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For example, consider the axiom or initial equation that is shown below:
For example, consider the axiom or initial equation that is shown below:
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| [[Image:Logical_Graph_Figure_3_Visible_Frame.jpg|500px]] || (3)
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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>(~(~)~)~=</math> ” or set off in a text display:
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 graph]]s on the plane-embedded graphs shown above, we get the following composite picture:
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| [[Image:Logical_Graph_Figure_4_Visible_Frame.jpg|500px]] || (4)
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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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| [[Image:Logical_Graph_Figure_5_Visible_Frame.jpg|500px]] || (5)
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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 graph]]s, 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>(\!</math>" or "<math>)\!</math>", that we happen to encounter in our travels.
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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 ''[[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.
First the plane-embedded maps:
First the plane-embedded maps:
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Next the plane maps and their dual trees superimposed:
Next the plane maps and their dual trees superimposed:
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| [[Image:Logical_Graph_Figure_8_Visible_Frame.jpg|500px]] || (8)
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Finally the dual trees by themselves:
Finally the dual trees by themselves:
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And here are the parse trees with their traversal strings indicated:
And here are the parse trees with their traversal strings indicated:
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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 graph]]s'' and ''[[existential graph]]s''.
At the next level of concreteness, a pointer→record data structure can be represented as follows:
At the next level of concreteness, a pointer→record data structure can be represented as follows:
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| [[Image:Logical_Graph_Figure_11_Visible_Frame.jpg|500px]] || (11)
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This portrays <math>index_0\!</math> as the address of a record that contains the following data:
This portrays <math>index_0\!</math> as the address of a record that contains the following data:
What makes it possible to represent graph-theoretical structures as data structures in computer memory is the fact that an address is just another datum, and so we may have a state of affairs like the following:
What makes it possible to represent graph-theoretical structures as data structures in computer memory is the fact that an address is just another datum, and so we may have a state of affairs like the following:
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Returning to the abstract level, it takes three nodes to represent the three data records illustrated above: one root node connected to a couple of adjacent nodes. The items of data that do not point any further up the tree are then treated as labels on the record-nodes where they reside, as shown below:
Returning to the abstract level, it takes three nodes to represent the three data records illustrated above: one root node connected to a couple of adjacent nodes. The items of data that do not point any further up the tree are then treated as labels on the record-nodes where they reside, as shown below:
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Notice that drawing the arrows is optional with rooted trees like these, since singling out a unique node as the root induces a unique orientation on all the edges of the tree, with ''up'' being the same direction as ''away from the root''.
Notice that drawing the arrows is optional with rooted trees like these, since singling out a unique node as the root induces a unique orientation on all the edges of the tree, with ''up'' being the same direction as ''away from the root''.
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.
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| [[Image:Logical_Graph_Figure_14_Banner.jpg|500px]] || (14)
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| [[Image:Logical_Graph_Figure_15_Banner.jpg|500px]] || (15)
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Let <math>S\!</math> be the set of rooted trees and let <math>S_0 \subset S</math> be the 2-element subset consisting of a rooted node and a rooted edge. We may express these definitions more briefly as <math>S = \{ \operatorname{rooted~trees} \}</math> and <math>S_0 = \{ \ominus, \vert \}.</math> Simple intuition, or a simple inductive proof, will assure us that any rooted tree can be reduced by means of the axioms of the primary arithmetic either to a root node <math>\ominus</math> or else to a rooted edge <math>\vert\,.</math>
Let <math>S\!</math> be the set of rooted trees and let <math>S_0 \subset S</math> be the 2-element subset consisting of a rooted node and a rooted edge. We may express these definitions more briefly as <math>S = \{ \operatorname{rooted~trees} \}</math> and <math>S_0 = \{ \ominus, \vert \}.</math> Simple intuition, or a simple inductive proof, will assure us that any rooted tree can be reduced by means of the axioms of the primary arithmetic either to a root node <math>\ominus</math> or else to a rooted edge <math>\vert\,.</math>
For example, consider the reduction that proceeds as follows:
For example, consider the reduction that proceeds as follows:
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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.
Our observer might think to summarize the results of many such observations by introducing a label or variable to signify any shape of branch whatever, writing something like the following:
Our observer might think to summarize the results of many such observations by introducing a label or variable to signify any shape of branch whatever, writing something like the following:
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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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| [[Image:Logical_Graph_Figure_19.jpg|500px]] || (19)
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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, ''a''( ) = ( ), 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, ''a''( ) = ( ), as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms.