Logical graph - Revision history

Jon Awbrey: spacing in TeX $a \texttt{( )} = \texttt{( )},~\!$

2015-11-06T14:26:22Z

spacing in TeX <math>a \texttt{( )} = \texttt{( )},~\!</math>

Jon Awbrey: /* Axioms */ sub $=\!=\!=\!=\!=\!=$ for $\overline{\underline{~~~~~~}}\!$

2015-11-06T14:21:59Z

Axioms: sub <math>=\!=\!=\!=\!=\!=</math> for <math>\overline{\underline{~~~~~~}}\!</math>

Jon Awbrey: /* Duality : logical and topological */ redo spaces -- TeX "~" doesn't work here anymore

2015-11-06T14:08:05Z

Duality : logical and topological: redo spaces -- TeX "~" doesn't work here anymore

Jon Awbrey: update

2015-11-06T13:56:03Z

update

Jon Awbrey: update

2013-12-01T16:24:46Z

update

Show changes

Jon Awbrey: + Category:Charles Sanders Peirce

2011-10-29T18:40:09Z

+ Category:Charles Sanders Peirce

Jon Awbrey: Revert Vandalism By User:BillieKelly

2011-04-28T11:44:08Z

Revert Vandalism By User:BillieKelly

BillieKelly: /* Duality : logical and topological */

2011-04-28T08:54:42Z

Duality : logical and topological

BillieKelly: /* Two-thirds majority function */

2011-04-28T08:53:50Z

Two-thirds majority function

BillieKelly: /* Two-thirds majority function */

2011-04-28T08:52:28Z

Two-thirds majority function

@@ Line 211: / Line 211: @@
 |}
-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.

@@ Line 277: / Line 277: @@
 |}
-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{13:56, 6 November 2015 (UTC)~}}\!</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===

@@ Line 45: / Line 45: @@
 |}
-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"
-| width="33%" | <math>\texttt{(~(~)~)}\!</math>
+| width="33%" | <math>\texttt{( ( ) )}\!</math>
 | width="34%" | <math>=\!</math>
 | width="33%" | &nbsp;
@@ Line 79: / Line 79: @@
 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:

@@ Line 277: / Line 277: @@
 |}
-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>\overline{\underline{13:56, 6 November 2015 (UTC)~}}\!</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===
@@ Line 713: / Line 713: @@
 ===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]
-{{col-break}}
+* [http://ref.subwiki.org/wiki/Logical_graph Logical Graph @ Subject Wikis]
 * [http://beta.wikiversity.org/wiki/Logical_graph Logical Graph @ Wikiversity Beta]
 ===Logical operators===
@@ Line 827: / Line 820: @@
 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.
-{{col-begin}}
+* [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://mathweb.org/wiki/Logical_graph Logical Graph], [http://mathweb.org/wiki/ MathWeb Wiki]
+* [http://planetmath.org/LogicalGraphIntroduction Logical Graph 1], [http://planetmath.org/ PlanetMath]
 * [http://semanticweb.org/wiki/Logical_graph Logical Graph], [http://semanticweb.org/ Semantic Web]
-{{col-break}}
+* [http://wikinfo.org/w/index.php/Logical_graph Logical Graph], [http://wikinfo.org/w/ Wikinfo]
-* [http://planetmath.org/LogicalGraphIntroduction Logical Graph 1], [http://planetmath.org/LogicalGraphFormalDevelopment Logical Graph 2], [http://planetmath.org/ PlanetMath]
+* [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]]

@@ Line 866: / Line 866: @@
 [[Category:Peer Educational Resource]]
 [[Category:Artificial Intelligence]]
 [[Category:Combinatorics]]
 [[Category:Computer Science]]

@@ Line 71: / Line 71: @@
 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.
-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 paren[http://www.bestdissertation.com/services/thesis.html thesis] 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.
 {| align="center" cellpadding="10"

@@ Line 606: / Line 606: @@
 |}
-The required [http://www.bestdissertation.com/services/thesis.html thesis] can be proven in the medium of logical graphs as shown in the following Figure.
+The required equation can be proven in the medium of logical graphs as shown in the following Figure.
 {| align="center" cellpadding="8"

Logical graph - Revision history

Jon Awbrey: spacing in TeX a \texttt{( )} = \texttt{( )},~\!

Jon Awbrey: /* Axioms */ sub =\!=\!=\!=\!=\!= for \overline{\underline{~~~~~~}}\!

Jon Awbrey: /* Duality : logical and topological */ redo spaces -- TeX "~" doesn't work here anymore

Jon Awbrey: update

Jon Awbrey: update

Jon Awbrey: + Category:Charles Sanders Peirce

Jon Awbrey: Revert Vandalism By User:BillieKelly

BillieKelly: /* Duality : logical and topological */

BillieKelly: /* Two-thirds majority function */

BillieKelly: /* Two-thirds majority function */

Jon Awbrey: spacing in TeX $a \texttt{( )} = \texttt{( )},~\!$

Jon Awbrey: /* Axioms */ sub $=\!=\!=\!=\!=\!=$ for $\overline{\underline{~~~~~~}}\!$