called ''traversing'' the tree, and the string read off is called the ''traversal string'' of the tree. The reverse ritual, that passes from the string to the tree, is called ''parsing'' the string, and the tree constructed is called the ''parse graph'' of the string. The speakers thereof tend to be a bit loose in this language, often using ''parse string'' to mean the string that gets parsed into the associated graph.
We have treated in some detail various forms of the initial equation or logical axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}.</math> For the sake of comparison, let's record the plane-embedded and topological dual forms of the axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~)(~)} = \texttt{(~)} {}^{\prime\prime}.</math>
We have treated in some detail various forms of the initial equation or logical axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{( ( ) )} = \quad {}^{\prime\prime}.~\!</math> For the sake of comparison, let's record the plane-embedded and topological dual forms of the axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{( )( )} = \texttt{( )} {}^{\prime\prime}.\!</math>
First the plane-embedded maps:
First the plane-embedded maps:
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We have at this point enough material to begin thinking about the forms of [[analogy]], [[iconicity]], [[metaphor]], [[morphism]], whatever you want to call them, that are pertinent to the use of logical graphs in their various logical interpretations, for instance, those that Peirce described as ''[[entitative graph]]s'' and ''[[existential graph]]s''.
We have at this point enough material to begin thinking about the forms of analogy, iconicity, metaphor, morphism, whatever you want to call them, that are pertinent to the use of logical graphs in their various logical interpretations, for instance, those that Peirce described as ''entitative graphs'' and ''existential graphs''.
==Computational representation==
==Computational representation==
Though it may not seem too exciting, logically speaking, there are many reasons to make oneself at home with the system of forms that is represented indifferently, topologically speaking, by rooted trees, balanced strings of parentheses, or finite sets of non-intersecting simple closed curves in the plane.
Though it may not seem too exciting, logically speaking, there are many reasons to make oneself at home with the system of forms that is represented indifferently, topologically speaking, by rooted trees, balanced strings of parentheses, or finite sets of non-intersecting simple closed curves in the plane.
:* One reason is that it gives us a respectable example of a sign domain on which to cut our semiotic teeth, non-trivial in the sense that it contains a [[countable]] [[infinity]] of signs.
:* One reason is that it gives us a respectable example of a sign domain on which to cut our semiotic teeth, non-trivial in the sense that it contains a countable infinity of signs.
:* Another reason is that it allows us to study a simple form of [[computation]] that is recognizable as a species of ''[[semiosis]]'', or sign-transforming process.
:* Another reason is that it allows us to study a simple form of computation that is recognizable as a species of ''[[semiosis]]'', or sign-transforming process.
This space of forms, along with the two axioms that induce its [[partition of a set|partition]] into exactly two [[equivalence class]]es, is what [[George Spencer Brown]] called the ''primary arithmetic''.
This space of forms, along with the two axioms that induce its partition into exactly two equivalence classes, is what George Spencer Brown called the ''primary arithmetic''.
The axioms of the primary arithmetic are shown below, as they appear in both graph and string forms, along with pairs of names that come in handy for referring to the two opposing directions of applying the axioms.
The axioms of the primary arithmetic are shown below, as they appear in both graph and string forms, along with pairs of names that come in handy for referring to the two opposing directions of applying the axioms.
| <math>S_0\!</math>
| <math>S_0\!</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>\{ \ominus, \vert \} = \{</math>[[Image:Rooted Node.jpg|16px]], [[Image:Rooted Edge.jpg|12px]]<math>\}\!</math>
| <math>\{ \ominus, \vert \} = \{\!</math>[[Image:Rooted Node.jpg|16px]], [[Image:Rooted Edge.jpg|12px]]<math>\}\!</math>
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Regarded as a semiotic process, this amounts to a sequence of signs, every one after the first serving as the ''[[interpretant]]'' of its predecessor, ending in a final sign that may be taken as the canonical sign for their common object, in the upshot being the result of the computation process. Simple as it is, this exhibits the main features of any computation, namely, a semiotic process that proceeds from an obscure sign to a clear sign of the same object, being in its aim and effect an action on behalf of clarification.
Regarded as a semiotic process, this amounts to a sequence of signs, every one after the first serving as the ''interpretant'' of its predecessor, ending in a final sign that may be taken as the canonical sign for their common object, in the upshot being the result of the computation process. Simple as it is, this exhibits the main features of any computation, namely, a semiotic process that proceeds from an obscure sign to a clear sign of the same object, being in its aim and effect an action on behalf of clarification.
