# User:Jon Awbrey/DIFF/A

Differential Logic : Series A

## Note 1

One of the first things that you can do, once you have a really decent calculus for boolean functions or propositional logic, whatever you want to call it, is to compute the differentials of these functions or propositions.

Now there are many ways to dance around this idea, and I feel like I have tried them all, before one gets down to acting on it, and there many issues of interpretation and justification that we will have to clear up after the fact, that is, before we can be sure that it all really makes any sense, but I think this time I'll just jump in, and show you the form in which this idea first came to me.

Start with a proposition of the form x & y, which I graph as two labels attached to a root node, so:

```o-------------------------------------------------o
|                                                 |
|                       x y                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                     x and y                     |
o-------------------------------------------------o
```

Written as a string, this is just the concatenation x y.

The proposition xy may be taken as a boolean function f(xy) having the abstract type f : B × B → B, where B = {0, 1} is read in such a way that 0 means false and 1 means true.

In this style of graphical representation, the value true looks like a blank label and the value false looks like an edge.

```o-------------------------------------------------o
|                                                 |
|                                                 |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                      true                       |
o-------------------------------------------------o
```
```o-------------------------------------------------o
|                                                 |
|                        o                        |
|                        |                        |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                      false                      |
o-------------------------------------------------o
```

Back to the proposition xy. Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition xy is true, as pictured:

```o-------------------------------------------------o
|                                                 |
|                                                 |
|          o-----------o   o-----------o          |
|         /             \ /             \         |
|        /               o               \        |
|       /               /%\               \       |
|      /               /%%%\               \      |
|     o               o%%%%%o               o     |
|     |               |%%%%%|               |     |
|     |               |%%%%%|               |     |
|     |       x       |%%%%%|       y       |     |
|     |               |%%%%%|               |     |
|     |               |%%%%%|               |     |
|     o               o%%%%%o               o     |
|      \               \%%%/               /      |
|       \               \%/               /       |
|        \               o               /        |
|         \             / \             /         |
|          o-----------o   o-----------o          |
|                                                 |
|                                                 |
o-------------------------------------------------o
```

Now ask yourself: What is the value of the proposition xy at a distance of dx and dy from the cell xy where you are standing?

Don't think about it -- just compute:

```o-------------------------------------------------o
|                                                 |
|                   dx o   o dy                   |
|                     / \ / \                     |
|                  x o---@---o y                  |
|                                                 |
o-------------------------------------------------o
|              (x + dx) and (y + dy)              |
o-------------------------------------------------o
```

To make future graphs easier to draw in Ascii land, I will use devices like `@=@=@` and `o=o=o` to identify several nodes into one, as in this next redrawing:

```o-------------------------------------------------o
|                                                 |
|                   x  dx y  dy                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
|              (x + dx) and (y + dy)              |
o-------------------------------------------------o
```

However you draw it, these expressions follow because the expression x + dx, where the plus sign indicates (mod 2) addition in B, and thus corresponds to an exclusive-or in logic, parses to a graph of the following form:

```o-------------------------------------------------o
|                                                 |
|                     x    dx                     |
|                      o---o                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                      x + dx                     |
o-------------------------------------------------o
```

Next question: What is the difference between the value of the proposition xy "over there" and the value of the proposition xy where you are, all expressed as general formula, of course? Here 'tis:

```o-------------------------------------------------o
|                                                 |
|             x  dx y  dy                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /                           |
|                \| |/         x y                |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|           ((x + dx) & (y + dy)) - xy            |
o-------------------------------------------------o
```

Oh, I forgot to mention: Computed over B, plus and minus are the very same operation. This will make the relationship between the differential and the integral parts of the resulting calculus slightly stranger than usual, but never mind that now.

Last question, for now: What is the value of this expression from your current standpoint, that is, evaluated at the point where xy is true? Well, substituting 1 for x and 1 for y in the graph amounts to the same thing as erasing those labels:

```o-------------------------------------------------o
|                                                 |
|            dx    dy                             |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /                           |
|                \| |/                            |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|           ((1 + dx) & (1 + dy)) - 1&1           |
o-------------------------------------------------o
```

And this is equivalent to the following graph:

```o-------------------------------------------------o
|                                                 |
|                     dx   dy                     |
|                      o   o                      |
|                       \ /                       |
|                        o                        |
|                        |                        |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                    dx or dy                     |
o-------------------------------------------------o
```

Enough for the moment. Explanation to follow.

## Note 2

We have just met with the fact that the differential of the "and" is the "or" of the differentials.

x and y --Diff--> dx or dy.
```o-------------------------------------------------o
|                                                 |
|                                    dx   dy      |
|                                     o   o       |
|                                      \ /        |
|                                       o         |
|        x y                            |         |
|         @          --Diff-->          @         |
|                                                 |
o-------------------------------------------------o
|        x y         --Diff-->     ((dx) (dy))    |
o-------------------------------------------------o
```

It will be necessary to develop a more refined analysis of this statement directly, but that is roughly the nub of it.

If the form of the above statement reminds you of De Morgan's rule, it is no accident, as differentiation and negation turn out to be closely related operations. Indeed, one can find discussions of logical difference calculus in the Boole-De Morgan correspondence and C.S. Peirce also made use of differential operators in a logical context, but the exploration of these ideas has been hampered by a number of factors, not the least of which being a syntax adequate to handle the complexity of expressions that evolve.

For my part, it was definitely a case of the calculus being smarter than the calculator thereof. The graphical pictures were catalytic in their power over my thinking process, leading me so quickly past so many obstructions that I did not have time to think about all of the difficulties that would otherwise have inhibited the derivation. It did eventually became necessary to write all this up in a linear script, and to deal with the various problems of interpretation and justification that I could imagine, but that took another 120 pages, and so, if you don't like this intuitive approach, then let that be your sufficient notice.

Let us run through the initial example again, this time attempting to interpret the formulas that develop at each stage along the way.

We begin with a proposition or a boolean function f(xy) = xy.

```o-------------------------------------------------o
|                                                 |
|                                                 |
|          o-----------o   o-----------o          |
|         /             \ /             \         |
|        /               o               \        |
|       /               /`\               \       |
|      /               /```\               \      |
|     o               o`````o               o     |
|     |               |`````|               |     |
|     |               |`````|               |     |
|     |       x       |``f``|       y       |     |
|     |               |`````|               |     |
|     |               |`````|               |     |
|     o               o`````o               o     |
|      \               \```/               /      |
|       \               \`/               /       |
|        \               o               /        |
|         \             / \             /         |
|          o-----------o   o-----------o          |
|                                                 |
|                                                 |
o-------------------------------------------------o
|                                                 |
|                       x y                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| f =                   x y                       |
o-------------------------------------------------o
```

A function like this has an abstract type and a concrete type. The abstract type is what we invoke when we write things like f : B × B → B or f : B2 → B. The concrete type takes into account the qualitative dimensions or the "units" of the case, which can be explained as follows.

• Let X be the set of values {(x), x} = {not x, x}.
• Let Y be the set of values {(y), y} = {not y, y}.

Then interpret the usual propositions about x, y as functions of the concrete type f : X × Y → B.

We are going to consider various "operators" on these functions. Here, an operator F is a function that takes one function f into another function Ff.

The first couple of operators that we need to consider are logical analogues of those that occur in the classical "finite difference calculus", namely:

• The difference operator Δ, written here as D.
• The enlargement operator Ε, written here as E.

These days, E is more often called the shift operator.

In order to describe the universe in which these operators operate, it will be necessary to enlarge our original universe of discourse. We mount up from the space U = X  × Y to its differential extension, EU = U  × dU = X × Y × dX  × dY, with dX = {(dx), dx} and dY = {(dy), dy}. The interpretations of these new symbols can be diverse, but the easiest for now is just to say that dx means "change x" and dy means "change y". To draw the differential extension EU of our present universe U = X  × Y as a venn diagram, it would take us four logical dimensions X, Y, dX, dY, but we can project a suggestion of what it's about on the universe X  × Y by drawing arrows that cross designated borders, labeling the arrows as dx when crossing the border between x and (x) and as dy when crossing the border between y and (y), in either direction, in either case.

```o-------------------------------------------------o
|                                                 |
|                                                 |
|          o-----------o   o-----------o          |
|         /             \ /             \         |
|        /       x       o       y       \        |
|       /               /`\               \       |
|      /               /```\               \      |
|     o               o`````o               o     |
|     |               |`````|               |     |
|     |         dy    |`````|    dx         |     |
|     |     <---------|--o--|--------->     |     |
|     |               |`````|               |     |
|     |               |`````|               |     |
|     o               o`````o               o     |
|      \               \```/               /      |
|       \               \`/               /       |
|        \               o               /        |
|         \             / \             /         |
|          o-----------o   o-----------o          |
|                                                 |
|                                                 |
o-------------------------------------------------o
```

We can form propositions from these differential variables in the same way that we would any other logical variables, for instance, interpreting the proposition (dx (dy)) to say "dx ⇒ dy", in other words, however you wish to take it, whether indicatively or injunctively, as saying something to the effect that there is "no change in x without a change in y".

