Discussion of the topic continues at Logical Graphs • Formal Development.
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Speaking of algebra, and having just encountered 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.
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 we noticed first, “a ( ) = ( )”, 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 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 it is of course 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 both their motive and their justification from the observation and summarization of patterns which are visible in the arithmetic spectrum.
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Experience teaches that complex objects are best approached in a gradual, laminar, modular fashion, one step, one layer, one piece at a time, especially when that complexity is irreducible, when all our articulations and all our representations will be cloven at joints disjoint from the structure of the object itself, 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, instanced here by the primary arithmetic, a level of formal structure Peirce verged on intuiting at numerous points and times in his work on logical graphs but Spencer Brown named and brought more completely to life.
Another reason for lingering a while longer in these primitive forests is that an acquaintance with “bare trees”, those unadorned with literal or numerical labels, will provide a basis for understanding what’s really at issue in oft‑debated questions about the “ontological status of variables”.
It is probably best to illustrate the theme in the setting of a concrete case. To do that let’s look again at the previous example of reductive evaluation taking place in the primary arithmetic.
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After we’ve seen a few sign-transformations of roughly that shape we’ll most likely notice it doesn’t really matter what other branches are rooted next to the lone edge off to the side — the end result will always be the same. Eventually it will occur to us 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 and germinated by their summarizations, are the root of all algebra.
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Let 
For example, consider the reduction which 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 an interpretant of its predecessor, ending in a final sign which 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, the sequence exhibits the main features of any computation, namely, a semiotic process proceeding from an obscure sign to a clear sign of the same object, being in its aim and effect an action on behalf of clarification.
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The axioms of the primary arithmetic are shown below, as they appear in both graph and string forms, along with pairs of names which come in handy for referring to the two opposing directions of applying the axioms.
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This much preparation allows us to take up the founding axioms or initial equations which determine the entire system of logical graphs.
Though it may not seem too exciting, logically speaking, there are many reasons to make oneself at home with the system of forms represented indifferently, topologically speaking, by rooted trees, balanced strings of parentheses, and finite sets of non‑intersecting simple closed curves in the plane.
This space of forms, along with the pair of axioms which divide it into two formal equivalence classes, is what Spencer Brown called the primary arithmetic.
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At the next level of concreteness, a pointer‑record data structure can be represented as follows.
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This portrays index0 as the address of a record which contains the following data.
datum1, datum2, datum3, …, and so on.
What makes it possible to represent graph‑theoretic forms as dynamic data structures is the fact that an address is just another datum to be stored on a record, 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. Items of data not pointing any further up the tree are 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 direction away from the root.
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The parse graphs we’ve been looking at so far bring us one step closer to the pointer graphs it takes to make the above types of maps and trees live in computer memory but they are still a couple of steps too abstract to detail the concrete species of dynamic data structures we need. The time has come to flesh out the skeletons we have drawn up to this point.
Nodes in a graph represent records in computer memory. A record is a collection of data conceived to reside at a specific address. The address of a record is analogous to a demonstrative pronoun, a word like this or that, on which account programmers call it a pointer and semioticians recognize it as a type of sign called an index.
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We have now treated in some detail various forms of the axiom or initial equation which is formulated in string form as “( ( ) ) = ”. For the sake of comparison, let’s record the planar and dual forms of the axiom which is formulated in string form as “( )( ) = ( )”.
First the plane-embedded maps:
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Next the plane maps and their dual trees superimposed:
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Finally the rooted trees by themselves:
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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 we may call them, which bear on the use of logical graphs in their various incarnations, for example, those Peirce described as entitative graphs and existential graphs.
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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.
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Inquiry Into Inquiry 