===Primary algebra as pattern calculus===
===Primary algebra as pattern calculus===
Experience teaches that complex objects are best approached in a gradual, laminar, [[module|modular]] fashion, one step, one layer, one piece at a time, and it's just as much the case when the complexity of the object is irreducible, that is, when the articulations of the representation are necessarily at joints that are cloven disjointedly from nature, with some assembly required in the synthetic integrity of the intuition.
Experience teaches that complex objects are best approached in a gradual, laminar, modular fashion, one step, one layer, one piece at a time, and it's just as much the case when the complexity of the object is irreducible, that is, when the articulations of the representation are necessarily at joints that are cloven disjointedly from nature, with some assembly required in the synthetic integrity of the intuition.
That's one good reason for spending so much time on the first half of [[zeroth order logic]], represented here by the primary arithmetic, a level of formal structure that C.S. Peirce verged on intuiting at numerous points and times in his work on logical graphs, and that Spencer Brown named and brought more completely to life.
That's one good reason for spending so much time on the first half of [[zeroth order logic]], represented here by the primary arithmetic, a level of formal structure that C.S. Peirce verged on intuiting at numerous points and times in his work on logical graphs, and that Spencer Brown named and brought more completely to life.
There is one other reason for lingering a while longer in these primitive forests, and this is that an acquaintance with "bare trees", those as yet unadorned with literal or numerical labels, will provide a firm basis for understanding what's really at issue in such problems as the "ontological status of variables".
There is one other reason for lingering a while longer in these primitive forests, and this is that an acquaintance with “bare trees”, those as yet unadorned with literal or numerical labels, will provide a firm basis for understanding what's really at issue in such problems as the “ontological status of variables”.
It is probably best to illustrate this theme in the setting of a concrete case, which we can do by reviewing the previous example of reductive evaluation shown in Figure 16.
It is probably best to illustrate this theme in the setting of a concrete case, which we can do by reviewing the previous example of reductive evaluation shown in Figure 16.
Observations like that, made about an arithmetic of any variety, germinated by their summarizations, are the root of all algebra.
Observations like that, made about an arithmetic of any variety, germinated by their summarizations, are the root of all algebra.
Speaking of algebra, and having encountered already one example of an algebraic law, we might as well introduce the axioms of the ''primary algebra'', once again deriving their substance and their name from the works of [[Charles Sanders Peirce]] and [[George Spencer Brown]], respectively.
Speaking of algebra, and having encountered already one example of an algebraic law, we might as well introduce the axioms of the ''primary algebra'', once again deriving their substance and their name from the works of Charles Sanders Peirce and George Spencer Brown, respectively.
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The choice of axioms for any formal system is to some degree a matter of aesthetics, as it is commonly the case that many different selections of formal rules will serve as axioms to derive all the rest as theorems. As it happens, the example of an algebraic law that we noticed first, <math>a(~) = (~),</math> as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms.
The choice of axioms for any formal system is to some degree a matter of aesthetics, as it is commonly the case that many different selections of formal rules will serve as axioms to derive all the rest as theorems. As it happens, the example of an algebraic law that we noticed first, <math>a \texttt{( )} = \texttt{( )},~\!</math> as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms.
We might also notice at this point a subtle difference between the primary arithmetic and the primary algebra with respect to the grounds of justification that we have naturally if tacitly adopted for their respective sets of axioms.
We might also notice at this point a subtle difference between the primary arithmetic and the primary algebra with respect to the grounds of justification that we have naturally if tacitly adopted for their respective sets of axioms.
The arithmetic axioms were introduced by fiat, in a quasi-[[apriori]] fashion, though of course it is only long prior experience with the practical uses of comparably developed generations of formal systems that would actually induce us to such a quasi-primal move. The algebraic axioms, in contrast, can be seen to derive their motive and their justice from the observation and summarization of patterns that are visible in the arithmetic spectrum.
The arithmetic axioms were introduced by fiat, in an ''a priori'' fashion, though of course it is only long prior experience with the practical uses of comparably developed generations of formal systems that would actually induce us to such a quasi-primal move. The algebraic axioms, in contrast, can be seen to derive their motive and their justice from the observation and summarization of patterns that are visible in the arithmetic spectrum.
==Formal development==
==Formal development==
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All of the axioms in this set have the form of equations. This means that all of the inference steps that they allow are reversible. The proof annotation scheme employed below makes use of a double bar <math>\overline{\underline{~~~~~~}}</math> to mark this fact, although it will often be left to the reader to decide which of the two possible directions is the one required for applying the indicated axiom.