Given the proposition f(x, y) in U = X  × Y, the (first order) enlargement of f is the proposition Ef in EU that is defined by the formula Ef(x, y, dx, dy) = f(x + dx, y + dy).

In the example f(x, y) = xy, we obtain:

Ef(x, y, dx, dy) = (x + dx)(y + dy).
```o-------------------------------------------------o
|                                                 |
|                   x  dx y  dy                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
| Ef =            (x, dx) (y, dy)                 |
o-------------------------------------------------o
```

Given the proposition f(x, y) in U = X × Y, the (first order) difference of f is the proposition Df in EU that is defined by the formula Df = Ef – f, or, written out in full, Df(x, y, dx, dy) = f(x + dx, y + dy) – f(x, y).

In the example f(x, y) = xy, the result is:

• Df(x, y, dx, dy) = (x + dx)(y + dy) – xy.
```o-------------------------------------------------o
|                                                 |
|             x  dx y  dy                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /                           |
|                \| |/         x y                |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df =           ((x, dx)(y, dy), xy)             |
o-------------------------------------------------o
```

We did not yet go through the trouble to interpret this (first order) difference of conjunction fully, but were happy simply to evaluate it with respect to a single location in the universe of discourse, namely, at the point picked out by the singular proposition xy, in as much as if to say, at the place where x = 1 and y = 1. This evaluation is written in the form Df|xy or Df|<1, 1>, and we arrived at the locally applicable law that states that f = xy = x & yDf|xy = ((dx)(dy)) = dx or dy.

```o-------------------------------------------------o
|                                                 |
|                                                 |
|          o-----------o   o-----------o          |
|         /             \ /             \         |
|        /       x       o       y       \        |
|       /               /`\               \       |
|      /               /```\               \      |
|     o               o`````o               o     |
|     |               |`````|               |     |
|     |      dy (dx)  |`````|  dx (dy)      |     |
|     |   o<----------|--o--|---------->o   |     |
|     |               |``|``|               |     |
|     |               |``|``|               |     |
|     o               o``|``o               o     |
|      \               \`|`/               /      |
|       \               \|/               /       |
|        \               |               /        |
|         \             /|\             /         |
|          o-----------o | o-----------o          |
|                        |                        |
|                      dx|dy                      |
|                        |                        |
|                        v                        |
|                        o                        |
|                                                 |
o-------------------------------------------------o
|                                                 |
|                     dx   dy                     |
|                      o   o                      |
|                       \ /                       |
|                        o                        |
|                        |                        |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df|xy =           ((dx) (dy))                   |
o-------------------------------------------------o
```

The picture illustrates the analysis of the inclusive disjunction ((dx)(dy)) into the exclusive disjunction: dx(dy) + dy(dx) + dx dy, a proposition that may be interpreted to say "change x or change y or both". And this can be recognized as just what you need to do if you happen to find yourself in the center cell and desire a detailed description of ways to depart it.

## Note 3

Last time we computed what will variously be called the difference map, the difference proposition, or the local proposition Dfp for the proposition f(xy) = xy at the point p where x = 1 and y = 1.

In the universe U = X × Y, the four propositions xy, x(y), (x)y, (x)(y) that indicate the "cells", or the smallest regions of the venn diagram, are called singular propositions. These serve as an alternative notation for naming the points <1, 1>, <1, 0>, <0, 1>, <0, 0>, respectively.

Thus, we can write Dfp = Df|p = Df|<1, 1> = Df|xy, so long as we know the frame of reference in force.

Sticking with the example f(xy) = xy, let us compute the value of the difference proposition Df at all of the points.

```o-------------------------------------------------o
|                                                 |
|             x  dx y  dy                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /                           |
|                \| |/         x y                |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df =        ((x, dx)(y, dy), xy)                |
o-------------------------------------------------o
```
```o-------------------------------------------------o
|                                                 |
|                dx    dy                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /                           |
|                \| |/                            |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df|xy =           ((dx) (dy))                   |
o-------------------------------------------------o
```
```o-------------------------------------------------o
|                                                 |
|                   o                             |
|                dx |  dy                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /         o                 |
|                \| |/          |                 |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df|x(y) =          (dx) dy                      |
o-------------------------------------------------o
```
```o-------------------------------------------------o
|                                                 |
|             o                                   |
|             |  dx    dy                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /         o                 |
|                \| |/          |                 |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df|(x)y =            dx (dy)                    |
o-------------------------------------------------o
```
```o-------------------------------------------------o
|                                                 |
|             o     o                             |
|             |  dx |  dy                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /       o   o               |
|                \| |/         \ /                |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df|(x)(y) =          dx dy                      |
o-------------------------------------------------o
```

The easy way to visualize the values of these graphical expressions is just to notice the following equivalents:

```o-------------------------------------------------o
|                                                 |
|  x                                              |
|  o-o-o-...-o-o-o                                |
|   \           /                                 |
|    \         /                                  |
|     \       /                                   |
|      \     /                          x         |
|       \   /                           o         |
|        \ /                            |         |
|         @              =              @         |
|                                                 |
o-------------------------------------------------o
|  (x, , ... , , )       =             (x)        |
o-------------------------------------------------o
```
```o-------------------------------------------------o
|                                                 |
|                o                                |
| x_1 x_2   x_k  |                                |
|  o---o-...-o---o                                |
|   \           /                                 |
|    \         /                                  |
|     \       /                                   |
|      \     /                                    |
|       \   /                                     |
|        \ /                       x_1 ... x_k    |
|         @              =              @         |
|                                                 |
o-------------------------------------------------o
|  (x_1, ..., x_k, ())   =         x_1 ... x_k    |
o-------------------------------------------------o
```

Laying out the arrows on the augmented venn diagram, one gets a picture of a differential vector field.

```o-------------------------------------------------o
|                                                 |
|                        o                        |
|                        |                        |
|                      dx|dy                      |
|                        |                        |
|          o-----------o | o-----------o          |
|         /             \|/             \         |
|        /       x       |       y       \        |
|       /               /|\               \       |
|      /               /`|`\               \      |
|     o               o``|``o               o     |
|     |      dy (dx)  |``v``|  dx (dy)      |     |
|     |   o-----------|->o<-|-----------o   |     |
|     |               |`````|               |     |
|     |   o<----------|--o--|---------->o   |     |
|     |      dy (dx)  |``|``|  dx (dy)      |     |
|     o               o``|``o               o     |
|      \               \`|`/               /      |
|       \               \|/               /       |
|        \               |               /        |
|         \             /|\             /         |
|          o-----------o | o-----------o          |
|                        |                        |
|                      dx|dy                      |
|                        |                        |
|                        v                        |
|                        o                        |
|                                                 |
o-------------------------------------------------o
```

This really just constitutes a depiction of the interpretations in EU = X × Y × dX × dY that satisfy the difference proposition Df, namely, these:

```1.   x  y  dx  dy
2.   x  y  dx (dy)
3.   x  y (dx) dy
4.   x (y)(dx) dy
5.  (x) y  dx (dy)
6.  (x)(y) dx  dy
```

By inspection, it is fairly easy to understand Df as telling you what you have to do from each point of U in order to change the value borne by f(xy).

## Note 4

We have been studying the action of the difference operator D, also known as the localization operator, on the proposition f : X × Y → B that is commonly known as the conjunction xy. We described Df as a (first order) differential proposition, that is, a proposition of the type Df : X × Y × dX × dY → B. Abstracting from the augmented venn diagram that illustrates how the models, or the satisfying interpretations, of Df distribute within the extended universe EU = X × Y × dX × dY, we can depict Df in the form of a digraph or directed graph, one whose points are labeled with the elements of U = X × Y and whose arrows are labeled with the elements of dU = dX × dY.

```o-------------------------------------------------o
|  f =                  x y                       |
o-------------------------------------------------o
|                                                 |
| Df =              x  y  ((dx)(dy))              |
|                                                 |
|           +       x (y)  (dx) dy                |
|                                                 |
|           +      (x) y    dx (dy)               |
|                                                 |
|           +      (x)(y)   dx  dy                |
|                                                 |
o-------------------------------------------------o
|                                                 |
|                       x y                       |
|  x (y) o<------------->o<------------->o (x) y  |
|             (dx) dy    ^    dx (dy)             |
|                        |                        |
|                        |                        |
|                     dx | dy                     |
|                        |                        |
|                        |                        |
|                        v                        |
|                        o                        |
|                     (x) (y)                     |
|                                                 |
o-------------------------------------------------o
```

Any proposition worth its salt has many equivalent ways to view it, any one of which may reveal some unsuspected aspect of its meaning. We will encounter more and more of these variant readings as we go.

