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Futures Of Logical Graphs

MyWikiBiz, Author Your Legacy — Friday August 29, 2008

< Directory:Jon Awbrey | Papers(Redirected from Futures Of Logical Graphs)

This article develops an extension of Charles Sanders Peirce's Logical Graphs.

1 Introduction
2 Categories of structured individuals
3 Weed and seed theorem
4 Case analysis-synthesis theorem
5 Logic as sign transformation
6 Analyzing contingent truths
7 Proof as semiosis
8 References
9 See also
- 9.1 Related essays and projects
- 9.2 Related concepts and topics
10 Document history
- 10.1 Futures Of Logical Graphs (2005)
- 10.2 Futures Of Logical Graphs : Discussion (2005)
11 Aficionados

Introduction

I think I am finally ready to speculate on the futures of logical graphs that will be able to rise to the challenge of embodying the fundamental logical insights of Peirce.

For the sake of those who may be unfamiliar with it, let us first divert ourselves with an exposition of a standard way that graphs of the order that Peirce considered, those embedded in a continuous manifold like a plane sheet of paper, without or without the paper bridges that Peirce used to augment his genus, can be represented as parse-strings in Asciiish and sculpted into pointer-structures in computer memory.

A blank sheet of paper can be represented as a blank space in a line of text, 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 drawn in box form below:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o-----------o ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | o-------o | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | | ` ` ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | | ` ` ` | | ` ` ` = ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | | ` ` ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | o-------o | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o-----------o ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

We can write this in line as "(()) = " or set it off as follows:

(( )) =

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 graphs on the plane-embedded graphs shown above, we get the following composite picture:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o-----------o ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | o-------o | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | | ` ` ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | | ` o ` | | ` ` ` = ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | | ` | ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | o---|---o | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | ` ` o ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o-----|-----o ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` @ ` ` ` ` ` ` = ` ` ` ` @ ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

For a number of pragmatic reasons it is convenient 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, indicated in the above Figure by the amphora or at sign, "@".

Extracting the dual graph from its composite matrix, we get this picture:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` @ ` ` ` ` ` = ` ` ` ` ` @ ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

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 trees 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 particular case either "(" or ")", that we happen to encounter in our travels.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ( | ) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ( | ) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` (( )) ` ` ` = ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

This ritual is called traversing the tree, and the string read off is often 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 often called the parse graph of the string. I 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 much preparation allows us to present the two most basic axioms of logical graphs, shown in graph and string forms below, along with handy names for referring to the different directions of applying the axioms.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `( ) ( )` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Axiom I_1.` ` Distract <--- | ---> Condense ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` (( )) ` ` ` = ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Axiom I_2.` ` ` Unfold <--- | ---> Refold ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

The parse graphs that we've been looking at so far are one step toward the pointer graphs that it takes to make trees live in computer memory, but they are still a couple of steps too abstract to properly suggest in much concrete detail the species of dynamic data structures that we need. I now proceed to flesh out the skeleton that I've drawn up to this point.

Nodes in a graph depict records in computer memory. A record is a collection of data that can be thought to reside at a specific address. For semioticians, an address can be recognized as a type of index, and is commonly spoken of, on analogy with demonstrative pronouns, as a pointer, even among computer programmers who are otherwise innocent of semiotics.

At the next level of concretion, a record/node can be represented as follows:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o-----------------------------o ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | datum_1 datum_2 datum_3 ... | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o-----------------------------o ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ^ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | index_0 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

This depicts the circumstance that index₀ is the address of the record in question, which record contains the data: datum₁, datum₂, datum₃, …, and so on.

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 can have a circumstance like this:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o-----o o-----o ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ... | | ... | ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o-----o o-----o ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ^ ` ` ` ^ ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o---------------------|-------|-----------o ` ` ` `
` ` ` ` ` | datum_1 datum_2 ... index_1 index_2 ... | ` ` ` `
` ` ` ` ` o-----------------------------------------o ` ` ` `
` ` ` ` ` ^ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | index_0 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Back at the abstract level, it takes three nodes to represent the three data records, with a root node connected to two other nodes. The ordinary bits of data are then treated as labels on the nodes:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` |/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` @ datum_1 datum_2 ... ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Notice that, with rooted trees like these, drawing the arrows is optional, since singling out a unique node as the root induces a unique orientation on all the edges of the tree, up being the same as away from the root.

We have already seen various forms of the axiom that is formulated in string form as "(( )) = ". For the sake of comparison, let's record the planar and dual forms of the axiom that is formulated in string form as "( )( ) = ( )".

First the planar form:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` o-------o ` ` ` o-------o ` ` ` ` ` ` ` o-------o ` ` `
` ` ` | ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` ` ` | ` ` ` | ` ` `
` ` ` | ` ` ` | ` ` ` | ` ` ` | ` ` ` = ` ` ` | ` ` ` | ` ` `
` ` ` | ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` ` ` | ` ` ` | ` ` `
` ` ` o-------o ` ` ` o-------o ` ` ` ` ` ` ` o-------o ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Next the planar and dual forms superimposed:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` o-------o ` ` ` o-------o ` ` ` ` ` ` ` o-------o ` ` `
` ` ` | ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` ` ` | ` ` ` | ` ` `
` ` ` | ` o ` | ` ` ` | ` o ` | ` ` ` = ` ` ` | ` o ` | ` ` `
` ` ` | ` `\` | ` ` ` | `/` ` | ` ` ` ` ` ` ` | ` | ` | ` ` `
` ` ` o-----\-o ` ` ` o-/-----o ` ` ` ` ` ` ` o---|---o ` ` `
` ` ` ` ` ` `\` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` = ` ` ` ` ` @ ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Finally the dual form by itself:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` `
` ` ` ` ` `\` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` \ ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` `\` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` = ` ` ` ` ` @ ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Categories of structured individuals

We have at this point enough material to begin thinking about the forms of analogy, iconicity, metaphor, morphism, whatever you want to call it, that are pertinent to the use of logical graphs in their various logical interpretations, for instance, those that Peirce described as entitative and existential.

By way of providing a conceptual-technical framework for organizing that discussion, let me introduce the concept of a category of structured individuals (COSI). There may be some cause for some readers to rankle at the very idea of a structured individual, for taking the notion of an individual in its strictest etymology would render it absurd that an atom could have parts, but all we mean by individual in this context is an individual by dint of some conversational convention currently in play, not an individual on account of its intrinsic indivisibility. Incidentally, though, it will also be convenient to take in the case of a class or collection of individuals with no pertinent inner structure as a trivial case of a COSI.

For some reason — that I won't stop to examine right now — I tend to think category of structured individuals when the individuals in question have a whole lot of structure, but collection of structured items when the individuals have a minimal amount of internal structure. For example, any set is a COSI, so any relation in extension is a COSI, but a 1-adic relation is just a set of 1-tuples, that are in some lights indiscernible from their single components, and so its structured individuals have far less structure than the k-tuples of k-adic relations, when k exceeds one. This spectrum of differentiations among relational models will be useful to bear in mind when the time comes to say what distinguishes relational thinking proper from 1-adic and 2-adic thinking, that constitute its degenerate cases.

Still on our way to saying what brands of iconicity are worth buying, at least when it comes to graphical systems of logic, it will useful to introduce one more distinction that affects the types of mappings that can be formed between two COSI's.

One type of structure-preserving map is a system-wide iconic map (SWIM). The other is a type that we might call a pointwise-restricted iconic map, or a pointedly rigid iconic map (PRIM). I tried to make this nomenclature as self-explanatory as I could, but failing that I will explain it next time.