All of the axioms in this set have the form of equations. This means that all of the inference steps that they allow are reversible. The proof annotation scheme employed below makes use of a double bar <math>=\!=\!=\!=\!=\!=</math> to mark this fact, although it will often be left to the reader to decide which of the two possible directions is the one required for applying the indicated axiom.
===Frequently used theorems===
===Frequently used theorems===
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The proof that follows is adapted from the one that was given by [[George Spencer Brown]] in his book ''Laws of Form'' (LOF) and credited to two of his students, John Dawes and D.A. Utting.
The proof that follows is adapted from the one that was given by George Spencer Brown in his book ''Laws of Form'' (LOF) and credited to two of his students, John Dawes and D.A. Utting.
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| (25)
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====Peirce's law====
====Peirce's law====
: ''Main article'' : [http://mywikibiz.com/Peirce's_law Peirce's law]
: ''Main article'' : [[Peirce's law]]
Peirce's law is commonly written in the following form:
Peirce's law is commonly written in the following form:
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| <math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p</math>
| <math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p\!</math>
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The existential graph representation of Peirce's law is shown in Figure 34.
The existential graph representation of Peirce's law is shown in Figure 33.
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| [[Image:Logical_Graph_Figure_31.jpg|500px]] || (34)
| [[Image:Peirce's Law 1.0 Splash Page.png|500px]] || (33)
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A graphical proof of Peirce's law is shown in Figure 35.
A graphical proof of Peirce's law is shown in Figure 34.
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| [[Image:Logical_Graph_Figure_32.jpg|500px]] || (35)
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| (34)
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The following animation replays the steps of the proof.
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| [[Image:Peirce's Law 2.0 Animation.gif]]
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| (35)
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====Praeclarum theorema====
====Praeclarum theorema====
An illustrious example of a propositional theorem is the ''praeclarum theorema'', the ''admirable'', ''shining'', or ''splendid'' theorem of [[Leibniz]].
An illustrious example of a propositional theorem is the ''praeclarum theorema'', the ''admirable'', ''shining'', or ''splendid'' theorem of Leibniz.
<blockquote>
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<p>If ''a'' is ''b'' and ''d'' is ''c'', then ''ad'' will be ''bc''.</p>
<p>If ''a'' is ''b'' and ''d'' is ''c'', then ''ad'' will be ''bc''.</p>
<p>''ad'' is ''bd'', and ''bd'' is ''bc'', therefore ''ad'' is ''bc''. Q.E.D.</p>
<p>''ad'' is ''bd'', and ''bd'' is ''bc'', therefore ''ad'' is ''bc''. Q.E.D.</p>
<p>([[Leibniz]], ''Logical Papers'', p. 41).</p>
<p>(Leibniz, ''Logical Papers'', p. 41).</p>
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Under the existential interpretation, the praeclarum theorema is represented by means of the following logical graph.
Under the existential interpretation, the praeclarum theorema is represented by means of the following logical graph.
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| [[Image:Logical_Graph_Figure_33.jpg|500px]] || (36)
| [[Image:Praeclarum Theorema 1.0 Splash Page.png|500px]] || (36)
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And here's a neat proof of that nice theorem.
And here's a neat proof of that nice theorem.
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| [[Image:Logical_Graph_Figure_34.jpg|500px]] || (37)
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| [[Image:Equational Inference Rule Reflect ad(bc).png|500px]]
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| [[Image:Equational Inference Rule Weed a, d.png|500px]]
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| [[Image:Equational Inference Rule Weed bc.png|500px]]
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| [[Image:Equational Inference Rule Quit abcd.png|500px]]
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| [[Image:Equational Inference Marquee QED.png|500px]]
|}
| (37)
|}
The steps of the proof are replayed in the following animation.