## Note 5

The enlargement operator E, also known as the shift operator, has many interesting and very useful properties in its own right, so let us not fail to observe a few of the more salient features that play out on the surface of our simple example, f(xy) = xy.

Introduce a suitably generic definition of the extended universe of discourse:

Let U = X1 × … × Xk and EU = U × dU = X1 × … × Xk × dX1 × … × dXk.

For a proposition f : X1 × … × Xk → B, the (first order) enlargement of f is the proposition Ef : EU → B that is defined by:

Ef(x1, …, xk, dx1, …, dxk) = f(x1 + dx1, …, xk + dxk).

It should be noted that the so-called differential variables dxj are really just the same kind of boolean variables as the other xj. It is conventional to give the additional variables these brands of inflected names, but whatever extra connotations we might choose to attach to these syntactic conveniences are wholly external to their purely algebraic meanings.

For the example f(x, y) = xy, we obtain:

Ef(x, y, dx, dy) = (x + dx)(y + dy).

Given that this expression uses nothing more than the boolean ring operations of addition (+) and multiplication (·), it is permissible to multiply things out in the usual manner to arrive at the result:

Ef(x, y, dx, dy) = x y + x dy + y dx + dx dy

To understand what this means in logical terms, for instance, as expressed in a boolean expansion or a disjunctive normal form (DNF), it is perhaps a little better to go back and analyze the expression the same way that we did for Df. Thus, let us compute the value of the enlarged proposition Ef at each of the points in the universe of discourse U = X × Y.

```o-------------------------------------------------o
|                                                 |
|                   x  dx y  dy                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
| Ef =            (x, dx) (y, dy)                 |
o-------------------------------------------------o
```
```o-------------------------------------------------o
|                                                 |
|                      dx    dy                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
| Ef|xy =            (dx) (dy)                    |
o-------------------------------------------------o
```
```o-------------------------------------------------o
|                                                 |
|                         o                       |
|                      dx |  dy                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
| Ef|x(y) =          (dx)  dy                     |
o-------------------------------------------------o
```
```o-------------------------------------------------o
|                                                 |
|                   o                             |
|                   |  dx    dy                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
| Ef|(x)y =           dx  (dy)                    |
o-------------------------------------------------o
```
```o-------------------------------------------------o
|                                                 |
|                   o     o                       |
|                   |  dx |  dy                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
| Ef|(x)(y) =         dx   dy                     |
o-------------------------------------------------o
```

Given the sort of data that arises from this form of analysis, we can now fold the disjoined ingredients back into a boolean expansion or a DNF that is equivalent to the proposition Ef.

Ef = xy Efxy + x(y) Efx(y) + (x)y Ef(x)y + (x)(y) Ef(x)(y)

Here is a summary of the result, illustrated by means of a digraph picture, where the "no change" element (dx)(dy) is drawn as a loop at the point x y.

```o-------------------------------------------------o
|  f =                  x y                       |
o-------------------------------------------------o
|                                                 |
| Ef =              x  y   (dx)(dy)               |
|                                                 |
|           +       x (y)  (dx) dy                |
|                                                 |
|           +      (x) y    dx (dy)               |
|                                                 |
|           +      (x)(y)   dx  dy                |
|                                                 |
o-------------------------------------------------o
|                                                 |
|                    (dx) (dy)                    |
|                    .--->---.                    |
|                     \     /                     |
|                      \x y/                      |
|                       \ /                       |
|  x (y) o-------------->o<--------------o (x) y  |
|             (dx) dy    ^    dx (dy)             |
|                        |                        |
|                        |                        |
|                     dx | dy                     |
|                        |                        |
|                        |                        |
|                        |                        |
|                        o                        |
|                     (x) (y)                     |
|                                                 |
o-------------------------------------------------o
```

We may understand the enlarged proposition Ef as telling us all the different ways to reach a model of f from any point of the universe U.

## Note 6

To broaden our experience with simple examples, let us now contemplate the sixteen functions of concrete type X × Y → B and abstract type B × B → B. For future reference, I will set here a few tables that detail the actions of E and D and on each of these functions, allowing us to view the results in several different ways.

By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions in a number of different languages for zeroth order logic.

Table 1. Propositional Forms on Two Variables
L1 L2 L3 L4 L5 L6
x : 1 1 0 0
y : 1 0 1 0
f0 f0000 0 0 0 0 ( ) false 0
f1 f0001 0 0 0 1 (x)(y) neither x nor y ¬x ∧ ¬y
f2 f0010 0 0 1 0 (x) y y and not x ¬x ∧ y
f3 f0011 0 0 1 1 (x) not x ¬x
f4 f0100 0 1 0 0 x (y) x and not y x ∧ ¬y
f5 f0101 0 1 0 1 (y) not y ¬y
f6 f0110 0 1 1 0 (x, y) x not equal to y x ≠ y
f7 f0111 0 1 1 1 (x y) not both x and y ¬x ∨ ¬y
f8 f1000 1 0 0 0 x y x and y x ∧ y
f9 f1001 1 0 0 1 ((x, y)) x equal to y x = y
f10 f1010 1 0 1 0 y y y
f11 f1011 1 0 1 1 (x (y)) not x without y x → y
f12 f1100 1 1 0 0 x x x
f13 f1101 1 1 0 1 ((x) y) not y without x x ← y
f14 f1110 1 1 1 0 ((x)(y)) x or y x ∨ y
f15 f1111 1 1 1 1 (( )) true 1