Because I plan this time around a somewhat leisurely excursion through the primordial wilds of logics that were so intrepidly explored by C.S. Peirce and again in recent times revisited by George Spencer Brown, let me just give a few extra pointers to those who wish to run on ahead of this torturous tortoise pace:

Jon Awbrey, "Propositional Equation Reasoning Systems (PERS)", Eprint.
Lou Kauffman, "Box Algebra, Boundary Mathematics, Logic, and Laws of Form", Eprint.

Two paces back I used the word "category" in a way that will turn out to be not too remote a cousin of its present day mathematical bearing, but also in way that's not unrelated to Peirce's theory of categories.

When I call to mind a category of structured individuals (COSI), I get a picture of a certain form, with blanks to be filled in as the thought of it develops, that can be sketched at first like so:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` Category` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/`\` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` \ ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` `\` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `
` Individuals ` ` ` ` ` ` o ` `...` ` o ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` `/`\` ` ` ` `/`\` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` / ` \ ` ` ` / ` \ ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` `/` ` `\` ` `/` ` `\` ` ` ` ` ` ` ` ` `
` Structures` ` ` ` ` o->-o->-o ` o->-o->-o ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

The various glyphs of this picturesque hierarchy serve to remind us that a COSI in general consists of many individuals, which in spite of their calling as such may have specific structures involving the ordering of their component parts. Of course, this generic picture may have degenerate realizations, as when we have a 1-adic relation, that may be viewed in most settings as nothing different than a set:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` Category` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/`\` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` \ ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` `\` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `
` Individuals ` ` ` ` ` ` o ` `...` ` o ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `
` Structures` ` ` ` ` ` ` o ` `...` ` o ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

The practical use of Peirce's categories is simply to organize our thoughts about what sorts of formal models are demanded by a material situation, for instance, a domain of phenomena from atoms to biology to culture. To say that "k-ness" is involved in a phenomenon is simply to say that we need k-adic relations to model it adequately, and that the phenomenon itself appears to demand nothing less. Aside from this, Peirce's realization that k-ness for k = 1, 2, 3 affords us with a sufficient basis for all that we need to model is a formal fact that depends on a particular theorem in the logic of relatives. If it weren't for that, there would hardly be any reason to single out three.

In order to discuss the various forms of iconicity that might be involved in the application of Peirce's logical graphs and their kind to the object domain of logic itself, we will need to bring out two or three "categories of structured individuals" (COSI's), depending on how we count. These are called the "object domain", the "sign domain", and the "interpretant sign domain", which may be written O, S, I, respectively, or X, Y, Z, respectively, depending on the style that fits the current frame of discussion.

For the time being, I will be considering systems where the sign domain and the interpretant domain are the same sets of entities, although, of course, their roles in a given "sign relation", say, L ⊆ O × S × I or L ⊆ X × Y × Z, remain as distinct as ever. I will tend to use the term "semiotic domain" for the common set of elements that constitute the signs and the interpretant signs in any setting where the sign domain and the interpretant domain are equal as sets.

At the "alpha level", "primary arithmetic", or "zeroth order" of consideration that we have so far introduced, the sign domain is any one of the several "formal languages" that we have placed in one-to-one correspondence with each other, namely, the languages of non-intersecting plane closed curves, well-formed parenthesis strings, or rooted trees. The interpretant sign domain will for the present be taken to be any one of the same languages, and so I'll refer to any of them indifferently as the "semiotic domain".

Briefly if roughly put, icons are signs that denote their objects by virtue of sharing properties with them. To put it a bit more fully, icons are signs that receive their interpretant signs on account of having specific properties in common with their objects.

The family of related relationships that fall under the headings of analogy, icon, metaphor, model, simile, simulation, and so on forms an extremely important complex of ideas in mathematics, there being recognized under the generic idea of "structure-preserving mappings" and commonly formalized in the language of homomorphisms, morphisms, or "arrows", depending on the operative level of abstraction that's in play.

To consider how a system of logical graphs, taken together as a semiotic domain, might bear an iconic relationship to a system of logical objects that make up our object domain, we will next need to consider what our logical objects are.

A popular answer, if by popular one means that both Peirce and Frege agreed on it, is to say that our ultimate logical objects are without loss of generality most conveniently referred to as Truth and Falsity. If nothing else, it serves the end of beginning simply to go along with this thought for a while, and so we can start with an object domain that consists of just two objects or "values", to wit, O = B = {false, true}.

Given those two categories of structured individuals, namely, O = B = {false, true} and S = {logical graphs}, the next task is to consider the brands of morphisms from S to O that we might reasonably have in mind when we speak of the "arrows of interpretation".

With the aim of embedding our consideration of logical graphs, as seems most fitting, within Peirce's theory of triadic sign relations, we have declared the first layers of our object, sign, and interpretant domains. As we often do in formal studies, we've taken the sign and interpretant domains to be the same set, S = I, calling it the semiotic domain, or, as I see that I've done in some other notes, the syntactic domain.

Truth and Falsity, the objects that we've so far declared, are recognizable as abstract objects, and like so many other hypostatic abstractions that we use they have their use in moderating between a veritable profusion of more concrete objects and more concrete signs, in factoring complexity as some people say, despite the fact that some complexities are irreducible in fact.

That much of a stake in the ground will have to do as a philosophical tether for now, since we are about to play out the syntactic line just about as far as we can stretch it, and it can happen that some will forget this home port.

As agents of systems, whether that system is our own physiology or our own society, we move through what we commonly imagine to be a continuous manifold of states, but with distinctions being drawn in that space that are every bit as compelling to us, and often quite literally, as the difference between life and death. So the relation of discretion to continuity is not one of those issues that we can take lightly, or simply dissolve by choosing a side and ignoring the other, as we may imagine in abstraction. I'll try to get back to this point later, one in a long list of cautionary notes that experience tells me has to be attached to every tale of our pilgrimage, but for now we must get under way.

Returning to En and Ex, the two most popular interpretations of logical graphs, that happen to be dual to each other in a certain sense, let's see how they fly as "hermeneutic arrows" from the syntactical domain S to the objective domain O, at any rate, as their trajectories can be spied in the radar of what George Spencer Brown called the "primary arithmetic".

Taking En and Ex as arrows of the form En, Ex : S → O, at the level of arithmetic taking S = {rooted trees} and O = {Falsity, Truth}, let's factor each arrow across the domain of formal constants S₀ = {O, |}, the domain that consists of a single rooted node plus a single rooted edge. As a strategic tactic, this allows each arrow to be broken into a purely syntactic part En_syn, Ex_syn : S → S₀ and its purely semantic part En_sem, Ex_sem : S₀ → O.

As things work out, the syntactic factors are formally the same, leaving our dualing interpretations to differ in their semantic components alone. Specifically, we have the following mappings:

En_sem : @ ~> false, | ~> true

Ex_sem : @ ~> true, | ~> false

On the other side of the ledger, because the syntactic factors, En_syn and Ex_syn, are indiscernible from each other, there is a syntactic contribution to the overall interpretation process that can be most readily investigated on purely formal grounds. That will be the task to face when next we meet on these lists.