{| align="center" cellpadding="8"
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
|-
| [[Image:Praeclarum Theorema 2.0 Animation.gif]]
|}
| (38)
|}
|}
a b + a c + b c
a b + a c + b c
\end{matrix}</math>
\end{matrix}</math>
| (38)
|
|}
|}
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
|-
|-
| [[Image:Majority Function Example Proof 1 Title.png|500px]]
| [[Image:Two-Thirds Majority Eq 1 Pf 1 Banner Title.png|500px]]
|-
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 1.jpg|500px]]
| [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 1.png|500px]]
|-
|-
| [[Image:Equational Inference Reflect ab, ac, bc.png|500px]]
| [[Image:Equational Inference Bar Reflect ab, ac, bc.png|500px]]
|-
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 2.jpg|500px]]
| [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 2.png|500px]]
|-
|-
| [[Image:Equational Inference Distribute (abc).png|500px]]
| [[Image:Equational Inference Bar Distribute (abc).png|500px]]
|-
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 3.jpg|500px]]
| [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 3.png|500px]]
|-
|-
| [[Image:Equational Inference Collect ab, ac, bc.png|500px]]
| [[Image:Equational Inference Bar Collect ab, ac, bc.png|500px]]
|-
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 4.jpg|500px]]
| [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 4.png|500px]]
|-
|-
| [[Image:Equational Inference Quit (a), (b), (c).png|500px]]
| [[Image:Equational Inference Bar Quit (a), (b), (c).png|500px]]
|-
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 5.jpg|500px]]
| [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 5.png|500px]]
|-
|-
| [[Image:Equational Inference Cancel (( )).png|500px]]
| [[Image:Equational Inference Bar Cancel (( )).png|500px]]
|-
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 6.jpg|500px]]
| [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 6.png|500px]]
|-
|-
| [[Image:Equational Inference Weed ab, ac, bc.png|500px]]
| [[Image:Equational Inference Bar Weed ab, ac, bc.png|500px]]
|-
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 7.jpg|500px]]
| [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 7.png|500px]]
|-
|-
| [[Image:Equational Inference Delete a, b, c.png|500px]]
| [[Image:Equational Inference Bar Delete a, b, c.png|500px]]
|-
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 8.jpg|500px]]
| [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 8.png|500px]]
|-
|-
| [[Image:Equational Inference Cancel (( )).png|500px]]
| [[Image:Equational Inference Bar Cancel (( )).png|500px]]
|-
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 9.jpg|500px]]
| [[Image:Two-Thirds Majority 2.0 Eq 1 Pf 1 Storyboard 9.png|500px]]
|-
|-
| [[Image:Equational Inference QED.png|500px]]
| [[Image:Equational Inference Banner QED.png|500px]]
|}
|}
| (39)
| (39)
==Bibliography==
==Bibliography==
* [[Gottfried Leibniz|Leibniz, G.W.]] (1679–1686 ?), "Addenda to the Specimen of the Universal Calculus", pp. 40–46 in G.H.R. Parkinson (ed. and trans., 1966), ''Leibniz : Logical Papers'', Oxford University Press, London, UK.
* Leibniz, G.W. (1679–1686 ?), “Addenda to the Specimen of the Universal Calculus”, pp. 40–46 in G.H.R. Parkinson (ed. and trans., 1966), ''Leibniz : Logical Papers'', Oxford University Press, London, UK.
* Peirce, C.S. (''c.'' 1886), "Qualitative Logic", MS 736. Published as pp. 101–115 in Carolyn Eisele (ed., 1976), ''The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy'', Mouton, The Hague.
* Peirce, C.S. (1931–1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as (CP volume.paragraph).
* Peirce, C.S. (1886 a), "Qualitative Logic", MS 582. Published as pp. 323–371 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884–1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993.
* Peirce, C.S. (1981–), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN. Cited as (CE volume, page).
* Peirce, C.S. (1886 b), "The Logic of Relatives : Qualitative and Quantitative", MS 584. Published as pp. 372–378 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884–1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993.
* Peirce, C.S. (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, ''American Journal of Mathematics'' 7 (1885), 180–202. Reprinted as CP 3.359–403 and CE 5, 162–190.
* [[George Spencer Brown|Spencer Brown, George]] (1969), ''[[Laws of Form]]'', George Allen and Unwin, London, UK.
* Peirce, C.S. (''c.'' 1886), “Qualitative Logic”, MS 736. Published as pp. 101–115 in Carolyn Eisele (ed., 1976), ''The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy'', Mouton, The Hague.
* Peirce, C.S. (1886 a), “Qualitative Logic”, MS 582. Published as pp. 323–371 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884–1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993.
* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, “Propositional Equation Reasoning Systems”]
* Peirce, C.S. (1886 b), “The Logic of Relatives : Qualitative and Quantitative”, MS 584. Published as pp. 372–378 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884–1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993.
* Spencer Brown, George (1969), ''Laws of Form'', George Allen and Unwin, London, UK.