The next four Tables expand the expressions of Ef and Df in two different ways, for each of the sixteen functions. Notice that the functions are given in a different order, here being collected into a set of seven natural classes.

```Table 2.  Ef Expanded Over Ordinary Features {x, y}
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      |  Ef | xy   | Ef | x(y)  | Ef | (x)y  | Ef | (x)(y)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (x)(y)   |   dx  dy   |   dx (dy)  |  (dx) dy   |  (dx)(dy)  |
|      |            |            |            |            |            |
| f_2  |   (x) y    |   dx (dy)  |   dx  dy   |  (dx)(dy)  |  (dx) dy   |
|      |            |            |            |            |            |
| f_4  |    x (y)   |  (dx) dy   |  (dx)(dy)  |   dx  dy   |   dx (dy)  |
|      |            |            |            |            |            |
| f_8  |    x  y    |  (dx)(dy)  |  (dx) dy   |   dx (dy)  |   dx  dy   |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (x)      |   dx       |   dx       |  (dx)      |  (dx)      |
|      |            |            |            |            |            |
| f_12 |    x       |  (dx)      |  (dx)      |   dx       |   dx       |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (x, y)   |  (dx, dy)  | ((dx, dy)) | ((dx, dy)) |  (dx, dy)  |
|      |            |            |            |            |            |
| f_9  |  ((x, y))  | ((dx, dy)) |  (dx, dy)  |  (dx, dy)  | ((dx, dy)) |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (y)   |       dy   |      (dy)  |       dy   |      (dy)  |
|      |            |            |            |            |            |
| f_10 |       y    |      (dy)  |       dy   |      (dy)  |       dy   |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (x  y)   | ((dx)(dy)) | ((dx) dy)  |  (dx (dy)) |  (dx  dy)  |
|      |            |            |            |            |            |
| f_11 |   (x (y))  | ((dx) dy)  | ((dx)(dy)) |  (dx  dy)  |  (dx (dy)) |
|      |            |            |            |            |            |
| f_13 |  ((x) y)   |  (dx (dy)) |  (dx  dy)  | ((dx)(dy)) | ((dx) dy)  |
|      |            |            |            |            |            |
| f_14 |  ((x)(y))  |  (dx  dy)  |  (dx (dy)) | ((dx) dy)  | ((dx)(dy)) |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
```
```Table 3.  Df Expanded Over Ordinary Features {x, y}
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      |  Df | xy   | Df | x(y)  | Df | (x)y  | Df | (x)(y)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (x)(y)   |   dx  dy   |   dx (dy)  |  (dx) dy   | ((dx)(dy)) |
|      |            |            |            |            |            |
| f_2  |   (x) y    |   dx (dy)  |   dx  dy   | ((dx)(dy)) |  (dx) dy   |
|      |            |            |            |            |            |
| f_4  |    x (y)   |  (dx) dy   | ((dx)(dy)) |   dx  dy   |   dx (dy)  |
|      |            |            |            |            |            |
| f_8  |    x  y    | ((dx)(dy)) |  (dx) dy   |   dx (dy)  |   dx  dy   |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (x)      |   dx       |   dx       |   dx       |   dx       |
|      |            |            |            |            |            |
| f_12 |    x       |   dx       |   dx       |   dx       |   dx       |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (x, y)   |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |
|      |            |            |            |            |            |
| f_9  |  ((x, y))  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (y)   |       dy   |       dy   |       dy   |       dy   |
|      |            |            |            |            |            |
| f_10 |       y    |       dy   |       dy   |       dy   |       dy   |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (x  y)   | ((dx)(dy)) |  (dx) dy   |   dx (dy)  |   dx  dy   |
|      |            |            |            |            |            |
| f_11 |   (x (y))  |  (dx) dy   | ((dx)(dy)) |   dx  dy   |   dx (dy)  |
|      |            |            |            |            |            |
| f_13 |  ((x) y)   |   dx (dy)  |   dx  dy   | ((dx)(dy)) |  (dx) dy   |
|      |            |            |            |            |            |
| f_14 |  ((x)(y))  |   dx  dy   |   dx (dy)  |  (dx) dy   | ((dx)(dy)) |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
```
```Table 4.  Ef Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      |   T_11 f   |   T_10 f   |   T_01 f   |   T_00 f   |
|      |            |            |            |            |            |
|      |            | Ef| dx dy  | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (x)(y)   |    x  y    |    x (y)   |   (x) y    |   (x)(y)   |
|      |            |            |            |            |            |
| f_2  |   (x) y    |    x (y)   |    x  y    |   (x)(y)   |   (x) y    |
|      |            |            |            |            |            |
| f_4  |    x (y)   |   (x) y    |   (x)(y)   |    x  y    |    x (y)   |
|      |            |            |            |            |            |
| f_8  |    x  y    |   (x)(y)   |   (x) y    |    x (y)   |    x  y    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (x)      |    x       |    x       |   (x)      |   (x)      |
|      |            |            |            |            |            |
| f_12 |    x       |   (x)      |   (x)      |    x       |    x       |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (x, y)   |   (x, y)   |  ((x, y))  |  ((x, y))  |   (x, y)   |
|      |            |            |            |            |            |
| f_9  |  ((x, y))  |  ((x, y))  |   (x, y)   |   (x, y)   |  ((x, y))  |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (y)   |       y    |      (y)   |       y    |      (y)   |
|      |            |            |            |            |            |
| f_10 |       y    |      (y)   |       y    |      (y)   |       y    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (x  y)   |  ((x)(y))  |  ((x) y)   |   (x (y))  |   (x  y)   |
|      |            |            |            |            |            |
| f_11 |   (x (y))  |  ((x) y)   |  ((x)(y))  |   (x  y)   |   (x (y))  |
|      |            |            |            |            |            |
| f_13 |  ((x) y)   |   (x (y))  |   (x  y)   |  ((x)(y))  |  ((x) y)   |
|      |            |            |            |            |            |
| f_14 |  ((x)(y))  |   (x  y)   |   (x (y))  |  ((x) y)   |  ((x)(y))  |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|                   |            |            |            |            |
| Fixed Point Total |      4     |      4     |      4     |     16     |
|                   |            |            |            |            |
o-------------------o------------o------------o------------o------------o
```
```Table 5.  Df Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      | Df| dx dy  | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (x)(y)   |  ((x, y))  |    (y)     |    (x)     |     ()     |
|      |            |            |            |            |            |
| f_2  |   (x) y    |   (x, y)   |     y      |    (x)     |     ()     |
|      |            |            |            |            |            |
| f_4  |    x (y)   |   (x, y)   |    (y)     |     x      |     ()     |
|      |            |            |            |            |            |
| f_8  |    x  y    |  ((x, y))  |     y      |     x      |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (x)      |    (())    |    (())    |     ()     |     ()     |
|      |            |            |            |            |            |
| f_12 |    x       |    (())    |    (())    |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (x, y)   |     ()     |    (())    |    (())    |     ()     |
|      |            |            |            |            |            |
| f_9  |  ((x, y))  |     ()     |    (())    |    (())    |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (y)   |    (())    |     ()     |    (())    |     ()     |
|      |            |            |            |            |            |
| f_10 |       y    |    (())    |     ()     |    (())    |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (x  y)   |  ((x, y))  |     y      |     x      |     ()     |
|      |            |            |            |            |            |
| f_11 |   (x (y))  |   (x, y)   |    (y)     |     x      |     ()     |
|      |            |            |            |            |            |
| f_13 |  ((x) y)   |   (x, y)   |     y      |    (x)     |     ()     |
|      |            |            |            |            |            |
| f_14 |  ((x)(y))  |  ((x, y))  |    (y)     |    (x)     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
```

If the medium truly is the message, the blank slate is the innate idea.

## Note 7

If you think that I linger in the realm of logical difference calculus out of sheer vacillation about getting down to the differential proper, it is probably out of a prior expectation that you derive from the art or the long-engrained practice of real analysis. But the fact is that ordinary calculus only rushes on to the sundry orders of approximation because the strain of comprehending the full import of E and D at once whelm over its discrete and finite powers to grasp them. But here, in the fully serene idylls of ZOL, we find ourselves fit with the compass of a wit that is all we'd ever wish to explore their effects with care.

So let us do just that.

I will first rationalize the novel grouping of propositional forms in the last set of Tables, as that will extend a gentle invitation to the mathematical subject of group theory, and demonstrate its relevance to differential logic in a strikingly apt and useful way. The data for that account is contained in Table 4.

```Table 4.  Ef Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      |   T_11 f   |   T_10 f   |   T_01 f   |   T_00 f   |
|      |            |            |            |            |            |
|      |            | Ef| dx dy  | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (x)(y)   |    x  y    |    x (y)   |   (x) y    |   (x)(y)   |
|      |            |            |            |            |            |
| f_2  |   (x) y    |    x (y)   |    x  y    |   (x)(y)   |   (x) y    |
|      |            |            |            |            |            |
| f_4  |    x (y)   |   (x) y    |   (x)(y)   |    x  y    |    x (y)   |
|      |            |            |            |            |            |
| f_8  |    x  y    |   (x)(y)   |   (x) y    |    x (y)   |    x  y    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (x)      |    x       |    x       |   (x)      |   (x)      |
|      |            |            |            |            |            |
| f_12 |    x       |   (x)      |   (x)      |    x       |    x       |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (x, y)   |   (x, y)   |  ((x, y))  |  ((x, y))  |   (x, y)   |
|      |            |            |            |            |            |
| f_9  |  ((x, y))  |  ((x, y))  |   (x, y)   |   (x, y)   |  ((x, y))  |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (y)   |       y    |      (y)   |       y    |      (y)   |
|      |            |            |            |            |            |
| f_10 |       y    |      (y)   |       y    |      (y)   |       y    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (x  y)   |  ((x)(y))  |  ((x) y)   |   (x (y))  |   (x  y)   |
|      |            |            |            |            |            |
| f_11 |   (x (y))  |  ((x) y)   |  ((x)(y))  |   (x  y)   |   (x (y))  |
|      |            |            |            |            |            |
| f_13 |  ((x) y)   |   (x (y))  |   (x  y)   |  ((x)(y))  |  ((x) y)   |
|      |            |            |            |            |            |
| f_14 |  ((x)(y))  |   (x  y)   |   (x (y))  |  ((x) y)   |  ((x)(y))  |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|                   |            |            |            |            |
| Fixed Point Total |      4     |      4     |      4     |     16     |
|                   |            |            |            |            |
o-------------------o------------o------------o------------o------------o
```

The shift operator E can be understood as enacting a substitution operation on the proposition that is given as its argument. In our immediate example, we have the following data and definition:

E : (UB) → (EUB),
E : f(x, y) → Ef(x, y, dx, dy),
Ef(x, y, dx, dy) = f(x + dx, y + dy).

Therefore, if we evaluate Ef at particular values of dx and dy, for example, dx = i and dy = j, where i, j are in B, we obtain:

Eij : (UB) → (UB),
Eij : f → : Eijf,
Eijf = Ef|<dx = i, dy = j> = f(x + i, y + j).

The notation is a little bit awkward, but the data of the Table should make the sense clear. The important thing to observe is that Eij has the effect of transforming each proposition f : U → B into some other proposition f´ : U → B. As it happens, the action is one-to-one and onto for each Eij, so the gang of four operators {Eij : i, j in B} is an example of what is called a transformation group on the set of sixteen propositions. Bowing to a longstanding local and linear tradition, I will therefore redub the four elements of this group as T00, T01, T10, T11, to bear in mind their transformative character, or nature, as the case may be. Abstractly viewed, this group of order four has the following operation table:

```o----------o----------o----------o----------o----------o
|          %          |          |          |          |
|    *     %   T_00   |   T_01   |   T_10   |   T_11   |
|          %          |          |          |          |
o==========o==========o==========o==========o==========o
|          %          |          |          |          |
|   T_00   %   T_00   |   T_01   |   T_10   |   T_11   |
|          %          |          |          |          |
o----------o----------o----------o----------o----------o
|          %          |          |          |          |
|   T_01   %   T_01   |   T_00   |   T_11   |   T_10   |
|          %          |          |          |          |
o----------o----------o----------o----------o----------o
|          %          |          |          |          |
|   T_10   %   T_10   |   T_11   |   T_00   |   T_01   |
|          %          |          |          |          |
o----------o----------o----------o----------o----------o
|          %          |          |          |          |
|   T_11   %   T_11   |   T_10   |   T_01   |   T_00   |
|          %          |          |          |          |
o----------o----------o----------o----------o----------o
```

It happens that there are just two possible groups of 4 elements. One is the cyclic group Z4 (German Zyklus), which this is not. The other is Klein's four-group V4 (German Vier), which it is.