Cast into the form of a 3-adic sign relation, the situation before us can now be given the following shape:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` Y ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `Semiotic Domain` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o-------------------o ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` Rooted  Trees ` | ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o-------------------o ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` Syntactic ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` Reduction ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` v ` ` ` ` ` ` ` ` `
` ` ` ` o-----------o ` ` En_sem` ` o-----------o ` ` ` ` ` `
` ` ` ` | ` ` ` ` ` |<--------------| ` ` ` o ` | ` ` ` ` ` `
` ` ` ` | ` F ` T ` | ` ` ` ` ` ` ` | ` O ` | ` | ` ` ` ` ` `
` ` ` ` | ` ` ` ` ` |<--------------| ` ` ` @ ` | ` ` ` ` ` `
` ` ` ` o-----------o ` ` Ex_sem` ` o-----------o ` ` ` ` ` `
` ` ` ` ` ` ` X ` ` ` ` ` ` ` ` ` ` ` ` `Y_0` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` Canonical ` ` ` ` ` ` `
` ` ` ` Object Domain ` ` ` ` ` ` ` `Sign Domain` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

The interpretation maps En, Ex : Y → X are factored into a shared syntactic part:

En_syn = Ex_syn = E_syn : Y → Y₀

and a couple of differential semantic parts:

En_sem, Ex_sem : Y₀ → X

The functional images of the syntactic reduction map E_syn : Y → Y₀ are the two simplest signs or the most reduced pair of expressions, regarded as rooted trees taking the shapes @ and |, and these may be treated as the canonical representatives of their respective equivalence classes.

The more Peirce-systent among you, on contemplating that last picture, will 1st or 2nd or 3rd-naturally ask, "What happened to the irreducible 3-adicity of sign relations in this portrayal of logical graphs?"

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` Y ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `Semiotic Domain` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o-------------------o ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` Rooted  Trees ` | ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o-------------------o ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` Syntactic ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` Reduction ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` v ` ` ` ` ` ` ` ` `
` ` ` ` o-----------o ` ` En_sem` ` o-----------o ` ` ` ` ` `
` ` ` ` | ` ` ` ` ` |<--------------| ` ` ` o ` | ` ` ` ` ` `
` ` ` ` | ` F ` T ` | ` ` ` ` ` ` ` | ` O ` | ` | ` ` ` ` ` `
` ` ` ` | ` ` ` ` ` |<--------------| ` ` ` @ ` | ` ` ` ` ` `
` ` ` ` o-----------o ` ` Ex_sem` ` o-----------o ` ` ` ` ` `
` ` ` ` ` ` ` X ` ` ` ` ` ` ` ` ` ` ` ` `Y_0` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` Canonical ` ` ` ` ` ` `
` ` ` ` Object Domain ` ` ` ` ` ` ` `Sign Domain` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

The answer is that the last bastion of 3-adic irreducibility presidios precisely in the duality of the dual interpretations En_sem and Ex_sem. To see this, consider the consequences of there being, contrary to all that we've assumed up to this point, some ultimately compelling reason to assert that the clean slate, the empty medium, the vacuum potential, whatever one wants to call it, is inherently more meaningful of either Falsity or Truth. This would issue in a conviction forthwith that the 3-adic sign relation involved in this case decomposes as a composition of a couple of functions, that is to say, reduces to a 2-adic relation.

The duality of interpretation for logical graphs tells us that the empty medium, the tabula rasa, what Peirce called the Sheet of Assertion (SA) is a genuine symbol, not to be found among the degenerate species of signs that make up icons and indices, nor, as the SA has no parts, can it number icons or indices among its parts. What goes for the medium must go for all of the signs that it mediates. Thus we have the kinds of signs that Peirce in one place called "pure symbols", naming a selection of signs for basic logical operators specifically among them.

Every word is a symbol. Every sentence is a symbol. Every book is a symbol. Every representamen depending upon conventions is a symbol. Just as a photograph is an index having an icon incorporated into it, that is, excited in the mind by its force, so a symbol may have an icon or an index incorporated into it, that is, the active law that it is may require its interpretation to involve the calling up of an image, or a composite photograph of many images of past experiences, as ordinary common nouns and verbs do; or it may require its interpretation to refer to the actual surrounding circumstances of the occasion of its embodiment, like such words as 'that', 'this', 'I', 'you', 'which', 'here', 'now', 'yonder', etc. Or it may be pure symbol, neither 'iconic' nor 'indicative', like the words 'and', 'or', 'of', etc. (C.S. Peirce, Collected Papers, CP 4.447).

Some will recall the many animadversions that we had on this topic, starting here:

PS. Pure Symbols
- http://stderr.org/pipermail/inquiry/2005-March/thread.html#2465
- http://stderr.org/pipermail/inquiry/2005-April/thread.html#2517

PS. Pure Symbols : Discussion

And some will find an ethical principle in this freedom of interpretation. The act of interpretation bears within it an inalienable degree of freedom. In consequence of this truth, as far as the activity of interpretation goes, freedom and responsibility are the very same thing. We cannot blame objects for what we say or what we think. We cannot blame symbols for what we do. We cannot escape our response ability. We cannot escape our freedom.

Though it may not seem too exciting, logically speaking, there are many good reasons for getting comfortable with the system of forms that is represented indifferently, topologically speaking, by rooted trees, well-formed strings of parentheses, or finite sets of non-intersecting simple closed curves in the plane. One reason is that it provides us with a respectable example of a sign domain to cut our semiotic teeth on, being non-trivial in the sense that it contains a countable infinity of signs. Another reason is that it allows us to study a simple form of computation that is recognizable as a species of semiotic process.

This space of forms, along with the two axioms that result in its being partitioned into just two equivalence classes, is what George Spencer Brown called the primary arithmetic.

Here are the two arithmetic axioms:

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `( ) ( )` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Axiom I_1.` ` Distract <--- | ---> Condense ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` (( )) ` ` ` = ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Axiom I_2.` ` ` Unfold <--- | ---> Refold ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

Taking S = {rooted trees} and S₀ = {O, |}, simple intuition, or a simple inductive proof, will assure us that any rooted tree can be reduced by means of these axioms to either a root node or else a rooted edge.

For example, consider the reduction that proceeds as follows:

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` o o o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\| | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` o o o ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\|/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o===========================================================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` o o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` o o o ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\|/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o===========================================================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` o o ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` |/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o===========================================================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

Regarded as a semiotic process, this amounts to a sequence of signs, every one after the first being the interpretant of its predecessor, ending in a sign that we may regard as the canonical sign for their common object, in the upshot, the result of the computation process. Simple as it is, this exhibits the main features of all computation, specifically, a semiotic process that proceeds from an obscure sign to a clear sign of the same object, in sum, a case of clarification.

Hard experience teaches that complex objects are best approached in a gradual, laminar, modular fashion, one step, one layer, one piece at a time, and it's just as much the way when the complexity of the object is irreducible, when the articulations of the representation will necessarily be joints that are cloven disjointedly from nature, some assembly required in the synthetic integrity of our intuitions.

That's my excuse, and I'm persistent about it, for spending so much time on the first half of zeroth order logic, that is, the primary arithmetic, that C.S. Peirce verged on intuiting at numerous points and times in his work on logical graphs, and that Spencer Brown named and brought to life.

Before it slips from mind, there is one other reason for lingering a bit longer in these forests primeval, and this is that our acquaintance with bare trees, those as yet unornamented with numerous and literal labels, will repay us at later stages of the game when we come to worry, as most folks do eventually, over such problems as the ontological status of variables.

It will be best to illustrate this theme in the setting of a concrete case, which we can do by revisiting the previous example of reductive evaluation:

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` o o o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\| | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` o o o ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\|/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o===========================================================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` o o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` o o o ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\|/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o===========================================================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` o o ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` |/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |`
o===========================================================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

The observation of several semioses of roughly this shape will most probably lead an observer with any observational facility whatever to notice that it doesn't really matter what sorts of branches happen to sprout from the side of the root aside from the lone edge that also grows there — the end will all be one.

