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→Duality : logical and topological: TeX formats + del redundant wiki links
==Duality : logical and topological==
==Duality : logical and topological==
There are two types of [[duality (mathematics)|duality]] that have to be kept separately mind in the use of logical graphs — [[De Morgan's laws|logical duality]] and [[topology|topological]] [[dual graph|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 [[genus (mathematics)|topological genus]] — can be represented in linear text as what are called ''[[parsing|parse string]]s'' or ''[[tree traversal|traversal string]]s'' and parsed into ''[[data structure|pointer structure]]s'' 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>(~(~)~)~=</math> ” or set off in a text display:
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>(~(~)~)</math>
| width="33%" | <math>\texttt{(~(~)~)}</math>
| width="34%" | <math>=\!</math>
| width="34%" | <math>=\!</math>
| width="33%" |
| width="33%" |
|}
|}
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 [[duality (mathematics)|topological dual]]s. 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 graph]]s 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:
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
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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 graph]]s, 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 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 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>(\!</math>" or "<math>)\!</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"
{| align="center" cellpadding="10"
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This ritual is called ''[[tree traversal|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 tree|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 now treated in some detail various forms of the axiom or initial equation that is formulated in string form as " <math>((~))~=</math> ". For the sake of comparison, let's record the planar and dual forms of the axiom that is formulated in string form as "<math>(~)(~)~=~(~)</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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Next the planar maps and their dual trees superimposed:
Next the plane-embedded maps and their dual trees superimposed:
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"