More concretely viewed, the group as a whole pushes the set of sixteen propositions around in such a way that they fall into seven natural classes, called orbits. One says that the orbits are preserved by the action of the group. There is an Orbit Lemma of immense utility to "those who count" which, depending on your upbringing, you may associate with the names of Burnside, Cauchy, Frobenius, or some subset or superset of these three, vouching that the number of orbits is equal to the mean number of fixed points, in other words, the total number of points (in our case, propositions) that are left unmoved by the separate operations, divided by the order of the group. In this instance, T00 operates as the group identity, fixing all 16 propositions, while the other three group elements fix 4 propositions each, and so we get:

Number of orbits = (4 + 4 + 4 + 16) ÷ 4 = 7.

Amazing!

## Note 8

We have been contemplating functions of the type f : U → B, studying the action of the operators E and D on this family. These functions, that we may identify for our present aims with propositions, inasmuch as they capture their abstract forms, are logical analogues of scalar potential fields. These are the sorts of fields that are so picturesquely presented in elementary calculus and physics textbooks by images of snow-covered hills and parties of skiers who trek down their slopes like least action heroes. The analogous scene in propositional logic presents us with forms more reminiscent of plateaunic idylls, being all plains at one of two levels, the mesas of verity and falsity, as it were, with nary a niche to inhabit between them, restricting our options for a sporting gradient of downhill dynamics to just one of two, standing still on level ground or falling off a bluff.

We are still working well within the logical analogue of the classical finite difference calculus, taking in the novelties that the logical transmutation of familiar elements is able to bring to light. Soon we will take up several different notions of approximation relationships that may be seen to organize the space of propositions, and these will allow us to define several different forms of differential analysis applying to propositions. In time we will find reason to consider more general types of maps, having concrete types of the form X1 × … × Xk → Y1 × … × Yn and abstract types Bk → Bn. We will think of these mappings as transforming universes of discourse into themselves or into others, in short, as transformations of discourse.

Before we continue with this intinerary, however, I would like to highlight another sort of differential aspect that concerns the boundary operator or the marked connective that serves as one of the two basic connectives in the cactus language for ZOL.

For example, consider the proposition f of concrete type f : X × Y × Z → B and abstract type f : B3 → B that is written `(x, y, z)` in cactus syntax. Taken as an assertion in what Peirce called the existential interpretation, `(x, y, z)` says that just one of x, y, z is false. It is useful to consider this assertion in relation to the conjunction xyz of the features that are engaged as its arguments. A venn diagram of `(x, y, z)` looks like this:

```o-----------------------------------------------------------o
| U                                                         |
|                                                           |
|                      o-------------o                      |
|                     /               \                     |
|                    /                 \                    |
|                   /                   \                   |
|                  /                     \                  |
|                 /                       \                 |
|                o            x            o                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|             o--o----------o   o----------o--o             |
|            /    \%%%%%%%%%%\ /%%%%%%%%%%/    \            |
|           /      \%%%%%%%%%%o%%%%%%%%%%/      \           |
|          /        \%%%%%%%%/ \%%%%%%%%/        \          |
|         /          \%%%%%%/   \%%%%%%/          \         |
|        /            \%%%%/     \%%%%/            \        |
|       o              o--o-------o--o              o       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       o        y        o%%%%%%%o        z        o       |
|        \                 \%%%%%/                 /        |
|         \                 \%%%/                 /         |
|          \                 \%/                 /          |
|           \                 o                 /           |
|            \               / \               /            |
|             o-------------o   o-------------o             |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
```

In relation to the center cell indicated by the conjunction xyz, the region indicated by `(x, y, z)` is comprised of the adjacent or the bordering cells. Thus they are the cells that are just across the boundary of the center cell, as if reached by way of Leibniz's minimal changes from the point of origin, here, xyz.

The same form of boundary relationship is exhibited for any cell of origin that one might elect to indicate, say, by means of the conjunction of positive and negative basis features u1 … uk, where uj = xj or uj = (xj), for j = 1 to k. The proposition (u1, …, uk) indicates the disjunctive region consisting of the cells that are "just next door" to the cell u1 … uk.

## Note 9

Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have. Then, your conception of those effects is the whole of your conception of the object.

Charles Sanders Peirce, "The Maxim of Pragmatism, CP 5.438.

One other subject that it would be opportune to mention at this point, while we have an object example of a mathematical group fresh in mind, is the relationship between the pragmatic maxim and what are commonly known in mathematics as representation principles. As it turns out, with regard to its formal characteristics, the pragmatic maxim unites the aspects of a representation principle with the attributes of what would ordinarily be known as a closure principle. We will consider the form of closure that is invoked by the pragmatic maxim on another occasion, focusing here and now on the topic of group representations.

Let us return to the example of the so-called four-group V4. We encountered this group in one of its concrete representations, namely, as a transformation group that acts on a set of objects, in this particular case a set of sixteen functions or propositions. Forgetting about the set of objects that the group transforms among themselves, we may take the abstract view of the group's operational structure, say, in the form of the group operation table copied here:

```o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    .    %    e    |    f    |    g    |    h    |
|         %         |         |         |         |
o=========o=========o=========o=========o=========o
|         %         |         |         |         |
|    e    %    e    |    f    |    g    |    h    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    f    %    f    |    e    |    h    |    g    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    g    %    g    |    h    |    e    |    f    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    h    %    h    |    g    |    f    |    e    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o
```

This table is abstractly the same as, or isomorphic to, the versions with the Eij operators and the Tij transformations that we discussed earlier. That is to say, the story is the same — only the names have been changed. An abstract group can have a multitude of significantly and superficially different representations. Even after we have long forgotten the details of the particular representation that we may have come in with, there are species of concrete representations, called the regular representations, that are always readily available, as they can be generated from the mere data of the abstract operation table itself.

For example, select a group element from the top margin of the Table, and "consider its effects" on each of the group elements as they are listed along the left margin. We may record these effects as Peirce usually did, as a logical "aggregate" of elementary dyadic relatives, that is to say, a disjunction or a logical sum whose terms represent the ordered pairs of <input : output> transactions that are produced by each group element in turn. This yields what is usually known as one of the regular representations of the group, specifically, the first, the post-, or the right regular representation. It has long been conventional to organize the terms in the form of a matrix:

Reading "+" as a logical disjunction:

```   G  =  e  +  f  +  g  + h,
```

And so, by expanding effects, we get:

```   G  =  e:e  +  f:f  +  g:g  +  h:h

+  e:f  +  f:e  +  g:h  +  h:g

+  e:g  +  f:h  +  g:e  +  h:f

+  e:h  +  f:g  +  g:f  +  h:e
```

More on the pragmatic maxim as a representation principle later.

## Note 10

Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have. Then, your conception of those effects is the whole of your conception of the object.

Peirce, "Maxim of Pragmaticism", Collected Papers, CP 5.438.

The genealogy of this conception of pragmatic representation is very intricate. I will delineate some details that I presently fancy I remember clearly enough, subject to later correction. Without checking historical accounts, I will not be able to pin down anything like a real chronology, but most of these notions were standard furnishings of the 19th Century mathematical study, and only the last few items date as late as the 1920's.

The idea about the regular representations of a group is universally known as Cayley's Theorem, usually in the form: "Every group is isomorphic to a subgroup of Aut(X), the group of automorphisms of an appropriate set X". There is a considerable generalization of these regular representations to a broad class of relational algebraic systems in Peirce's earliest papers. The crux of the whole idea is this:

```   Contemplate the effects of the symbol
whose meaning you wish to investigate
as they play out on all the stages of
conduct on which you have the ability
to imagine that symbol playing a role.
```

This idea of contextual definition is basically the same as Jeremy Bentham's notion of paraphrasis, a "method of accounting for fictions by explaining various purported terms away" (Quine, in Van Heijenoort, p. 216). Today we'd call these constructions term models. This, again, is the big idea behind Schönfinkel's combinators {S, K, I}, and hence of lambda calculus, and I reckon you know where that leads.

## Note 11

Let me return to Peirce's early papers on the algebra of relatives to pick up the conventions that he used there, and then rewrite my account of regular representations in a way that conforms to those.

Peirce expresses the action of an "elementary dual relative" like so:

[Let] A:B be taken to denote the elementary relative which multiplied into B gives A. (Peirce, CP 3.123).