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:

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `a`/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o===========================================================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

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 seen 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 Charles S. Peirce and G. Spencer Brown, respectively.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a(a)` ` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Axiom J_1.` ` ` Insert <--- | ---> Delete ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `ab ` ac` ` ` ` ` ` ` b ` c ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` a @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `((ab)(ac)) ` ` = ` ` a((b)(c)) ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Axiom J_2.` Distribute <--- | ---> Collect` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

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 that we noticed first, 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 that 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 of course it is 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 their motive and their justice from the observation and summarization of patterns that are visible in the arithmetic spectrum.

Starting once again with the primary arithmetic, let us count the ways that this formal system might be iconic of our negative and positive logical objects, that which we'd avoid and that which we'd approach, or Falsity and Truth, respectively.

Before we go any further we need to observe that there is no fact of the matter as to whether a given sign is an icon of a given object, that is to say, in any way that fails to refer to the conduct of a given interpreter, which conduct is most conveniently and formally summed up in the fashion of an interpretant sign.

Thus, if you find youself in an argument with another interpreter who swears to the influence of some quality common to the object and the sign and that really does affect his or her conduct in regard to the two of them, then that argument is almost certainly bound to be utterly futile. I am sure we've all been there.

When I first became acquainted with the Entish and Extish hermenautics of logical graphs, back in the late great 1960's, I was struck in the spirit of those times by what I imagined to be their Zen and Zenoic sensibilities, the tao is silent wit of the Zen mind being the empty mind, that seems to go along with the Ex interpretation, and the way from the way that's marked is not the true way to the mark that's marked is not the remarkable mark and to the sign that's signed is not the significant sign of the En interpretation, reminding us that the sign is not the object, no matter how apt the image. And later, when my discovery of the cactus graph extension of logical graphs led to the leimons of neural pools, where En says that truth is an active condition, while Ex says that sooth is a quiescent mind, all these themes got reinforced more still.

We hold these truths to be self-iconic, but they come in complementary couples, in consort to the flip-side of the tao.

In light of the foregoing reflections on the forms of iconicity worth having, I will leave it to the reading tastes of the given hermenaut whether to read the uncut page as more iconic of falsity or truth. Once that choice is made, then it's perfectly natural for the chooser to think that the choice was the chosen one, and there is very little reason to become disillusioned about it.

But there are other orders of analogy, iconicity, metaphor, morphism, etc. that we need to attend to in the way that a system of signs can represent a system of objects. At the level of the primary arithmetic, this refers to the way that the distinction between falsity and truth, not the values alone, can be represented in the distinction between one sort of sign and another sort of sign.

A sort of signs is more formally known as an equivalence class (EC). There are in general many sorts of sorts of signs that we might wish to consider in this inquiry, but let's begin with the sort of signs all of whose members denote the same object as their referent, a sort of signs to be henceforth referred to as a referential equivalence class (REC).

Cf: FOLG 5. http://stderr.org/pipermail/inquiry/2005-October/003113.html
In: FOLG. http://stderr.org/pipermail/inquiry/2005-October/thread.html#3104

Toward the outset of this excursion, I mentioned the distinction between a pointwise-restricted iconic map or a pointedly rigid iconic map (PRIM) and a system-wide iconic map (SWIM). The time has come to make use of that mention.

We are once again concerned with categories of structured items (COSI's) and the categories of mappings between them, indeed, the two ideas are all but inseparable, there being many good reasons to consider the very notion of structure to be most clearly defined in terms of the brands of "arrows", maps, or morphisms between items that are admitted to the category in view.

At the level of the primary arithmetic (PAR), we have a set-up like this:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
Categories` ` ` ` `!O!` ` ` ` ` ` ` ` ` ` ` ` `!S!` ` ` ` ` `
` ` ` ` ` ` ` ` ` `/`\` ` ` ` ` ` ` ` ` ` ` ` `/`\` ` ` ` ` `
` ` ` ` ` ` ` ` ` / ` \ ` ` ` ` ` ` ` ` ` ` ` / ` \ ` ` ` ` `
` ` ` ` ` ` ` ` `/` ` `\` ` ` ` ` ` ` ` ` ` `/` ` `\` ` ` ` `
` ` ` ` ` ` ` ` / ` ` ` \ ` ` ` ` ` ` ` ` ` / ` ` ` \ ` ` ` `
` ` ` ` ` ` ` `/` ` ` ` `\` ` Denotes ` ` `/` ` ` ` `\` ` ` `
Individuals `{F}` ` ` ` `{T}` <------ ` {...} ` ` ` {...} ` `
` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` `/|\` ` ` ` `/|\` ` `
` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` / | \ ` ` ` / | \ ` `
` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` `/` | `\` ` `/` | `\` `
Structures` ` F ` ` ` ` ` T ` ` ` ` ` o ` o ` o ` o ` o ` o `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o o ` o ` ` ` ` o `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | `\`/` ` ` ` ` | `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` o ` o o ` o o `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` | ` | `\`/` | `
` ` ` ` ` ` ` F ` ` ` ` ` T ` ` ` ` ` @ ` @ ` @ ` @ ` @ ` @ `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

The object domain O is the boolean domain B = {Falsity, Truth}, the semiotic domain S is any of the spaces isomorphic to the set of rooted trees, matched-up parentheses, or unlabeled alpha graphs, and we treat a couple of denotation maps D_en, D_ex : S → O.

Either one of the denotation maps induces the same partition of S into REC's, a partition whose structure is suggested by the following two sets of strings:

{" ", "(( ))", "(( )( ))", ...},

{"( )", "( )( )", "((( )))", ...}.

These are of course the parenthesis strings that correspond to the rooted trees that are shown in the lower right corner of the Figure.

In thinking about mappings between categories of structured individuals, we can take each mapping in two parts. At the first level of analysis, there is the part that maps individuals to individuals. At the second level of analysis, there is the part that maps the structural parts of each individual to the structural parts of the individual that forms its counterpart under the first part of the mapping in question.

The general scheme of things is suggested by the following Figure, where the mapping f from COSI U to COSI V is analyzed in terms of a mapping g that takes individuals to individuals, ignoring their inner structures, and a set of mappings h_j, where j ranges over the individuals of COSI U, and where h_j specifies just how the parts of j map to the parts of g(j), its counterpart under g.

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` U ` ` ` ` ` ` ` f ` ` ` ` ` ` ` V ` ` ` ` ` ` `
` ` ` ` ` ` ` @ ` ` --------------------> ` ` @ ` ` ` ` ` ` `
` ` ` ` ` ` `/`\` ` ` ` ` ` ` ` ` ` ` ` ` ` `/`\` ` ` ` ` ` `
` ` ` ` ` ` / ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` / ` \ ` ` ` ` ` `
` ` ` ` ` `/` ` `\` ` ` ` ` ` ` ` ` ` ` ` `/` ` `\` ` ` ` ` `
` ` ` ` ` / ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` / ` ` ` \ ` ` ` ` `
` ` ` ` `/` ` ` ` `\` ` ` ` ` g ` ` ` ` `/` ` ` ` `\` ` ` ` `
` ` ` ` o ` `...` ` o ` ` --------> ` ` o ` `...` ` o ` ` ` `
` ` ` `/`\` ` ` ` `/`\` ` ` ` ` ` ` ` `/`\` ` ` ` `/`\` ` ` `
` ` ` / ` \ ` ` ` / ` \ ` ` ` ` ` ` ` / ` \ ` ` ` / ` \ ` ` `
` ` `/` ` `\` ` `/` ` `\` ` ` ` ` ` `/` ` `\` ` `/` ` `\` ` `
` ` o->-o->-o ` o->-o->-o ` ` ` ` ` o->-o->-o ` o->-o->-o ` `
` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` `\` ` ` ` h_j ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o-------->----------o ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Next time we'll apply this general scheme to the En and Ex interpretations of logical graphs, and see how it helps us to sort out the varieties of iconic mapping that are involved in that setting.