* [[Ampheck]]
==Resources==
==Resources==
* [http://planetmath.org/ PlanetMath]
* [http://planetmath.org/ PlanetMath]
** [http://planetmath.org/encyclopedia/LogicalGraph.html Logical Graph : Introduction]
** [http://planetmath.org/LogicalGraphIntroduction Logical Graph : Introduction]
** [http://planetmath.org/encyclopedia/LogicalGraphFormalDevelopment.html Logical Graph : Formal Development]
** [http://planetmath.org/LogicalGraphFormalDevelopment Logical Graph : Formal Development]
* Bergman and Paavola (eds.), [http://www.helsinki.fi/science/commens/dictionary.html Commens Dictionary of Peirce's Terms]
* Bergman and Paavola (eds.), [http://www.helsinki.fi/science/commens/dictionary.html Commens Dictionary of Peirce's Terms]
* [http://dr-dau.net/index.shtml Dau, Frithjof]
* [http://dr-dau.net/index.shtml Dau, Frithjof]
** [http://dr-dau.net/eg_readings.shtml Peirce's Existential Graphs : Readings and Links]
** [http://dr-dau.net/eg_readings.shtml Peirce's Existential Graphs : Readings and Links]
** [http://dr-dau.net/pc.shtml Existential Graphs as Moving Pictures of Thought] — Computer Animated Proof of Leibniz's Praeclarum Theorema
** [http://web.archive.org/web/20070706192257/http://dr-dau.net/pc.shtml Existential Graphs as Moving Pictures of Thought] — Computer Animated Proof of Leibniz's Praeclarum Theorema
* [http://www.math.uic.edu/~kauffman/ Kauffman, Louis H.]
* [http://www.math.uic.edu/~kauffman/ Kauffman, Louis H.]
* [http://pt.wikipedia.org/wiki/Grafo_l%C3%B3gico Grafo lógico], [http://pt.wikipedia.org/ Portuguese Wikipedia].
* [http://pt.wikipedia.org/wiki/Grafo_l%C3%B3gico Grafo lógico], [http://pt.wikipedia.org/ Portuguese Wikipedia].
==Document history==
==Syllabus==
===Focal nodes===
* [[Inquiry Live]]
* [[Logic Live]]
===Peer nodes===
* [http://intersci.ss.uci.edu/wiki/index.php/Logical_graph Logical Graph @ InterSciWiki]
* [http://mywikibiz.com/Logical_graph Logical Graph @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Logical_graph Logical Graph @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Logical_graph Logical Graph @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Logical_graph Logical Graph @ Wikiversity Beta]
===Logical operators===
{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}
===Related topics===
{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}
===Relational concepts===
{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}
===Information, Inquiry===
{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}
===Related articles===
{{col-begin}}
{{col-begin}}
{{col-break}}
{{col-break}}
* [http://mywikibiz.com/Logical_graph Logical Graph], [http://mywikibiz.com/ MyWikiBiz]
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://beta.wikiversity.org/wiki/Logical_graph Logical Graph], [http://beta.wikiversity.org/ Beta Wikiversity]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://www.proofwiki.org/wiki/Definition:Logical_Graph Logical Graph], [http://www.proofwiki.org/ ProofWiki]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
{{col-break}}
* [http://knol.google.com/k/logical-graphs-1 Logical Graphs : 1], [http://knol.google.com/ Google Knol]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://knol.google.com/k/logical-graphs-2 Logical Graphs : 2], [http://knol.google.com/ Google Knol]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://www.getwiki.net/-Logical_Graph Logical Graph], [http://www.getwiki.net/ GetWiki]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
{{col-break}}
* [http://www.textop.org/wiki/index.php?title=Logical_graph Logical Graph], [http://www.textop.org/wiki/ Textop Wiki]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://www.wikinfo.org/index.php/Logical_graph Logical Graph], [http://www.wikinfo.org/ Wikinfo]
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://en.wikipedia.org/wiki/Logical_graph Logical Graph], [http://en.wikipedia.org/ Wikipedia]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}
{{col-end}}
==Document history==
Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.
* [http://intersci.ss.uci.edu/wiki/index.php/Logical_graph Logical Graph], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Logical_graph Logical Graph], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/LogicalGraphIntroduction Logical Graph 1], [http://planetmath.org/ PlanetMath]
* [http://planetmath.org/LogicalGraphFormalDevelopment Logical Graph 2], [http://planetmath.org/ PlanetMath]
* [http://semanticweb.org/wiki/Logical_graph Logical Graph], [http://semanticweb.org/ Semantic Web]
* [http://wikinfo.org/w/index.php/Logical_graph Logical Graph], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Logical_graph Logical Graph], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Logical_graph Logical Graph], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Logical_graph&oldid=67277491 Logical Graph], [http://en.wikipedia.org/ Wikipedia]
[[Category:Artificial Intelligence]]
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