And though he is well aware that it is not at all necessary to arrange elementary relatives into arrays, matrices, or tables, when he does so he tends to prefer organizing dyadic relations in the following manner:

```   [  A:A   A:B   A:C  |
|                   |
|  B:A   B:B   B:C  |
|                   |
|  C:A   C:B   C:C  ]
```

That conforms to the way that the last school of thought I matriculated into stipulated that we tabulate material:

```   [  e_11  e_12  e_13  |
|                    |
|  e_21  e_22  e_23  |
|                    |
|  e_31  e_32  e_33  ]
```

So, for example, let us suppose that we have the small universe {A, B, C}, and the 2-adic relation m = mover of that is represented by this matrix:

```   m  =

[  m_AA (A:A)   m_AB (A:B)   m_AC (A:C)  |
|                                        |
|  m_BA (B:A)   m_BB (B:B)   m_BC (B:C)  |
|                                        |
|  m_CA (C:A)   m_CB (C:B)   m_CC (C:C)  ]
```

Also, let m be such that:

```   A is a mover of A and B,
B is a mover of B and C,
C is a mover of C and A.
```

In sum:

```   m  =

[  1 * (A:A)   1 * (A:B)   0 * (A:C)  |
|                                     |
|  0 * (B:A)   1 * (B:B)   1 * (B:C)  |
|                                     |
|  1 * (C:A)   0 * (C:B)   1 * (C:C)  ]
```

For the sake of orientation and motivation, compare with Peirce's notation in CP 3.329.

I think that will serve to fix notation and set up the remainder of the account.

## Note 12

```It is common in algebra to switch around
between different conventions of display,
as the momentary fancy happens to strike,
and I see that Peirce is no different in
this sort of shiftiness than anyone else.
A changeover appears to occur especially
whenever he shifts from logical contexts
to algebraic contexts of application.

In the paper "On the Relative Forms of Quaternions" (CP 3.323),
we observe Peirce providing the following sorts of explanation:

| If X, Y, Z denote the three rectangular components of a vector, and W denote
| numerical unity (or a fourth rectangular component, involving space of four
| dimensions), and (Y:Z) denote the operation of converting the Y component
| of a vector into its Z component, then
|
|     1  =  (W:W) + (X:X) + (Y:Y) + (Z:Z)
|
|     i  =  (X:W) - (W:X) - (Y:Z) + (Z:Y)
|
|     j  =  (Y:W) - (W:Y) - (Z:X) + (X:Z)
|
|     k  =  (Z:W) - (W:Z) - (X:Y) + (Y:X)
|
| In the language of logic (Y:Z) is a relative term whose relate is
| a Y component, and whose correlate is a Z component.  The law of
| multiplication is plainly (Y:Z)(Z:X) = (Y:X), (Y:Z)(X:W) = 0,
| and the application of these rules to the above values of
| 1, i, j, k gives the quaternion relations
|
|     i^2  =  j^2  =  k^2  =  -1,
|
|     ijk  =  -1,
|
|     etc.
|
| The symbol a(Y:Z) denotes the changing of Y to Z and the
| multiplication of the result by 'a'.  If the relatives be
| arranged in a block
|
|     W:W     W:X     W:Y     W:Z
|
|     X:W     X:X     X:Y     X:Z
|
|     Y:W     Y:X     Y:Y     Y:Z
|
|     Z:W     Z:X     Z:Y     Z:Z
|
| then the quaternion w + xi + yj + zk
| is represented by the matrix of numbers
|
|     w       -x      -y      -z
|
|     x        w      -z       y
|
|     y        z       w      -x
|
|     z       -y       x       w
|
| The multiplication of such matrices follows the same laws as the
| multiplication of quaternions.  The determinant of the matrix =
| the fourth power of the tensor of the quaternion.
|
| The imaginary x + y(-1)^(1/2) may likewise be represented by the matrix
|
|      x      y
|
|     -y      x
|
| and the determinant of the matrix = the square of the modulus.
|
| C.S. Peirce, 'Collected Papers', CP 3.323, (1882).
|'Johns Hopkins University Circulars', No. 13, p. 179.

This way of talking is the mark of a person who opts
to multiply his matrices "on the right", as they say.
Yet Peirce still continues to call the first element
of the ordered pair (i:j) its "relate" while calling
the second element of the pair (i:j) its "correlate".
That doesn't comport very well, so far as I can tell,
with his customary reading of relative terms, suited
more to the multiplication of matrices "on the left".

So I still have a few wrinkles to iron out before
I can give this story a smooth enough consistency.
```

## Note 13

```Let us make up the model universe \$1\$ = A + B + C and the 2-adic relation
n = "noter of", as when "X is a data record that contains a pointer to Y".
That interpretation is not important, it's just for the sake of intuition.
In general terms, the 2-adic relation n can be represented by this matrix:

n  =

[  n_AA (A:A)   n_AB (A:B)   n_AC (A:C)  |
|                                        |
|  n_BA (B:A)   n_BB (B:B)   n_BC (B:C)  |
|                                        |
|  n_CA (C:A)   n_CB (C:B)   n_CC (C:C)  ]

Also, let n be such that:

A is a noter of A and B,
B is a noter of B and C,
C is a noter of C and A.

Filling in the instantial values of the "coefficients" n_ij,
as the indices i and j range over the universe of discourse:

n  =

[  1 * (A:A)   1 * (A:B)   0 * (A:C)  |
|                                     |
|  0 * (B:A)   1 * (B:B)   1 * (B:C)  |
|                                     |
|  1 * (C:A)   0 * (C:B)   1 * (C:C)  ]

In Peirce's time, and even in some circles of mathematics today,
the information indicated by the elementary relatives (i:j), as
i, j range over the universe of discourse, would be referred to
as the "umbral elements" of the algebraic operation represented
by the matrix, though I seem to recall that Peirce preferred to
call these terms the "ingredients".  When this ordered basis is
understood well enough, one will tend to drop any mention of it
from the matrix itself, leaving us nothing but these bare bones:

n  =

[  1  1  0  |
|           |
|  0  1  1  |
|           |
|  1  0  1  ]

However the specification may come to be written, this
is all just convenient schematics for stipulating that:

n  =  A:A  +  B:B  +  C:C  +  A:B  +  B:C  +  C:A

Recognizing !1! = A:A + B:B + C:C to be the identity transformation,
the 2-adic relation n = "noter of" may be represented by an element
!1! + A:B + B:C + C:A of the so-called "group ring", all of which
just makes this element a special sort of linear transformation.

Up to this point, we are still reading the elementary relatives of
the form i:j in the way that Peirce reads them in logical contexts:
i is the relate, j is the correlate, and in our current example we
read i:j, or more exactly, n_ij = 1, to say that i is a noter of j.
This is the mode of reading that we call "multiplying on the left".

In the algebraic, permutational, or transformational contexts of
application, however, Peirce converts to the alternative mode of
reading, although still calling i the relate and j the correlate,
the elementary relative i:j now means that i gets changed into j.
In this scheme of reading, the transformation A:B + B:C + C:A is
a permutation of the aggregate \$1\$ = A + B + C, or what we would
now call the set {A, B, C}, in particular, it is the permutation
that is otherwise notated as:

( A B C )
<       >
( B C A )

This is consistent with the convention that Peirce uses in
the paper "On a Class of Multiple Algebras" (CP 3.324-327).
```

## Note 14

```We have been contemplating the virtues and the utilities of
the pragmatic maxim as a standard heuristic in hermeneutics,
that is, as a principle of interpretation that guides us in
finding clarifying representations for a problematic corpus
of symbols by means of their actions on other symbols or in
terms of their effects on the syntactic contexts wherein we
discover them or where we might conceive to distribute them.

I began this excursion by taking off from the moving platform
of differential logic and passing by way of the corresponding
transformation groups, as they act on propositions, and on to
an exercise in applying the pragmatic maxim, by contemplating
the regular representations of groups as giving us one of the
simplest conceivable, relatively concrete applications of the
general principle of representation in question.

There are a few problems of implementation that have to be worked out
in practice, most of which are cleared up by keeping in mind which of
several possible conventions we have chosen to follow at a given time.

But there does appear to remain this rather more substantial question:
Are the effects we seek relates or correlates, or does it even matter?

I will have to leave that question as it is for now,
in hopes that a solution will evolve itself in time.
```

## Note 15

```Obstacles to Applying the Pragmatic Maxim

No sooner do you get a good idea and try to apply it
than you find that a motley array of obstacles arise.

It would be good if we could in practice more consistently
apply the pragmatic maxim to the purpose for which it was
purportedly intended by its author.  That aim would be
the clarification of concepts, that is, intellectual
symbols or mental signs, to the point where their
inherent senses, or their lacks thereof, would
be rendered manifest to suitable interpreters.

There are big obstacles and little obstacles to applying the pragmatic maxim.
In good subgoaling fashion, I will merely mention a few of the bigger blocks,
as if in passing, but not really getting past them, and then I will get down
to the details of the problems that more immediately obstruct our advance.

Obstacle 1.  People do not always read the instructions very carefully.
There is a tendency in readers of particular prior persuasions to blow
the problem all out of proportion, to think that the maxim is meant to
reveal the absolutely positive and the totally unique meaning of every
preconception to which they might deign or elect to apply it.  Reading
the maxim with an even minimal attention, you can see that it promises
no such finality of unindexed sense, but ties what you conceive to you.
I have lately come to wonder at the tenacity of this misinterpretation.
Perhaps people reckon that nothing less would be worth their attention.
I am not sure.  I can only say the achievement of more modest goals is
the sort of thing on which our daily life depends, and there can be no
final end to inquiry nor any ultimate community without a continuation
of life, and that means life on a day to day basis.  All of which only
brings me back to the point of persisting with local meantime examples,
because if we can't apply the maxim there, we can't apply it anywhere.
```