Corresponding to the Entitative and Existential interpretations of the primary arithmetic, there are two distinct mappings from the sign domain S, containing the topological equivalents of bare and rooted trees, onto the object domain O, containing the two objects whose conventional, ordinary, or meta-language names are Falsity and Truth, respectively.

The next two Figures suggest how one might view the interpretation maps as mappings from a COSI S to a COSI O. Here I have placed names of categories at the bottom, indices of individuals at the next level, and extended upward from there whatever structures the individuals may have.

Here is the Figure for the Entitative interpretation:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` o o ` o ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` | `\`/` ` `
Structures` ` F ` ` ` ` ` T ` ` ` ` ` @ ` @ `...` @ ` @ `...`
` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` `\` | `/` ` `\` | `/` `
` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` \ | / ` ` ` \ | / ` `
` ` ` ` ` ` ` | ` ` ` ` ` | ` ` En` ` ` `\|/` ` ` ` `\|/` ` `
Individuals ` o ` ` ` ` ` o ` <-----` ` ` o ` ` ` ` ` o ` ` `
` ` ` ` ` ` ` `\` ` ` ` `/` ` ` ` ` ` ` ` `\` ` ` ` `/` ` ` `
` ` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` \ ` ` ` / ` ` ` `
` ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` `\` ` `/` ` ` ` `
` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` `
` ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` `\`/` ` ` ` ` `
Categories` ` ` ` `!O!` ` ` ` ` ` ` ` ` ` ` ` `!S!` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Here is the Figure for the Existential interpretation:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o o ` o ` ` ` ` o ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | `\`/` ` ` ` ` | ` ` `
Structures` ` F ` ` ` ` ` T ` ` ` ` ` @ ` @ `...` @ ` @ `...`
` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` `\` | `/` ` `\` | `/` `
` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` \ | / ` ` ` \ | / ` `
` ` ` ` ` ` ` | ` ` ` ` ` | ` ` Ex` ` ` `\|/` ` ` ` `\|/` ` `
Individuals ` o ` ` ` ` ` o ` <-----` ` ` o ` ` ` ` ` o ` ` `
` ` ` ` ` ` ` `\` ` ` ` `/` ` ` ` ` ` ` ` `\` ` ` ` `/` ` ` `
` ` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` \ ` ` ` / ` ` ` `
` ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` `\` ` `/` ` ` ` `
` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` `
` ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` `\`/` ` ` ` ` `
Categories` ` ` ` `!O!` ` ` ` ` ` ` ` ` ` ` ` `!S!` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Note that the structure of a tree begins at its root, marked by an "O". The objects in O have no further structure to speak of, so there is nothing much happening in the object domain O between the level of individuals and the level of structures. In the sign domain S, the individuals are the parts of the partition into referential equivalence classes, each part of which contains a countable infinity of syntactic structures, rooted trees, or whatever form one views their structures taking. The sense of the Figures is that the interpretation under consideration maps the individual on the left (right) side of S to the individual on the left (right) side of O.

An iconic mapping, that gets formalized in mathematical terms as a morphism, is said to be a structure-preserving map. This does not mean that all of the structure of the source domain is preserved in the map images of the target domain, but only some of the structure, that is, specific types of relation that are defined among the elements of the source and target, respectively.

For example, let's start with the archetype of all morphisms, namely, a linear function or a linear mapping f : X → Y.

To say that the function f is linear is to say that we have already got in mind a couple of relations on X and Y that have forms roughly analogous to "addition tables", so let's signify their operation by means of the symbols "#" for "addition in X" and "+" for "addition in Y".

More exactly, the use of "#" refers to a 3-adic relation L_X ⊆ X × X × X that licenses the formula "a # b = c" just when <a, b, c> is in L_X, and the use of "+" refers to a 3-adic relation L_Y ⊆ Y × Y × Y that licenses the formula "p + q = r" just when <p, q, r> is in L_Y.

In this setting, the mapping f : X → Y is said to be linear, and to preserve the structure of L_X in the structure of L_Y, if and only if f(a # b) = f(a) + f(b), for all pairs a, b in X. In other words, f distributes over the additions # to +, just as if it were a form of multiplication, like m(a + b) = ma + mb.

Writing this more directly in terms of the 3-adic relations L_X and L_Y instead of via their operation symbols, we would say that f : X → Y is linear with regard to L_X and L_Y if and only if <a, b, c> being in the relation L_X determines that its map image <f(a), f(b), f(c)> be in L_Y. To see this, observe that <a, b, c> being in L_X implies that c = a # b, and <f(a), f(b), f(c)> being in L_Y implies that f(c) = f(a) + f(b), so we have that f(a # b) = f(c) = f(a) + f(b), and the two notions are one.

The idea of mappings that preserve 3-adic relations should ring a few bells here.

Once again into the breach between the interpretations En, Ex : S → O, drawing but a single Figure in the sand and relying on the reader to recall:

En maps every tree on the left (right) of S to the left (right) of O.

Ex maps every tree on the left (right) of S to the right (left) of O.

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` o o ` o ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` | `\`/` ` `
Structures` ` F ` ` ` ` ` T ` ` ` ` ` @ ` @ `...` @ ` @ `...`
` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` `\` | `/` ` `\` | `/` `
` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` \ | / ` ` ` \ | / ` `
` ` ` ` ` ` ` | ` ` ` ` ` | ` ` En` ` ` `\|/` ` ` ` `\|/` ` `
Individuals ` o ` ` ` ` ` o ` <-----` ` ` o ` ` ` ` ` o ` ` `
` ` ` ` ` ` ` `\` ` ` ` `/` ` ` Ex` ` ` ` `\` ` ` ` `/` ` ` `
` ` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` \ ` ` ` / ` ` ` `
` ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` `\` ` `/` ` ` ` `
` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` `
` ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` `\`/` ` ` ` ` `
Categories` ` ` ` `!O!` ` ` ` ` ` ` ` ` ` ` ` `!S!` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Now, those who wish to say that these logical signs are iconic of their logical objects must not only find some reason that logic itself singles out one interpretation over the other, but, even if they succeed in that, they must then make us believe that every sign for Truth is iconic of Truth, while every sign for Falsity is iconic of Falsity. Well, I confess that it strains my imagination, if not the over-abundant resources of theirs.

One of the questions that arises at this point, where we have a very small object domain O = {Falsity, Truth} and a very large sign domain S ≈ {rooted trees}, is the following:

Why do we have so many ways of saying the same thing?

In other words, what possible utility is there in a language having so many signs to denote the same object? Why not just restrict the language to a canonical collection of signs, each of which denotes one and only one object, exclusively and uniquely?

Indeed, language reformers from time to time have proposed the design of languages that have just this property, but I think this is one of those places where natural evolution has luckily hit on a better plan than the sorts of intentional design that inexperienced designers typically craft.

The answer to the puzzle of semiotic multiplicity appears to have something to do with the use of language in interacting with a complex external world. The objective world throws its multiplicity of problems at us, and the first duty of language is to provide some expression of their structure, on the fly, as quickly as possible, in real time, as they come in, no matter how obscurely our quick and dirty expressions of the problematic situation might otherwise be.

Of course, very little of this can be apparent at the level of primary arithmetic, but I think it should become a little more obvious as we enter the primary algebra.

I will now give a reference version of the CSP-GSB axioms for the abstract calculus that is formally recognizable in several senses as giving form to propositional logic.

