Inquiry Into Inquiry

Primary Arithmetic as Semiotic System (concl.)

Let be the set of rooted trees and let be the 2‑element subset consisting of a rooted node and a rooted edge.  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 to either a root node or a rooted edge.

For example, consider the reduction which proceeds as follows.

(16)

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.

Resources

• Logical Graphs
• Survey of Animated Logical Graphs

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

Primary Arithmetic as Semiotic System (cont.)

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.

Resources

• Logical Graphs
• Survey of Animated Logical Graphs

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Quick Tour of the Neighborhood

This much preparation allows us to take up the founding axioms or initial equations which determine the entire system of logical graphs.

Primary Arithmetic as Semiotic System

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.

• For one thing it gives us a non‑trivial example of a sign domain on which to cut our semiotic teeth, non‑trivial in the sense that it contains a countable infinity of signs.
• In addition it allows us to study a simple form of computation recognizable as a species of semiosis or sign‑transforming process.

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.

Resources

• Logical Graphs
• Survey of Animated Logical Graphs

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Computational Representation (concl.)

At the next level of concreteness, a pointer‑record data structure can be represented as follows.

(11)

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.

(12)

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.

(13)

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.

Resources

• Logical Graphs
• Survey of Animated Logical Graphs

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Computational Representation

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.

Resources

• Logical Graphs
• Survey of Animated Logical Graphs

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Duality : Logical and Topological (concl.)

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:

(7)

Next the plane maps and their dual trees superimposed:

(8)

Finally the rooted trees by themselves:

(9)

And here are the parse trees with their traversal strings indicated:

(10)

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.

Resources

• Logical Graphs
• Survey of Animated Logical Graphs

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

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.

(6)

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

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Duality : Logical and Topological (cont.)

Last time we took up the axiom or initial equation shown below.

(3)

We noted it could be written inline as “( ( ) ) =    ” or set off in the following text display.

( ( ) ) =    

When we turn to representing the corresponding expressions in computer memory, where they can be manipulated with the greatest of ease, we begin by transforming the planar graphs into their topological duals.  Planar regions of the original graph become nodes or points of the dual graph and boundaries between planar regions of the original graph become edges or lines between nodes of the dual graph.

For example, overlaying the corresponding dual graphs on the plane‑embedded graphs shown above, we get the following composite picture.

(4)

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 matrix, we get the following equation.

(5)

Resources

• Logical Graphs
• Survey of Animated Logical Graphs

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Duality : Logical and Topological

In using logical graphs there are two types of duality to consider, logical duality and topological duality.

Graphs of the order Peirce considered, those embedded in a continuous manifold like that represented by a plane sheet of paper, can be represented in linear text by what are called 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.

For example, consider the axiom or initial equation shown below.

(3)

That can be written inline as “( ( ) ) =    ” or set off in the following text display.

( ( ) ) =    

Resources

• Logical Graphs
• Survey of Animated Logical Graphs

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Abstract Point of View

The bird’s eye view in question is more formally known as the perspective of formal equivalence, from which remove one overlooks many distinctions which appear momentous in more concrete settings.  Expressions inscribed in different formalisms whose syntactic structures are algebraically or topologically isomorphic are not recognized as being different from each other in any significant sense.  An eye to historical detail will note in passing that C.S. Peirce used a streamer‑cross symbol where Spencer Brown used a carpenter’s square marker to roughly the same formal purpose, to give just one example, but the main theme of interest at the level of pure form is indifferent to variations of that order.

In Lieu of a Beginning

We may start by contemplating the following two formal equations.

	(1)
	(2)

Resources

• Logical Graphs
• Survey of Animated Logical Graphs

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science