## Note 16

```Obstacles to Applying the Pragmatic Maxim (cont.)

Obstacle 2.  Applying the pragmatic maxim, even with a moderate aim, can be hard.
I think that my present example, deliberately impoverished as it is, affords us
with an embarassing richness of evidence of just how complex the simple can be.

All the better reason for me to see if I can finish it up before moving on.

Expressed most simply, the idea is to replace the question of "what it is",
which modest people know is far too difficult for them to answer right off,
with the question of "what it does", which most of us know a modicum about.

In the case of regular representations of groups we found
a non-plussing surplus of answers to sort our way through.
So let us track back one more time to see if we can learn
any lessons that might carry over to more realistic cases.

Here is is the operation table of V_4 once again:

Table 1.  Klein Four-Group V_4
o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    .    %    e    |    f    |    g    |    h    |
|         %         |         |         |         |
o=========o=========o=========o=========o=========o
|         %         |         |         |         |
|    e    %    e    |    f    |    g    |    h    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    f    %    f    |    e    |    h    |    g    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    g    %    g    |    h    |    e    |    f    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o
|         %         |         |         |         |
|    h    %    h    |    g    |    f    |    e    |
|         %         |         |         |         |
o---------o---------o---------o---------o---------o

A group operation table is really just a device for
recording a certain 3-adic relation, to be specific,
the set of triples of the form <x, y, z> satisfying
the equation x.y = z, where "." signifies the group
operation, usually omitted as understood in context.

In the case of V_4 = (G, .), where G is the "underlying set"
{e, f, g, h}, we have the 3-adic relation L(V_4) c G x G x G
whose triples are listed below:

<e, e, e>
<e, f, f>
<e, g, g>
<e, h, h>

<f, e, f>
<f, f, e>
<f, g, h>
<f, h, g>

<g, e, g>
<g, f, h>
<g, g, e>
<g, h, f>

<h, e, h>
<h, f, g>
<h, g, f>
<h, h, e>

It is part of the definition of a group that the 3-adic
relation L c G^3 is actually a function L : G x G -> G.
It is from this functional perspective that we can see
an easy way to derive the two regular representations.
Since we have a function of the type L : G x G -> G,
we can define a couple of substitution operators:

1.  Sub(x, <_, y>) puts any specified x into
the empty slot of the rheme <_, y>, with
the effect of producing the saturated
rheme <x, y> that evaluates to xy.

2.  Sub(x, <y, _>) puts any specified x into
the empty slot of the rheme <y, _>, with
the effect of producing the saturated
rheme <y, x> that evaluates to yx.

In (1), we consider the effects of each x in its
practical bearing on contexts of the form <_, y>,
as y ranges over G, and the effects are such that
x takes <_, y> into xy, for y in G, all of which
is summarily notated as x = {(y : xy) : y in G}.
The pairs (y : xy) can be found by picking an x
from the left margin of the group operation table
and considering its effects on each y in turn as
these run across the top margin.  This aspect of
pragmatic definition we recognize as the regular
ante-representation:

e  =  e:e  +  f:f  +  g:g  +  h:h

f  =  e:f  +  f:e  +  g:h  +  h:g

g  =  e:g  +  f:h  +  g:e  +  h:f

h  =  e:h  +  f:g  +  g:f  +  h:e

In (2), we consider the effects of each x in its
practical bearing on contexts of the form <y, _>,
as y ranges over G, and the effects are such that
x takes <y, _> into yx, for y in G, all of which
is summarily notated as x = {(y : yx) : y in G}.
The pairs (y : yx) can be found by picking an x
from the top margin of the group operation table
and considering its effects on each y in turn as
these run down the left margin.  This aspect of
pragmatic definition we recognize as the regular
post-representation:

e  =  e:e  +  f:f  +  g:g  +  h:h

f  =  e:f  +  f:e  +  g:h  +  h:g

g  =  e:g  +  f:h  +  g:e  +  h:f

h  =  e:h  +  f:g  +  g:f  +  h:e

If the ante-rep looks the same as the post-rep,
now that I'm writing them in the same dialect,
that is because V_4 is abelian (commutative),
and so the two representations have the very
same effects on each point of their bearing.
```

## Note 17

```So long as we're in the neighborhood, we might as well take in
some more of the sights, for instance, the smallest example of
a non-abelian (non-commutative) group.  This is a group of six
elements, say, G = {e, f, g, h, i, j}, with no relation to any
other employment of these six symbols being implied, of course,
and it can be most easily represented as the permutation group
on a set of three letters, say, X = {A, B, C}, usually notated
as G = Sym(X) or more abstractly and briefly, as Sym(3) or S_3.
Here are the permutation (= substitution) operations in Sym(X):

Table 1.  Permutations or Substitutions in Sym_{A, B, C}
o---------o---------o---------o---------o---------o---------o
|         |         |         |         |         |         |
|    e    |    f    |    g    |    h    |    i    |    j    |
|         |         |         |         |         |         |
o=========o=========o=========o=========o=========o=========o
|         |         |         |         |         |         |
|  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |
|         |         |         |         |         |         |
|  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |
|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |
|         |         |         |         |         |         |
|  A B C  |  C A B  |  B C A  |  A C B  |  C B A  |  B A C  |
|         |         |         |         |         |         |
o---------o---------o---------o---------o---------o---------o

Here is the operation table for S_3, given in abstract fashion:

Table 2.  Symmetric Group S_3

|                        ^
|                     e / \ e
|                      /   \
|                     /  e  \
|                  f / \   / \ f
|                   /   \ /   \
|                  /  f  \  f  \
|               g / \   / \   / \ g
|                /   \ /   \ /   \
|               /  g  \  g  \  g  \
|            h / \   / \   / \   / \ h
|             /   \ /   \ /   \ /   \
|            /  h  \  e  \  e  \  h  \
|         i / \   / \   / \   / \   / \ i
|          /   \ /   \ /   \ /   \ /   \
|         /  i  \  i  \  f  \  j  \  i  \
|      j / \   / \   / \   / \   / \   / \ j
|       /   \ /   \ /   \ /   \ /   \ /   \
|      (  j  \  j  \  j  \  i  \  h  \  j  )
|       \   / \   / \   / \   / \   / \   /
|        \ /   \ /   \ /   \ /   \ /   \ /
|         \  h  \  h  \  e  \  j  \  i  /
|          \   / \   / \   / \   / \   /
|           \ /   \ /   \ /   \ /   \ /
|            \  i  \  g  \  f  \  h  /
|             \   / \   / \   / \   /
|              \ /   \ /   \ /   \ /
|               \  f  \  e  \  g  /
|                \   / \   / \   /
|                 \ /   \ /   \ /
|                  \  g  \  f  /
|                   \   / \   /
|                    \ /   \ /
|                     \  e  /
|                      \   /
|                       \ /
|                        v

By the way, we will meet with the symmetric group S_3 again
when we return to take up the study of Peirce's early paper
"On a Class of Multiple Algebras" (CP 3.324-327), and also
his late unpublished work "The Simplest Mathematics" (1902)
(CP 4.227-323), with particular reference to the section
that treats of "Trichotomic Mathematics" (CP 4.307-323).
```

## Note 18

```By way of collecting a short-term pay-off for all the work that we
did on the regular representations of the Klein 4-group V_4, let us
write out as quickly as possible in "relative form" a minimal budget
of representations for the symmetric group on three letters, Sym(3).
After doing the usual bit of compare and contrast among the various
representations, we will have enough concrete material beneath our
abstract belts to tackle a few of the presently obscur'd details
of Peirce's early "Algebra + Logic" papers.

Table 1.  Permutations or Substitutions in Sym {A, B, C}
o---------o---------o---------o---------o---------o---------o
|         |         |         |         |         |         |
|    e    |    f    |    g    |    h    |    i    |    j    |
|         |         |         |         |         |         |
o=========o=========o=========o=========o=========o=========o
|         |         |         |         |         |         |
|  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |
|         |         |         |         |         |         |
|  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |
|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |
|         |         |         |         |         |         |
|  A B C  |  C A B  |  B C A  |  A C B  |  C B A  |  B A C  |
|         |         |         |         |         |         |
o---------o---------o---------o---------o---------o---------o

Writing this table in relative form generates
the following "natural representation" of S_3.

e  =  A:A + B:B + C:C

f  =  A:C + B:A + C:B

g  =  A:B + B:C + C:A

h  =  A:A + B:C + C:B

i  =  A:C + B:B + C:A

j  =  A:B + B:A + C:C

I have without stopping to think about it written out this natural
representation of S_3 in the style that comes most naturally to me,
to wit, the "right" way, whereby an ordered pair configured as X:Y
constitutes the turning of X into Y.  It is possible that the next
time we check in with CSP that we will have to adjust our sense of
direction, but that will be an easy enough bridge to cross when we
come to it.
```