The first order of business is to give the exact forms of the axioms that I use, devolving from Peirce's "Logical Graphs" via Spencer-Brown's Laws of Form (LOF). In formal proofs, I will use a variation of the annotation scheme from LOF to mark each step of the proof according to which axiom, or initial, is being invoked to justify the corresponding step of syntactic transformation, whether it applies to graphs or to strings.

The axioms are just four in number, and they come in a couple of flavors: the arithmetic initials I₁ and I₂, and the algebraic initials J₁ and J₂.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `( ) ( )` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Axiom I_1.` ` Distract <--- | ---> Condense ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` (( )) ` ` ` = ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Axiom I_2.` ` ` Unfold <--- | ---> Refold ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a(a)` ` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Axiom J_1.` ` ` Insert <--- | ---> Delete ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `ab ` ac` ` ` ` ` ` ` b ` c ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` a @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `((ab)(ac)) ` ` = ` ` a((b)(c)) ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Axiom J_2.` Distribute <--- | ---> Collect` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

Notice that all of the axioms in this set have the form of equations. This means that all of the inference steps they allow are reversible. In the proof annotation scheme below, I will use a double bar "=====" to mark this fact, but I may at times leave it to the reader to pick which direction is the one required for applying the indicated axiom.

People who busy themselves with working out difficult proofs can tell you that the actual business of proof is a far more strategic affair than the simple cranking of inference rules might suggest. Part of the reason for this lies in the fact that the usual brands of inferences rules combine the moving forward of a state of inquiry with the losing of information along the way that doesn't appear to be immediately relevant, at least, not as viewed in the local focus and the short run of the moment to moment proceedings of the proof in question. Over the long haul, this has the pernicious side-effect that one is forever strategically required to reconstruct much of the information that one had strategically thought to forget in earlier stages of the proof, if "before the proof started" can also be counted as an earlier stage of the proof in view.

This is just one of the reasons that it can be very instructive to study equational inference rules of the sort that our axioms have just provided. Although equational forms of reasoning are paramount in mathematics, they are less familiar to the student of the usual logic textbooks, who may find a few surprises here.

By way of gaining a minimal experience with how equational proofs look in the present forms of syntax, I will lay out the proofs of a few essential theorems in the primary algebra.

The first theorem is known as the Double Negation Theorem (DNT).

o-----------------------------------------------------------o
| C_1.` Double Negation Theorem ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` a ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ((a)) ` ` ` = ` ` ` ` a ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` Reflect <---- | ----> Reflect ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

The proof that follows it is derived from the one that was given by George Spencer Brown in his book Laws of Form, and credited to two of his students, John Dawes and D.A. Utting. This result is annotated as Consequence 1 (C₁) or as Reflection in LOF.

o-----------------------------------------------------------o
| C_1.` Double Negation Theorem.` Proof.` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< I2. Unfold "(())" >=========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` a o ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< J1. Insert "(a)" >==========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `a o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` a o ` a o ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< J2. Distribute "((a))" >====o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` a o ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` `o` `a o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< J1. Delete "(a)" >==========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< J1. Insert "a" >============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` `o a` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o ` ` o a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< J2. Collect "a" >===========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` `o a` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< J1. Delete "((a))" >========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< I2. Refold "(())" >=========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< QED >=======================o

Weed and seed theorem

One theorem of frequent use is what I like to call the Weed and Seed Theorem. The proof is just an inductive exercise, so I will let it go till later, but it says that a label can be freely distributed or freely erased (retracted or withdrawn) anywhere in a subtree whose root is labeled with that label. The second theorem on the list to be shown here is actually the base case of this Weed and Seed theorem. I talk about it under the LOF name of Generation.

o-----------------------------------------------------------o
| C_2.` Generation Theorem` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` b o ` ` ` ` ` ` ` ` a o b ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` = ` ` ` a @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a(b)` ` ` ` = ` ` ` a(ab) ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` `Degenerate <---- | ----> Regenerate` ` ` ` ` ` |
o-----------------------------------------------------------o

Here is a proof of the Generation Theorem.

o-----------------------------------------------------------o
| C_2.` Generation Theorem. `Proof. ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` b o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< C1. Reflect "a(b)" >========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` b o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< I2. Unfold "(())" >=========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` b o ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` |/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< J1. Insert "a" >============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` b o ` o a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o o a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` |/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< J2. Collect "a" >===========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` b o ` o a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` |/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< C1. Reflect "a", "b" >======o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o b ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< QED >=======================o

Now that we've seen a few — very simple but still non-trivial — examples of semiotic processes, namely, ones which fall under the headings of logical computation, evaluation, and proof, there are a number of questions that typically arise with respect to the relationship between sign relations and sign processes.

For concreteness, let's consider the example of logical evaluation that we looked at in Note 15.

Cf: FOLG 15. http://stderr.org/pipermail/inquiry/2005-October/003134.html In: FOLG. http://stderr.org/pipermail/inquiry/2005-October/thread.html#3104

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` o o o ` ` ` ` ` ` o o ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| ` `\| | ` ` ` ` ` ` | | ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` o o o ` ` ` ` ` o o o ` ` ` ` ` ` o o ` ` ` ` ` ` o ` |
| ` ` `\|/` ` ` ` ` ` `\|/` ` ` ` ` ` ` |/` ` ` ` ` ` ` | ` |
| ` ` ` @ ` ` ` = ` ` ` @ ` ` ` = ` ` ` @ ` ` ` = ` ` ` @ ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| (()())(())()` = ` (())(())()` = ` ` (())()` ` = ` ` `( )` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

What sorts of sign relation are implicated in this sign process? For simplicity, let's answer for the existential interpretation.

In Ex, all four of the listed signs are expressions of Falsity, and, viewed within the special type of semiotic procedure that is being considered here, each sign interprets its predecessor in the sequence. Thus we might begin by drawing up this Table:

o-------------------o-------------------o-------------------o
| Object` ` ` ` ` ` | Sign` ` ` ` ` ` ` | Interpretant` ` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "(()())(())()"` ` | "(())(())()"` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "(())(())()"` ` ` | "(())()"` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "(())()"` ` ` ` ` | "()"` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o-------------------o-------------------o

That much of a sign relation is enough to cover the case before us, but of course it is only a small sample from the larger population of triples of the form <o, s, i> that is implied by the definition of the primary arithmetic.

Let's take another look at the semiotic sequence associated with a logical evaluation and the corresponding sample of a sign relation that we were looking at last time.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` o o o ` ` ` ` ` ` o o ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| ` `\| | ` ` ` ` ` ` | | ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` o o o ` ` ` ` ` o o o ` ` ` ` ` ` o o ` ` ` ` ` ` o ` |
| ` ` `\|/` ` ` ` ` ` `\|/` ` ` ` ` ` ` |/` ` ` ` ` ` ` | ` |
| ` ` ` @ ` ` ` = ` ` ` @ ` ` ` = ` ` ` @ ` ` ` = ` ` ` @ ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| (()())(())()` = ` (())(())()` = ` ` (())()` ` = ` ` `( )` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

o-------------------o-------------------o-------------------o
| Object` ` ` ` ` ` | Sign` ` ` ` ` ` ` | Interpretant` ` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "(()())(())()"` ` | "(())(())()"` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "(())(())()"` ` ` | "(())()"` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "(())()"` ` ` ` ` | "()"` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o-------------------o-------------------o

The sign of equality "=", interpreted as logical equivalence "⇔", that marked our steps in the process of conducting the evaluation, is evidently intended to denote an equivalence relation, and this is a 2-adic relation that is reflexive, symmetric, and transitive. If we then pass to the reflexive, symmetric, transitive closure of the <s, i> pairs that occur in our initial sample, attaching the constant reference to Falsity in the object domain, we will sweep out a more complete selection of the sign relation that inheres in the definition of the primary logical arithmetic.

