Duality : Logical and Topological (cont.)

It is easy to see the relation between the parenthetical expressions of Peirce’s logical graphs, showing their contents in order of containment, and the corresponding dual graphs, forming a species of rooted trees to be described in greater detail below.

In the case of our last example, a moment’s contemplation of the following picture will lead us to see how 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 “(” or “)”, we happen to encounter in our travels.

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The above ritual is called traversing the tree, and the string read off is called the traversal string of the tree.  The reverse ritual, which 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 users of that jargon tend to use it loosely, often using parse string to mean the string whose parsing creates the associated graph.

Resources

• Logical Graphs
• Survey of Animated Logical Graphs

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cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science