## Note 19

```To construct the regular representations of S_3,
we pick up from the data of its operation table:

Table 1.  Symmetric Group S_3

|                        ^
|                     e / \ e
|                      /   \
|                     /  e  \
|                  f / \   / \ f
|                   /   \ /   \
|                  /  f  \  f  \
|               g / \   / \   / \ g
|                /   \ /   \ /   \
|               /  g  \  g  \  g  \
|            h / \   / \   / \   / \ h
|             /   \ /   \ /   \ /   \
|            /  h  \  e  \  e  \  h  \
|         i / \   / \   / \   / \   / \ i
|          /   \ /   \ /   \ /   \ /   \
|         /  i  \  i  \  f  \  j  \  i  \
|      j / \   / \   / \   / \   / \   / \ j
|       /   \ /   \ /   \ /   \ /   \ /   \
|      (  j  \  j  \  j  \  i  \  h  \  j  )
|       \   / \   / \   / \   / \   / \   /
|        \ /   \ /   \ /   \ /   \ /   \ /
|         \  h  \  h  \  e  \  j  \  i  /
|          \   / \   / \   / \   / \   /
|           \ /   \ /   \ /   \ /   \ /
|            \  i  \  g  \  f  \  h  /
|             \   / \   / \   / \   /
|              \ /   \ /   \ /   \ /
|               \  f  \  e  \  g  /
|                \   / \   / \   /
|                 \ /   \ /   \ /
|                  \  g  \  f  /
|                   \   / \   /
|                    \ /   \ /
|                     \  e  /
|                      \   /
|                       \ /
|                        v

Just by way of staying clear about what we are doing,

It is part of the definition of a group that the 3-adic
relation L c G^3 is actually a function L : G x G -> G.
It is from this functional perspective that we can see
an easy way to derive the two regular representations.

Since we have a function of the type L : G x G -> G,
we can define a couple of substitution operators:

1.  Sub(x, <_, y>) puts any specified x into
the empty slot of the rheme <_, y>, with
the effect of producing the saturated
rheme <x, y> that evaluates to xy.

2.  Sub(x, <y, _>) puts any specified x into
the empty slot of the rheme <y, _>, with
the effect of producing the saturated
rheme <y, x> that evaluates to yx.

In (1), we consider the effects of each x in its
practical bearing on contexts of the form <_, y>,
as y ranges over G, and the effects are such that
x takes <_, y> into xy, for y in G, all of which
is summarily notated as x = {(y : xy) : y in G}.
The pairs (y : xy) can be found by picking an x
from the left margin of the group operation table
and considering its effects on each y in turn as
these run along the right margin.  This produces
the regular ante-representation of S_3, like so:

e   =   e:e  +  f:f  +  g:g  +  h:h  +  i:i  +  j:j

f   =   e:f  +  f:g  +  g:e  +  h:j  +  i:h  +  j:i

g   =   e:g  +  f:e  +  g:f  +  h:i  +  i:j  +  j:h

h   =   e:h  +  f:i  +  g:j  +  h:e  +  i:f  +  j:g

i   =   e:i  +  f:j  +  g:h  +  h:g  +  i:e  +  j:f

j   =   e:j  +  f:h  +  g:i  +  h:f  +  i:g  +  j:e

In (2), we consider the effects of each x in its
practical bearing on contexts of the form <y, _>,
as y ranges over G, and the effects are such that
x takes <y, _> into yx, for y in G, all of which
is summarily notated as x = {(y : yx) : y in G}.
The pairs (y : yx) can be found by picking an x
on the right margin of the group operation table
and considering its effects on each y in turn as
these run along the left margin.  This generates
the regular post-representation of S_3, like so:

e   =   e:e  +  f:f  +  g:g  +  h:h  +  i:i  +  j:j

f   =   e:f  +  f:g  +  g:e  +  h:i  +  i:j  +  j:h

g   =   e:g  +  f:e  +  g:f  +  h:j  +  i:h  +  j:i

h   =   e:h  +  f:j  +  g:i  +  h:e  +  i:g  +  j:f

i   =   e:i  +  f:h  +  g:j  +  h:f  +  i:e  +  j:g

j   =   e:j  +  f:i  +  g:h  +  h:g  +  i:f  +  j:e

If the ante-rep looks different from the post-rep,
it is just as it should be, as S_3 is non-abelian
(non-commutative), and so the two representations
differ in the details of their practical effects,
though, of course, being representations of the
same abstract group, they must be isomorphic.
```

## Note 20

```| the way of heaven and earth
| is to be long continued
| in their operation
| without stopping
|
| i ching, hexagram 32

You may be wondering what happened to the announced subject of "Differential Logic".
If you think that we have been taking a slight excursion my reply to the charge of
a scenic rout would be both "yes and no".  What happened was this.  We chanced to
make the observation that the shift operators E_ij form a transformation group
that acts on the set of propositions of the form f : B^2 -> B.  Group theory
is a very attractive subject, but it did not have the effect of drawing us
so far off our initial course as one might at first think.  For one thing,
groups, in particular, the special family of groups that have come to be
named after the Norwegian mathematician Marius Sophus Lie, turn out to
be of critical importance in the solution of differential equations.
For another thing, group operations afford us examples of 3-adic
relations that have been extremely well-studied over the years,
and thus they supply us with no small bit of guidance in the
study of sign relations, another class of 3-adic relations
that have significance for logical studies, in our brief
acquaintance with which we have scarcely even begun to
break the ice.  Finally, I could not resist taking up
the connection between group representations, which
constitute a very generic class of logical models,
and the all-important pragmatic maxim.

Biographical Data for Marius Sophus Lie (1842-1899):
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html
```

## Note 21

```We've seen a couple of groups, V_4 and S_3, represented in various ways, and
we've seen their representations presented in a variety of different manners.
Let us look at one other stylistic variant for presenting a representation
that is frequently seen, the so-called "matrix representation" of a group.

Recalling the manner of our acquaintance with the symmetric group S_3,
we began with the "bigraph" (bipartite graph) picture of its natural
representation as the set of all permutations or substitutions on
the set X = {A, B, C}.

Table 1.  Permutations or Substitutions in Sym {A, B, C}
o---------o---------o---------o---------o---------o---------o
|         |         |         |         |         |         |
|    e    |    f    |    g    |    h    |    i    |    j    |
|         |         |         |         |         |         |
o=========o=========o=========o=========o=========o=========o
|         |         |         |         |         |         |
|  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |
|         |         |         |         |         |         |
|  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |
|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |
|         |         |         |         |         |         |
|  A B C  |  C A B  |  B C A  |  A C B  |  C B A  |  B A C  |
|         |         |         |         |         |         |
o---------o---------o---------o---------o---------o---------o

Then we rewrote these permutations -- since they are
functions f : X -> X they can also be recognized as
2-adic relations f c X x X -- in "relative form",
in effect, in the manner to which Peirce would
a relative half-a-chance:

e  =  A:A + B:B + C:C

f  =  A:C + B:A + C:B

g  =  A:B + B:C + C:A

h  =  A:A + B:C + C:B

i  =  A:C + B:B + C:A

j  =  A:B + B:A + C:C

These days one is much more likely to encounter the natural representation
of S_3 in the form of a "linear representation", that is, as a family of
linear transformations that map the elements of a suitable vector space
into each other, all of which would in turn usually be represented by
a set of matrices like these:

Table 2.  Matrix Representations of the Permutations in Sym(3)
o---------o---------o---------o---------o---------o---------o
|         |         |         |         |         |         |
|    e    |    f    |    g    |    h    |    i    |    j    |
|         |         |         |         |         |         |
o=========o=========o=========o=========o=========o=========o
|         |         |         |         |         |         |
|  1 0 0  |  0 0 1  |  0 1 0  |  1 0 0  |  0 0 1  |  0 1 0  |
|  0 1 0  |  1 0 0  |  0 0 1  |  0 0 1  |  0 1 0  |  1 0 0  |
|  0 0 1  |  0 1 0  |  1 0 0  |  0 1 0  |  1 0 0  |  0 0 1  |
|         |         |         |         |         |         |
o---------o---------o---------o---------o---------o---------o

The key to the mysteries of these matrices is revealed by noting that their
coefficient entries are arrayed and overlayed on a place mat marked like so:

[ A:A  A:B  A:C |
| B:A  B:B  B:C |
| C:A  C:B  C:C ]

Of course, the place-settings of convenience at different symposia may vary.
```