o-------------------o-------------------o-------------------o
| Object` ` ` ` ` ` | Sign` ` ` ` ` ` ` | Interpretant` ` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "(()())(())()"` ` | "(()())(())()"` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "(()())(())()"` ` | "(())(())()"` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "(()())(())()"` ` | "(())()"` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "(()())(())()"` ` | "()"` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "(())(())()"` ` ` | "(()())(())()"` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "(())(())()"` ` ` | "(())(())()"` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "(())(())()"` ` ` | "(())()"` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "(())(())()"` ` ` | "()"` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "(())()"` ` ` ` ` | "(()())(())()"` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "(())()"` ` ` ` ` | "(())(())()"` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "(())()"` ` ` ` ` | "(())()"` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "(())()"` ` ` ` ` | "()"` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "()"` ` ` ` ` ` ` | "(()())(())()"` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "()"` ` ` ` ` ` ` | "(())(())()"` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "()"` ` ` ` ` ` ` | "(())()"` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Falsity ` ` ` ` ` | "()"` ` ` ` ` ` ` | "()"` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o-------------------o-------------------o

Earnest contemplation of the relationship between semiotic trajectories and the infrastructure of sign relations that is needed to support them may bring the seeker to a state of enlightenment about a motley crew of old knots in the semiotic web of maya, most pointedly the one that goes about raveling and reveiling the world in the name of infinite semiosis.

To see how the variety of misunderstandings about infinite semiosis got started, it may help to refresh our memories with regard to one of Peirce's last, best definitions of a sign relation:

A sign is something, A, which brings something, B, its interpretant sign determined or created by it, into the same sort of correspondence with something, C, its object, as that in which itself stands to C. (C.S. Peirce, NEM 4, pp. 20-21, cf. p. 54 (1902)).

Now it's true that Peirce's definition of a sign relation requires that every sign in a sign relation creates or determines an interpretant sign that serves as a sign in the very same sign relation, and which therefore creates or determines its own interpretant sign, and so on, 'ad infinitum'. But there is nothing that keeps this "infinite semiosis" from being bounded in the nutshell of a finite sign relation, because nothing says that all of the signs must be distinct, and nothing says that this formal determination has to be extended in a temporal sequence, though of course that may happen.

In sum, we may view the sign relation as a generative structure, as a matrix that funds the generation of many possible semioses.

Before we leave it for richer coasts — not to say we won't find ourselves returning eternally — let's note one other feature of our randomly chosen microcosm, one I suspect we'll see many echoes of in the macrocosm of our future wanderings.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` o o o ` ` ` ` ` ` o o ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| ` `\| | ` ` ` ` ` ` | | ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` o o o ` ` ` ` ` o o o ` ` ` ` ` ` o o ` ` ` ` ` ` o ` |
| ` ` `\|/` ` ` ` ` ` `\|/` ` ` ` ` ` ` |/` ` ` ` ` ` ` | ` |
| ` ` ` @ ` ` ` = ` ` ` @ ` ` ` = ` ` ` @ ` ` ` = ` ` ` @ ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| (()())(())()` = ` (())(())()` = ` ` (())()` ` = ` ` `( )` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

One of the things that makes this sign sequence so special, amidst the generations of other sign sequences that can be generated from the sign relation of the primary arithmetic, is that it goes from a relatively obscure and verbose sign to an optimally clear and succinct sign for the same thing. For all its simplicity, then, it possesses a property that is characteristic of a semiotic process known as inquiry.

The third theorem that we will have need of here is one that GSB annotates as Integration, but I tend to count it as being more a question of Dominance and Recession among formal expressions.

o-----------------------------------------------------------o
| C_3.` Dominant Form Theorem ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a( )` ` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` `Remark <---- | ----> Recess ` ` ` ` ` ` ` `|
o-----------------------------------------------------------o

Here is a proof of the Dominant Form Theorem.

o-----------------------------------------------------------o
| C_3.` Dominant Form Theorem.` Proof.` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< C2. Regenerate "a" >========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< J1. Delete "a" >============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< QED >=======================o

Now to take up a more interesting example, here is the statement and a proof of the Splendid Theorem from Leibniz that was brought to my attention by John Sowa.

| If 'a' is 'b' and 'd' is 'c', then 'ad' will be 'bc'.
| This is a fine theorem, which is proved in this way:
|
| 'a' is 'b', therefore 'ad' is 'bd' (by what precedes),
|
| 'd' is 'c', therefore 'bd' is 'bc' (again by what precedes),
|
| 'ad' is 'bd', and 'bd' is 'bc', therefore 'ad' is 'bc'.  Q.E.D.
|
| Leibniz, 'Logical Papers', page 41.
|
| Leibniz, G.W., "Addenda to the Specimen of the Universal Calculus",
| pages 40-46 in Parkinson, G.H.R. (ed.), 'Leibniz:  Logical Papers',
| Oxford University Press, London, UK, 1966.   (Gerhardt, 7, p. 223).

o-----------------------------------------------------------o
| Praeclarum Theorema (Leibniz) ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` ` o bc` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` a o ` o d ` ` o ad` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o---------o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` = ` ` ` ` ` ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| `((a(b))(d(c))((ad(bc)))) ` = ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

And here's a neat proof of that nice theorem.

o-----------------------------------------------------------o
| Praeclarum Theorema (Leibniz).` Proof.` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` ` o bc` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` a o ` o d ` ` o ad` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o---------o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< C1. Reflect "ad(bc)" >======o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` a o ` o d ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `ad o---------o bc` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Weed "a", "d" >=============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `ad o---------o bc` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< C1. Reflect "b", "c" >======o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `abcd o---------o bc` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Weed "bc" >=================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `abcd o---------o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< C3. Recess "abcd" >=========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o---------o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< I2. Refold "(())" >=========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< QED >=======================o

The relation between the primary arithmetic and the primary algebra is founded on the idea that a variable name in the algebra indicates the contemplated absence or presence as an operand of any expression in the arithmetic, with the understanding that each appearance of the same variable name indicates the same state of contemplation with respect to the same expression of the arithmetic.

For example, consider the following expression:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` a ` ` a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` o-----o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

We may regard this algebraic expression as a general expression for an infinite set of arithmetic expressions, starting like so:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` o ` ` ` ` ` o ` o o ` o
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` `\`/` `\`/`
` ` ` ` ` ` o ` ` o o ` o o ` o o ` ` o o o o o o o o ` ` o `
` ` ` ` ` ` | ` ` | `\`/` `\`/` | ` ` | `\|/` `\|/` | ` ` | `
` o-----o ` o-----o ` o-----o ` o-----o ` o-----o ` o-----o `
` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `
` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `
` @ ` ` ` ` @ ` ` ` ` @ ` ` ` ` @ ` ` ` ` @ ` ` ` ` @ ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Now consider what this says about the following algebraic law:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` a ` ` a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` o-----o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

It permits us to understand the algebraic law as saying, in effect, that every one of the arithmetic expressions of the contemplated pattern evaulates to the very same canonical expression as the upshot of that evaluation. This is, as far as I know, just about as close as we can come to a conceptually and ontologically minimal way of understanding the relation between an algebra and its corresponding arithmetic.

There was a wobbly construction in my last paragraph, so let me try to shore it up in this re-construction:

The suggested perspective on algebraic expressions allows us to understand the algebraic law as saying, in effect, that every one of the arithmetic expressions that fits the indicated algebraic pattern evaluates to the very same canonical expression for the end result of its evaluation. This is, as far as I can tell, just about as close as we can come to a conceptually and ontologically minimal way of understanding the relation between an algebra and its corresponding arithmetic.

Of course, it is not really necessary to consider every possible substitution of arithmetic expressions for the algebraic variables, since only the value of each arithmetic expression can make any difference to the end result. Nevertheless, taking an algebraic expression as a syntactic mechanism for singling out a particular subset of the primary arithmetic is a move that suggests very fruitful directions of generalization.

In particular, this point of view helps us to sidestep many of the mysteries that encumber particular mechanisms of substitution, which it takes all the rigors of combinator calculus and lambda calculus even to begin clearing up, and it also provides us with an alternative way of approaching the puzzles of so-called imaginary values.

But I will have to leave it with those hints for now, as there is still much to do at the elementary level.

In lieu of a field study requirement for my bachelor's degree program I spent a couple of years in several state and university libraries reading everything I could find by and about Peirce, poring most memorably through the 38 reels of microfilmed manuscripts that Michigan State had at the time, all in trying to track down some hint of a clue to some puzzling passages that I read in Peirce's "Simplest Mathematics", most acutely coming to a head with that bizzare line of type in CP 4.306, that the editors of CP, no doubt compromised by the typographer's resistance to cutting new symbols, transmogrified into an averse verse that's even more cryptic than the manuscript hieroglyphic.

The first key to the mystery is discovered in Peirce's use of operator variables, that he and his students Christine Ladd-Franklin and O.H. Mitchell explored in some depth. I will shortly discuss this theme as it affects logical graphs, but it may be useful to give a shorter and sweeter explanation of how the basic idea typically arises in common logical practice.

Think of De Morgan's rules:

~[A ∧ B] = ~A ∨ ~B

~[A ∨ B] = ~A ∧ ~B

We could capture the common form of these two rules in a single formula by letting "X" and "Y" be variable names that range over a pre-selected set of logical operators, and then by asking what X and Y would satisfy:

~[A X B] = ~A Y ~B

We already know two solutions to this operator equation, specifically, <X, Y> = <∧, ∨> and <X, Y> = <∨, ∧>. Wouldn't it be just like Peirce to ask if there are others? I will leave that as an exercise for the reader.

Now that it's come to logical operator variables (LOV's), I can find no grander way to elope than Peirce himself did:

I shall not further enlarge upon this matter at this point, although the conception mentioned opens a wide field; because it cannot be set in its proper light without overstepping the limits of dichotomic mathematics. (C.S. Peirce, Collected Papers, CP 4.306).

And that goes treble for me. The further exploration of operator laws and operator variables, touching as it does on the ground that was classically called second intentional logic, like the man said, "opens a wide field", one that we may have hopes of revisiting in a less whirlwind touristy way one of these days. For now I will tend to that corner of the field where this particular variety of logical graphs grows, communing with a few of the ways that operative variations and operative themes sprout therein.

To begin with a concrete case that's as easy as possible, let's examine this extremely simple algebraic expression:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

In this context the variable name "a" appears as an operand name. In functional terms, "a" is called an argument name, but we are probably well advised to avoid the confusing connotations of the word "argument" here, as it also refers in logical discussions to a more or less specific pattern of reasoning. In syntactic terms, this same "a" would be classified as a terminal sign.

As we've already discussed, the algebraic variable name indicates the contemplated absence or presence of any arithmetic expression taking its place in the surrounding template, which expression is proxied well enough by its value, of which values we know but two. Thus, the given algebraic expression varies between these choices:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` @ ` ` ` ` , ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

The above selection of arithmetic expressions is what it means to contemplate the absence or presence of the operand "a" in the algebraic expression "(a)". But what would it mean to contemplate the absence or presence of the operator "(_)" in the algebraic expression "(a)"?

We had been contemplating the penultimately simple algebraic expression "(a)" as a name for a set of arithmetic expressions, namely, (a) = {(), (())}, taking the equality sign in the appropriate sense.

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` `
` ` ` ` ` a ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o ` ` ` ` ` ` ` ` ` o ` ` ` ` o ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` @ ` ` ` ` = ` ` { ` @ ` `,` ` @ ` } ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Then we asked the corresponding question about the operator "(_)": The above selection of arithmetic expressions is what it means to contemplate the absence or presence of the operand "a" in the algebraic expression "(a)". But what would it mean to contemplate the absence or presence of the operator "(_)" in the algebraic expression "(a)"?

Clearly, a variation between the absence and the presence of the operator "(_)" in the algebraic expression "(a)" refers to a variation between the algebraic expressions "a" and "(a)", respectively, somewhat as pictured here:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` a ` ` ` ` ` ` ` ` ` ` ` ` ` ` a ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ? ` ` ` ` ` ` ` ` ` a ` ` ` ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` @ ` ` ` ` = ` ` { ` @ ` `,` ` @ ` } ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

But how shall we signify such variations in a coherent calculus?

In the days when I scribbled these things on the backs of computer punchcards, I think that the first thing I tried was drawing big loopy script characters, placing some inside the loops of others. Lower case alphas, betas, gammas, deltas, and so on worked the best, but here in Ascii I will ty to convey something approaching the same general impression by using p's and q's.

Here is how we might suggest an algebraic expression of the form "(q)" where the absence or presence of the operator "(_)" depends on the value of the algebraic expression "p", the operator "(_)" being absent whenever p is unmarked and present when whenever p is marked.

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o---o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | q | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o---o ` ` ` = ` ` ` { ` q ` , ` (q) ` } ` ` ` ` ` `
` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

It was obvious to me from the very outset that this sort of tactic would need some work to become a usable calculus, especially when it became time to feed those punchcards back into the computer.

One of the other tactics of syntax that I tried at this time — somewhere in the 70's ... when did we quit using punchcards? — by way of porting operator variables into logical graphs and the laws of form, was to hollow out a leg of Spencer-Brown's crosses, gnomons, markers, whatever you call them, like this:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ----------o---o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` | ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` q ` ` | p | ` ` = ` ` { ` q ` , ` (q) ` } ` ` `
` ` ` ` ` ` ` ` ` ` | ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

The initial idea that I had in mind here was the same as before, that the operator over q is counted as absent when p evaluates to a space and counted as present when p evaluates to a cross.

However, much in the same way that operators with a shade of negativity to them tend to be more generative than the purely "positivistic" brand, it turned out to be slightly more useful to reverse this initial polarity of operation, letting the operator over q be counted as absent whenever p evaluates to a cross and be counted as present whenever p evaluates to a space.

So that is the convention that I shall adopt from here on.

A funny thing just happened. Let's see if we can tell where. We started with the algebraic expression "(a)", in which the operand "a" suggests the contemplated absence or presence of any arithmetic expression or its value, then we contemplated the absence of presence of the operator "(_)" in "(a)" to be indicated by a cross or a space, respectively, for the value of a newly introduced variable, "b", placed in a new slot of a newly extended operator form, as suggested by this picture:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ----------o---o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` | ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` a ` ` | b | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` | ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

The funny thing that just happened — it's an acquired sense of humor — is that our contemplation of an operator variable just as quickly got converted into the contemplation of a newly introjected but otherwise quite ordinary operand variable, albeit within a newly-fanged formula. In its interpretation for logic, the new form of operation that forms here may be viewed as an extension of ordinary negation, specifically, a negation of the first variable that is "controlled" by the value of the second variable. Thus, we may regard this development as marking a form of "controlled reflection", or a form of "reflective control". B

Futures Of Logical Graphs

MyWikiBiz, Author Your Legacy — Friday August 29, 2008

Contents

Introduction

Categories of structured individuals

Weed and seed theorem